Mathematics related structures written in Java
Fractions, (equiv. Rational Numbers, q ∈ ℚ) are the set of all numbers q = a / b, for a and b in the integers, b ≠ 0. Binary operations include addition, subtraction, multiplication, division. Exponentiation is defined for a rational base and integer power. The unary operation, the reciprocal, is defined for all non-zero rational numbers.
Complex numbers, one of three hypercomplex number systems, are an extension of the real numbers, (z ∈ ℂ = ℝ + ℝi) by adjoining the imaginary unit i, where i2 = -1.
Dual numbers, one of three hypercomplex number systems, are an extension of the real numbers, (ℝ + ℝε) by adjoining the dual unit ε, where ε2 = 0.
Hyperbolic numbers, one of three hypercomplex number systems, are an extension of the real numbers, (ℝ + ℝj) by adjoining the hyperbolic unit j, where j2 = 1.
Quaternions, often denoted as, h = a + bi + cj + dk ∈ ℍ, where i, j, and k are unit vectors in ℝ3. This is often described as an extension of the complex numbers (ℍ = ℂ + ℂj) because taking complex numbers z1 = a+bi and z2 = c+di, (a+bi) + (c+di)j = a+bi + cj + dij = a+bi + cj + dk.
This can also be redefined in terms of a scalar added to a vector in ℝ3:
a + bi + cj + dk = r + v⃗, s.t. r = a, v⃗ = bi + cj + dk.
The definition for Quaternion addition via the scalar and vector method is:
(r1, v⃗1) + (r2, v⃗2) = (r1 + r2, v⃗1 + v⃗2)
The definition for Quaternion multiplication via the scalar and vector method is:
(r1, v⃗1)(r2, v⃗2) = (r1r2 - v⃗1 ⋅ v⃗2, r1v⃗2 + r2v⃗1 + v⃗1 ⨯ v⃗2)
The definition for Quaternion reciprocation via the scalar and vector method is:
(r, v⃗)-1 = (r / (r2 + v⃗ ⋅ v⃗), -v⃗ / (r2 + v⃗ ⋅ v⃗))
Matrices are 2-dimensional grids of numbers. These are often used for linear transformations from ℝm to ℝn, where the matrix is n by m in size. Square matrices have special properties, such as the existence of the determinant, and in special cases, can be deemed invertible (determinant ≠ 0) or nilpotent (mn = 0 for some n, m ∈ M, n ∈ ℕ). These can also be used to solve systems of linear equations.
Vectors represent points, directions and other components in n-space (n being the dimension of the vector). These can be thought of as column matrices, or a m by 1 matrix. These are used primarily as a tool to describe a direction with a particular magnitude. Various other operations are defined for vectors, such as inner and outer products, or cross products.
| Structure | Add. | Sub. | Mult. | Div. | Non-Int Pow | Exponent |
|---|---|---|---|---|---|---|
| Real (double) | Yes | Yes | Yes | Yes | Yes | Yes |
| Fraction | Yes | Yes | Yes | Yes | No | No |
| Complex | Yes | Yes | Yes | Yes | Yes | Yes |
| Dual | Yes | Yes | Yes | Yes | Yes | Yes |
| Hyperbolic | Yes | Yes | Yes | Yes | Yes | Yes |
| Quaternions | Yes | Yes | Yes | Yes | No | No |
| Matrices | Yes | Yes | Yes | No | Sometimes1 | Sometimes2 |
| Vectors | Yes | Yes | Yes | No | No | No |
1 Matrix must have non-zero determinant.
2 Matrix must be nilpotent (mn = 0 for some n, m ∈ M, n ∈ ℕ) or diagonalizable (M = PDP-1 for some P and D diagonal).