units.h

//--------------------------------------------------------------------------------------------------
// 
//	Units: A compile-time c++14 unit conversion library with no dependencies
//
//--------------------------------------------------------------------------------------------------
//
// The MIT License (MIT)
//
// Permission is hereby granted, free of charge, to any person obtaining a copy of this software 
// and associated documentation files (the "Software"), to deal in the Software without 
// restriction, including without limitation the rights to use, copy, modify, merge, publish, 
// distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the 
// Software is furnished to do so, subject to the following conditions:
// 
// The above copyright notice and this permission notice shall be included in all copies or 
// substantial portions of the Software.
// 
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING 
// BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND 
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, 
// DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING 
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
//
//--------------------------------------------------------------------------------------------------
//
// ATTRIBUTION:
// Parts of this work have been adapted from: 
// http://stackoverflow.com/questions/35069778/create-comparison-trait-for-template-classes-whose-parameters-are-in-a-different
// http://stackoverflow.com/questions/28253399/check-traits-for-all-variadic-template-arguments/28253503
// http://stackoverflow.com/questions/36321295/rational-approximation-of-square-root-of-stdratio-at-compile-time?noredirect=1#comment60266601_36321295
//
//--------------------------------------------------------------------------------------------------
//
/// @file	units.h
/// @brief	Complete implementation of `units` - a compile-time, header-only, unit conversion 
///			library built on c++14 with no dependencies.
//
//--------------------------------------------------------------------------------------------------
//
// CHANGE LOG:
//
// - v2.0.0 (04/03/2016): Initial
//
//--------------------------------------------------------------------------------------------------
// 
// Copyright (c) 2016 Nic Holthaus
// 
//--------------------------------------------------------------------------------------------------

#ifndef units_h__
#define units_h__

#if defined(_MSC_VER) && _MSC_VER < 1800
#	pragma warning(push)
#	pragma warning(disable : 4520)
#endif

#ifndef UNIT_LIB_DEFAULT_TYPE
#define UNIT_LIB_DEFAULT_TYPE double
#endif

//--------------------
//	INCLUDES
//--------------------

#include <ratio>
#include <type_traits>
#include <cstdint>
#include <cmath>
#include <iostream>
#include <limits>

//------------------------------
//	MACROS
//------------------------------

#if !defined(_MSC_VER) || _MSC_VER > 1800

	/** 
	 * @def		UNIT_ADD(nameSingular, namePlural, abbreviation, definition)
	 * @brief	Macro for generating the boiler-plate code needed for a new unit.
	 * @details	The macro generates singular, plural, and abbreviated forms
	 *			of the unit definition (e.g. `meter`, `meters`, and `m`), as well as the
	 *			appropriately named unit container (e.g. `meter_t`). A literal suffix is created
	 *			using the abbreviation (e.g. `10.0_m`). It also defines a class-specific
	 *			cout function which prints both the value and abbreviation of the unit when invoked.
	 * @param	namespaceName namespace in which the new units will be encapsulated. All literal values
	 *			are placed in the `units::literals` namespace.
	 * @param	nameSingular singular version of the unit name, e.g. 'meter'
	 * @param	namePlural - plural version of the unit name, e.g. 'meters'
	 * @param	abbreviation - abbreviated unit name, e.g. 'm'
	 * @param	definition - the variadic parameter is used for the definition of the unit 
	 *			(e.g. `unit<std::ratio<1>, units::category::length_unit>`)
	 * @note	a variadic template is used for the definition to allow templates with
	 *			commas to be easily expanded. All the variadic 'arguments' should together
	 *			comprise the unit definition.
	 */
	#define UNIT_ADD(namespaceName, nameSingular, namePlural, abbreviation, /*definition*/...)\
	namespace namespaceName\
	{\
		/** @name Units (full names plural) */ /** @{ */ using namePlural = __VA_ARGS__; /** @} */\
		/** @name Units (full names singular) */ /** @{ */ using nameSingular = namePlural; /** @} */\
		/** @name Units (abbreviated) */ /** @{ */ using abbreviation = namePlural; /** @} */\
		/** @name Unit Containers */ /** @{ */ using nameSingular ## _t = unit_t<nameSingular>; /** @} */\
	}\
	inline std::ostream& operator<<(std::ostream& os, const namespaceName::nameSingular ## _t& obj) { os << obj() << " "#abbreviation; return os; };\
	namespace literals\
	{\
		inline namespaceName::nameSingular ## _t operator""_ ## abbreviation (long double d) { return namespaceName::nameSingular ## _t(d); };\
		inline namespaceName::nameSingular ## _t operator""_ ## abbreviation (unsigned long long d) { return namespaceName::nameSingular ## _t((long double)d); };	/* may want to think of something better than this cast.*/\
	}

	/** 
	 * @def		UNIT_ADD_DECIBEL(namespaceName, nameSingular, abbreviation)
	 * @brief	Macro to create decibel container and literals for an existing unit type.
	 * @details	This macro generates the decibel unit container, cout overload, and literal definitions.
	 * @param	namespaceName namespace in which the new units will be encapsulated. All literal values
	 *			are placed in the `units::literals` namespace.
	 * @param	nameSingular singular version of the base unit name, e.g. 'watt'
	 * @param	abbreviation - abbreviated decibel unit name, e.g. 'dBW'
	 */
	#define UNIT_ADD_DECIBEL(namespaceName, nameSingular, abbreviation)\
	namespace namespaceName\
	{\
		/** @name Unit Containers */ /** @{ */ using abbreviation ## _t = unit_t<nameSingular, UNIT_LIB_DEFAULT_TYPE, units::decibel_scale>; /** @} */\
	}\
	inline std::ostream& operator<<(std::ostream& os, const namespaceName::abbreviation ## _t& obj) { os << obj() << " "#abbreviation; return os; };\
	namespace literals\
	{\
		inline namespaceName::abbreviation ## _t operator""_ ## abbreviation (long double d) { return namespaceName::abbreviation ## _t(d); };\
		inline namespaceName::abbreviation ## _t operator""_ ## abbreviation (unsigned long long d) { return namespaceName::abbreviation ## _t((long double)d); };	/* may want to think of something better than this cast.*/\
	}

	/** 
	 * @def		UNIT_ADD_CATEGORY_TRAIT(unitCategory, baseUnit)
	 * @brief	Macro to create the `is_category_unit` type trait.
	 * @details	This trait allows users to test whether a given type matches
	 *			an intended category. This macro comprises all the boiler-plate
	 *			code necessary to do so.
	 * @param	unitCategory The name of the category of unit, e.g. length or mass.
	 * @param	baseUnit The categories base unit (typically SI base unit). This is
	 *			necessary for compatibility with compilers which are not fully c++11
	 *			compliant.
	 */
	#define UNIT_ADD_CATEGORY_TRAIT(unitCategory, baseUnit)\
	namespace traits\
	{\
		/** @cond */\
		namespace detail\
		{\
			template<typename T> struct is_ ## unitCategory ## _unit_impl : std::false_type {};\
			template<typename C, typename U, typename P, typename T>\
			struct is_ ## unitCategory ## _unit_impl<units::unit<C, U, P, T>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_traits<units::unit<C, U, P, T>>::base_unit_type>, units::category::unitCategory ## _unit>::type {};\
			template<typename U, typename S, template<typename> class N>\
			struct is_ ## unitCategory ## _unit_impl<units::unit_t<U, S, N>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_t_traits<units::unit_t<U, S, N>>::unit_type>, units::category::unitCategory ## _unit>::type {};\
		}\
		/** @endcond */\
		\
		/** @ingroup	TypeTraits*/\
		/** @brief		Trait which tests whether a type represents a unit of unitCategory*/\
		/** @details	Inherits from `std::true_type` or `std::false_type`. Use `is_ ## unitCategory ## _unit<T>::value` to test the unit represents a unitCategory quantity.*/\
		/** @tparam		T	one or more types to test*/\
		template<typename... T> struct is_ ## unitCategory ## _unit : std::integral_constant<bool, units::all_true<units::traits::detail::is_ ## unitCategory ## _unit_impl<typename std::decay<T>::type>::value...>::value> {};\
	}

//------------------------------
//	MACROS (VS2013)
//------------------------------

#else

	/**
	 * @def		UNIT_ADD(namespaceName, nameSingular, namePlural, abbreviation, definition)
	 * @brief	Macro for generating the boiler-plate code needed for a new unit.
	 * @details	This macro should be used within an appropriate namespace for the unit
	 *			category. The macro generates singular, plural, and abbreviated forms
	 *			of the unit definition (e.g. `meter`, `meters`, and `m`), as well as the
	 *			appropriately named unit container (e.g. `meter_t`). It also defines a class-specific
	 *			cout function which prints both the value and abbreviation of the unit when invoked.
	 * @param	namespaceName namespace in which the new units will be encapsulated. All literal values
	 *			are placed in the `units::literals` namespace.
	 * @param	nameSingular singular version of the unit name, e.g. 'meter'
	 * @param	namePlural - plural version of the unit name, e.g. 'meters'
	 * @param	abbreviation - abbreviated unit name, e.g. 'm'
	 * @param	definition - the variadic parameter is used for the definition of the unit
	 *			(e.g. `unit<std::ratio<1>, units::category::length_unit>`)
	 * @note	a variadic template is used for the definition to allow templates with
	 *			commas to be easily expanded. All the variadic 'arguments' should together
	 *			comprise the unit definition.
	 */
	#define UNIT_ADD(namespaceName, nameSingular, namePlural, abbreviation, /*definition*/...)\
	namespace namespaceName\
	{\
		/** @name Units (full names plural) */ /** @{ */ using namePlural = __VA_ARGS__; /** @} */\
		/** @name Units (full names singular) */ /** @{ */ using nameSingular = namePlural; /** @} */\
		/** @name Units (abbreviated) */ /** @{ */ using abbreviation = namePlural; /** @} */\
		/** @name Unit Containers */ /** @{ */ using nameSingular ## _t = unit_t<nameSingular>; /** @} */\
	}\
	inline std::ostream& operator<<(std::ostream& os, const namespaceName::nameSingular ## _t& obj) { os << obj() << " "#abbreviation; return os; };\

	/**
	 * @def		UNIT_ADD_DECIBEL(namespaceName, nameSingular, abbreviation)
	 * @brief	Macro to create decibel container and literals for an existing unit type.
	 * @details	This macro generates the decibel unit container, cout overload, and literal definitions.
	 * @param	namespaceName namespace in which the new units will be encapsulated. All literal values
	 *			are placed in the `units::literals` namespace.
	 * @param	nameSingular singular version of the base unit name, e.g. 'watt'
	 * @param	abbreviation - abbreviated decibel unit name, e.g. 'dBW'
	 */
	#define UNIT_ADD_DECIBEL(namespaceName, nameSingular, abbreviation)\
	namespace namespaceName\
	{\
		/** @name Unit Containers */ /** @{ */ using abbreviation ## _t = unit_t<nameSingular, UNIT_LIB_DEFAULT_TYPE, units::decibel_scale>; /** @} */\
	}\
	inline std::ostream& operator<<(std::ostream& os, const namespaceName::abbreviation ## _t& obj) { os << obj() << " "#abbreviation; return os; };\

	/**
	 * @def		UNIT_ADD_CATEGORY_TRAIT(unitCategory, baseUnit)
	 * @brief	Macro to create the `is_category_unit` type trait.
	 * @details	This trait allows users to test whether a given type matches
	 *			an intended category. This macro comprises all the boiler-plate
	 *			code necessary to do so.
	 * @param	unitCategory The name of the category of unit, e.g. length or mass.
	 * @param	baseUnit The categories base unit (typically SI base unit). This is
	 *			necessary for compatibility with compilers which are not fully c++11
	 *			compliant.
	 */
	#define UNIT_ADD_CATEGORY_TRAIT(unitCategory, baseUnit)\
	namespace traits\
	{\
		/** @cond */\
		namespace detail\
		{\
			template<typename T> struct is_ ## unitCategory ## _unit_impl : std::false_type {};\
			template<typename C, typename U, typename P, typename T>\
			struct is_ ## unitCategory ## _unit_impl<units::unit<C, U, P, T>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_traits<units::unit<C, U, P, T>>::base_unit_type>, units::category::unitCategory ## _unit>::type {};\
			template<typename U, typename S, template<typename> class N>\
			struct is_ ## unitCategory ## _unit_impl<units::unit_t<U, S, N>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_t_traits<units::unit_t<U, S, N>>::unit_type>, units::category::unitCategory ## _unit>::type {};\
		}\
		/** @endcond */\
		\
	  		 /** @ingroup		TypeTraits*/\
	  		 /** @brief			Trait which tests whether a type represents a unit of #unitCategory*/\
	  		 /** @details		Inherits from `std::true_type` or `std::false_type`. Use `is_#unitCategory_unit<T>::value` to test*/\
	  		 /**				the unit represents a  #unitCategory quantity.*/\
	  		 /** @tparam		T	one or more types to test*/\
			template<typename T1, typename T2 = T1, typename T3 = T1>\
			struct is_ ## unitCategory ## _unit : std::integral_constant<bool, units::traits::detail::is_ ## unitCategory ## _unit_impl<typename std::decay<T1>::type>::value && units::traits::detail::is_ ## unitCategory ## _unit_impl<typename std::decay<T2>::type>::value && units::traits::detail::is_ ## unitCategory ## _unit_impl<typename std::decay<T3>::type>::value>{};\
	}
#endif

/**
 * @def		UNIT_ADD_WITH_METRIC_PREFIXES(nameSingular, namePlural, abbreviation, definition)
 * @brief	Macro for generating the boiler-plate code needed for a new unit, including its metric
 *			prefixes from fempto to peta.
 * @details	This macro should be used within an appropriate namespace for the unit
 *			category. The macro generates singular, plural, and abbreviated forms
 *			of the unit definition (e.g. `meter`, `meters`, and `m`), as well as the
 *			appropriately named unit container (e.g. `meter_t`). A literal suffix is created
 *			using the abbreviation (e.g. `10.0_m`). It also defines a class-specific
 *			cout function which prints both the value and abbreviation of the unit when invoked.
 * @param	namespaceName namespace in which the new units will be encapsulated. All literal values
 *			are placed in the `units::literals` namespace.
 * @param	nameSingular singular version of the unit name, e.g. 'meter'
 * @param	namePlural - plural version of the unit name, e.g. 'meters'
 * @param	abbreviation - abbreviated unit name, e.g. 'm'
 * @param	definition - the variadic parameter is used for the definition of the unit
 *			(e.g. `unit<std::ratio<1>, units::category::length_unit>`)
 * @note	a variadic template is used for the definition to allow templates with
 *			commas to be easily expanded. All the variadic 'arguments' should together
 *			comprise the unit definition.
 */
#define UNIT_ADD_WITH_METRIC_PREFIXES(namespaceName, nameSingular, namePlural, abbreviation, /*definition*/...)\
	UNIT_ADD(namespaceName, nameSingular, namePlural, abbreviation, __VA_ARGS__)\
	UNIT_ADD(namespaceName, femto ## nameSingular, femto ## namePlural, f ## abbreviation, femto<namePlural>)\
	UNIT_ADD(namespaceName, pico ## nameSingular, pico ## namePlural, p ## abbreviation, pico<namePlural>)\
	UNIT_ADD(namespaceName, nano ## nameSingular, nano ## namePlural, n ## abbreviation, nano<namePlural>)\
	UNIT_ADD(namespaceName, micro ## nameSingular, micro ## namePlural, u ## abbreviation, micro<namePlural>)\
	UNIT_ADD(namespaceName, milli ## nameSingular, milli ## namePlural, m ## abbreviation, milli<namePlural>)\
	UNIT_ADD(namespaceName, centi ## nameSingular, centi ## namePlural, c ## abbreviation, centi<namePlural>)\
	UNIT_ADD(namespaceName, deci ## nameSingular, deci ## namePlural, d ## abbreviation, deci<namePlural>)\
	UNIT_ADD(namespaceName, deca ## nameSingular, deca ## namePlural, da ## abbreviation, deca<namePlural>)\
	UNIT_ADD(namespaceName, hecto ## nameSingular, hecto ## namePlural, h ## abbreviation, hecto<namePlural>)\
	UNIT_ADD(namespaceName, kilo ## nameSingular, kilo ## namePlural, k ## abbreviation, kilo<namePlural>)\
	UNIT_ADD(namespaceName, mega ## nameSingular, mega ## namePlural, M ## abbreviation, mega<namePlural>)\
	UNIT_ADD(namespaceName, giga ## nameSingular, giga ## namePlural, G ## abbreviation, giga<namePlural>)\
	UNIT_ADD(namespaceName, tera ## nameSingular, tera ## namePlural, T ## abbreviation, tera<namePlural>)\
	UNIT_ADD(namespaceName, peta ## nameSingular, peta ## namePlural, P ## abbreviation, peta<namePlural>)

//--------------------
//	UNITS NAMESPACE
//--------------------

/**
 * @namespace units
 * @brief Unit Conversion Library namespace
 */
namespace units
{
	//----------------------------------
	//	DOXYGEN
	//----------------------------------

	/**
	 * @defgroup	UnitContainers Unit Containers
	 * @brief		Defines a series of classes which contain dimensioned values. Unit containers
	 *				store a value, and support various arithmetic operations.
	 */

	/**
	 * @defgroup	UnitTypes Unit Types
	 * @brief		Defines a series of classes which represent units. These types are tags used by
	 *				the conversion function, to create compound units, or to create `unit_t` types.
	 *				By themselves, they are not containers and have no stored value.
	 */

	/**
	 * @defgroup	UnitManipulators Unit Manipulators
	 * @brief		Defines a series of classes used to manipulate unit types, such as `inverse<>`, `squared<>`, and metric prefixes. 
	 *				Unit manipulators can be chained together, e.g. `inverse<squared<pico<time::seconds>>>` to
	 *				represent picoseconds^-2.
	 */

	 /**
	  * @defgroup	CompileTimeUnitManipulators Compile-time Unit Manipulators
	  * @brief		Defines a series of classes used to manipulate `unit_value_t` types at compile-time, such as `unit_value_add<>`, `unit_value_sqrt<>`, etc.
	  *				Compile-time manipulators can be chained together, e.g. `unit_value_sqrt<unit_value_add<unit_value_power<a, 2>, unit_value_power<b, 2>>>` to
	  *				represent `c = sqrt(a^2 + b^2).
	  */

	 /**
	 * @defgroup	UnitMath Unit Math
	 * @brief		Defines a collection of unit-enabled, strongly-typed versions of `<cmath>` functions.
	 * @details		Includes most c++11 extensions.
	 */

	/**
	 * @defgroup	Conversion Explicit Conversion
	 * @brief		Functions used to convert values of one logical type to another.
	 */

	/**
	 * @defgroup	TypeTraits Type Traits
	 * @brief		Defines a series of classes to obtain unit type information at compile-time.
	 */

	//------------------------------
	//	FORWARD DECLARATIONS
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace constants
	{
		static const double PI = 3.14159265358979323846264338327950288419716939937510;
	}
	/** @endcond */	// END DOXYGEN IGNORE

	//------------------------------
	//	RATIO TRAITS
	//------------------------------

	/**
	 * @ingroup TypeTraits
	 * @{
	 */

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/// has_num implementation.
		template<class T>
		struct has_num_impl
		{
			template<class U>
			static auto test(U*)->std::is_integral<decltype(U::num)>;
			template<typename>
			static std::false_type test(...);

			using type = decltype(test<T>(0));
		};
	}

	/**
	 * @brief		Trait which checks for the existence of a static numerator.
	 * @details		Inherits from `std::true_type` or `std::false_type`. Use `has_num<T>::value` to test
	 *				whether `class T` has a numerator static member.
	 */
	template<class T>
	struct has_num : units::detail::has_num_impl<T>::type {};

	namespace detail
	{
		/// has_den implementation.
		template<class T>
		struct has_den_impl
		{
			template<class U>
			static auto test(U*)->std::is_integral<decltype(U::den)>;
			template<typename>
			static std::false_type test(...);

			using type = decltype(test<T>(0));
		};
	}

	/**
	 * @brief		Trait which checks for the existence of a static denominator.
	 * @details		Inherits from `std::true_type` or `std::false_type`. Use `has_den<T>::value` to test
	 *				whether `class T` has a denominator static member.
	 */
	template<class T>
	struct has_den : units::detail::has_den_impl<T>::type {};

	/** @endcond */	// END DOXYGEN IGNORE

	namespace traits
	{
		/**
		 * @brief		Trait that tests whether a type represents a std::ratio.
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_ratio<T>::value` to test
		 *				whether `class T` implements a std::ratio.
		 */
		template<class T>
		struct is_ratio : std::integral_constant<bool,
			has_num<T>::value &&
			has_den<T>::value>
		{};
	}

	//------------------------------
	//	UNIT TRAITS
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	/**
	 * @brief		void type.
	 * @details		Helper class for creating type traits.
	 */
	template<class ...>
	struct void_t { typedef void type; };

	/**
	 * @brief		parameter pack for boolean arguments.
	 */
	template<bool...> struct bool_pack {};

	/**
	 * @brief		Trait which tests that a set of other traits are all true.
	 */
	template<bool... Args>
	struct all_true : std::is_same<units::bool_pack<true, Args...>, units::bool_pack<Args..., true>> {};
	/** @endcond */	// DOXYGEN IGNORE
	
	/** 
	 * @brief namespace representing type traits which can access the properties of types provided by the units library.
	 */
	namespace traits
	{
#ifdef FOR_DOXYGEN_PURPOSES_ONLY
		/**
		 * @ingroup		TypeTraits
		 * @brief		Traits class defining the properties of units.
		 * @details		The units library determines certain properties of the units passed to 
		 *				them and what they represent by using the members of the corresponding 
		 *				unit_traits instantiation.
		 */
		template<class T>
		struct unit_traits
		{
			typedef typename T::base_unit_type base_unit_type;											///< Unit type that the unit was derived from. May be a `base_unit` or another `unit`. Use the `base_unit_of` trait to find the SI base unit type. This will be `void` if type `T` is not a unit.
			typedef typename T::conversion_ratio conversion_ratio;										///< `std::ratio` representing the conversion factor to the `base_unit_type`. This will be `void` if type `T` is not a unit.
			typedef typename T::pi_exponent_ratio pi_exponent_ratio;									///< `std::ratio` representing the exponent of pi to be used in the conversion. This will be `void` if type `T` is not a unit.
			typedef typename T::translation_ratio translation_ratio;									///< `std::ratio` representing a datum translation to the base unit (i.e. degrees C to degrees F conversion). This will be `void` if type `T` is not a unit.
		};
#endif
		/** @cond */	// DOXYGEN IGNORE
		/**
		* @brief		unit traits implementation for classes which are not units.
		*/
		template<class T, typename = void>
		struct unit_traits
		{
			typedef void base_unit_type;
			typedef void conversion_ratio;
			typedef void pi_exponent_ratio;
			typedef void translation_ratio;
		};

		template<class T>
		struct unit_traits
			<T, typename void_t<
			typename T::base_unit_type,
			typename T::conversion_ratio,
			typename T::pi_exponent_ratio,
			typename T::translation_ratio>::type>
		{
			typedef typename T::base_unit_type base_unit_type;											///< Unit type that the unit was derived from. May be a `base_unit` or another `unit`. Use the `base_unit_of` trait to find the SI base unit type. This will be `void` if type `T` is not a unit.
			typedef typename T::conversion_ratio conversion_ratio;										///< `std::ratio` representing the conversion factor to the `base_unit_type`. This will be `void` if type `T` is not a unit.
			typedef typename T::pi_exponent_ratio pi_exponent_ratio;									///< `std::ratio` representing the exponent of pi to be used in the conversion. This will be `void` if type `T` is not a unit.
			typedef typename T::translation_ratio translation_ratio;									///< `std::ratio` representing a datum translation to the base unit (i.e. degrees C to degrees F conversion). This will be `void` if type `T` is not a unit.
		};
		/** @endcond */	// END DOXYGEN IGNORE
	}

