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A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
Complex modulus may refer to: Modulus of complex number , in mathematics, the norm or absolute value, of a complex number: | x + i y | = x 2 + y 2 {\displaystyle |x+iy|={\sqrt {x^{2}+y^{2}}}} Dynamic modulus , in materials engineering, the ratio of stress to strain under vibratory conditions
The tiling of the Poincaré disk is given in a natural way by the J-invariant, which is invariant under the modular group, and attains every complex number once in each triangle of these regions. This tessellation can be refined slightly, dividing each region into two halves (conventionally colored black and white), by adding an orientation ...
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 ...
A modest extension of the version of de Moivre's formula given in this article can be used to find the n-th roots of a complex number for a non-zero integer n. (This is equivalent to raising to a power of 1 / n). If z is a complex number, written in polar form as = ( + ),
A complex number z may be viewed as the position of a point P in a 2-dimensional space, called the complex plane. The absolute value (or modulus) of z may be thought of as the distance of P from the origin of that space.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801.