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In polar form, if and are real numbers then the conjugate of is . This can be shown using Euler's formula . The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}} (or r 2 {\displaystyle r^{2}} in polar coordinates ).
A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle.
The complex number z can be represented in rectangular form as = + where i is the imaginary unit, or can alternatively be written in polar form as = ( + ) and from there, by Euler's formula, [14] as = = . where e is Euler's number, and φ, expressed in radians, is the principal value of the complex number function arg applied to x + iy ...
A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers.
x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, e ix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries.
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 = ( + ),
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 ...
The polar decomposition of a matrix can be seen as the matrix analog of the polar form of a complex number as =, where is its absolute value (a non-negative real number), and is a complex number with unit norm (an element of the circle group).