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
		 * @brief		helper type to identify base units.
		 * @details		A non-templated base class for `base_unit` which enables RTTI testing.
		 */
		struct _base_unit_t {};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	namespace traits
	{
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests if a class is a `base_unit` type.
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_base_unit<T>::value` to test
		 *				whether `class T` implements a `base_unit`.
		 */
		template<class T>
		struct is_base_unit : std::is_base_of<units::detail::_base_unit_t, T> {};
	}

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
		 * @brief		helper type to identify units.
		 * @details		A non-templated base class for `unit` which enables RTTI testing.
		 */
		struct _unit {};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	namespace traits
	{
		/**
		 * @ingroup		TypeTraits
		 * @brief		Traits which tests if a class is a `unit`
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_unit<T>::value` to test
		 *				whether `class T` implements a `unit`.
		 */
		template<class T>
		struct is_unit : std::is_base_of<units::detail::_unit, T>::type {};
	}

	/** @} */ // end of TypeTraits

	//------------------------------
	//	BASE UNIT CLASS
	//------------------------------

	/**
	 * @ingroup		UnitTypes
	 * @brief		Class representing SI base unit types.
	 * @details		Base units are represented by a combination of `std::ratio` template parameters, each
	 *				describing the exponent of the type of unit they represent. Example: meters per second
	 *				would be described by a +1 exponent for meters, and a -1 exponent for seconds, thus:
	 *				`base_unit<std::ratio<1>, std::ratio<0>, std::ratio<-1>>`
	 * @tparam		Meter		`std::ratio` representing the exponent value for meters.
	 * @tparam		Kilogram	`std::ratio` representing the exponent value for kilograms.
	 * @tparam		Second		`std::ratio` representing the exponent value for seconds.
	 * @tparam		Radian		`std::ratio` representing the exponent value for radians. Although radians are not SI base units, they are included because radians are described by the SI as m * m^-1, which would make them indistinguishable from scalars.
	 * @tparam		Ampere		`std::ratio` representing the exponent value for amperes.
	 * @tparam		Kelvin		`std::ratio` representing the exponent value for Kelvin.
	 * @tparam		Mole		`std::ratio` representing the exponent value for moles.
	 * @tparam		Candela		`std::ratio` representing the exponent value for candelas.
	 * @sa			category	 for type aliases for SI base_unit types.
	 */
	template<class Meter = std::ratio<0>,
	class Kilogram = std::ratio<0>,
	class Second = std::ratio<0>,
	class Radian = std::ratio<0>,
	class Ampere = std::ratio<0>,
	class Kelvin = std::ratio<0>,
	class Mole = std::ratio<0>,
	class Candela = std::ratio < 0 >>
	struct base_unit : units::detail::_base_unit_t
	{
		static_assert(traits::is_ratio<Meter>::value, "Template parameter `Meter` must be a `std::ratio` representing the exponent of meters the unit has");
		static_assert(traits::is_ratio<Kilogram>::value, "Template parameter `Kilogram` must be a `std::ratio` representing the exponent of kilograms the unit has");
		static_assert(traits::is_ratio<Second>::value, "Template parameter `Second` must be a `std::ratio` representing the exponent of seconds the unit has");
		static_assert(traits::is_ratio<Ampere>::value, "Template parameter `Ampere` must be a `std::ratio` representing the exponent of amperes the unit has");
		static_assert(traits::is_ratio<Kelvin>::value, "Template parameter `Kelvin` must be a `std::ratio` representing the exponent of kelvin the unit has");
		static_assert(traits::is_ratio<Candela>::value, "Template parameter `Candela` must be a `std::ratio` representing the exponent of candelas the unit has");
		static_assert(traits::is_ratio<Mole>::value, "Template parameter `Mole` must be a `std::ratio` representing the exponent of moles the unit has");
		static_assert(traits::is_ratio<Radian>::value, "Template parameter `Radian` must be a `std::ratio` representing the exponent of radians the unit has");
	};

	//------------------------------
	//	UNIT CATEGORIES
	//------------------------------

	/**
	 * @brief		namespace representing the implemented base and derived unit types. These will not generally be needed by library users.
	 * @sa			base_unit for the definition of the category prarameters.
	 */
	namespace category
	{
		// SCALAR (DIMENSIONLESS) TYPES
		using scalar_unit = base_unit<>;			///< Represents a quantity with no dimension.
		using dimensionless_unit = base_unit<>;		///< Represents a quantity with no dimension.

		// SI BASE UNIT TYPES	--------------------		METERS			KILOGRAMS		SECONDS			RADIANS			AMPERES			KELVIN			MOLE			CANDELA			
		using	length_unit						=	base_unit<std::ratio<1>>;																														///< Represents an SI base unit of length
		using	mass_unit						=	base_unit<std::ratio<0>,	std::ratio<1>>;																										///< Represents an SI base unit of mass
		using	time_unit						=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;																						///< Represents an SI base unit of time
		using	angle_unit						=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;																		///< Represents an SI base unit of angle
		using	current_unit					=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;														///< Represents an SI base unit of current
		using	temperature_unit				=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;										///< Represents an SI base unit of temperature
		using	substance_unit					=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;						///< Represents an SI base unit of amount of substance
		using	luminous_intensity_unit			=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;		///< Represents an SI base unit of luminous intensity

		// SI DERIVED UNIT TYPES	---------------				METERS			KILOGRAMS		SECONDS			RADIANS			AMPERES			KELVIN			MOLE			CANDELA			
		using	solid_angle_unit				=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<2>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>>;		///< Represents an SI derived unit of solid angle
		using	frequency_unit					=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<-1>>;																					///< Represents an SI derived unit of frequency
		using	velocity_unit					=	base_unit<std::ratio<1>,	std::ratio<0>,	std::ratio<-1>>;																					///< Represents an SI derived unit of velocity
		using	angular_velocity_unit			=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<-1>,	std::ratio<1>>;																		///< Represents an SI derived unit of angular velocity
		using	acceleration_unit				=	base_unit<std::ratio<1>,	std::ratio<0>,	std::ratio<-2>>;																					///< Represents an SI derived unit of acceleration
		using	force_unit						=	base_unit<std::ratio<1>,	std::ratio<1>,	std::ratio<-2>>;																					///< Represents an SI derived unit of force
		using	pressure_unit					=	base_unit<std::ratio<-1>,	std::ratio<1>,	std::ratio<-2>>;																					///< Represents an SI derived unit of pressure
		using	charge_unit						=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<1>,	std::ratio<0>,	std::ratio<1>>;														///< Represents an SI derived unit of charge
		using	energy_unit						=	base_unit<std::ratio<2>,	std::ratio<1>,	std::ratio<-2>>;																					///< Represents an SI derived unit of energy
		using	power_unit						=	base_unit<std::ratio<2>,	std::ratio<1>,	std::ratio<-3>>;																					///< Represents an SI derived unit of power
		using	voltage_unit					=	base_unit<std::ratio<2>,	std::ratio<1>,	std::ratio<-3>,	std::ratio<0>,	std::ratio<-1>>;													///< Represents an SI derived unit of voltage
		using	capacitance_unit				=	base_unit<std::ratio<-2>,	std::ratio<-1>,	std::ratio<4>,	std::ratio<0>,	std::ratio<2>>;														///< Represents an SI derived unit of capacitance
		using	impedance_unit					=	base_unit<std::ratio<2>,	std::ratio<1>,	std::ratio<-3>,	std::ratio<0>,	std::ratio<-2>>;													///< Represents an SI derived unit of impedance
		using	conductance_unit				=	base_unit<std::ratio<-2>,	std::ratio<-1>,	std::ratio<3>,	std::ratio<0>,	std::ratio<2>>;														///< Represents an SI derived unit of conductance
		using	magnetic_flux_unit				=	base_unit<std::ratio<2>,	std::ratio<1>,	std::ratio<-2>,	std::ratio<0>,	std::ratio<-1>>;													///< Represents an SI derived unit of magnetic flux
		using	magnetic_field_strength_unit	=	base_unit<std::ratio<0>,	std::ratio<1>,	std::ratio<-2>,	std::ratio<0>,	std::ratio<-1>>;													///< Represents an SI derived unit of magnetic field strength
		using	inductance_unit					=	base_unit<std::ratio<2>,	std::ratio<1>,	std::ratio<-2>,	std::ratio<0>,	std::ratio<-2>>;													///< Represents an SI derived unit of inductance
		using	luminous_flux_unit				=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<2>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;		///< Represents an SI derived unit of luminous flux
		using	illuminance_unit				=	base_unit<std::ratio<-2>,	std::ratio<0>,	std::ratio<0>,	std::ratio<2>,	std::ratio<0>,	std::ratio<0>,	std::ratio<0>,	std::ratio<1>>;		///< Represents an SI derived unit of illuminance
		using	radioactivity_unit				=	base_unit<std::ratio<0>,	std::ratio<0>,	std::ratio<-1>>;																					///< Represents an SI derived unit of radioactivity

		// OTHER UNIT TYPES			---------------				METERS			KILOGRAMS		SECONDS			RADIANS			AMPERES			KELVIN			MOLE			CANDELA			
		using	torque_unit						=	base_unit<std::ratio<2>,	std::ratio<1>,	std::ratio<-2>>;																					///< Represents an SI derived unit of torque
		using	area_unit						=	base_unit<std::ratio<2>>;																														///< Represents an SI derived unit of area
		using	volume_unit						=	base_unit<std::ratio<3>>;																														///< Represents an SI derived unit of volume
		using	density_unit					=	base_unit<std::ratio<-3>,	std::ratio<1>>;																										///< Represents an SI derived unit of density
		using	concentration_unit				=	base_unit<>;
	}

	//------------------------------
	//	UNIT CLASSES
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	/**
	 * @brief		unit type template specialization for units derived from base units.
	 */
	template <class, class, class, class> struct unit;
	template<class Conversion, class... Exponents, class PiExponent, class Translation>
	struct unit<Conversion, base_unit<Exponents...>, PiExponent, Translation> : units::detail::_unit
	{
		static_assert(traits::is_ratio<Conversion>::value, "Template parameter `Conversion` must be a `std::ratio` representing the conversion factor to `BaseUnit`.");
		static_assert(traits::is_ratio<PiExponent>::value, "Template parameter `PiExponent` must be a `std::ratio` representing the exponents of Pi the unit has.");
		static_assert(traits::is_ratio<Translation>::value, "Template parameter `Translation` must be a `std::ratio` representing an additive translation required by the unit conversion.");

		typedef typename units::base_unit<Exponents...> base_unit_type;
		typedef Conversion conversion_ratio;
		typedef Translation translation_ratio;
		typedef PiExponent pi_exponent_ratio;
	};
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @brief		Type representing an arbitrary unit.
	 * @ingroup		UnitTypes
	 * @details		`unit` types are used as tags for the `conversion` function. They are *not* containers
	 *				(see `unit_t` for a  container class). Each unit is defined by:
	 *
	 *				- A `std::ratio` defining the conversion factor to the base unit type. (e.g. `std::ratio<1,12>` for inches to feet)
	 *				- A base unit that the unit is derived from (or a unit category. Must be of type `unit` or `base_unit`)
	 *				- An exponent representing factors of PI required by the conversion. (e.g. `std::ratio<-1>` for a radians to degrees conversion)
	 *				- a ratio representing a datum translation required for the conversion (e.g. `std::ratio<32>` for a farenheit to celsius conversion)
	 *
	 *				Typically, a specific unit, like `meters`, would be implemented as a type alias
	 *				of `unit`, i.e. `using meters = unit<std::ratio<1>, units::category::length_unit`, or
	 *				`using inches = unit<std::ratio<1,12>, feet>`.
	 * @tparam		Conversion	std::ratio representing scalar multiplication factor.
	 * @tparam		BaseUnit	Unit type which this unit is derived from. May be a `base_unit`, or another `unit`.
	 * @tparam		PiExponent	std::ratio representing the exponent of pi required by the conversion.
	 * @tparam		Translation	std::ratio representing any datum translation required by the conversion.
	 */
	template<class Conversion, class BaseUnit, class PiExponent = std::ratio<0>, class Translation = std::ratio<0>>
	struct unit : units::detail::_unit
	{
		static_assert(traits::is_unit<BaseUnit>::value, "Template parameter `BaseUnit` must be a `unit` type.");
		static_assert(traits::is_ratio<Conversion>::value, "Template parameter `Conversion` must be a `std::ratio` representing the conversion factor to `BaseUnit`.");
		static_assert(traits::is_ratio<PiExponent>::value, "Template parameter `PiExponent` must be a `std::ratio` representing the exponents of Pi the unit has.");

		typedef typename units::traits::unit_traits<BaseUnit>::base_unit_type base_unit_type;
		typedef typename std::ratio_multiply<typename BaseUnit::conversion_ratio, Conversion> conversion_ratio;
		typedef typename std::ratio_add<typename BaseUnit::pi_exponent_ratio, PiExponent> pi_exponent_ratio;
		typedef typename std::ratio_add<std::ratio_multiply<typename BaseUnit::conversion_ratio, Translation>, typename BaseUnit::translation_ratio> translation_ratio;
	};

	//------------------------------
	//	BASE UNIT MANIPULATORS
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
		 * @brief		base_unit_of trait implementation
		 * @details		recursively seeks base_unit type that a unit is derived from. Since units can be
		 *				derived from other units, the `base_unit_type` typedef may not represent this value.
		 */
		template<class> struct base_unit_of_impl;
		template<class Conversion, class BaseUnit, class PiExponent, class Translation>
		struct base_unit_of_impl<unit<Conversion, BaseUnit, PiExponent, Translation>> : base_unit_of_impl<BaseUnit> {};
		template<class... Exponents>
		struct base_unit_of_impl<base_unit<Exponents...>>
		{
			typedef base_unit<Exponents...> type;
		};
		template<>
		struct base_unit_of_impl<void>
		{
			typedef void type;
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	namespace traits
	{
		/**
		 * @brief		Trait which returns the `base_unit` type that a unit is originally derived from.
		 * @details		Since units can be derived from other `unit` types in addition to `base_unit` types,
		 *				the `base_unit_type` typedef will not always be a `base_unit` (or unit category).
		 *				Since compatible
		 */
		template<class U>
		using base_unit_of = typename units::detail::base_unit_of_impl<U>::type;
	}

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
		 * @brief		implementation of base_unit_multiply
		 * @details		'multiples' (adds exponent ratios of) two base unit types. Base units can be found
		 *				using `base_unit_of`.
		 */
		template<class, class> struct base_unit_multiply_impl;
		template<class... Exponents1, class... Exponents2>
		struct base_unit_multiply_impl<base_unit<Exponents1...>, base_unit<Exponents2...>> {
			using type = base_unit<std::ratio_add<Exponents1, Exponents2>...>;
		};

		/**
		 * @brief		represents type of two base units multiplied together
		 */
		template<class U1, class U2>
		using base_unit_multiply = typename base_unit_multiply_impl<U1, U2>::type;

		/**
		 * @brief		implementation of base_unit_divide
		 * @details		'dived' (subtracts exponent ratios of) two base unit types. Base units can be found
		 *				using `base_unit_of`.
		 */
		template<class, class> struct base_unit_divide_impl;
		template<class... Exponents1, class... Exponents2>
		struct base_unit_divide_impl<base_unit<Exponents1...>, base_unit<Exponents2...>> {
			using type = base_unit<std::ratio_subtract<Exponents1, Exponents2>...>;
		};

		/**
		 * @brief		represents the resulting type of `base_unit` U1 divided by U2.
		 */
		template<class U1, class U2>
		using base_unit_divide = typename base_unit_divide_impl<U1, U2>::type;

		/**
		 * @brief		implementation of inverse_base
		 * @details		multiplies all `base_unit` exponent ratios by -1. The resulting type represents
		 *				the inverse base unit of the given `base_unit` type.
		 */
		template<class> struct inverse_base_impl;

		template<class... Exponents>
		struct inverse_base_impl<base_unit<Exponents...>> {
			using type = base_unit<std::ratio_multiply<Exponents, std::ratio<-1>>...>;
		};

		/**
		 * @brief		represent the inverse type of `class U`
		 * @details		E.g. if `U` is `length_unit`, then `inverse<U>` will represent `length_unit^-1`.
		 */
		template<class U> using inverse_base = typename inverse_base_impl<U>::type;

		/**
		 * @brief		implementation of `squared_base`
		 * @details		multiplies all the exponent ratios of the given class by 2. The resulting type is
		 *				equivalent to the given type squared.
		 */
		template<class U> struct squared_base_impl;
		template<class... Exponents>
		struct squared_base_impl<base_unit<Exponents...>> {
			using type = base_unit<std::ratio_multiply<Exponents, std::ratio<2>>...>;
		};

		/**
		 * @brief		represents the type of a `base_unit` squared.
		 * @details		E.g. `squared<length_unit>` will represent `length_unit^2`.
		 */
		template<class U> using squared_base = typename squared_base_impl<U>::type;

		/**
		 * @brief		implementation of `cubed_base`
		 * @details		multiplies all the exponent ratios of the given class by 3. The resulting type is
		 *				equivalent to the given type cubed.
		 */
		template<class U> struct cubed_base_impl;
		template<class... Exponents>
		struct cubed_base_impl<base_unit<Exponents...>> {
			using type = base_unit<std::ratio_multiply<Exponents, std::ratio<3>>...>;
		};

		/**
		 * @brief		represents the type of a `base_unit` cubed.
		 * @details		E.g. `cubed<length_unit>` will represent `length_unit^3`.
		 */
		template<class U> using cubed_base = typename cubed_base_impl<U>::type;

		/**
		 * @brief		implementation of `sqrt_base`
		 * @details		divides all the exponent ratios of the given class by 2. The resulting type is
		 *				equivalent to the square root of the given type.
		 */
		template<class U> struct sqrt_base_impl;
		template<class... Exponents>
		struct sqrt_base_impl<base_unit<Exponents...>> {
			using type = base_unit<std::ratio_divide<Exponents, std::ratio<2>>...>;
		};

		/**
		 * @brief		represents the square-root type of a `base_unit`.
		 * @details		E.g. `sqrt<length_unit>` will represent `length_unit^(1/2)`.
		 */
		template<class U> using sqrt_base = typename sqrt_base_impl<U>::type;

		/**
		 * @brief		implementation of `cbrt_base`
		 * @details		divides all the exponent ratios of the given class by 3. The resulting type is
		 *				equivalent to the given type's cube-root.
		 */
		template<class U> struct cbrt_base_impl;
		template<class... Exponents>
		struct cbrt_base_impl<base_unit<Exponents...>> {
			using type = base_unit<std::ratio_multiply<Exponents, std::ratio<3>>...>;
		};

		/**
		 * @brief		represents the cube-root type of a `base_unit` .
		 * @details		E.g. `cbrt<length_unit>` will represent `length_unit^(1/3)`.
		 */
		template<class U> using cbrt_base = typename cbrt_base_impl<U>::type;
	}
	/** @endcond */	// END DOXYGEN IGNORE

	//------------------------------
	//	UNIT MANIPULATORS
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
		 * @brief		implementation of `unit_multiply`.
		 * @details		multiplies two units. The base unit becomes the base units of each with their exponents
		 *				added together. The conversion factors of each are multiplied by each other. Pi exponent ratios
		 *				are added, and datum translations are removed.
		 */
		template<class Unit1, class Unit2>
		struct unit_multiply_impl
		{
			using type = unit < std::ratio_multiply<typename Unit1::conversion_ratio, typename Unit2::conversion_ratio>,
				base_unit_multiply <traits::base_unit_of<typename Unit1::base_unit_type>, traits::base_unit_of<typename Unit2::base_unit_type>>,
				std::ratio_add<typename Unit1::pi_exponent_ratio, typename Unit2::pi_exponent_ratio>,
				std::ratio < 0 >> ;
		};

		/**
		 * @brief		represents the type of two units multiplied together.
		 * @details		recalculates conversion and exponent ratios at compile-time.
		 */
		template<class U1, class U2>
		using unit_multiply = typename unit_multiply_impl<U1, U2>::type;

		/**
		 * @brief		implementation of `unit_divide`.
		 * @details		divides two units. The base unit becomes the base units of each with their exponents
		 *				subtracted from each other. The conversion factors of each are divided by each other. Pi exponent ratios
		 *				are subtracted, and datum translations are removed.
		 */
		template<class Unit1, class Unit2>
		struct unit_divide_impl
		{
			using type = unit < std::ratio_divide<typename Unit1::conversion_ratio, typename Unit2::conversion_ratio>,
				base_unit_divide<traits::base_unit_of<typename Unit1::base_unit_type>, traits::base_unit_of<typename Unit2::base_unit_type>>,
				std::ratio_subtract<typename Unit1::pi_exponent_ratio, typename Unit2::pi_exponent_ratio>,
				std::ratio < 0 >> ;
		};

		/**
		 * @brief		represents the type of two units divided by each other.
		 * @details		recalculates conversion and exponent ratios at compile-time.
		 */
		template<class U1, class U2>
		using unit_divide = typename unit_divide_impl<U1, U2>::type;

		/**
		 * @brief		implementation of `inverse`
		 * @details		inverts a unit (equivalent to 1/unit). The `base_unit` and pi exponents are all multiplied by
		 *				-1. The conversion ratio numerator and denominator are swapped. Datum translation
		 *				ratios are removed.
		 */
		template<class Unit>
		struct inverse_impl
		{
			using type = unit < std::ratio<Unit::conversion_ratio::den, Unit::conversion_ratio::num>,
				inverse_base<traits::base_unit_of<typename units::traits::unit_traits<Unit>::base_unit_type>>,
				std::ratio_multiply<typename units::traits::unit_traits<Unit>::pi_exponent_ratio, std::ratio<-1>>,
				std::ratio < 0 >> ;	// inverses are rates or change, the translation factor goes away.
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @brief		represents the inverse unit type of `class U`.
	 * @ingroup		UnitManipulators
	 * @tparam		U	`unit` type to invert.
	 * @details		E.g. `inverse<meters>` will represent meters^-1 (i.e. 1/meters).
	 */
	template<class U> using inverse = typename units::detail::inverse_impl<U>::type;

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
		 * @brief		implementation of `squared`
		 * @details		Squares the conversion ratio, `base_unit` exponents, pi exponents, and removes
		 *				datum translation ratios.
		 */
		template<class Unit>
		struct squared_impl
		{
			static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
			using Conversion = typename Unit::conversion_ratio;
			using type = unit < std::ratio_multiply<Conversion, Conversion>,
				squared_base<traits::base_unit_of<typename Unit::base_unit_type>>,
				std::ratio_multiply<typename Unit::pi_exponent_ratio, std::ratio<2>>,
				std::ratio < 0 >> ;
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @brief		represents the unit type of `class U` squared
	 * @ingroup		UnitManipulators
	 * @tparam		U	`unit` type to square.
	 * @details		E.g. `square<meters>` will represent meters^2.
	 */
	template<class U>
	using squared = typename units::detail::squared_impl<U>::type;

