: Mathematics: complex conjugate

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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 opposite in sign. That is, if ${\\displaystyle a}$ and ${\\displaystyle b}$ are real numbers then the complex conjugate of ${\\displaystyle a+bi}$ is ${\\displaystyle a-bi.}$ The complex conjugate of ${\\displaystyle z}$ is often denoted as ${\\displaystyle {\\overline {z}}}$ or ${\\displaystyle z^{*}}$ . In polar form, if ${\\displaystyle r}$ and ${\\displaystyle \\varphi }$ are real numbers then the conjugate of ${\\displaystyle re^{i\\varphi }}$ is ${\\displaystyle re^{-i\\varphi }.}$ This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: ${\\displaystyle a^{2}+b^{2}}$ (or ${\\displaystyle r^{2}}$ in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Complex_conjugate)