stylistic:tone, structural:summarization, linguistic:paraphrasing
Complex conjugate, absolute value and argument

The complex conjugate of the complex number z = x + yi is defined as z ¯ = x − y i. {\displaystyle {\overline {z}}=x-yi.} It is also denoted by some authors by z ∗ {\displaystyle z^{*}}. Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: z ¯ ¯ = z. {\displaystyle {\overline {\overline {z}}}=z.} A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

For any complex number z = x + yi, the product

is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root | z | = x 2 + y 2. {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.} By Pythagoras' theorem, | z | {\displaystyle |z|} is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that | z | = 1 {\displaystyle |z|=1}. If z = x = x + 0 i {\displaystyle z=x=x+0i} is a real number, then | z | = | x | {\displaystyle |z|=|x|}: its absolute value as a complex number and as a real number are equal.

Using the conjugate, the reciprocal of a nonzero complex number z = x + y i {\displaystyle z=x+yi} can be computed to be

1 z = z ¯ z z ¯ = z ¯ | z | 2 = x − y i x 2 + y 2 = x x 2 + y 2 − y x 2 + y 2 i. {\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.} More generally, the division of an arbitrary complex number w = u + v i {\displaystyle w=u+vi} by a non-zero complex number z = x + y i {\displaystyle z=x+yi} equals w z = w z ¯ | z | 2 = (u + v i) (x − i y) x 2 + y 2 = u x + v y x 2 + y 2 + v x − u y x 2 + y 2 i. {\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.} This process is sometimes called " rationalization " of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

The argument of z (sometimes called the "phase" φ) is the angle of the radius Oz with the positive real axis, and is written as arg z, expressed in radians in this article. The angle is defined only up to adding integer multiples of 2 π {\displaystyle 2\pi }, since a rotation by 2 π {\displaystyle 2\pi } (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval (− π, π ] {\displaystyle (-\pi,\pi ]}, which is referred to as the principal value. The argument can be computed from the rectangular form x + yi by means of the arctan (inverse tangent) function.