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
			 * @brief		implementation of `cubed`
			 * @details		Cubes the conversion ratio, `base_unit` exponents, pi exponents, and removes
			 *				datum translation ratios.
			 */
		template<class Unit>
		struct cubed_impl
		{
			static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
			using Conversion = typename Unit::conversion_ratio;
			using type = unit < std::ratio_multiply<Conversion, std::ratio_multiply<Conversion, Conversion>>,
				cubed_base<traits::base_unit_of<typename Unit::base_unit_type>>,
				std::ratio_multiply<typename Unit::pi_exponent_ratio, std::ratio<3>>,
				std::ratio < 0 >> ;
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @brief		represents the type of `class U` cubed.
	 * @ingroup		UnitManipulators
	 * @tparam		U	`unit` type to cube.
	 * @details		E.g. `cubed<meters>` will represent meters^3.
	 */
	template<class U>
	using cubed = typename units::detail::cubed_impl<U>::type;

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		//----------------------------------
		//	RATIO_SQRT IMPLEMENTATION
		//----------------------------------

		using Zero = std::ratio<0>;
		using One = std::ratio<1>;
		template <typename R> using Square = std::ratio_multiply<R, R>;

		// Find the largest std::integer N such that Predicate<N>::value is true.
		template <template <std::intmax_t N> class Predicate, typename enabled = void>
		struct BinarySearch {
			template <std::intmax_t N>
			struct SafeDouble_ {
				static const std::intmax_t value = 2 * N;
				static_assert(value > 0, "Overflows when computing 2 * N");
			};

			template <intmax_t Lower, intmax_t Upper, typename Condition1 = void, typename Condition2 = void>
			struct DoubleSidedSearch_ : DoubleSidedSearch_<Lower, Upper,
				typename std::conditional<(Upper - Lower == 1), std::true_type, std::false_type>::type,
				typename std::conditional<((Upper - Lower>1 && Predicate<Lower + (Upper - Lower) / 2>::value)), std::true_type, std::false_type>::type> {};

			template <intmax_t Lower, intmax_t Upper>
			struct DoubleSidedSearch_<Lower, Upper, std::false_type, std::false_type> : DoubleSidedSearch_<Lower, Lower + (Upper - Lower) / 2> {};

			template <intmax_t Lower, intmax_t Upper, typename Condition2>
			struct DoubleSidedSearch_<Lower, Upper, std::true_type, Condition2> : std::integral_constant<intmax_t, Lower>{};

			template <intmax_t Lower, intmax_t Upper, typename Condition1>
			struct DoubleSidedSearch_<Lower, Upper, Condition1, std::true_type> : DoubleSidedSearch_<Lower + (Upper - Lower) / 2, Upper>{};

			template <std::intmax_t Lower, class enabled1 = void>
			struct SingleSidedSearch_ : SingleSidedSearch_<Lower, typename std::conditional<Predicate<SafeDouble_<Lower>::value>::value, std::true_type, std::false_type>::type>{};

			template <std::intmax_t Lower>
			struct SingleSidedSearch_<Lower, std::false_type> : DoubleSidedSearch_<Lower, SafeDouble_<Lower>::value> {};

			template <std::intmax_t Lower>
			struct SingleSidedSearch_<Lower, std::true_type> : SingleSidedSearch_<SafeDouble_<Lower>::value>{};

			const static std::intmax_t value = SingleSidedSearch_<1>::value;
 		};

		template <template <std::intmax_t N> class Predicate>
		struct BinarySearch<Predicate, typename std::enable_if<!Predicate<1>::value>::type> : std::integral_constant<std::intmax_t, 0>{};

		// Find largest std::integer N such that N<=sqrt(R)
		template <typename R>
		struct Integer {
			template <std::intmax_t N> using Predicate_ = std::ratio_less_equal<std::ratio<N>, std::ratio_divide<R, std::ratio<N>>>;
			const static std::intmax_t value = BinarySearch<Predicate_>::value;
		};

		template <typename R>
		struct IsPerfectSquare {
			const static std::intmax_t DenSqrt_ = Integer<std::ratio<R::den>>::value;
			const static std::intmax_t NumSqrt_ = Integer<std::ratio<R::num>>::value;
			const static bool value =( DenSqrt_ * DenSqrt_ == R::den && NumSqrt_ * NumSqrt_ == R::num);
			using Sqrt = std::ratio<NumSqrt_, DenSqrt_>;
		};

		// Represents sqrt(P)-Q.
		template <typename Tp, typename Tq>
		struct Remainder {
			using P = Tp;
			using Q = Tq;
		};

		// Represents 1/R = I + Rem where R is a Remainder.
		template <typename R>
		struct Reciprocal {
			using P_ = typename R::P;
			using Q_ = typename R::Q;
			using Den_ = std::ratio_subtract<P_, Square<Q_>>;
			using A_ = std::ratio_divide<Q_, Den_>;
			using B_ = std::ratio_divide<P_, Square<Den_>>;
			const static std::intmax_t I_ = (A_::num + Integer<std::ratio_multiply<B_, Square<std::ratio<A_::den>>>>::value) / A_::den;
			using I = std::ratio<I_>;
			using Rem = Remainder<B_, std::ratio_subtract<I, A_>>;
		};

		// Expands sqrt(R) to continued fraction:
		// f(x)=C1+1/(C2+1/(C3+1/(...+1/(Cn+x)))) = (U*x+V)/(W*x+1) and sqrt(R)=f(Rem).
		// The error |f(Rem)-V| = |(U-W*V)x/(W*x+1)| <= |U-W*V|*Rem <= |U-W*V|/I' where
		// I' is the std::integer part of reciprocal of Rem.
		template <typename Tr, std::intmax_t N>
		struct ContinuedFraction {
			template <typename T>
			using Abs_ = typename std::conditional<std::ratio_less<T, Zero>::value, std::ratio_subtract<Zero, T>, T>::type;

			using R = Tr;
			using Last_ = ContinuedFraction<R, N - 1>;
			using Reciprocal_ = Reciprocal<typename Last_::Rem>;
			using Rem = typename Reciprocal_::Rem;
			using I_ = typename Reciprocal_::I;
			using Den_ = std::ratio_add<typename Last_::W, I_>;
			using U = std::ratio_divide<typename Last_::V, Den_>;
			using V = std::ratio_divide<std::ratio_add<typename Last_::U, std::ratio_multiply<typename Last_::V, I_>>, Den_>;
			using W = std::ratio_divide<One, Den_>;
			using Error = Abs_<std::ratio_divide<std::ratio_subtract<U, std::ratio_multiply<V, W>>, typename Reciprocal<Rem>::I>>;
		};

		template <typename Tr>
		struct ContinuedFraction<Tr, 1> {
			using R = Tr;
			using U = One;
			using V = std::ratio<Integer<R>::value>;
			using W = Zero;
			using Rem = Remainder<R, V>;
			using Error = std::ratio_divide<One, typename Reciprocal<Rem>::I>;
		};

		template <typename R, typename Eps, std::intmax_t N = 1, typename enabled = void>
		struct Sqrt_ : Sqrt_<R, Eps, N + 1> {};

		template <typename R, typename Eps, std::intmax_t N>
		struct Sqrt_<R, Eps, N, typename std::enable_if<std::ratio_less_equal<typename ContinuedFraction<R, N>::Error, Eps>::value>::type> {
			using type = typename ContinuedFraction<R, N>::V;
		};

		template <typename R, typename Eps, typename enabled = void>
		struct Sqrt {
			static_assert(std::ratio_greater_equal<R, Zero>::value, "R can't be negative");
		};

		template <typename R, typename Eps>
		struct Sqrt<R, Eps, typename std::enable_if<std::ratio_greater_equal<R, Zero>::value && IsPerfectSquare<R>::value>::type> {
			using type = typename IsPerfectSquare<R>::Sqrt;
		};

		template <typename R, typename Eps>
		struct Sqrt<R, Eps, typename std::enable_if<(std::ratio_greater_equal<R, Zero>::value && !IsPerfectSquare<R>::value)>::type> : Sqrt_<R, Eps>{};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @ingroup		TypeTraits
	 * @brief		Calculate square root of a ratio at compile-time
	 * @details		Calculates a rational approximation of the square root of the ratio. The error
	 *				in the calculation is bounded by 1/epsilon (Eps). E.g. for the default value
	 *				of 10000000000, the maximum error will be a/10000000000, or 1e-8, or said another way,
	 *				the error will be on the order of 10^-9. Since these calculations are done at 
	 *				compile time, it is advisable to set epsilon to the highest value that does not
	 *				cause an integer overflow in the calculation. If you can't compile `ratio_sqrt` 
	 *				due to overflow errors, reducing the value of epsilon sufficiently will correct
	 *				the problem.\n\n
	 *				`ratio_sqrt` is guaranteed to converge for all values of `Ratio` which do not
	 *				overflow. 
	 * @note		This function provides a rational approximation, _NOT_ an exact value.
	 * @tparam		Ratio	ratio to take the square root of. This can represent any rational value,
	 *						_not_ just integers or values with integer roots.
	 * @tparam		Eps		Value of epsilon, which represents the inverse of the maximum allowable
	 *						error. This value should be chosen to be as high as possible before
	 *						integer overflow errors occur in the compiler.
	 */
	template<typename Ratio, std::intmax_t Eps = 10000000000>
	using ratio_sqrt = typename  units::detail::Sqrt<Ratio, std::ratio<1, Eps>>::type;

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
		 * @brief		implementation of `sqrt`
		 * @details		square roots the conversion ratio, `base_unit` exponents, pi exponents, and removes
		 *				datum translation ratios.
		 */
		template<class Unit, std::intmax_t Eps>
		struct sqrt_impl
		{
			static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
			using Conversion = typename Unit::conversion_ratio;
			using type = unit <ratio_sqrt<Conversion, Eps>,
				sqrt_base<traits::base_unit_of<typename Unit::base_unit_type>>,
				std::ratio_divide<typename Unit::pi_exponent_ratio, std::ratio<2>>,
				std::ratio < 0 >>;
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**	 
	 * @ingroup		UnitManipulators
	 * @brief		represents the square root of type `class U`.
	 * @details		Calculates a rational approximation of the square root of the unit. The error
	 *				in the calculation is bounded by 1/epsilon (Eps). E.g. for the default value
	 *				of 10000000000, the maximum error will be a/10000000000, or 1e-8, or said another way,
	 *				the error will be on the order of 10^-9. Since these calculations are done at
	 *				compile time, it is advisable to set epsilon to the highest value that does not
	 *				cause an integer overflow in the calculation. If you can't compile `ratio_sqrt`
	 *				due to overflow errors, reducing the value of epsilon sufficiently will correct
	 *				the problem.\n\n
	 *				`ratio_sqrt` is guaranteed to converge for all values of `Ratio` which do not
	 *				overflow.
	 * @tparam		U	`unit` type to take the square root of.
	 * @tparam		Eps	Value of epsilon, which represents the inverse of the maximum allowable
	 *					error. This value should be chosen to be as high as possible before
	 *					integer overflow errors occur in the compiler. 
	 * @note		USE WITH CAUTION. The is an approximate value. In general, squared<sqrt<meter>> != meter,
	 *				i.e. the operation is not reversible, and it will result in propogated approximations.
	 *				Use only when absolutely necessary.
	 */
	template<class U, std::intmax_t Eps = 10000000000>
	using square_root = typename units::detail::sqrt_impl<U, Eps>::type;

	//------------------------------
	//	COMPOUND UNITS
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
			 * @brief		implementation of compound_unit
			 * @details		multiplies a variadic list of units together, and is inherited from the resulting
			 *				type.
			 */
		template<class U, class... Us> struct compound_impl;
		template<class U> struct compound_impl<U> { using type = U; };
		template<class U1, class U2, class...Us>
		struct compound_impl<U1, U2, Us...>
			: compound_impl<unit_multiply<U1, U2>, Us...> {};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @brief		Represents a unit type made up from other units.
	 * @details		Compound units are formed by multiplying the units of all the types provided in
	 *				the template argument. Types provided must inherit from `unit`. A compound unit can
	 *				be formed from any number of other units, and unit manipulators like `inverse` and
	 *				`squared` are supported. E.g. to specify acceleration, on could create
	 *				`using acceleration = compound_unit<length::meters, inverse<squared<seconds>>;`
	 * @tparam		U...	units which, when multiplied together, form the desired compound unit.
	 * @ingroup		UnitTypes
	 */
	template<class U, class... Us>
	using compound_unit = typename units::detail::compound_impl<U, Us...>::type;

	//------------------------------
	//	PREFIXES
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/**
			 * @brief		prefix applicator.
			 * @details		creates a unit type from a prefix and a unit
			 */
		template<class Ratio, class Unit>
		struct prefix
		{
			static_assert(traits::is_ratio<Ratio>::value, "Template parameter `Ratio` must be a `std::ratio`.");
			static_assert(traits::is_unit<Unit>::value, "Template parameter `Unit` must be a `unit` type.");
			typedef typename units::unit<Ratio, Unit> type;
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @ingroup UnitManipulators
	 * @{
	 * @ingroup Prefixes
	 * @{
	 */
	template<class U> using atto = typename units::detail::prefix<std::atto, U>::type;								///< Represents the type of `class U` with the metric 'atto' prefix appended.  @details E.g. atto<meters> represents meters*10^-21	@tparam U unit type to apply the prefix to.
	template<class U> using femto = typename units::detail::prefix<std::femto, U>::type;							///< Represents the type of `class U` with the metric 'atto' prefix appended.  @details E.g. femto<meters> represents meters*10^-18	@tparam U unit type to apply the prefix to.
	template<class U> using pico = typename units::detail::prefix<std::pico, U>::type;								///< Represents the type of `class U` with the metric 'femto' prefix appended. @details E.g. pico<meters> represents meters*10^-15	@tparam U unit type to apply the prefix to.
	template<class U> using nano = typename units::detail::prefix<std::nano, U>::type;								///< Represents the type of `class U` with the metric 'pico' prefix appended.  @details E.g. nano<meters> represents meters*10^-12	@tparam U unit type to apply the prefix to.
	template<class U> using micro = typename units::detail::prefix<std::micro, U>::type;							///< Represents the type of `class U` with the metric 'nano' prefix appended.  @details E.g. micro<meters> represents meters*10^-9	@tparam U unit type to apply the prefix to.
	template<class U> using milli = typename units::detail::prefix<std::milli, U>::type;							///< Represents the type of `class U` with the metric 'micro' prefix appended. @details E.g. milli<meters> represents meters*10^-6	@tparam U unit type to apply the prefix to.
	template<class U> using centi = typename units::detail::prefix<std::centi, U>::type;							///< Represents the type of `class U` with the metric 'milli' prefix appended. @details E.g. centi<meters> represents meters*10^-3	@tparam U unit type to apply the prefix to.
	template<class U> using deci = typename units::detail::prefix<std::deci, U>::type;								///< Represents the type of `class U` with the metric 'centi' prefix appended. @details E.g. deci<meters> represents meters*10^-2	@tparam U unit type to apply the prefix to.
	template<class U> using deca = typename units::detail::prefix<std::deca, U>::type;								///< Represents the type of `class U` with the metric 'deci' prefix appended.  @details E.g. deca<meters> represents meters*10^-1	@tparam U unit type to apply the prefix to.
	template<class U> using hecto = typename units::detail::prefix<std::hecto, U>::type;							///< Represents the type of `class U` with the metric 'deca' prefix appended.  @details E.g. hecto<meters> represents meters*10^1	@tparam U unit type to apply the prefix to.
	template<class U> using kilo = typename units::detail::prefix<std::kilo, U>::type;								///< Represents the type of `class U` with the metric 'hecto' prefix appended. @details E.g. kilo<meters> represents meters*10^2	@tparam U unit type to apply the prefix to.
	template<class U> using mega = typename units::detail::prefix<std::mega, U>::type;								///< Represents the type of `class U` with the metric 'kilo' prefix appended.  @details E.g. mega<meters> represents meters*10^3	@tparam U unit type to apply the prefix to.
	template<class U> using giga = typename units::detail::prefix<std::giga, U>::type;								///< Represents the type of `class U` with the metric 'mega' prefix appended.  @details E.g. giga<meters> represents meters*10^6	@tparam U unit type to apply the prefix to.
	template<class U> using tera = typename units::detail::prefix<std::tera, U>::type;								///< Represents the type of `class U` with the metric 'giga' prefix appended.  @details E.g. tera<meters> represents meters*10^9	@tparam U unit type to apply the prefix to.
	template<class U> using peta = typename units::detail::prefix<std::peta, U>::type;								///< Represents the type of `class U` with the metric 'tera' prefix appended.  @details E.g. peta<meters> represents meters*10^12	@tparam U unit type to apply the prefix to.
	template<class U> using exa = typename units::detail::prefix<std::exa, U>::type;								///< Represents the type of `class U` with the metric 'peta' prefix appended.  @details E.g. exa<meters> represents meters*10^15	@tparam U unit type to apply the prefix to.
	/** @} @} */

	//------------------------------
	//	CONVERSION TRAITS
	//------------------------------

	namespace traits
	{
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which checks whether two units can be converted to each other
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_convertible_unit<U1, U2>::value` to test
		 *				whether `class U1` is convertible to `class U2`. Note: convertible has both the semantic meaning,
		 *				(i.e. meters can be converted to feet), and the c++ meaning of conversion (type meters can be
		 *				converted to type feet). Conversion is always symmetric, so if U1 is convertible to U2, then
		 *				U2 will be convertible to U1.
		 * @tparam		U1 Unit to convert from.
		 * @tparam		U2 Unit to convert to.
		 * @sa			is_convertible_unit_t
		 */
		template<class U1, class U2>
		struct is_convertible_unit : std::is_same <traits::base_unit_of<typename units::traits::unit_traits<U1>::base_unit_type>,
			base_unit_of<typename units::traits::unit_traits<U2>::base_unit_type >> {};
	}

	//------------------------------
	//	CONVERSION FUNCTION
	//------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/// convert dispatch for units which are both the same
		template<class, class, typename T>
		static inline T convert(const T& value, std::true_type, std::false_type, std::false_type)
		{
			return value;
		}

		/// convert dispatch for units which are both the same
		template<class, class, typename T>
		static inline T convert(const T& value, std::true_type, std::false_type, std::true_type)
		{
			return value;
		}

		/// convert dispatch for units which are both the same
		template<class, class, typename T>
		static inline T convert(const T& value, std::true_type, std::true_type, std::false_type)
		{
			return value;
		}

		/// convert dispatch for units which are both the same
		template<class, class, typename T>
		static inline T convert(const T& value, std::true_type, std::true_type, std::true_type)
		{
			return value;
		}

		/// convert dispatch for units of different types w/ no translation and no PI
		template<class UnitFrom, class UnitTo, typename T>
		static inline T convert(const T& value, std::false_type, std::false_type, std::false_type)
		{
			using Ratio = std::ratio_divide<typename UnitFrom::conversion_ratio, typename UnitTo::conversion_ratio>;
			return ((UNIT_LIB_DEFAULT_TYPE)(Ratio::num) * value / Ratio::den);
		}

		/// convert dispatch for units of different types w/ no translation, but has PI
		template<class UnitFrom, class UnitTo, typename T>
		static inline T convert(const T& value, std::false_type, std::true_type, std::false_type)
		{
			using Ratio = std::ratio_divide<typename UnitFrom::conversion_ratio, typename UnitTo::conversion_ratio>;
			using PiRatio = std::ratio_subtract<typename UnitFrom::pi_exponent_ratio, typename UnitTo::pi_exponent_ratio>;
			return (((UNIT_LIB_DEFAULT_TYPE)(Ratio::num) * value / Ratio::den) * std::pow(constants::PI, ((UNIT_LIB_DEFAULT_TYPE)(PiRatio::num) / PiRatio::den)));
		}

		/// convert dispatch for units of different types with a translation, but no PI
		template<class UnitFrom, class UnitTo, typename T>
		static inline T convert(const T& value, std::false_type, std::false_type, std::true_type)
		{
			using Ratio = std::ratio_divide<typename UnitFrom::conversion_ratio, typename UnitTo::conversion_ratio>;
			using Translation = std::ratio_divide<std::ratio_subtract<typename UnitFrom::translation_ratio, typename UnitTo::translation_ratio>, typename UnitTo::conversion_ratio>;
			return (((UNIT_LIB_DEFAULT_TYPE)(Ratio::num) * value / Ratio::den) + ((UNIT_LIB_DEFAULT_TYPE)(Translation::num) / Translation::den));
		}

		/// convert dispatch for units of different types with a translation AND PI
		template<class UnitFrom, class UnitTo, typename T>
		static inline T convert(const T& value, std::false_type, std::true_type, std::true_type)
		{
			using Ratio = std::ratio_divide<typename UnitFrom::conversion_ratio, typename UnitTo::conversion_ratio>;
			using Translation = std::ratio_divide<std::ratio_subtract<typename UnitFrom::translation_ratio, typename UnitTo::translation_ratio>, typename UnitTo::conversion_ratio>;
			using PiRatio = std::ratio_subtract<typename UnitFrom::pi_exponent_ratio, typename UnitTo::pi_exponent_ratio>;
			return (((UNIT_LIB_DEFAULT_TYPE)(Ratio::num) * value / Ratio::den) * std::pow(constants::PI, ((UNIT_LIB_DEFAULT_TYPE)(PiRatio::num) / PiRatio::den)) + ((UNIT_LIB_DEFAULT_TYPE)(Translation::num) / Translation::den));
		}
	}
	/** @endcond */	// END DOXYGEN IGNORE


	/**
	 * @ingroup		Conversion
	 * @brief		converts a <i>value</i> from one type to another.
	 * @details		Converts a <i>value</i> of a built-in arithmetic type to another unit. This does not change
	 *				the type of <i>value</i>, only what it contains. E.g. @code double result = convert<length::meters, length::feet>(1.0);	// result == 3.28084 @endcode
	 * @sa			unit_t	for implicit conversion of unit containers.
	 * @tparam		UnitFrom unit tag to convert <i>value</i> from. Must be a `unit` type (i.e. is_unit<UnitFrom>::value == true),
	 *				and must be convertible to `UnitTo` (i.e. is_converitble_unit<UnitFrom, UnitTo>::value == true).
	 * @tparam		UnitTo unit tag to convert <i>value</i> to. Must be a `unit` type (i.e. is_unit<UnitTo>::value == true),
	 *				and must be convertible from `UnitFrom` (i.e. is_converitble_unit<UnitFrom, UnitTo>::value == true).
	 * @tparam		T type of <i>value</i>. It is infered from <i>value</i>, and is expected to be a built-in arethmetic type.
	 * @param[in]	value Arithmetic value to convert from `UnitFrom` to `UnitTo`. The value should represent
	 *				a quantity in units of `UnitFrom`.
	 * @returns		value, converted from units of `UnitFrom` to `UnitTo`.
	 */
	template<class UnitFrom, class UnitTo, typename T = UNIT_LIB_DEFAULT_TYPE>
	static inline T convert(const T& value)
	{
		static_assert(traits::is_unit<UnitFrom>::value, "Template parameter `UnitFrom` must be a `unit` type.");
		static_assert(traits::is_unit<UnitTo>::value, "Template parameter `UnitTo` must be a `unit` type.");
		static_assert(traits::is_convertible_unit<UnitFrom, UnitTo>::value, "Units are not compatible.");

		using isSame = typename std::is_same<typename std::decay<UnitFrom>::type, typename std::decay<UnitTo>::type>::type;
		using piRequired = std::integral_constant<bool, !(std::is_same<std::ratio<0>, typename UnitFrom::pi_exponent_ratio>::value &&
			std::is_same<std::ratio<0>, typename UnitTo::pi_exponent_ratio>::value)>;
		using translationRequired = std::integral_constant<bool, !(std::is_same<std::ratio<0>, typename UnitFrom::translation_ratio>::value &&
			std::is_same<std::ratio<0>, typename UnitTo::translation_ratio>::value)>;

		return units::detail::convert<UnitFrom, UnitTo, T>(value, isSame{}, piRequired{}, translationRequired{});
	};

