Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but with opposite sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is ⁠ a − b i {\displaystyle a-bi} ⁠. The complex conjugate of z {\displaystyle z} is often denoted as z ¯ {\displaystyle {\overline {z}}} or ⁠ z ∗ {\displaystyle z^{*}} ⁠. In polar form, if r {\displaystyle r} and φ {\displaystyle \varphi } are real numbers then the conjugate of r e i φ {\displaystyle re^{i\varphi }} is ⁠ r e − i φ {\displaystyle re^{-i\varphi }} ⁠. 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). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.