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For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division: [ref 1] + ¯ = ¯ + ¯, ¯ = ¯ ¯, ¯ = ¯ ¯, ¯ = ¯ ¯, A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real.
A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a + 0i = a, 0 + bi = bi, and a + (−b)i = a − bi; for example, 3 + (−4)i = 3 − 4i.
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis. Although the Greek mathematician and engineer Heron of Alexandria is noted as the first to present a calculation involving the square root of a negative number, [6] [7] it was Rafael Bombelli who first set down the rules for multiplication of complex numbers in 1572.
The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a ...
Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation. Multiplication of a scalar and a vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did ...
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. [1] Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
The relation to complex numbers becomes clearer, too: in 2D, with two vector directions σ 1 and σ 2, there is only one bivector basis element σ 1 σ 2, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements σ 2 σ 3, σ 3 σ 1, σ 1 σ 2, so three imaginaries.
Bicomplex numbers feature two distinct imaginary units. Multiplication being associative and commutative, the product of these imaginary units must have positive one for its square. Such an element as this product has been called a hyperbolic unit. [1]