	//----------------------------------
	//	NON-LINEAR SCALE TRAITS
	//----------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace traits
	{
		namespace detail
		{
			/**
			* @brief		implementation of has_operator_parenthesis
			* @details		checks that operator() returns the same type as `Ret`
			*/
			template<class T, class Ret>
			struct has_operator_parenthesis_impl
			{
				template<class U>
				static auto test(U*) -> decltype(std::declval<U>()());
				template<typename>
				static std::false_type test(...);

				using type = typename std::is_same<Ret, decltype(test<T>(0))>::type;
			};
		}

		/**
		 * @brief		checks that `class T` has an `operator()` member which returns `Ret`
		 * @details		used as part of the linear_scale concept.
		 */
		template<class T, class Ret>
		struct has_operator_parenthesis : traits::detail::has_operator_parenthesis_impl<T, Ret>::type {};
	}

	namespace traits
	{
		namespace detail
		{
			/**
			* @brief		implementation of has_value_member
			* @details		checks for a member named `m_member` with type `Ret`
			*/
			template<class T, class Ret>
			struct has_value_member_impl
			{
				template<class U>
				static auto test(U* p) -> decltype(p->m_value);
				template<typename>
				static auto test(...)->std::false_type;

				using type = typename std::is_same<typename std::decay<Ret>::type, typename std::decay<decltype(test<T>(0))>::type>::type;
			};
		}

		/**
		 * @brief		checks for a member named `m_member` with type `Ret`
		 * @details		used as part of the linear_scale concept checker.
		 */
		template<class T, class Ret>
		struct has_value_member : traits::detail::has_value_member_impl<T, Ret>::type {};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	namespace traits
	{
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests that `class T` meets the requirements for a non-linear scale
		 * @details		A non-linear scale must:
		 *				- be default constructible
		 *				- have an `operator()` member which returns the non-linear value stored in the scale
		 *				- have an accessible `m_value` member type which stores the linearized value in the scale.
		 *
		 *				Linear/nonlinear scales are used by `units::unit` to store values and scale them
		 *				if they represent things like dB.
		 */
		template<class T, class Ret>
		struct is_nonlinear_scale : std::integral_constant<bool,
			std::is_default_constructible<T>::value &&
			has_operator_parenthesis<T, Ret>::value &&
			has_value_member<T, Ret>::value &&
			std::is_trivial<T>::value>
		{};
	}

	//------------------------------
	//	UNIT_T TYPE TRAITS
	//------------------------------

	namespace traits
	{
#ifdef FOR_DOXYGEN_PURPOSOES_ONLY
		/**
		* @ingroup		TypeTraits
		* @brief		Trait for accessing the publically defined types of `units::unit_t`
		* @details		The units library determines certain properties of the unit_t types passed to them
		*				and what they represent by using the members of the corresponding unit_t_traits instantiation.
		*/
		template<typename T>
		struct unit_t_traits
		{
			typedef typename T::non_linear_scale_type non_linear_scale_type;	///< Type of the unit_t non_linear_scale (e.g. linear_scale, decibel_scale). This property is used to enable the proper linear or logatirhmic arithmetic functions.
			typedef typename T::underlying_type underlying_type;				///< Underlying storage type of the `unit_t`, e.g. `double`.
			typedef typename T::unit_type unit_type;							///< Type of unit the `unit_t` represents, e.g. `meters`
		};
#endif

		/** @cond */	// DOXYGEN IGNORE
		/**
		 * @brief		unit_t_traits specialization for things which are not unit_t
		 * @details
		 */
		template<typename T, typename = void>
		struct unit_t_traits
		{
			typedef void non_linear_scale_type;
			typedef void underlying_type;
			typedef void unit_type;
		};
	
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait for accessing the publically defined types of `units::unit_t`
		 * @details
		 */
		template<typename T>
		struct unit_t_traits <T, typename void_t<
			typename T::non_linear_scale_type,
			typename T::underlying_type,
			typename T::unit_type>::type>
		{
			typedef typename T::non_linear_scale_type non_linear_scale_type;
			typedef typename T::underlying_type underlying_type;
			typedef typename T::unit_type unit_type;
		};
		/** @endcond */	// END DOXYGEN IGNORE
	}

	namespace traits
	{
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests whether two container types derived from `unit_t` are convertible to each other
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_convertible_unit_t<U1, U2>::value` to test
		 *				whether `class U1` is convertible to `class U2`. Note: convertible has both the semantic meaning,
		 *				(i.e. meters can be converted to feet), and the c++ meaning of conversion (type meters can be
		 *				converted to type feet). Conversion is always symmetric, so if U1 is convertible to U2, then
		 *				U2 will be convertible to U1.
		 * @tparam		U1 Unit to convert from.
		 * @tparam		U2 Unit to convert to.
		 * @sa			is_convertible_unit
		 */
		template<class U1, class U2>
		struct is_convertible_unit_t : std::integral_constant<bool,
			is_convertible_unit<typename units::traits::unit_t_traits<U1>::unit_type, typename units::traits::unit_t_traits<U2>::unit_type>::value>
		{};
	}

	//---------------------------------- 
	//	UNIT TYPE
	//----------------------------------

	/** @cond */	// DOXYGEN IGNORE
	// forward declaration
	template<typename T> struct linear_scale;

	namespace detail
	{
		/**
		* @brief		helper type to identify units.
		* @details		A non-templated base class for `unit` which enables RTTI testing.
		*/
		struct _unit_t {};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	namespace traits
	{
		// forward declaration
		#if !defined(_MSC_VER) || _MSC_VER > 1800 // bug in VS2013 prevents this from working
		template<typename... T> struct is_scalar_unit;
		#else
		template<typename, typename, typename> struct is_scalar_unit;
		#endif

		/**
		 * @ingroup		TypeTraits
		 * @brief		Traits which tests if a class is a `unit`
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_unit<T>::value` to test
		 *				whether `class T` implements a `unit`.
		 */
		template<class T>
		struct is_unit_t : std::is_base_of<units::detail::_unit_t, T>::type {};
	}

	/**
	 * @ingroup		UnitContainers
	 * @brief		Container for values which represent quantities of a given unit.
	 * @details		Stores a value which represents a quantity in the given units. Unit containers
	 *				(except scalar values) are *not* convertible to built-in c++ types, in order to
	 *				provide type safety in dimensional analysis. Unit containers *are* implicitely
	 *				convertible to other compatible unit container types. Unit containers support
	 *				various types of arithmetic operations, depending on their scale type.
	 *
	 *				The value of a `unit_t` can only be changed on construction, or by assignment
	 *				from another `unit_t` type. If necessary, the underlying value can be accessed
	 *				using `operator()`: @code
	 *				meter_t m(5.0);
	 *				double val = m(); // val == 5.0	@endcode.
	 * @tparam		Units unit tag for which type of units the `unit_t` represents (e.g. meters)
	 * @tparam		T underlying type of the storage. Defaults to double.
	 * @tparam		NonLinearScale optional scale class for the units. Defaults to linear (i.e. does
	 *				not scale the unit value). Examples of non-linear scales could be logarithmic,
	 *				decibel, or richter scales. Non-linear scales must adhere to the non-linear-scale
	 *				concept, i.e. `is_nonlinear_scale<...>::value` must be `true`.
	 * @sa
	 *				- \ref lengthContainers "length unit containers"
	 *				- \ref massContainers "mass unit containers"
	 *				- \ref timeContainers "time unit containers"
	 *				- \ref angleContainers "angle unit containers"
	 *				- \ref currentContainers "current unit containers"
	 *				- \ref temperatureContainers "temperature unit containers"
	 *				- \ref substanceContainers "substance unit containers"
	 *				- \ref luminousIntensityContainers "luminous intensity unit containers"
	 *				- \ref solidAngleContainers "solid angle unit containers"
	 *				- \ref frequencyContainers "frequency unit containers"
	 *				- \ref velocityContainers "velocity unit containers"
	 *				- \ref angularVelocityContainers "angular velocity unit containers"
	 *				- \ref accelerationContainers "acceleration unit containers"
	 *				- \ref forceContainers "force unit containers"
	 *				- \ref pressureContainers "pressure unit containers"
	 *				- \ref chargeContainers "charge unit containers"
	 *				- \ref energyContainers "energy unit containers"
	 *				- \ref powerContainers "power unit containers"
	 *				- \ref voltageContainers "voltage unit containers"
	 *				- \ref capacitanceContainers "capacitance unit containers"
	 *				- \ref impedanceContainers "impedance unit containers"
	 *				- \ref magneticFluxContainers "magnetic flux unit containers"
	 *				- \ref magneticFieldStrengthContainers "magnetic field strength unit containers"
	 *				- \ref inductanceContainers "inductance unit containers"
	 *				- \ref luminousFluxContainers "luminous flux unit containers"
	 *				- \ref illuminanceContainers "illuminance unit containers"
	 *				- \ref radiationContainers "radiation unit containers"
	 *				- \ref torqueContainers "torque unit containers"
	 *				- \ref areaContainers "area unit containers"
	 *				- \ref volumeContainers "volume unit containers"
	 *				- \ref densityContainers "density unit containers"
	 *				- \ref concentrationContainers "concentration unit containers"
	 *				- \ref constantContainers "constant unit containers"
	 */
	template<class Units, typename T = UNIT_LIB_DEFAULT_TYPE, template<typename> class NonLinearScale = linear_scale>
	class unit_t : public NonLinearScale<T>, units::detail::_unit_t
	{
		static_assert(traits::is_unit<Units>::value, "Template parameter `Units` must be a unit tag. Check that you aren't using a unit type (_t).");
		static_assert(traits::is_nonlinear_scale<NonLinearScale<T>, T>::value, "Template parameter `NonLinearScale` does not conform to the `is_nonlinear_scale` concept.");

	protected:

		using nls = NonLinearScale<T>;
		using nls::m_value;

	public:

		typedef NonLinearScale<T> non_linear_scale_type;											///< Type of the non-linear scale of the unit_t (e.g. linear_scale)
		typedef T underlying_type;																	///< Type of the underlying storage of the unit_t (e.g. double)
		typedef Units unit_type;																	///< Type of `unit` the `unit_t` represents (e.g. meters)

		/**
		 * @ingroup		Constructors
		 * @brief		default constructor.
		 */
		unit_t() = default;

		/**
		 * @brief		constructor
		 * @details		constructs a new unit_t using the non-linear scale's constructor.
		 * @param[in]	value	unit value magnitude.
		 * @param[in]	args	additional constructor arguments are forwarded to the non-linear scale constructor. Which
		 *						args are required depends on which scale is used. For the default (linear) scale,
		 *						no additional args are necessary.
		 */
		template<class... Args>
		inline explicit unit_t(T value, const Args&... args) : nls(value, args...) {};

		/**
		 * @brief		constructor
		 * @details		enable implicit conversions from T types ONLY for linear scalar units
		 * @param[in]	value value of the unit_t
		 */
		template<class Ty, class = typename std::enable_if<traits::is_scalar_unit<Units>::value && std::is_arithmetic<Ty>::value>::type>
		inline unit_t(Ty value) : nls(value) {};

		/**
		 * @brief		copy constructor
		 * @details		performs implicit unit conversions if required.
		 * @param[in]	rhs unit to copy.
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
		inline unit_t(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs)
		{
			nls::m_value = units::convert<UnitsRhs, Units, T>(rhs.m_value);
		};

		/**
		 * @brief		assignment
		 * @details		performs implicit unit conversions if required
		 * @param[in]	rhs unit to copy.
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
		inline unit_t& operator=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs)
		{
			nls::m_value = units::convert<UnitsRhs, Units, T>(rhs.m_value);
			return *this;
		}

		/**
		* @brief		assignment
		* @details		performs implicit conversions from built-in types ONLY for scalar units
		* @param[in]	rhs value to copy.
		*/
		template<class Ty, class = typename std::enable_if<traits::is_scalar_unit<Units>::value && std::is_arithmetic<Ty>::value>::type>
		inline unit_t& operator=(Ty rhs)
		{
			nls::m_value = rhs;
			return *this;
		}

		/**
		 * @brief		less-than
		 * @details		compares the linearized value of two units. Performs unit conversions if necessary.
		 * @param[in]	rhs right-hand side unit for the comparison
		 * @returns		true IFF the value of `this` is less than the value of `rhs`
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
		inline bool operator<(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const
		{
			return (nls::m_value < units::convert<UnitsRhs, Units>(rhs.m_value));
		}

		/**
		 * @brief		less-than or equal
		 * @details		compares the linearized value of two units. Performs unit conversions if necessary.
		 * @param[in]	rhs right-hand side unit for the comparison
		 * @returns		true IFF the value of `this` is less than or equal to the value of `rhs`
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
		inline bool operator<=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const
		{
			return std::islessequal(nls::m_value, units::convert<UnitsRhs, Units>(rhs.m_value));
		}

		/**
		 * @brief		greater-than
		 * @details		compares the linearized value of two units. Performs unit conversions if necessary.
		 * @param[in]	rhs right-hand side unit for the comparison
		 * @returns		true IFF the value of `this` is greater than the value of `rhs`
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
		inline bool operator>(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const
		{
			return (nls::m_value > units::convert<UnitsRhs, Units>(rhs.m_value));
		}

		/**
		 * @brief		greater-than or equal
		 * @details		compares the linearized value of two units. Performs unit conversions if necessary.
		 * @param[in]	rhs right-hand side unit for the comparison
		 * @returns		true IFF the value of `this` is greater than or equal to the value of `rhs`
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
		inline bool operator>=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const
		{
			return std::isgreaterequal(nls::m_value, units::convert<UnitsRhs, Units>(rhs.m_value));
		}

		/**
		 * @brief		equality
		 * @details		compares the linearized value of two units. Performs unit conversions if necessary.
		 * @param[in]	rhs right-hand side unit for the comparison
		 * @returns		true IFF the value of `this` exactly equal to the value of rhs.
		 * @note		This may not be suitable for all applications when the underlying_type of unit_t is a double.
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs, typename std::enable_if<std::is_floating_point<T>::value || std::is_floating_point<Ty>::value, int>::type = 0>
		inline bool operator==(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const
		{
			auto x = nls::m_value;
			auto y = units::convert<UnitsRhs, Units>(rhs.m_value);
			return std::abs(x - y) < std::numeric_limits<T>::epsilon() * std::abs(x + y) ||
				std::abs(x - y) < std::numeric_limits<T>::min();
		}

		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs, typename std::enable_if<std::is_integral<T>::value && std::is_integral<Ty>::value, int>::type = 0>
		inline bool operator==(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const
		{
			auto x = nls::m_value;
			auto y = units::convert<UnitsRhs, Units>(rhs.m_value);
			return x == y;
		}

		/**
		 * @brief		inequality
		 * @details		compares the linearized value of two units. Performs unit conversions if necessary.
		 * @param[in]		rhs right-hand side unit for the comparison
		 * @returns		true IFF the value of `this` is not equal to the value of rhs.
		 * @note		This may not be suitable for all applications when the underlying_type of unit_t is a double.
		 */
		template<class UnitsRhs, typename Ty, template<typename> class NlsRhs>
		inline bool operator!=(const unit_t<UnitsRhs, Ty, NlsRhs>& rhs) const
		{
			return !(*this == rhs);
		}

		/**
		 * @brief		unit value
		 * @returns		value of the unit in it's underlying, non-safe type.
		 */
		template<typename Ty, class = typename std::enable_if<std::is_arithmetic<Ty>::value>::type>
		inline Ty to() const
		{
			return (Ty)(*this)();
		}

		/**
		 * @brief		linearized unit value
		 * @returns		linearized value of unit which has a non-linear scale. For `unit_t` types with
		 *			linear scales, this is equivalent to `value`.
		 */
		template<typename Ty, class = typename std::enable_if<std::is_arithmetic<Ty>::value>::type>
		inline Ty toLinearized() const
		{
			return (Ty)m_value;
		}

		/**
		 * @brief		conversion
		 * @details		Converts to a different unit container. Units can be converted to other containers
		 *				implicitly, but this can be used in cases where explicit notation of a conversion
		 *				is beneficial, or where an r-value container is needed.
		 * @tparam		U unit (not unit_t) to convert to
		 * @returns		a unit container with the specified units containing the equivalent value to
		 *				*this.
		 */
		template<class U>
		inline auto convert() const -> unit_t<U>
		{
			static_assert(traits::is_unit<U>::value, "Template parameter `U` must be a unit type.");
			return unit_t<U>(*this);
		}

		/**
		 * @brief		implicit type conversion.
		 * @details		only enabled for scalar unit types.
		 */
		template<class Ty, class = typename std::enable_if<traits::is_scalar_unit<Units>::value && std::is_arithmetic<Ty>::value>::type>
		operator Ty() const { return  units::convert<Units, unit<std::ratio<1>, units::category::scalar_unit>>(nls::m_value); }

	public:

		template<class U, typename Ty, template<typename> class Nlt>
		friend class unit_t;

	};

	template<class Units, typename T, template<typename> class NonLinearScale>
	inline std::ostream& operator<<(std::ostream& os, const unit_t<Units, T, NonLinearScale>& obj)
	{
		os << obj();
		return os;
	}

	template<class Units, typename T, template<typename> class NonLinearScale>
	unit_t<Units, T, NonLinearScale> operator-(const unit_t<Units, T, NonLinearScale>& val)
	{
		return unit_t<Units, T, NonLinearScale>(-val());
	}

	//------------------------------
	//	UNIT_CAST
	//------------------------------	
	
	/** 
	 * @ingroup		Conversion
	 * @brief		Casts a unit container to an arithmetic type.
	 * @details		unit_cast can be used to remove the strong typing from a unit class, and convert it
	 *				to a built-in arithmetic type. This may be useful for compatibility with libraries
	 *				and legacy code that don't support `unit_t` types. E.g 
	 * @code		meter_t unitVal(5);
	 *  double value = units::unit_cast<double>(unitVal);	// value = 5.0 
	 * @endcode
	 * @tparam		T		Type to cast the unit type to. Must be a built-in arithmetic type.
	 * @param		value	Unit value to cast.
	 * @sa			unit_t::to
	 */
	template<typename T, typename Units, class = typename std::enable_if<std::is_arithmetic<T>::value && traits::is_unit_t<Units>::value>::type>
	inline T unit_cast(const Units& value)
	{
		return static_cast<T>(value());
	}

	//------------------------------
	//	NON-LINEAR SCALE TRAITS
	//------------------------------

	// forward declaration
	template<typename T> struct decibel_scale;

	namespace traits
	{
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests whether a type is inherited from a linear scale.
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `has_linear_scale<U1 [, U2, ...]>::value` to test
		 *				one or more types to see if they represent unit_t's whose scale is linear.
		 * @tparam		T	one or more types to test.
		 */
#if !defined(_MSC_VER) || _MSC_VER > 1800	// bug in VS2013 prevents this from working
		template<typename... T>
		struct has_linear_scale : std::integral_constant<bool, units::all_true<std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T>::underlying_type>, T>::value...>::value > {};
#else
		template<typename T1, typename T2 = T1, typename T3 = T1>
		struct has_linear_scale : std::integral_constant<bool,
			std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T1>::underlying_type>, T1>::value &&
			std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T2>::underlying_type>, T2>::value &&
			std::is_base_of<units::linear_scale<typename units::traits::unit_t_traits<T3>::underlying_type>, T3>::value> {};
#endif

		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests whether a type is inherited from a decibel scale.
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `has_decibel_scale<U1 [, U2, ...]>::value` to test
		 *				one or more types to see if they represent unit_t's whose scale is in decibels.
		 * @tparam		T	one or more types to test.
		 */
#if !defined(_MSC_VER) || _MSC_VER > 1800	// bug in VS2013 prevents this from working
		template<typename... T>
		struct has_decibel_scale : std::integral_constant<bool,	units::all_true<std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T>::underlying_type>, T>::value...>::value> {};
#else
		template<typename T1, typename T2 = T1, typename T3 = T1>
		struct has_decibel_scale : std::integral_constant<bool,
			std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T1>::underlying_type>, T1>::value &&
			std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T2>::underlying_type>, T2>::value &&
			std::is_base_of<units::decibel_scale<typename units::traits::unit_t_traits<T2>::underlying_type>, T3>::value> {};
#endif

		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests whether two types has the same non-linear scale.
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_same_scale<U1 , U2>::value` to test
		 *				whether two types have the same non-linear scale.
		 * @tparam		T1	left hand type.
		 * @tparam		T2	right hand type
		 */
		template<typename T1, typename T2>
		struct is_same_scale : std::integral_constant<bool,
			std::is_same<typename units::traits::unit_t_traits<T1>::non_linear_scale_type, typename units::traits::unit_t_traits<T2>::non_linear_scale_type>::value>
		{};
	}

	//----------------------------------
	//	NON-LINEAR SCALES
	//----------------------------------

	// Non-linear transforms are used to pre and post scale units which are defined in terms of non-
	// linear functions of their current value. A good example of a non-linear scale would be a 
	// logarithmic or decibel scale

	//------------------------------
	//	LINEAR SCALE
	//------------------------------

	/**
	 * @brief		unit_t scale which is linear
	 * @details		Represents units on a linear scale. This is the appropriate unit_t scale for almost
	 *				all units almost all of the time.
	 * @tparam		T	underlying storage type
	 * @sa			unit_t
	 */
	template<typename T>
	struct linear_scale
	{
		inline linear_scale() = default;															///< default constructor.
		inline linear_scale(T value) : m_value(value) {}											///< constructor.
		inline T operator()() const { return m_value; }												///< returns value.

		T m_value;																					///< linearized value.	
	};

	//----------------------------------
	//	SCALAR (LINEAR) UNITS
	//----------------------------------

	// Scalar units are the *ONLY* units implicitly convertible to/from built-in types.
	namespace dimensionless
	{
		using scalar = unit<std::ratio<1>, units::category::scalar_unit>;
		using dimensionless = unit<std::ratio<1>, units::category::dimensionless_unit>;

		using scalar_t = unit_t<scalar>;
		using dimensionless_t = scalar_t;
	}

	namespace traits
	{
		/** @cond */	// DOXYGEN IGNORE
		namespace detail
		{
			template<class T> struct is_scalar_unit_impl : std::false_type {};
			template<typename C, typename U, typename P, typename T>
			struct is_scalar_unit_impl<units::unit<C, U, P, T>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_traits<units::unit<C, U, P, T>>::base_unit_type>, units::category::scalar_unit>::type{};
			template<typename U, typename S, template<typename> class N>
			struct is_scalar_unit_impl<units::unit_t<U, S, N>> : std::is_same<units::traits::base_unit_of<typename units::traits::unit_t_traits<units::unit_t<U, S, N>>::unit_type>, units::category::scalar_unit>::type{};
		}
		/** @endcond */	// END DOXYGEN IGNORE

		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests whether one or more types derived from `unit_t` represent scalar values.
		 * @details		Inherits from `std::true_type` or `std::false_type`. Use `is_scalar_unit<U1 [, U2, ...]>::value` to test
		 *				one or more types to see if they represent scalar value containers. A scalar unit is one which has no
		 *				dimensions (e.g. PI).
		 * @tparam		T	one or more types to test.
		 */
#if !defined(_MSC_VER) || _MSC_VER > 1800 // bug in VS2013 prevents this from working
		template<typename... T> struct is_scalar_unit : std::integral_constant<bool, units::all_true<units::traits::detail::is_scalar_unit_impl<typename std::decay<T>::type>::value...>::value> {};
#else
		template<typename T1, typename T2 = units::dimensionless::scalar , typename T3 = units::dimensionless::scalar> 
		struct is_scalar_unit : std::integral_constant<bool, units::traits::detail::is_scalar_unit_impl<typename std::decay<T1>::type>::value && units::traits::detail::is_scalar_unit_impl<typename std::decay<T2>::type>::value && units::traits::detail::is_scalar_unit_impl<typename std::decay<T3>::type>::value> {};
#endif
	}

	//------------------------------
	//	LINEAR ARITHMETIC
	//------------------------------

	template<class UnitTypeLhs, class UnitTypeRhs, typename std::enable_if<!traits::is_same_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
	inline int operator+(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs)
	{
		static_assert(traits::is_same_scale<UnitTypeLhs, UnitTypeRhs>::value, "Cannot add units with different linear/non-linear scales.");
		return 0;
	}

	/// Addition operator for unit_t types with a linear_scale.
	template<class UnitTypeLhs, class UnitTypeRhs, typename std::enable_if<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
	inline UnitTypeLhs operator+(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs)
	{
		return UnitTypeLhs(lhs() + convert<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type, typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>(rhs()));
	}

	/// Addition operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
	template<typename T, typename std::enable_if<std::is_arithmetic<T>::value, int>::type = 0>
	inline dimensionless::scalar_t operator+(const dimensionless::scalar_t& lhs, T rhs)
	{
		return dimensionless::scalar_t(lhs() + rhs);
	}

	/// Addition operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
	template<typename T, typename std::enable_if<std::is_arithmetic<T>::value, int>::type = 0>
	inline dimensionless::scalar_t operator+(T lhs, const dimensionless::scalar_t& rhs)
	{
		return dimensionless::scalar_t(lhs + rhs());
	}

	/// Subtraction operator for unit_t types with a linear_scale.
	template<class UnitTypeLhs, class UnitTypeRhs, typename std::enable_if<traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
	inline UnitTypeLhs operator-(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs)
	{
		return UnitTypeLhs(lhs() - convert<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type, typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>(rhs()));
	}

	/// Subtraction operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
	template<typename T, typename std::enable_if<std::is_arithmetic<T>::value, int>::type = 0>
	inline dimensionless::scalar_t operator-(const dimensionless::scalar_t& lhs, T rhs)
	{
		return dimensionless::scalar_t(lhs() - rhs);
	}

	/// Subtraction operator for scalar unit_t types with a linear_scale. Scalar types can be implicitly converted to built-in types.
	template<typename T, typename std::enable_if<std::is_arithmetic<T>::value, int>::type = 0>
	inline dimensionless::scalar_t operator-(T lhs, const dimensionless::scalar_t& rhs)
	{
		return dimensionless::scalar_t(lhs - rhs());
	}

	/// Multiplication type for convertible unit_t types with a linear scale. @returns the multiplied value, with the same type as left-hand side unit.
	template<class UnitTypeLhs, class UnitTypeRhs,
		typename std::enable_if<traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
		inline auto operator*(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) -> unit_t<compound_unit<squared<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>>>
	{
		return  unit_t<compound_unit<squared<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>>>
			(lhs() * convert<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type, typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>(rhs()));
	}

	/// Multiplication type for convertible unit_t types with a linear scale. @returns the multiplied value, whose type is a compound unit of the left and right hand side values.
	template<class UnitTypeLhs, class UnitTypeRhs,
		typename std::enable_if<!traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
		inline auto operator*(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) -> unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
	{
		return unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
			(lhs() * rhs());
	}

	/// Multiplication by a scalar for unit_t types with a linear scale.
	template<class UnitTypeLhs, typename T,
		typename std::enable_if<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeLhs>::value, int>::type = 0>
		inline UnitTypeLhs operator*(const UnitTypeLhs& lhs, T rhs)
	{
		return UnitTypeLhs(lhs() * rhs);
	}

	/// Multiplication by a scalar for unit_t types with a linear scale.
	template<class UnitTypeRhs, typename T,
		typename std::enable_if<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeRhs>::value, int>::type = 0>
		inline UnitTypeRhs operator*(T lhs, const UnitTypeRhs& rhs)
	{
		return UnitTypeRhs(lhs * rhs());
	}

	/// Division for convertible unit_t types with a linear scale. @returns the lhs divided by rhs value, whose type is a scalar
	template<class UnitTypeLhs, class UnitTypeRhs,
		typename std::enable_if<traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
		inline dimensionless::scalar_t operator/(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs)
	{
		return dimensionless::scalar_t(lhs() / convert<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type, typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>(rhs()));
	}

	/// Division for non-convertible unit_t types with a linear scale. @returns the lhs divided by the rhs, with a compound unit type of lhs/rhs 
	template<class UnitTypeLhs, class UnitTypeRhs,
		typename std::enable_if<!traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value && traits::has_linear_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
		inline auto operator/(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) ->  unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>>
	{
		return unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>>
			(lhs() / rhs());
	}

	/// Division by a scalar for unit_t types with a linear scale
	template<class UnitTypeLhs, typename T,
		typename std::enable_if<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeLhs>::value, int>::type = 0>
		inline UnitTypeLhs operator/(const UnitTypeLhs& lhs, T rhs)
	{
		return UnitTypeLhs(lhs() / rhs);
	}

	/// Division of a scalar  by a unit_t type with a linear scale
	template<class UnitTypeRhs, typename T,
		typename std::enable_if<std::is_arithmetic<T>::value && traits::has_linear_scale<UnitTypeRhs>::value, int>::type = 0>
		inline auto operator/(T lhs, const UnitTypeRhs& rhs) -> unit_t<inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
	{
		return unit_t<inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>
			(lhs / rhs());
	}

	//----------------------------------
	//	SCALAR COMPARISONS
	//----------------------------------

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator==(UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs)
	{
		auto x = lhs;
		auto y = rhs();
		return std::abs(x-y) < std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::epsilon() * std::abs(x+y) ||
			std::abs(x-y) < std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::min();
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator==(const Units& lhs, UNIT_LIB_DEFAULT_TYPE rhs)
	{
		auto x = lhs();
		auto y = rhs;
		return std::abs(x-y) < std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::epsilon() * std::abs(x+y) ||
			std::abs(x-y) < std::numeric_limits<UNIT_LIB_DEFAULT_TYPE>::min();
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator!=(UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs)
	{
		return!(lhs == rhs);
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator!=(const Units& lhs, UNIT_LIB_DEFAULT_TYPE rhs)
	{
		return !(lhs == rhs);
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator>=(UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs)
	{
		return std::isgreaterequal(lhs, rhs());
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator>=(const Units& lhs, UNIT_LIB_DEFAULT_TYPE rhs)
	{
		return std::isgreaterequal(lhs(), rhs);
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator>(UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs)
	{
		return lhs > rhs();
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator>(const Units& lhs, UNIT_LIB_DEFAULT_TYPE rhs)
	{
		return lhs() > rhs;
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator<=(UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs)
	{
		return std::islessequal(lhs, rhs());
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator<=(const Units& lhs, UNIT_LIB_DEFAULT_TYPE rhs)
	{
		return std::islessequal(lhs(), rhs);
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator<(UNIT_LIB_DEFAULT_TYPE lhs, const Units& rhs)
	{
		return lhs < rhs();
	}

	template<typename Units, class = typename std::enable_if<units::traits::is_scalar_unit<Units>::value>::type>
	bool operator<(const Units& lhs, UNIT_LIB_DEFAULT_TYPE rhs)
	{
		return lhs() < rhs;
	}

	//----------------------------------
	//	POW
	//----------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		/// recursive exponential implementation
		template <int N, class U> 
		struct power_of_ratio
		{
			typedef std::ratio_multiply<U, typename power_of_ratio<N - 1, U>::type> type;
		};

		/// End recursion
		template <class U>
		struct power_of_ratio<1, U>
		{
			typedef U type;
		};

		/// recursive exponential implementation
		template <int N, class U> struct power_of_unit
		{
			typedef typename units::detail::unit_multiply<U, typename power_of_unit<N - 1, U>::type> type;
		};

		/// End recursion
		template <class U> struct power_of_unit<1, U>
		{
			typedef U type;
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	namespace math
	{
		/**
		 * @brief		computes the value of <i>value</i> raised to the <i>power</i>
		 * @details		Only implemented for linear_scale units. <i>Power</i> must be known at compile time, so the resulting unit type can be deduced.
		 * @tparam		power exponential power to raise <i>value</i> by.
		 * @param[in]	value `unit_t` derived type to raise to the given <i>power</i>
		 * @returns		new unit_t, raised to the given exponent
		 */
		template<int power, class UnitType, class = typename std::enable_if<traits::has_linear_scale<UnitType>::value, int>>
		inline auto pow(const UnitType& value) -> unit_t<typename units::detail::power_of_unit<power, typename units::traits::unit_t_traits<UnitType>::unit_type>::type, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
		{
			return unit_t<typename units::detail::power_of_unit<power, typename units::traits::unit_t_traits<UnitType>::unit_type>::type, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
				(std::pow(value(), power));
		}
	}

	//------------------------------
	//	DECIBEL SCALE
	//------------------------------

	/**
	* @brief		unit_t scale for representing decibel values.
	* @details		internally stores linearized values. `operator()` returns the value in dB.
	* @tparam		T	underlying storage type
	* @sa			unit_t
	*/
	template<typename T>
	struct decibel_scale
	{
		inline decibel_scale() = default;
		inline decibel_scale(T value) { m_value = std::pow(10, value / 10); }
		inline T operator()() const { return 10 * std::log10(m_value); }

		T m_value;	///< linearized value	
	};

	//------------------------------
	//	SCALAR (DECIBEL) UNITS
	//------------------------------

	/**
	 * @brief		namespace for unit types and containers for units that have no dimension (scalar units)
	 * @sa			See unit_t for more information on unit type containers.
	 */
	namespace dimensionless
	{
		using dB_t = unit_t<scalar, UNIT_LIB_DEFAULT_TYPE, decibel_scale>;
		inline std::ostream& operator<<(std::ostream& os, const dB_t& obj) { os << obj() << " dB"; return os; };

		using dBi_t = dB_t;
	}

	//------------------------------
	//	DECIBEL ARITHMETIC
	//------------------------------

	/// Addition for convertible unit_t types with a decibel_scale
	template<class UnitTypeLhs, class UnitTypeRhs,
		typename std::enable_if<traits::has_decibel_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
		inline auto operator+(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) -> unit_t<compound_unit<squared<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>>, typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type, decibel_scale>
	{
		using LhsUnits = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
		using RhsUnits = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
		using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;

		unit_t<compound_unit<squared<LhsUnits>>, underlying_type, decibel_scale> ret;
		reinterpret_cast<decibel_scale<underlying_type>&>(ret).m_value = lhs.template toLinearized<underlying_type>() * convert<RhsUnits, LhsUnits>(rhs.template toLinearized<underlying_type>());
		return ret;
	}

	/// Addition between unit_t types with a decibel_scale and dimensionless dB units
	template<class UnitTypeLhs, typename std::enable_if<traits::has_decibel_scale<UnitTypeLhs>::value && !traits::is_scalar_unit<UnitTypeLhs>::value, int>::type = 0>
	inline UnitTypeLhs operator+(const UnitTypeLhs& lhs, const dimensionless::dB_t& rhs)
	{
		using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;

		UnitTypeLhs ret;
		reinterpret_cast<decibel_scale<underlying_type>&>(ret).m_value = lhs.template toLinearized<underlying_type>() * rhs.template toLinearized<underlying_type>();
		return ret;
	}

	/// Addition between unit_t types with a decibel_scale and dimensionless dB units
	template<class UnitTypeRhs, typename std::enable_if<traits::has_decibel_scale<UnitTypeRhs>::value && !traits::is_scalar_unit<UnitTypeRhs>::value, int>::type = 0>
	inline UnitTypeRhs operator+(const dimensionless::dB_t& lhs, const UnitTypeRhs& rhs)
	{
		using underlying_type = typename units::traits::unit_t_traits<UnitTypeRhs>::underlying_type;

		UnitTypeRhs ret;
		reinterpret_cast<decibel_scale<underlying_type>&>(ret).m_value = lhs.template toLinearized<underlying_type>() * rhs.template toLinearized<underlying_type>();
		return ret;
	}

	/// Subtraction for convertible unit_t types with a decibel_scale
	template<class UnitTypeLhs, class UnitTypeRhs, typename std::enable_if<traits::has_decibel_scale<UnitTypeLhs, UnitTypeRhs>::value, int>::type = 0>
	inline auto operator-(const UnitTypeLhs& lhs, const UnitTypeRhs& rhs) -> unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>>, typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type, decibel_scale>
	{
		using LhsUnits = typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type;
		using RhsUnits = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
		using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;

		unit_t<compound_unit<LhsUnits, inverse<RhsUnits>>, underlying_type, decibel_scale> ret;
		reinterpret_cast<decibel_scale<underlying_type>&>(ret).m_value = lhs.template toLinearized<underlying_type>() / convert<RhsUnits, LhsUnits>(rhs.template toLinearized<underlying_type>());
		return ret;
	}

	/// Subtraction between unit_t types with a decibel_scale and dimensionless dB units
	template<class UnitTypeLhs, typename std::enable_if<traits::has_decibel_scale<UnitTypeLhs>::value && !traits::is_scalar_unit<UnitTypeLhs>::value, int>::type = 0>
	inline UnitTypeLhs operator-(const UnitTypeLhs& lhs, const dimensionless::dB_t& rhs)
	{
		using underlying_type = typename units::traits::unit_t_traits<UnitTypeLhs>::underlying_type;

		UnitTypeLhs ret;
		reinterpret_cast<decibel_scale<underlying_type>&>(ret).m_value = lhs.template toLinearized<underlying_type>() / rhs.template toLinearized<underlying_type>();
		return ret;
	}

	/// Subtraction between unit_t types with a decibel_scale and dimensionless dB units
	template<class UnitTypeRhs, typename std::enable_if<traits::has_decibel_scale<UnitTypeRhs>::value && !traits::is_scalar_unit<UnitTypeRhs>::value, int>::type = 0>
	inline auto operator-(const dimensionless::dB_t& lhs, const UnitTypeRhs& rhs) -> unit_t<inverse<typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type>, typename units::traits::unit_t_traits<UnitTypeRhs>::underlying_type, decibel_scale>
	{
		using RhsUnits = typename units::traits::unit_t_traits<UnitTypeRhs>::unit_type;
		using underlying_type = typename units::traits::unit_t_traits<RhsUnits>::underlying_type;

		unit_t<inverse<RhsUnits>, underlying_type, decibel_scale> ret;
		reinterpret_cast<decibel_scale<underlying_type>&>(ret).m_value = lhs.template toLinearized<underlying_type>() / rhs.template toLinearized<underlying_type>();
		return ret;
	}

	//----------------------------------
	//	UNIT RATIO CLASS
	//----------------------------------

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		template<class Units>
		struct _unit_value_t {};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	namespace traits
	{
#ifdef FOR_DOXYGEN_PURPOSES_ONLY
		/**
		* @ingroup		TypeTraits
		* @brief		Trait for accessing the publically defined types of `units::unit_value_t_traits`
		* @details		The units library determines certain properties of the `unit_value_t` types passed to 
		*				them and what they represent by using the members of the corresponding `unit_value_t_traits`
		*				instantiation.
		*/
		template<typename T>
		struct unit_value_t_traits
		{
			typedef typename T::unit_type unit_type;	///< Dimension represented by the `unit_value_t`.
			typedef typename T::ratio ratio;			///< Quantity represented by the `unit_value_t`, expressed as arational number.
		};
#endif

		/** @cond */	// DOXYGEN IGNORE
		/**
		 * @brief		unit_value_t_traits specialization for things which are not unit_t
		 * @details
		 */
		template<typename T, typename = void>
		struct unit_value_t_traits
		{
			typedef void unit_type;
			typedef void ratio;
		};
	
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait for accessing the publically defined types of `units::unit_value_t_traits`
		 * @details
		 */
		template<typename T>
		struct unit_value_t_traits <T, typename void_t<
			typename T::unit_type,
			typename T::ratio>::type>
		{
			typedef typename T::unit_type unit_type;
			typedef typename T::ratio ratio;
		};
		/** @endcond */	// END DOXYGEN IGNORE
	}

	//------------------------------------------------------------------------------
	//	COMPILE-TIME UNIT VALUES AND ARITHMETIC
	//------------------------------------------------------------------------------

	/**
	 * @ingroup		UnitContainers
	 * @brief		Stores a rational unit value as a compile-time constant
	 * @details		unit_value_t is useful for performing compile-time arithmetic on known 
	 *				unit quantities.
	 * @tparam		Units	units represented by the `unit_value_t`
	 * @tparam		Num		numerator of the represented value.
	 * @tparam		Denom	denominator of the represented value.
	 * @sa			unit_value_t_traits to access information about the properties of the class,
	 *				such as it's unit type and rational value.
	 * @note		This is intentionally identical in concept to a `std::ratio`.
	 *
	 */
	template<typename Units, std::uintmax_t Num, std::uintmax_t Denom = 1>
	struct unit_value_t : units::detail::_unit_value_t<Units>
	{
		typedef Units unit_type;
		typedef std::ratio<Num, Denom> ratio;

		static_assert(traits::is_unit<Units>::value, "Template parameter `Units` must be a unit type.");
		static const unit_t<Units> value() { return unit_t<Units>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den); }
	};

	namespace traits
	{
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests whether a type is a unit_value_t representing the given unit type.
		 * @details		e.g. `is_unit_value_t<meters, myType>::value` would test that `myType` is a 
		 *				`unit_value_t<meters>`.
		 * @tparam		Units	units that the `unit_value_t` is supposed to have.
		 * @tparam		T		type to test.
		 */
		template<typename T, typename Units = typename traits::unit_value_t_traits<T>::unit_type>
		struct is_unit_value_t : std::integral_constant<bool, 
			std::is_base_of<units::detail::_unit_value_t<Units>, T>::value>
		{};
	
		/**
		 * @ingroup		TypeTraits
		 * @brief		Trait which tests whether type T is a unit_value_t with a unit type in the given category.
		 * @details		e.g. `is_unit_value_t_category<units::category::length, unit_value_t<feet>>::value` would be true
		 */
		template<typename Category, typename T>
		struct is_unit_value_t_category : std::integral_constant<bool,
			std::is_same<units::traits::base_unit_of<typename traits::unit_value_t_traits<T>::unit_type>, Category>::value>
		{
			static_assert(is_base_unit<Category>::value, "Template parameter `Category` must be a `base_unit` type.");
		};
	}

	/** @cond */	// DOXYGEN IGNORE
	namespace detail
	{
		// base class for common arithmetic
		template<class U1, class U2>
		struct unit_value_arithmetic
		{
			static_assert(traits::is_unit_value_t<U1>::value, "Template parameter `U1` must be a `unit_value_t` type.");
			static_assert(traits::is_unit_value_t<U2>::value, "Template parameter `U1` must be a `unit_value_t` type.");

			using _UNIT1 = typename traits::unit_value_t_traits<U1>::unit_type;
			using _UNIT2 = typename traits::unit_value_t_traits<U2>::unit_type;
			using _CONV1 = typename units::traits::unit_traits<_UNIT1>::conversion_ratio;
			using _CONV2 = typename units::traits::unit_traits<_UNIT2>::conversion_ratio;
			using _RATIO1 = typename traits::unit_value_t_traits<U1>::ratio;
			using _RATIO2 = typename traits::unit_value_t_traits<U2>::ratio;
			using _RATIO2CONV = typename std::ratio_divide<std::ratio_multiply<_RATIO2, _CONV2>, _CONV1>;
			using _PI_EXP = std::ratio_subtract<typename units::traits::unit_traits<_UNIT2>::pi_exponent_ratio, typename units::traits::unit_traits<_UNIT1>::pi_exponent_ratio>;
		};
	}
	/** @endcond */	// END DOXYGEN IGNORE

	/**
	 * @ingroup		CompileTimeUnitManipulators
	 * @brief		adds two unit_value_t types at compile-time
	 * @details		The resulting unit will the the `unit_type` of `U1`
	 * @tparam		U1	left-hand `unit_value_t`
	 * @tparam		U2	right-hand `unit_value_t`
	 * @sa			unit_value_t_traits to access information about the properties of the class,
	 *				such as it's unit type and rational value.
	 * @note		very similar in concept to `std::ratio_add`
	 */
	template<class U1, class U2>
	struct unit_value_add : units::detail::unit_value_arithmetic<U1, U2>, units::detail::_unit_value_t<typename traits::unit_value_t_traits<U1>::unit_type>
	{
		/** @cond */	// DOXYGEN IGNORE
		using Base = units::detail::unit_value_arithmetic<U1, U2>;
		typedef typename Base::_UNIT1 unit_type;
		using ratio = std::ratio_add<typename Base::_RATIO1, typename Base::_RATIO2CONV>;

		static_assert(traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, "Unit types are not compatible.");
		/** @endcond */	// END DOXYGEN IGNORE


		/**
		 * @brief		Value of sum
		 * @details		Returns the calculated value of the sum of `U1` and `U2`, in the same
		 *				units as `U1`.
		 * @returns		Value of the sum in the appropriate units.
		 */
		 static const unit_t<unit_type> value()
		{
			using UsePi = typename std::conditional<Base::_PI_EXP::num != 0, std::true_type, std::false_type>::type;
			return value(UsePi());
		}

		/** @cond */	// DOXYGEN IGNORE
		// value if PI isn't involved
		static const unit_t<unit_type> value(std::false_type) 
		{ 
			return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
		}

		// value if PI *is* involved
		static const unit_t<unit_type> value(std::true_type)
		{
			return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO1::num / Base::_RATIO1::den) +
			((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO2CONV::num / Base::_RATIO2CONV::den) * std::pow(units::constants::PI, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
		}
		/** @endcond */	// END DOXYGEN IGNORE
	};

	/**
	 * @ingroup		CompileTimeUnitManipulators
	 * @brief		subtracts two unit_value_t types at compile-time
	 * @details		The resulting unit will the the `unit_type` of `U1`
	 * @tparam		U1	left-hand `unit_value_t`
	 * @tparam		U2	right-hand `unit_value_t`
	 * @sa			unit_value_t_traits to access information about the properties of the class,
	 *				such as it's unit type and rational value.
	 * @note		very similar in concept to `std::ratio_subtract`
	 */
	template<class U1, class U2>
	struct unit_value_subtract : units::detail::unit_value_arithmetic<U1, U2>, units::detail::_unit_value_t<typename traits::unit_value_t_traits<U1>::unit_type>
	{
		/** @cond */	// DOXYGEN IGNORE
		using Base = units::detail::unit_value_arithmetic<U1, U2>;

		typedef typename Base::_UNIT1 unit_type;
		using ratio = std::ratio_subtract<typename Base::_RATIO1, typename Base::_RATIO2CONV>;

		static_assert(traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, "Unit types are not compatible.");
		/** @endcond */	// END DOXYGEN IGNORE

		/**
		 * @brief		Value of difference
		 * @details		Returns the calculated value of the difference of `U1` and `U2`, in the same
		 *				units as `U1`.
		 * @returns		Value of the difference in the appropriate units.
		 */
		static const unit_t<unit_type> value()
		{
			using UsePi = typename std::conditional<Base::_PI_EXP::num != 0, std::true_type, std::false_type>::type;
			return value(UsePi());
		}

		/** @cond */	// DOXYGEN IGNORE
		// value if PI isn't involved
		static const unit_t<unit_type> value(std::false_type)
		{
			return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
		}

		// value if PI *is* involved
		static const unit_t<unit_type> value(std::true_type)
		{
			return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO1::num / Base::_RATIO1::den) - ((UNIT_LIB_DEFAULT_TYPE)Base::_RATIO2CONV::num / Base::_RATIO2CONV::den)
				* std::pow(units::constants::PI, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
		}
		/** @endcond */	// END DOXYGEN IGNORE	};
	};

	/**
	 * @ingroup		CompileTimeUnitManipulators
	 * @brief		multiplies two unit_value_t types at compile-time
	 * @details		The resulting unit will the the `unit_type` of `U1 * U2`
	 * @tparam		U1	left-hand `unit_value_t`
	 * @tparam		U2	right-hand `unit_value_t`
	 * @sa			unit_value_t_traits to access information about the properties of the class,
	 *				such as it's unit type and rational value.
	 * @note		very similar in concept to `std::ratio_multiply`
	 */
	template<class U1, class U2>
	struct unit_value_multiply : units::detail::unit_value_arithmetic<U1, U2>,
		units::detail::_unit_value_t<typename std::conditional<traits::is_convertible_unit<typename traits::unit_value_t_traits<U1>::unit_type,
			typename traits::unit_value_t_traits<U2>::unit_type>::value, compound_unit<squared<typename traits::unit_value_t_traits<U1>::unit_type>>, 
			compound_unit<typename traits::unit_value_t_traits<U1>::unit_type, typename traits::unit_value_t_traits<U2>::unit_type>>::type>
	{
		/** @cond */	// DOXYGEN IGNORE
		using Base = units::detail::unit_value_arithmetic<U1, U2>;
		
		using unit_type = typename std::conditional<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, compound_unit<squared<typename Base::_UNIT1>>, compound_unit<typename Base::_UNIT1, typename Base::_UNIT2>>::type;
		using ratio = typename std::conditional<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, std::ratio_multiply<typename Base::_RATIO1, typename Base::_RATIO2CONV>, std::ratio_multiply<typename Base::_RATIO1, typename Base::_RATIO2>>::type;
		/** @endcond */	// END DOXYGEN IGNORE

		/**
		 * @brief		Value of product
		 * @details		Returns the calculated value of the product of `U1` and `U2`, in units
		 *				of `U1 x U2`.
		 * @returns		Value of the product in the appropriate units.
		 */
		static const unit_t<unit_type> value()
		{
			using UsePi = typename std::conditional<Base::_PI_EXP::num != 0, std::true_type, std::false_type>::type;
			return value(UsePi());
		}

		/** @cond */	// DOXYGEN IGNORE
		// value if PI isn't involved
		static const unit_t<unit_type> value(std::false_type)
		{
			return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
		}

		// value if PI *is* involved
		static const unit_t<unit_type> value(std::true_type)
		{
			return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::PI, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
		}
		/** @endcond */	// END DOXYGEN IGNORE
	};

	/**
	 * @ingroup		CompileTimeUnitManipulators
	 * @brief		divides two unit_value_t types at compile-time
	 * @details		The resulting unit will the the `unit_type` of `U1`
	 * @tparam		U1	left-hand `unit_value_t`
	 * @tparam		U2	right-hand `unit_value_t`
	 * @sa			unit_value_t_traits to access information about the properties of the class,
	 *				such as it's unit type and rational value.
	 * @note		very similar in concept to `std::ratio_divide`
	 */
	template<class U1, class U2>
	struct unit_value_divide : units::detail::unit_value_arithmetic<U1, U2>,
		units::detail::_unit_value_t<typename std::conditional<traits::is_convertible_unit<typename traits::unit_value_t_traits<U1>::unit_type,
		typename traits::unit_value_t_traits<U2>::unit_type>::value, dimensionless::scalar, compound_unit<typename traits::unit_value_t_traits<U1>::unit_type, 
		inverse<typename traits::unit_value_t_traits<U2>::unit_type>>>::type>
	{
		/** @cond */	// DOXYGEN IGNORE
		using Base = units::detail::unit_value_arithmetic<U1, U2>;
		
		using unit_type = typename std::conditional<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, dimensionless::scalar, compound_unit<typename Base::_UNIT1, inverse<typename Base::_UNIT2>>>::type;
		using ratio = typename std::conditional<traits::is_convertible_unit<typename Base::_UNIT1, typename Base::_UNIT2>::value, std::ratio_divide<typename Base::_RATIO1, typename Base::_RATIO2CONV>, std::ratio_divide<typename Base::_RATIO1, typename Base::_RATIO2>>::type;
		/** @endcond */	// END DOXYGEN IGNORE

		/**
		 * @brief		Value of quotient
		 * @details		Returns the calculated value of the quotient of `U1` and `U2`, in units
		 *				of `U1 x U2`.
		 * @returns		Value of the quotient in the appropriate units.
		 */
		static const unit_t<unit_type> value()
		{
			using UsePi = typename std::conditional<Base::_PI_EXP::num != 0, std::true_type, std::false_type>::type;
			return value(UsePi());
		}

		/** @cond */	// DOXYGEN IGNORE
		// value if PI isn't involved
		static const unit_t<unit_type> value(std::false_type)
		{
			return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
		}

		// value if PI *is* involved
		static const unit_t<unit_type> value(std::true_type)
		{
			return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::PI, ((UNIT_LIB_DEFAULT_TYPE)Base::_PI_EXP::num / Base::_PI_EXP::den)));
		}
		/** @endcond */	// END DOXYGEN IGNORE
	};

	/**
	 * @ingroup		CompileTimeUnitManipulators
	 * @brief		raises unit_value_to a power at compile-time
	 * @details		The resulting unit will the `unit_type` of `U1` squared
	 * @tparam		U1	`unit_value_t` to take the exponentiation of.
	 * @sa			unit_value_t_traits to access information about the properties of the class,
	 *				such as it's unit type and rational value.
	 * @note		very similar in concept to `units::math::pow`
	 */
	template<class U1, int power>
	struct unit_value_power : units::detail::unit_value_arithmetic<U1, U1>, units::detail::_unit_value_t<typename units::detail::power_of_unit<power, typename traits::unit_value_t_traits<U1>::unit_type>::type>
	{
		/** @cond */	// DOXYGEN IGNORE
		using Base = units::detail::unit_value_arithmetic<U1, U1>;

		using unit_type = typename units::detail::power_of_unit<power, typename Base::_UNIT1>::type;
		using ratio = typename units::detail::power_of_ratio<power, typename Base::_RATIO1>::type;
		using pi_exponent = std::ratio_multiply<std::ratio<power>, typename Base::_UNIT1::pi_exponent_ratio>;
		/** @endcond */	// END DOXYGEN IGNORE

		/**
		 * @brief		Value of exponentiation
		 * @details		Returns the calculated value of the exponentiation of `U1`, in units
		 *				of `U1^power`.
		 * @returns		Value of the exponentiation in the appropriate units.
		 */
		static const unit_t<unit_type> value()
		{
			using UsePi = typename std::conditional<Base::_PI_EXP::num != 0, std::true_type, std::false_type>::type;
			return value(UsePi());
		}

		/** @cond */	// DOXYGEN IGNORE
		// value if PI isn't involved
		static const unit_t<unit_type> value(std::false_type)
		{
			return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
		}

		// value if PI *is* involved
		static const unit_t<unit_type> value(std::true_type)
		{
			return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::PI, ((UNIT_LIB_DEFAULT_TYPE)pi_exponent::num / pi_exponent::den)));
		}
		/** @endcond */	// END DOXYGEN IGNORE	};
	};

	/**
	 * @ingroup		CompileTimeUnitManipulators
	 * @brief		calculates square root of unit_value_t at compile-time
	 * @details		The resulting unit will the square root `unit_type` of `U1`	 
	 * @tparam		U1	`unit_value_t` to take the square root of.
	 * @sa			unit_value_t_traits to access information about the properties of the class,
	 *				such as it's unit type and rational value.
	 * @note		very similar in concept to `units::ratio_sqrt`
	 */
	template<class U1, std::intmax_t Eps = 10000000000>
	struct unit_value_sqrt : units::detail::unit_value_arithmetic<U1, U1>, units::detail::_unit_value_t<square_root<typename traits::unit_value_t_traits<U1>::unit_type, Eps>>
	{
		/** @cond */	// DOXYGEN IGNORE
		using Base = units::detail::unit_value_arithmetic<U1, U1>;

		using unit_type = square_root<typename Base::_UNIT1, Eps>;
		using ratio = ratio_sqrt<typename Base::_RATIO1, Eps>;
		using pi_exponent = ratio_sqrt<typename Base::_UNIT1::pi_exponent_ratio, Eps>;
		/** @endcond */	// END DOXYGEN IGNORE

		/**
		 * @brief		Value of square root
		 * @details		Returns the calculated value of the square root of `U1`, in units
		 *				of `U1^1/2`.
		 * @returns		Value of the square root in the appropriate units.
		 */
		static const unit_t<unit_type> value()
		{
			using UsePi = typename std::conditional<Base::_PI_EXP::num != 0, std::true_type, std::false_type>::type;
			return value(UsePi());
		}

		/** @cond */	// DOXYGEN IGNORE
		// value if PI isn't involved
		static const unit_t<unit_type> value(std::false_type)
		{
			return unit_t<unit_type>((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den);
		}

		// value if PI *is* involved
		static const unit_t<unit_type> value(std::true_type)
		{
			return unit_t<unit_type>(((UNIT_LIB_DEFAULT_TYPE)ratio::num / ratio::den) * std::pow(units::constants::PI, ((UNIT_LIB_DEFAULT_TYPE)pi_exponent::num / pi_exponent::den)));
		}
		/** @endcond */	// END DOXYGEN IGNORE
	};

	//------------------------------
	//	LITERALS
	//------------------------------

	/**
	 * @namespace	units::literals
	 * @brief		namespace for unit literal definitions of all categories.
	 * @details		Literals allow for declaring unit types using suffix values. For example, a type
	 *				of `meter_t(6.2)` could be declared as `6.2_m`. All literals use an underscore
	 *				followed by the abbreviation for the unit. To enable literal syntax in your code,
	 *				include the statement `using namespace units::literals`.
	 * @anchor		unitLiterals
	 * @sa			See unit_t for more information on unit type containers.
	 */

	//------------------------------
	//	LENGTH UNITS
	//------------------------------

	/**
	 * @namespace	units::length
	 * @brief		namespace for unit types and containers representing length values
	 * @details		The SI unit for length is `meters`, and the corresponding `base_unit` category is
	 *				`length_unit`.
	 * @anchor		lengthContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(length, meter, meters, m, unit<std::ratio<1>, units::category::length_unit>)
	UNIT_ADD(length, foot, feet, ft, unit<std::ratio<381, 1250>, meters>)
	UNIT_ADD(length, mil, mils, mil, unit<std::ratio<1000>, feet>)
	UNIT_ADD(length, inch, inches, in, unit<std::ratio<1, 12>, feet>)
	UNIT_ADD(length, mile,   miles,    mi,    unit<std::ratio<5280>, feet>)
	UNIT_ADD(length, nauticalMile, nauticalMiles, nmi, unit<std::ratio<1852>, meters>)
	UNIT_ADD(length, astronicalUnit, astronicalUnits, au, unit<std::ratio<149597870700>, meters>)
	UNIT_ADD(length, lightyear, lightyears, ly, unit<std::ratio<9460730472580800>, meters>)
	UNIT_ADD(length, parsec, parsecs, pc, unit<std::ratio<648000>, astronicalUnits, std::ratio<-1>>)
	UNIT_ADD(length, angstrom, angstroms, angstrom, unit<std::ratio<1, 10>, nanometers>)
	UNIT_ADD(length, cubit, cubits, cbt, unit<std::ratio<18>, inches>)
	UNIT_ADD(length, fathom, fathoms, ftm, unit<std::ratio<6>, feet>)
	UNIT_ADD(length, chain, chains, ch, unit<std::ratio<66>, feet>)
	UNIT_ADD(length, furlong, furlongs, fur, unit<std::ratio<10>, chains>)
	UNIT_ADD(length, hand, hands, hand, unit<std::ratio<4>, inches>)
	UNIT_ADD(length, league, leagues, lea, unit<std::ratio<3>, miles>)
	UNIT_ADD(length, nauticalLeague, nauticalLeagues, nl, unit<std::ratio<3>, nauticalMiles>)
	UNIT_ADD(length, yard, yards, yd, unit<std::ratio<3>, feet>)

	UNIT_ADD_CATEGORY_TRAIT(length, meter)

	//------------------------------
	//	MASS UNITS
	//------------------------------

	/**
	 * @namespace	units::mass
	 * @brief		namespace for unit types and containers representing mass values
	 * @details		The SI unit for mass is `kilograms`, and the corresponding `base_unit` category is
	 *				`mass_unit`.
	 * @anchor		massContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(mass, gram, grams, g, unit<std::ratio<1, 1000>, units::category::mass_unit>)
	UNIT_ADD(mass, metric_ton, metric_tons, t, unit<std::ratio<1000>, kilograms>)
	UNIT_ADD(mass, pound, pounds, lb, unit<std::ratio<45359237, 100000000>, kilograms>)
	UNIT_ADD(mass, long_ton, long_tons, ln_t, unit<std::ratio<2240>, pounds>)
	UNIT_ADD(mass, short_ton, short_tons, sh_t, unit<std::ratio<2000>, pounds>)
	UNIT_ADD(mass, stone, stone, st, unit<std::ratio<14>, pounds>)
	UNIT_ADD(mass, ounce, ounces, oz, unit<std::ratio<1, 16>, pounds>)
	UNIT_ADD(mass, carat, carats, ct, unit<std::ratio<200>, milligrams>)
	UNIT_ADD(mass, slug, slugs, slug, unit<std::ratio<145939029, 10000000>, kilograms>)

	UNIT_ADD_CATEGORY_TRAIT(mass, kilogram)

	//------------------------------
	//	TIME UNITS
	//------------------------------

	/**
	 * @namespace	units::time
	 * @brief		namespace for unit types and containers representing time values
	 * @details		The SI unit for time is `seconds`, and the corresponding `base_unit` category is
	 *				`time_unit`.
	 * @anchor		timeContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(time, second, seconds, s, unit<std::ratio<1>, units::category::time_unit>)
	UNIT_ADD(time, minute, minutes, min, unit<std::ratio<60>, seconds>)
	UNIT_ADD(time, hour, hours, hr, unit<std::ratio<60>, minutes>)
	UNIT_ADD(time, day, days, d, unit<std::ratio<24>, hours>)
	UNIT_ADD(time, week, weeks, wk, unit<std::ratio<7>, days>)
	UNIT_ADD(time, year, years, yr, unit<std::ratio<365>, days>)

	UNIT_ADD_CATEGORY_TRAIT(time, second)

	//------------------------------
	//	ANGLE UNITS
	//------------------------------

	/**
	 * @namespace	units::angle
	 * @brief		namespace for unit types and containers representing angle values
	 * @details		The SI unit for angle is `radians`, and the corresponding `base_unit` category is
	 *				`angle_unit`.
	 * @anchor		angleContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(angle, radian, radians, rad, unit<std::ratio<1>, units::category::angle_unit>)
	UNIT_ADD(angle, degree, degrees, deg, unit<std::ratio<1, 180>, radians, std::ratio<1>>)
	UNIT_ADD(angle, arcminute, arcminutes, arcmin, unit<std::ratio<1, 60>, degrees>)
	UNIT_ADD(angle, arcsecond, arcseconds, arcsec, unit<std::ratio<1, 60>, arcminutes>)
	UNIT_ADD(angle, milliarcsecond, milliarcseconds, mas, milli<arcseconds>)
	UNIT_ADD(angle, turn, turns, tr, unit<std::ratio<2>, radians, std::ratio<1>>)
	UNIT_ADD(angle, gradian, gradians, gon, unit<std::ratio<1, 400>, turns>)

	UNIT_ADD_CATEGORY_TRAIT(angle, radian)

	//------------------------------
	//	UNITS OF CURRENT
	//------------------------------
	/**
	 * @namespace	units::current
	 * @brief		namespace for unit types and containers representing current values
	 * @details		The SI unit for current is `amperes`, and the corresponding `base_unit` category is
	 *				`current_unit`.
	 * @anchor		currentContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(current, ampere, amperes, A, unit<std::ratio<1>, units::category::current_unit>)
	
	UNIT_ADD_CATEGORY_TRAIT(current, ampere)

	//------------------------------
	//	UNITS OF TEMPERATURE
	//------------------------------

	// NOTE: temperature units have special conversion overloads, since they
	// require translations and aren't a reversible transform.

	/**
	 * @namespace	units::temperature
	 * @brief		namespace for unit types and containers representing temperature values
	 * @details		The SI unit for temperature is `kelvin`, and the corresponding `base_unit` category is
	 *				`temperature_unit`.
	 * @anchor		temperatureContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(temperature, kelvin, kelvin, K, unit<std::ratio<1>, units::category::temperature_unit>)
	UNIT_ADD(temperature, celsius, celsius, degC, unit<std::ratio<1>, kelvin, std::ratio<0>, std::ratio<27315, 100>>)
	UNIT_ADD(temperature, fahrenheit, fahrenheit, degF, unit<std::ratio<5, 9>, celsius, std::ratio<0>, std::ratio<-160, 9>>)
	UNIT_ADD(temperature, reaumur, reaumur, Re, unit<std::ratio<10, 8>, celsius>)
	UNIT_ADD(temperature, rankine, rankine, Ra, unit<std::ratio<5, 9>, kelvin>)

	UNIT_ADD_CATEGORY_TRAIT(temperature, kelvin)

	//------------------------------
	//	UNITS OF AMOUNT OF SUBSTANCE
	//------------------------------

	/**
	 * @namespace	units::substance
	 * @brief		namespace for unit types and containers representing substance values
	 * @details		The SI unit for substance is `moles`, and the corresponding `base_unit` category is
	 *				`substance_unit`.
	 * @anchor		substanceContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(substance, mole, moles, mol, unit<std::ratio<1>, units::category::substance_unit>)
	
	UNIT_ADD_CATEGORY_TRAIT(substance, mole)

	//------------------------------
	//	UNITS OF LUMINOUS INTENSITY
	//------------------------------

	/**
	 * @namespace	units::luminous_intensity
	 * @brief		namespace for unit types and containers representing luminous_intensity values
	 * @details		The SI unit for luminous_intensity is `candelas`, and the corresponding `base_unit` category is
	 *				`luminous_intensity_unit`.
	 * @anchor		luminousIntensityContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(luminous_intensity, candela, candelas, cd, unit<std::ratio<1>, units::category::luminous_intensity_unit>)
	
	UNIT_ADD_CATEGORY_TRAIT(luminous_intensity, candela)

	//------------------------------
	//	UNITS OF SOLID ANGLE
	//------------------------------

	/**
	 * @namespace	units::solid_angle
	 * @brief		namespace for unit types and containers representing solid_angle values
	 * @details		The SI unit for solid_angle is `steradians`, and the corresponding `base_unit` category is
	 *				`solid_angle_unit`.
	 * @anchor		solidAngleContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(solid_angle, steradian, steradians, sr, unit<std::ratio<1>, units::category::solid_angle_unit>)
	UNIT_ADD(solid_angle, degree_squared, degrees_squared, sq_deg, squared<angle::degrees>)
	UNIT_ADD(solid_angle, spat, spats, sp, unit<std::ratio<4>, steradians, std::ratio<1>>)

	UNIT_ADD_CATEGORY_TRAIT(solid_angle, steradian)

	//------------------------------
	//	FREQUENCY UNITS
	//------------------------------

	/**
	 * @namespace	units::frequency
	 * @brief		namespace for unit types and containers representing frequency values
	 * @details		The SI unit for frequency is `hertz`, and the corresponding `base_unit` category is
	 *				`frequency_unit`.
	 * @anchor		frequencyContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(frequency, hertz, hertz, Hz, unit<std::ratio<1>, units::category::frequency_unit>)

	UNIT_ADD_CATEGORY_TRAIT(frequency, hertz)

	//------------------------------
	//	VELOCITY UNITS
	//------------------------------

	/**
	 * @namespace	units::velocity
	 * @brief		namespace for unit types and containers representing velocity values
	 * @details		The SI unit for velocity is `meters_per_second`, and the corresponding `base_unit` category is
	 *				`velocity_unit`.
	 * @anchor		velocityContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(velocity, meters_per_second, meters_per_second, mps, unit<std::ratio<1>, units::category::velocity_unit>)
	UNIT_ADD(velocity, feet_per_second, feet_per_second, fps, compound_unit<length::feet, inverse<time::seconds>>)
	UNIT_ADD(velocity, miles_per_hour, miles_per_hour, mph, compound_unit<length::miles, inverse<time::hour>>)
	UNIT_ADD(velocity, kilometers_per_hour, kilometers_per_hour, kph, compound_unit<length::kilometers, inverse<time::hour>>)
	UNIT_ADD(velocity, knot, knots, kts, compound_unit<length::nauticalMiles, inverse<time::hour>>)
	
	UNIT_ADD_CATEGORY_TRAIT(velocity, meters_per_second)

	//------------------------------
	//	ANGULAR VELOCITY UNITS
	//------------------------------

	/**
	 * @namespace	units::angular_velocity
	 * @brief		namespace for unit types and containers representing angular velocity values
	 * @details		The SI unit for angular velocity is `radians_per_second`, and the corresponding `base_unit` category is
	 *				`angular_velocity_unit`.
	 * @anchor		angularVelocityContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(angular_velocity, radians_per_second, radians_per_second, rad_per_s, unit<std::ratio<1>, units::category::angular_velocity_unit>)
	UNIT_ADD(angular_velocity, degrees_per_second, degrees_per_second, deg_per_s, compound_unit<angle::degrees, inverse<time::seconds>>)
	UNIT_ADD(angular_velocity, revolutions_per_minute, revolutions_per_minute, rpm, unit<std::ratio<2, 60>, radians_per_second, std::ratio<1>>)
	UNIT_ADD(angular_velocity, milliarcseconds_per_year, milliarcseconds_per_year, mas_per_yr, compound_unit<angle::milliarcseconds, inverse<time::year>>)

	UNIT_ADD_CATEGORY_TRAIT(angular_velocity, radians_per_second)

	//------------------------------
	//	UNITS OF ACCELERATION
	//------------------------------

	/**
	 * @namespace	units::acceleration
	 * @brief		namespace for unit types and containers representing acceleration values
	 * @details		The SI unit for acceleration is `meters_per_second_squared`, and the corresponding `base_unit` category is
	 *				`acceleration_unit`.
	 * @anchor		accelerationContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(acceleration, meters_per_second_squared, meters_per_second_squared, mps_sq, unit<std::ratio<1>, units::category::acceleration_unit>)
	UNIT_ADD(acceleration, feet_per_second_squared, feet_per_second_squared, fps_sq, compound_unit<length::feet, inverse<squared<time::seconds>>>)
	UNIT_ADD(acceleration, standard_gravity, standard_gravity, SG, unit<std::ratio<980665, 100000>, meters_per_second_squared>)

	UNIT_ADD_CATEGORY_TRAIT(acceleration, meters_per_second_squared)

	//------------------------------
	//	UNITS OF FORCE
	//------------------------------

	/**
	 * @namespace	units::force
	 * @brief		namespace for unit types and containers representing force values
	 * @details		The SI unit for force is `newtons`, and the corresponding `base_unit` category is
	 *				`force_unit`.
	 * @anchor		forceContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(force, newton, newtons, N, unit<std::ratio<1>, units::category::force_unit>)
	UNIT_ADD(force, pound, pounds, lbf, compound_unit<mass::slug, length::foot, inverse<squared<time::seconds>>>)
	UNIT_ADD(force, dyne, dynes, dyn, unit<std::ratio<1, 100000>, newtons>)
	UNIT_ADD(force, kilopond, kiloponds, kp, compound_unit<acceleration::standard_gravity, mass::kilograms>)
	UNIT_ADD(force, poundal, poundals, pdl, compound_unit<mass::pound, length::foot, inverse<squared<time::seconds>>>)

	UNIT_ADD_CATEGORY_TRAIT(force, newton)

	//------------------------------
	//	UNITS OF PRESSURE
	//------------------------------

	/**
	 * @namespace	units::pressure
	 * @brief		namespace for unit types and containers representing pressure values
	 * @details		The SI unit for pressure is `pascals`, and the corresponding `base_unit` category is
	 *				`pressure_unit`.
	 * @anchor		pressureContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(pressure, pascal, pascals, Pa, unit<std::ratio<1>, units::category::pressure_unit>)
	UNIT_ADD(pressure, bar, bars, bar, unit<std::ratio<100>, kilo<pascals>>)
	UNIT_ADD(pressure, atmosphere, atmospheres, atm, unit<std::ratio<101325>, pascals>)
	UNIT_ADD(pressure, pounds_per_square_inch, pounds_per_square_inch, psi, compound_unit<force::pounds, inverse<squared<length::inch>>>)
	UNIT_ADD(pressure, torr, torrs, torr, unit<std::ratio<1, 760>, atmospheres>)
	
	UNIT_ADD_CATEGORY_TRAIT(pressure, pascals)

	//------------------------------
	//	UNITS OF CHARGE
	//------------------------------

	/**
	 * @namespace	units::charge
	 * @brief		namespace for unit types and containers representing charge values
	 * @details		The SI unit for charge is `coulombs`, and the corresponding `base_unit` category is
	 *				`charge_unit`.
	 * @anchor		chargeContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(charge, coulomb, coulombs, C, unit<std::ratio<1>, units::category::charge_unit>)
	UNIT_ADD_WITH_METRIC_PREFIXES(charge, ampere_hour, ampere_hours, Ah, compound_unit<current::ampere, time::hours>)

	UNIT_ADD_CATEGORY_TRAIT(charge, coulombs)

	//------------------------------
	//	UNITS OF ENERGY
	//------------------------------

	/**
	 * @namespace	units::energy
	 * @brief		namespace for unit types and containers representing energy values
	 * @details		The SI unit for energy is `joules`, and the corresponding `base_unit` category is
	 *				`energy_unit`.
	 * @anchor		energyContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(energy, joule, joules, J, unit<std::ratio<1>, units::category::energy_unit>)
	UNIT_ADD_WITH_METRIC_PREFIXES(energy, calorie, calories, cal, unit<std::ratio<4184, 1000>, joules>)
	UNIT_ADD(energy, kilowatt_hour, kilowatt_hours, kWh, unit<std::ratio<36, 10>, megajoules>)
	UNIT_ADD(energy, watt_hour, watt_hours, Wh, unit<std::ratio<1, 1000>, kilowatt_hours>)
	UNIT_ADD(energy, british_thermal_unit, british_thermal_units, BTU, unit<std::ratio<105505585262, 100000000>, joules>)
	UNIT_ADD(energy, british_thermal_unit_iso, british_thermal_units_iso, BTU_iso, unit<std::ratio<1055056, 1000>, joules>)
	UNIT_ADD(energy, british_thermal_unit_59, british_thermal_units_59, BTU59, unit<std::ratio<1054804, 1000>, joules>)
	UNIT_ADD(energy, therm, therms, thm, unit<std::ratio<100000>, british_thermal_units_59>)
	UNIT_ADD(energy, foot_pound, foot_pounds, ftlbf, unit<std::ratio<13558179483314004, 10000000000000000>, joules>)

	UNIT_ADD_CATEGORY_TRAIT(energy, joule)

	//------------------------------
	//	UNITS OF POWER
	//------------------------------

	/**
	 * @namespace	units::power
	 * @brief		namespace for unit types and containers representing power values
	 * @details		The SI unit for power is `watts`, and the corresponding `base_unit` category is
	 *				`power_unit`.
	 * @anchor		powerContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(power, watt, watts, W, unit<std::ratio<1>, units::category::power_unit>)
	UNIT_ADD(power, horsepower, horsepower, hp, unit<std::ratio<7457, 10>, watts>)
	UNIT_ADD_DECIBEL(power, watt, dBW)
	UNIT_ADD_DECIBEL(power, milliwatt, dBm)
	
	UNIT_ADD_CATEGORY_TRAIT(power, watt)

	//------------------------------
	//	UNITS OF VOLTAGE
	//------------------------------

	/**
	 * @namespace	units::voltage
	 * @brief		namespace for unit types and containers representing voltage values
	 * @details		The SI unit for voltage is `volts`, and the corresponding `base_unit` category is
	 *				`voltage_unit`.
	 * @anchor		voltageContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(voltage, volt, volts, V, unit<std::ratio<1>, units::category::voltage_unit>)
	UNIT_ADD(voltage, statvolt, statvolts, statV, unit<std::ratio<1000000, 299792458>, volts>)
	UNIT_ADD(voltage, abvolt, abvolts, abV, unit<std::ratio<1, 100000000>, volts>)
	
	UNIT_ADD_CATEGORY_TRAIT(voltage, volts)

	//------------------------------
	//	UNITS OF CAPACITANCE
	//------------------------------

	/**
	 * @namespace	units::capacitance
	 * @brief		namespace for unit types and containers representing capacitance values
	 * @details		The SI unit for capacitance is `farads`, and the corresponding `base_unit` category is
	 *				`capacitance_unit`.
	 * @anchor		capacitanceContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(capacitance, farad, farads, F, unit<std::ratio<1>, units::category::capacitance_unit>)
	
	UNIT_ADD_CATEGORY_TRAIT(capacitance, farad)

	//------------------------------
	//	UNITS OF IMPEDANCE
	//------------------------------

	/**
	 * @namespace	units::impedance
	 * @brief		namespace for unit types and containers representing impedance values
	 * @details		The SI unit for impedance is `ohms`, and the corresponding `base_unit` category is
	 *				`impedance_unit`.
	 * @anchor		impedanceContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(impedance, ohm, ohms, Ohm, unit<std::ratio<1>, units::category::impedance_unit>)
	
	UNIT_ADD_CATEGORY_TRAIT(impedance, ohm)

	//------------------------------
	//	UNITS OF CONDUCTANCE
	//------------------------------

	/**
	 * @namespace	units::conductance
	 * @brief		namespace for unit types and containers representing conductance values
	 * @details		The SI unit for conductance is `siemens`, and the corresponding `base_unit` category is
	 *				`conductance_unit`.
	 * @anchor		conductanceContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(conductance, siemen, siemens, S, unit<std::ratio<1>, units::category::conductance_unit>)
	
	UNIT_ADD_CATEGORY_TRAIT(conductance, siemens)

	//------------------------------
	//	UNITS OF MAGNETIC FLUX
	//------------------------------

	/**
	 * @namespace	units::magnetic_flux
	 * @brief		namespace for unit types and containers representing magnetic_flux values
	 * @details		The SI unit for magnetic_flux is `webers`, and the corresponding `base_unit` category is
	 *				`magnetic_flux_unit`.
	 * @anchor		magneticFluxContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(magnetic_flux, weber, webers, Wb, unit<std::ratio<1>, units::category::magnetic_flux_unit>)
	UNIT_ADD(magnetic_flux, maxwell, maxwells, Mx, unit<std::ratio<1, 100000000>, webers>)

	UNIT_ADD_CATEGORY_TRAIT(magnetic_flux, webers)

	//----------------------------------------
	//	UNITS OF MAGNETIC FIELD STRENGTH
	//----------------------------------------

	/**
	 * @namespace	units::magnetic_field_strength
	 * @brief		namespace for unit types and containers representing magnetic_field_strength values
	 * @details		The SI unit for magnetic_field_strength is `teslas`, and the corresponding `base_unit` category is
	 *				`magnetic_field_strength_unit`.
	 * @anchor		magneticFieldStrengthContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	// Unfortunately `_T` is a WINAPI macro, so we have to use `_Te` as the tesla abbreviation.
	UNIT_ADD_WITH_METRIC_PREFIXES(magnetic_field_strength, tesla, teslas, Te, unit<std::ratio<1>, units::category::magnetic_field_strength_unit>)
	UNIT_ADD(magnetic_field_strength, gauss, gauss, G, compound_unit<magnetic_flux::maxwell, inverse<squared<length::centimeter>>>)
		
	UNIT_ADD_CATEGORY_TRAIT(magnetic_field_strength, tesla)

	//------------------------------
	//	UNITS OF INDUCTANCE
	//------------------------------

	/**
	 * @namespace	units::inductance
	 * @brief		namespace for unit types and containers representing inductance values
	 * @details		The SI unit for inductance is `henrys`, and the corresponding `base_unit` category is
	 *				`inductance_unit`.
	 * @anchor		inductanceContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(inductance, henry, henries, H, unit<std::ratio<1>, units::category::inductance_unit>)

	UNIT_ADD_CATEGORY_TRAIT(inductance, henry)

	//------------------------------
	//	UNITS OF LUMINOUS FLUX
	//------------------------------

	/**
	 * @namespace	units::luminous_flux
	 * @brief		namespace for unit types and containers representing luminous_flux values
	 * @details		The SI unit for luminous_flux is `lumens`, and the corresponding `base_unit` category is
	 *				`luminous_flux_unit`.
	 * @anchor		luminousFluxContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(luminous_flux, lumen, lumens, lm, unit<std::ratio<1>, units::category::luminous_flux_unit>)
	
	UNIT_ADD_CATEGORY_TRAIT(luminous_flux, lumen)

	//------------------------------
	//	UNITS OF ILLUMINANCE
	//------------------------------

	/**
	 * @namespace	units::illuminance
	 * @brief		namespace for unit types and containers representing illuminance values
	 * @details		The SI unit for illuminance is `luxes`, and the corresponding `base_unit` category is
	 *				`illuminance_unit`.
	 * @anchor		illuminanceContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(illuminance, lux, luxes, lx, unit<std::ratio<1>, units::category::illuminance_unit>)
	UNIT_ADD(illuminance, footcandle, footcandles, fc, compound_unit<luminous_flux::lumen, inverse<squared<length::foot>>>)
	UNIT_ADD(illuminance, lumens_per_square_inch, lumens_per_square_inch, lm_per_in_sq, compound_unit<luminous_flux::lumen, inverse<squared<length::inch>>>)
	UNIT_ADD(illuminance, phot, phots, ph, compound_unit<luminous_flux::lumens, inverse<squared<length::centimeter>>>)
	
	UNIT_ADD_CATEGORY_TRAIT(illuminance, lux)

	//------------------------------
	//	UNITS OF RADIATION
	//------------------------------

	/**
	 * @namespace	units::radiation
	 * @brief		namespace for unit types and containers representing radiation values
	 * @details		The SI units for radiation are:
	 *				- source activity:	becquerel
	 *				- absorbed dose:	gray
	 *				- equivalent dose:	sievert
	 * @anchor		radiationContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD_WITH_METRIC_PREFIXES(radiation, becquerel, becquerels, Bq, unit<std::ratio<1>, units::frequency::hertz>)
	UNIT_ADD_WITH_METRIC_PREFIXES(radiation, gray, grays, Gy, compound_unit<energy::joules, inverse<mass::kilogram>>)
	UNIT_ADD_WITH_METRIC_PREFIXES(radiation, sievert, sieverts, Sv, unit<std::ratio<1>, grays>)
	UNIT_ADD(radiation, curie, curies, Ci, unit<std::ratio<37>, gigabecquerels>)
	UNIT_ADD(radiation, rutherford, rutherfords, rd, unit<std::ratio<1>, megabecquerels>)
	UNIT_ADD(radiation, rad, rads, rads, unit<std::ratio<1>, centigrays>)

	UNIT_ADD_CATEGORY_TRAIT(radioactivity, becquerels)

	//------------------------------
	//	UNITS OF TORQUE
	//------------------------------

	/**
	 * @namespace	units::torque
	 * @brief		namespace for unit types and containers representing torque values
	 * @details		The SI unit for torque is `newton_meters`, and the corresponding `base_unit` category is
	 *				`torque_units`.
	 * @anchor		torqueContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(torque, newton_meter, newton_meters, Nm, unit<std::ratio<1>, units::energy::joule>)
	UNIT_ADD(torque, foot_pound, foot_pounds, ftlb, compound_unit<length::foot, force::pounds>)
	UNIT_ADD(torque, foot_poundal, foot_poundals, ftpdl, compound_unit<length::foot, force::poundal>)
	UNIT_ADD(torque, inch_pound, inch_pounds, inlb, compound_unit<length::inch, force::pounds>)
	UNIT_ADD(torque, meter_kilogram, meter_kilograms, mkgf, compound_unit<length::meter, force::kiloponds>)
	
	UNIT_ADD_CATEGORY_TRAIT(torque, newton_meter)

	//------------------------------
	//	AREA UNITS
	//------------------------------

	/**
	 * @namespace	units::area
	 * @brief		namespace for unit types and containers representing area values
	 * @details		The SI unit for area is `square_meters`, and the corresponding `base_unit` category is
	 *				`area_unit`.
	 * @anchor		areaContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(area, square_meter, square_meters, sq_m, unit<std::ratio<1>, units::category::area_unit>)
	UNIT_ADD(area, square_foot, square_feet, sq_ft, squared<length::feet>)
	UNIT_ADD(area, square_inch, square_inches, sq_in, squared<length::inch>)
	UNIT_ADD(area, square_mile, square_miles, sq_mi, squared<length::miles>)
	UNIT_ADD(area, square_kilometer, square_kilometers, sq_km, squared<length::kilometers>)
	UNIT_ADD(area, hectare, hectares, ha, unit<std::ratio<10000>, square_meters>)
	UNIT_ADD(area, acre, acres, acre, unit<std::ratio<43560>, square_feet>)
	
	UNIT_ADD_CATEGORY_TRAIT(area, square_meters)

	//------------------------------
	//	UNITS OF VOLUME
	//------------------------------

	/**
	 * @namespace	units::volume
	 * @brief		namespace for unit types and containers representing volume values
	 * @details		The SI unit for volume is `cubic_meters`, and the corresponding `base_unit` category is
	 *				`volume_unit`.
	 * @anchor		volumeContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(volume, cubic_meter, cubic_meters, cu_m, unit<std::ratio<1>, units::category::volume_unit>)
	UNIT_ADD(volume, cubic_millimeter, cubic_millimeters, cu_mm, cubed<length::millimeter>)
	UNIT_ADD(volume, cubic_kilometer, cubic_kilometers, cu_km, cubed<length::kilometer>)
	UNIT_ADD_WITH_METRIC_PREFIXES(volume, liter, liters, L, cubed<deci<length::meter>>)
	UNIT_ADD(volume, cubic_inch, cubic_inches, cu_in, cubed<length::inches>)
	UNIT_ADD(volume, cubic_foot, cubic_feet, cu_ft, cubed<length::feet>)
	UNIT_ADD(volume, cubic_yard, cubic_yards, cu_yd, cubed<length::yards>)
	UNIT_ADD(volume, cubic_mile, cubic_miles, cu_mi, cubed<length::miles>)
	UNIT_ADD(volume, gallon, gallons, gal, unit<std::ratio<231>, cubic_inches>)
	UNIT_ADD(volume, quart, quarts, qt, unit<std::ratio<1, 4>, gallons>)
	UNIT_ADD(volume, pint, pints, pt, unit<std::ratio<1, 2>, quarts>)
	UNIT_ADD(volume, cup, cups, c, unit<std::ratio<1, 2>, pints>)
	UNIT_ADD(volume, fluid_ounce, fluid_ounces, fl_oz, unit<std::ratio<1, 8>, cups>)
	UNIT_ADD(volume, barrel, barrels, bl, unit<std::ratio<42>, gallons>)
	UNIT_ADD(volume, bushel, bushels, bu, unit<std::ratio<215042, 100>, cubic_inches>)
	UNIT_ADD(volume, cord, cords, cord, unit<std::ratio<128>, cubic_feet>)
	UNIT_ADD(volume, cubic_fathom, cubic_fathoms, cu_fm, cubed<length::fathom>)
	UNIT_ADD(volume, tablespoon, tablespoons, tbsp, unit<std::ratio<1, 2>, fluid_ounces>)
	UNIT_ADD(volume, teaspoon, teaspoons, tsp, unit<std::ratio<1, 6>, fluid_ounces>)
	UNIT_ADD(volume, pinch, pinches, pinch, unit<std::ratio<1, 8>, teaspoons>)
	UNIT_ADD(volume, dash, dashes, dash, unit<std::ratio<1, 2>, pinches>)
	UNIT_ADD(volume, drop, drops, drop, unit<std::ratio<1, 360>, fluid_ounces>)
	UNIT_ADD(volume, fifth, fifths, fifth, unit<std::ratio<1, 5>, gallons>)
	UNIT_ADD(volume, dram, drams, dr, unit<std::ratio<1, 8>, fluid_ounces>)
	UNIT_ADD(volume, gill, gills, gi, unit<std::ratio<4>, fluid_ounces>)
	UNIT_ADD(volume, peck, pecks, pk, unit<std::ratio<1, 4>, bushels>)
	UNIT_ADD(volume, sack, sacks, sacks, unit<std::ratio<3>, bushels>)
	UNIT_ADD(volume, shot, shots, shots, unit<std::ratio<3, 2>, fluid_ounces>)
	UNIT_ADD(volume, strike, strikes, strikes, unit<std::ratio<2>, bushels>)

	UNIT_ADD_CATEGORY_TRAIT(volume, cubic_meter)

	//------------------------------
	//	UNITS OF DENSITY
	//------------------------------

	/**
	 * @namespace	units::density
	 * @brief		namespace for unit types and containers representing density values
	 * @details		The SI unit for density is `kilograms_per_cubic_meter`, and the corresponding `base_unit` category is
	 *				`density_unit`.
	 * @anchor		densityContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(density, kilograms_per_cubic_meter, kilograms_per_cubic_meter, kg_per_cu_m, unit<std::ratio<1>, units::category::density_unit>)
	UNIT_ADD(density, grams_per_milliliter, grams_per_milliliter, g_per_mL, compound_unit<mass::grams, inverse<volume::milliliter>>)
	UNIT_ADD(density, kilograms_per_liter, kilograms_per_liter, kg_per_L, unit<std::ratio<1>, compound_unit<mass::grams, inverse<volume::milliliter>>>)
	UNIT_ADD(density, ounces_per_cubic_foot, ounces_per_cubic_foot, oz_per_cu_ft, compound_unit<mass::ounces, inverse<volume::cubic_foot>>)
	UNIT_ADD(density, ounces_per_cubic_inch, ounces_per_cubic_inch, oz_per_cu_in, compound_unit<mass::ounces, inverse<volume::cubic_inch>>)
	UNIT_ADD(density, ounces_per_gallon, ounces_per_gallon, oz_per_gal, compound_unit<mass::ounces, inverse<volume::gallon>>)
	UNIT_ADD(density, pounds_per_cubic_foot, pounds_per_cubic_foot, lb_per_cu_ft, compound_unit<mass::pounds, inverse<volume::cubic_foot>>)
	UNIT_ADD(density, pounds_per_cubic_inch, pounds_per_cubic_inch, lb_per_cu_in, compound_unit<mass::pounds, inverse<volume::cubic_inch>>)
	UNIT_ADD(density, pounds_per_gallon, pounds_per_gallon, lb_per_gal, compound_unit<mass::pounds, inverse<volume::gallon>>)
	UNIT_ADD(density, slugs_per_cubic_foot, slugs_per_cubic_foot, slug_per_cu_ft, compound_unit<mass::slugs, inverse<volume::cubic_foot>>)

	UNIT_ADD_CATEGORY_TRAIT(density, kilograms_per_cubic_meter)

	//------------------------------
	//	UNITS OF CONCENTRATION
	//------------------------------

	/**
	 * @namespace	units::concentration
	 * @brief		namespace for unit types and containers representing concentration values
	 * @details		The SI unit for concentration is `parts_per_million`, and the corresponding `base_unit` category is
	 *				`scalar_unit`.
	 * @anchor		concentrationContainers
	 * @sa			See unit_t for more information on unit type containers.
	 */
	UNIT_ADD(concentration, ppm, parts_per_million, ppm, unit<std::ratio<1, 1000000>, units::category::scalar_unit>)
	UNIT_ADD(concentration, ppb, parts_per_billion, ppb, unit<std::ratio<1, 1000>, parts_per_million>)
	UNIT_ADD(concentration, ppt, parts_per_trillion, ppt, unit<std::ratio<1, 1000>, parts_per_billion>)
	UNIT_ADD(concentration, percent, percent, pct, unit<std::ratio<1, 100>, units::category::scalar_unit>)

	UNIT_ADD_CATEGORY_TRAIT(concentration, parts_per_million)

	//------------------------------
	//	CONSTANTS
	//------------------------------

	/**
	 * @brief		namespace for physical constants like PI and Avogadro's Number.
	 * @sa			See unit_t for more information on unit type containers.
	 */
	namespace constants
	{
		/**
		 * @name Unit Containers
		 * @anchor constantContainers
		 * @{
		 */
		static const unit_t<unit<std::ratio<1>, dimensionless::scalar, std::ratio<1>>>														pi(1.0);										///< Ratio of a circle's circumference to its diameter.
		static const velocity::meters_per_second_t																							c(299792458.0);									///< Speed of light in vacuum.
		static const unit_t<compound_unit<cubed<length::meters>, inverse<mass::kilogram>, inverse<squared<time::seconds>>>>					G(6.67408e-11);									///< Newtonian constant of gravitation.
		static const unit_t<compound_unit<energy::joule, time::seconds>>																	h(6.626070040e-34);								///< Planck constant.
		static const unit_t<compound_unit<force::newtons, inverse<squared<current::ampere>>>>												mu0(4.0e-7 * PI);								///< vacuum permeability.
		static const unit_t<compound_unit<capacitance::farad, inverse<length::meter>>>														epsilon0(1.0 / (mu0 * math::pow<2>(c)));		///< vacuum permitivity.
		static const impedance::ohm_t																										Z0(mu0 * c);									///< characteristic impedance of vacuum.
		static const unit_t<compound_unit<force::newtons, area::square_meter, inverse<squared<charge::coulomb>>>>							k_e(1.0 / (4 * pi * epsilon0));					///< Coulomb's constant.
		static const charge::coulomb_t																										e(1.602176565e-19);								///< elementary charge.
		static const mass::kilogram_t																										m_e(9.10938291e-31);							///< electron mass.
		static const mass::kilogram_t																										m_p(1.672621777e-27);							///< proton mass.
		static const unit_t<compound_unit<energy::joules, inverse<magnetic_field_strength::tesla>>>											mu_B(e * h / (4 * pi *m_e));					///< Bohr magneton.
		static const unit_t<inverse<substance::mol>>																						N_A(6.02214129e23);								///< Avagadro's Number.
		static const unit_t<compound_unit<energy::joules, inverse<temperature::kelvin>, inverse<substance::moles>>>							R(8.3144621);									///< Gas constant.
		static const unit_t<compound_unit<energy::joules, inverse<temperature::kelvin>>>													k_B(R / N_A);									///< Boltzmann constant.
		static const unit_t<compound_unit<charge::coulomb, inverse<substance::mol>>>														F(N_A * e);										///< Faraday constnat.
		static const unit_t<compound_unit<power::watts, inverse<area::square_meters>, inverse<squared<squared<temperature::kelvin>>>>>	sigma((2 * math::pow<5>(pi) * math::pow<4>(R)) / (15 * math::pow<3>(h) * math::pow<2>(c) * math::pow<4>(N_A)));	///< Stefan-Boltzmann constant.
		/** @} */
	}

	//----------------------------------
	//	UNIT-ENABLED CMATH FUNCTIONS
	//----------------------------------

	/**
	 * @brief		namespace for unit-enabled versions of the `<cmath>` library
	 * @details		Includes trigonometric functions, exponential/log functions, rounding functions, etc.
	 * @sa			See `unit_t` for more information on unit type containers.
	 */
	namespace math
	{

		//----------------------------------
		//	TRIGONOMETRIC FUNCTIONS
		//----------------------------------

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute cosine
		 * @details		The input value can be in any unit of angle, including radians or degrees.
		 * @tparam		AngleUnit	any `unit_t` type of `catgeory::angle_unit`. 
		 * @param[in]	angle		angle to compute the cosine of
		 * @returns		Returns the cosine of <i>angle</i>
		 */
		template<class AngleUnit>
		dimensionless::scalar_t cos(AngleUnit angle)
		{
			static_assert(traits::is_angle_unit<AngleUnit>::value, "Type `AngleUnit` must be a unit of angle derived from `unit_t`.");
			return dimensionless::scalar_t(std::cos(angle.template convert<angle::radian>()()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute sine
		 * @details		The input value can be in any unit of angle, including radians or degrees.
		 * @tparam		AngleUnit	any `unit_t` type of `catgeory::angle_unit`.
		 * @param[in]	angle		angle to compute the since of
		 * @returns		Returns the sine of <i>angle</i>
		 */
		template<class AngleUnit>
		dimensionless::scalar_t sin(AngleUnit angle)
		{
			static_assert(traits::is_angle_unit<AngleUnit>::value, "Type `AngleUnit` must be a unit of angle derived from `unit_t`.");
			return dimensionless::scalar_t(std::sin(angle.template convert<angle::radian>()()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute tangent
		 * @details		The input value can be in any unit of angle, including radians or degrees.
		 * @tparam		AngleUnit	any `unit_t` type of `catgeory::angle_unit`.
		 * @param[in]	angle		angle to compute the tangent of
		 * @returns		Returns the tangent of <i>angle</i>
		 */
		template<class AngleUnit>
		dimensionless::scalar_t tan(AngleUnit angle)
		{
			static_assert(traits::is_angle_unit<AngleUnit>::value, "Type `AngleUnit` must be a unit of angle derived from `unit_t`.");
			return dimensionless::scalar_t(std::tan(angle.template convert<angle::radian>()()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute arc cosine
		 * @details		Returns the principal value of the arc cosine of x, expressed in radians.
		 * @param[in]	x		Value whose arc cosine is computed, in the interval [-1,+1].
		 * @returns		Principal arc cosine of x, in the interval [0,pi] radians.
		 */
		template<class ScalarUnit>
		angle::radian_t acos(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return angle::radian_t(std::acos(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute arc sine
		 * @details		Returns the principal value of the arc sine of x, expressed in radians.
		 * @param[in]	x		Value whose arc sine is computed, in the interval [-1,+1].
		 * @returns		Principal arc sine of x, in the interval [-pi/2,+pi/2] radians.
		 */
		template<class ScalarUnit>
		angle::radian_t asin(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return angle::radian_t(std::asin(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute arc tangent
		 * @details		Returns the principal value of the arc tangent of x, expressed in radians. 
		 *				Notice that because of the sign ambiguity, the function cannot determine with 
		 *				certainty in which quadrant the angle falls only by its tangent value. See 
		 *				atan2 for an alternative that takes a fractional argument instead.
		 * @tparam		AngleUnit	any `unit_t` type of `catgeory::angle_unit`.
		 * @param[in]	x		Value whose arc tangent is computed, in the interval [-1,+1].
		 * @returns		Principal arc tangent of x, in the interval [-pi/2,+pi/2] radians.
		 */
		template<class ScalarUnit>
		angle::radian_t atan(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return angle::radian_t(std::atan(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute arc tangent with two parameters
		 * @details		To compute the value, the function takes into account the sign of both arguments in order to determine the quadrant.
		 * @param[in]	y		y-component of the triangle expressed.
		 * @param[in]	x		x-component of the triangle expressed.
		 * @returns		Returns the principal value of the arc tangent of <i>y/x</i>, expressed in radians.
		 */
		template<class Y, class X>
		angle::radian_t atan2(Y y, X x)
		{
			static_assert(traits::is_scalar_unit<decltype(y/x)>::value, "The quantity y/x must yield a dimensionless ratio.");

			// X and Y could be different length units, so normalize them
			return angle::radian_t(std::atan2(y.template convert<typename units::traits::unit_t_traits<X>::unit_type>()(), x()));
		}

		//----------------------------------
		//	HYPERBOLIC TRIG FUNCTIONS
		//----------------------------------

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute hyperbolic cosine
		 * @details		The input value can be in any unit of angle, including radians or degrees.
		 * @tparam		AngleUnit	any `unit_t` type of `catgeory::angle_unit`.
		 * @param[in]	angle		angle to compute the hyperbolic cosine of
		 * @returns		Returns the hyperbolic cosine of <i>angle</i>
		 */
		template<class AngleUnit>
		dimensionless::scalar_t cosh(AngleUnit angle)
		{
			static_assert(traits::is_angle_unit<AngleUnit>::value, "Type `AngleUnit` must be a unit of angle derived from `unit_t`.");
			return dimensionless::scalar_t(std::cosh(angle.template convert<angle::radian>()()));
		}

		/**
		* @ingroup		UnitMath
		* @brief		Compute hyperbolic sine
		* @details		The input value can be in any unit of angle, including radians or degrees.
		* @tparam		AngleUnit	any `unit_t` type of `catgeory::angle_unit`.
		* @param[in]	angle		angle to compute the hyperbolic sine of
		* @returns		Returns the hyperbolic sine of <i>angle</i>
		*/
		template<class AngleUnit>
		dimensionless::scalar_t sinh(AngleUnit angle)
		{
			static_assert(traits::is_angle_unit<AngleUnit>::value, "Type `AngleUnit` must be a unit of angle derived from `unit_t`.");
			return dimensionless::scalar_t(std::sinh(angle.template convert<angle::radian>()()));
		}

		/**
		* @ingroup		UnitMath
		* @brief		Compute hyperbolic tangent
		* @details		The input value can be in any unit of angle, including radians or degrees.
		* @tparam		AngleUnit	any `unit_t` type of `catgeory::angle_unit`.
		* @param[in]	angle		angle to compute the hyperbolic tangent of
		* @returns		Returns the hyperbolic tangent of <i>angle</i>
		*/
		template<class AngleUnit>
		dimensionless::scalar_t tanh(AngleUnit angle)
		{
			static_assert(traits::is_angle_unit<AngleUnit>::value, "Type `AngleUnit` must be a unit of angle derived from `unit_t`.");
			return dimensionless::scalar_t(std::tanh(angle.template convert<angle::radian>()()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute arc hyperbolic cosine
		 * @details		Returns the nonnegative arc hyperbolic cosine of x, expressed in radians.
		 * @param[in]	x	Value whose arc hyperbolic cosine is computed. If the argument is less
		 *					than 1, a domain error occurs.
		 * @returns		Nonnegative arc hyperbolic cosine of x, in the interval [0,+INFINITY] radians.
		 */
		template<class ScalarUnit>
		angle::radian_t acosh(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return angle::radian_t(std::acosh(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute arc hyperbolic sine
		 * @details		Returns the arc hyperbolic sine of x, expressed in radians.
		 * @param[in]	x	Value whose arc hyperbolic sine is computed.
		 * @returns		Arc hyperbolic sine of x, in radians.
		 */
		template<class ScalarUnit>
		angle::radian_t asinh(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return angle::radian_t(std::asinh(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute arc hyperbolic tangent
		 * @details		Returns the arc hyperbolic tangent of x, expressed in radians.
		 * @param[in]	x	Value whose arc hyperbolic tangent is computed, in the interval [-1,+1]. 
		 *					If the argument is out of this interval, a domain error occurs. For 
		 *					values of -1 and +1, a pole error may occur.
		 * @returns		units::angle::radian_t
		 */
		template<class ScalarUnit>
		angle::radian_t atanh(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return angle::radian_t(std::atanh(x()));
		}

		//----------------------------------
		//	TRANSCENDENTAL FUNCTIONS
		//----------------------------------

		// it makes NO SENSE to put dimensioned units into a transcendental function, and if you think it does you are
		// demonstrably wrong. https://en.wikipedia.org/wiki/Transcendental_function#Dimensional_analysis
		
		/**
		 * @ingroup		UnitMath
		 * @brief		Compute exponential function
		 * @details		Returns the base-e exponential function of x, which is e raised to the power x: ex.
		 * @param[in]	x	scalar value of the exponent.
		 * @returns		Exponential value of x.
		 *				If the magnitude of the result is too large to be represented by a value of the return type, the
		 *				function returns HUGE_VAL (or HUGE_VALF or HUGE_VALL) with the proper sign, and an overflow range error occurs
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t exp(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return dimensionless::scalar_t(std::exp(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute natural logarithm
		 * @details		Returns the natural logarithm of x.
		 * @param[in]	x	scalar value whose logarithm is calculated. If the argument is negative, a 
		 *					domain error occurs.
		 * @sa			log10 for more common base-10 logarithms
		 * @returns		Natural logarithm of x.
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t log(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return dimensionless::scalar_t(std::log(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute common logarithm
		 * @details		Returns the common (base-10) logarithm of x.
		 * @param[in]	x	Value whose logarithm is calculated. If the argument is negative, a 
		 *					domain error occurs.
		 * @returns		Common logarithm of x.
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t log10(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return dimensionless::scalar_t(std::log10(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Break into fractional and integral parts.
		 * @details		The integer part is stored in the object pointed by intpart, and the 
		 *				fractional part is returned by the function. Both parts have the same sign 
		 *				as x.
		 * @param[in]	x		scalar value to break into parts.
		 * @param[in]	intpart Pointer to an object (of the same type as x) where the integral part
		 *				is stored with the same sign as x.
		 * @returns		The fractional part of x, with the same sign.
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t modf(ScalarUnit x, ScalarUnit* intpart)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");

			UNIT_LIB_DEFAULT_TYPE intp;
			dimensionless::scalar_t fracpart = dimensionless::scalar_t(std::modf(x(), &intp));
			*intpart = intp;
			return fracpart;
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute binary exponential function
		 * @details		Returns the base-2 exponential function of x, which is 2 raised to the power x: 2^x.
		 * 2param[in]	x	Value of the exponent.
		 * @returns		2 raised to the power of x.
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t exp2(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return dimensionless::scalar_t(std::exp2(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute exponential minus one
		 * @details		Returns e raised to the power x minus one: e^x-1. For small magnitude values 
		 *				of x, expm1 may be more accurate than exp(x)-1.
		 * @param[in]	x	Value of the exponent.
		 * @returns		e raised to the power of x, minus one.
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t expm1(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return dimensionless::scalar_t(std::expm1(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute logarithm plus one
		 * @details		Returns the natural logarithm of one plus x. For small magnitude values of 
		 *				x, logp1 may be more accurate than log(1+x).
		 * @param[in]	x	Value whose logarithm is calculated. If the argument is less than -1, a 
		 *					domain error occurs.
		 * @returns		The natural logarithm of (1+x).
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t log1p(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return dimensionless::scalar_t(std::log1p(x()));
		}
		
		/**
		 * @ingroup		UnitMath
		 * @brief		Compute binary logarithm
		 * @details		Returns the binary (base-2) logarithm of x.
		 * @param[in]	x	Value whose logarithm is calculated. If the argument is negative, a 
		 *					domain error occurs.
		 * @returns		The binary logarithm of x: log2x.
		 */
		template<class ScalarUnit>
		dimensionless::scalar_t log2(ScalarUnit x)
		{
			static_assert(traits::is_scalar_unit<ScalarUnit>::value, "Type `ScalarUnit` must be a dimensionless unit derived from `unit_t`.");
			return dimensionless::scalar_t(std::log2(x()));
		}

		//----------------------------------
		//	POWER FUNCTIONS
		//----------------------------------
		
		/* pow is implemented earlier in the library since a lot of the unit definitions depend on it */

		/**
		 * @ingroup		UnitMath
		 * @brief		computes the square root of <i>value</i>
		 * @details		Only implemented for linear_scale units.
		 * @param[in]	value `unit_t` derived type to compute the square root of.
		 * @returns		new unit_t, whose units are the square root of value's. E.g. if values
		 *				had units of `square_meter`, then the return type will have units of
		 *				`meter`.
		 * @note		`sqrt` provides a _rational approximation_ of the square root of <i>value</i>.
		 *				In some cases, _both_ the returned value _and_ conversion factor of the returned
		 *				unit type may have errors no larger than `1e-10`.
		 */
		template<class UnitType, typename std::enable_if<units::traits::has_linear_scale<UnitType>::value, int>::type = 0>
		inline auto sqrt(const UnitType& value) -> unit_t<square_root<typename units::traits::unit_t_traits<UnitType>::unit_type>, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
		{
			return unit_t<square_root<typename units::traits::unit_t_traits<UnitType>::unit_type>, typename units::traits::unit_t_traits<UnitType>::underlying_type, linear_scale>
				(std::sqrt(value()));
		}

		//----------------------------------
		//	ROUNDING FUNCTIONS
		//----------------------------------

		/**
		 * @ingroup		UnitMath
		 * @brief		Round up value
		 * @details		Rounds x upward, returning the smallest integral value that is not less than x.
		 * @param[in]	x	Unit value to round up.
		 * @returns		The smallest integral value that is not less than x.
		 */
		template<class UnitType, class = typename std::enable_if<traits::is_unit_t<UnitType>::value>::type>
		UnitType ceil(UnitType x)
		{
			return UnitType(std::ceil(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Round down value
		 * @details		Rounds x downward, returning the largest integral value that is not greater than x.
		 * @param[in]	x	Unit value to round down.
		 * @returns		The value of x rounded downward.
		 */
		template<class UnitType, class = typename std::enable_if<traits::is_unit_t<UnitType>::value>::type>
		UnitType floor(UnitType x)
		{
			return UnitType(std::floor(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute remainder of division
		 * @details		Returns the floating-point remainder of numer/denom (rounded towards zero).
		 * @param[in]	numer	Value of the quotient numerator.
		 * @param[in]	denom	Value of the quotient denominator.
		 * @returns		The remainder of dividing the arguments.
		 */
		template<class UnitTypeLhs, class UnitTypeRhs, class = typename std::enable_if<traits::is_unit_t<UnitTypeLhs>::value && traits::is_unit_t<UnitTypeRhs>::value>::type>
		UnitTypeLhs fmod(UnitTypeLhs numer, UnitTypeRhs denom)
		{
			return UnitTypeLhs(std::fmod(numer(), denom.template convert<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>()()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Truncate value
		 * @details		Rounds x toward zero, returning the nearest integral value that is not 
		 *				larger in magnitude than x. Effectively rounds towards 0.
		 * @param[in]	x	Value to truncate
		 * @returns		The nearest integral value that is not larger in magnitude than x.
		 */
		template<class UnitType, class = typename std::enable_if<traits::is_unit_t<UnitType>::value>::type>
		UnitType trunc(UnitType x)
		{
			return UnitType(std::trunc(x()));
		}


		/**
		 * @ingroup		UnitMath
		 * @brief		Round to nearest
		 * @details		Returns the integral value that is nearest to x, with halfway cases rounded
		 *				away from zero.
		 * @param[in]	x	value to round.
		 * @returns		The value of x rounded to the nearest integral.
		 */
		template<class UnitType, class = typename std::enable_if<traits::is_unit_t<UnitType>::value>::type>
		UnitType round(UnitType x)
		{
			return UnitType(std::round(x()));
		}

		//----------------------------------
		//	FLOATING POINT MANIPULATION 
		//----------------------------------

		/**
		 * @ingroup		UnitMath
		 * @brief		Copy sign
		 * @details		Returns a value with the magnitude and dimension of x, and the sign of y. 
		 *				Values x and y do not have to be compatible units.
		 * @param[in]	x	Value with the magnitude of the resulting value.
		 * @param[in]	y	Value with the sign of the resulting value.
		 * @returns		value with the magnitude and dimension of x, and the sign of y.
		 */
		template<class UnitTypeLhs, class UnitTypeRhs, class = typename std::enable_if<traits::is_unit_t<UnitTypeLhs>::value && traits::is_unit_t<UnitTypeRhs>::value>::type>
		UnitTypeLhs copysign(UnitTypeLhs x, UnitTypeRhs y)
		{
			return UnitTypeLhs(std::copysign(x(), y()));	// no need for conversion to get the correct sign.
		}

		/// Overload to copy the sign from a raw double
		template<class UnitTypeLhs, class = typename std::enable_if<traits::is_unit_t<UnitTypeLhs>::value>::type>
		UnitTypeLhs copysign(UnitTypeLhs x, UNIT_LIB_DEFAULT_TYPE y)
		{
			return UnitTypeLhs(std::copysign(x(), y));
		}

		//----------------------------------
		//	MIN / MAX / DIFFERENCE 
		//----------------------------------
		
		/**
		 * @ingroup		UnitMath
		 * @brief		Positive difference
		 * @details		The function returns x-y if x>y, and zero otherwise, in the same units as x.
		 *				Values x and y do not have to be the same type of units, but they do have to
		 *				be compatible.
		 * @param[in]	x	Values whose difference is calculated.
		 * @param[in]	y	Values whose difference is calculated.
		 * @returns		The positive difference between x and y.
		 */
		template<class UnitTypeLhs, class UnitTypeRhs, class = typename std::enable_if<traits::is_unit_t<UnitTypeLhs>::value && traits::is_unit_t<UnitTypeRhs>::value>::type>
		UnitTypeLhs fdim(UnitTypeLhs x, UnitTypeRhs y)
		{
			static_assert(traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value, "Unit types are not compatible.");
			return UnitTypeLhs(std::fdim(x(), y.template convert<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>()()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Maximum value
		 * @details		Returns the larger of its arguments: either x or y, in the same units as x.
		 *				Values x and y do not have to be the same type of units, but they do have to
		 *				be compatible.
		 * @param[in]	x	Values among which the function selects a maximum.
		 * @param[in]	y	Values among which the function selects a maximum.
		 * @returns		The maximum numeric value of its arguments.
		 */
		template<class UnitTypeLhs, class UnitTypeRhs, class = typename std::enable_if<traits::is_unit_t<UnitTypeLhs>::value && traits::is_unit_t<UnitTypeRhs>::value>::type>
		UnitTypeLhs fmax(UnitTypeLhs x, UnitTypeRhs y)
		{
			static_assert(traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value, "Unit types are not compatible.");
			return UnitTypeLhs(std::fmax(x(), y.template convert<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>()()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Minimum value
		 * @details		Returns the smaller of its arguments: either x or y, in the same units as x.
		 *				If one of the arguments in a NaN, the other is returned.
		 *				Values x and y do not have to be the same type of units, but they do have to
		 *				be compatible.
		 * @param[in]	x	Values among which the function selects a minimum.
		 * @param[in]	y	Values among which the function selects a minimum.
		 * @returns		The minimum numeric value of its arguments.
		 */
		template<class UnitTypeLhs, class UnitTypeRhs, class = typename std::enable_if<traits::is_unit_t<UnitTypeLhs>::value && traits::is_unit_t<UnitTypeRhs>::value>::type>
		UnitTypeLhs fmin(UnitTypeLhs x, UnitTypeRhs y)
		{
			static_assert(traits::is_convertible_unit_t<UnitTypeLhs, UnitTypeRhs>::value, "Unit types are not compatible.");
			return UnitTypeLhs(std::fmin(x(), y.template convert<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type>()()));
		}

		//----------------------------------
		//	OTHER FUNCTIONS
		//----------------------------------
		
		/**
		 * @ingroup		UnitMath
		 * @brief		Compute absolute value
		 * @details		Returns the absolute value of x, i.e. |x|.
		 * @param[in]	x	Value whose absolute value is returned.
		 * @returns		The absolute value of x.
		 */
		template<class UnitType, class = typename std::enable_if<traits::is_unit_t<UnitType>::value>::type>
		UnitType fabs(UnitType x)
		{
			return UnitType(std::fabs(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Compute absolute value
		 * @details		Returns the absolute value of x, i.e. |x|.
		 * @param[in]	x	Value whose absolute value is returned.
		 * @returns		The absolute value of x.
		 */
		template<class UnitType, class = typename std::enable_if<traits::is_unit_t<UnitType>::value>::type>
		UnitType abs(UnitType x)
		{
			return UnitType(std::fabs(x()));
		}

		/**
		 * @ingroup		UnitMath
		 * @brief		Multiply-add
		 * @details		Returns x*y+z. The function computes the result without losing precision in 
		 *				any intermediate result. The resulting unit type is a compound unit of x* y.
		 * @param[in]	x	Values to be multiplied.
		 * @param[in]	y	Values to be multiplied.
		 * @param[in]	z	Value to be added.
		 * @returns		The result of x*y+z
		 */
		template<class UnitTypeLhs, class UnitMultiply, class UnitAdd, class = typename std::enable_if<traits::is_unit_t<UnitTypeLhs>::value && traits::is_unit_t<UnitMultiply>::value && traits::is_unit_t<UnitAdd>::value>::type>
		auto fma(UnitTypeLhs x, UnitMultiply y, UnitAdd z) -> decltype(x * y)
		{
			using resultType = decltype(x * y);
			static_assert(traits::is_convertible_unit_t<compound_unit<typename units::traits::unit_t_traits<UnitTypeLhs>::unit_type, typename units::traits::unit_t_traits<UnitMultiply>::unit_type>, typename units::traits::unit_t_traits<UnitAdd>::unit_type>::value, "Unit types are not compatible.");
			return resultType(std::fma(x(), y(), resultType(z)()));
		}

	}	// end namespace math

};	// end namespace units

#if defined(_MSC_VER) && _MSC_VER < 1800
#	pragma warning(pop)
#endif

#endif // units_h__

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