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Take A QuizIn 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 a {\displaystyle a} and b {\displaystyle b} are real, then) the complex conjugate of a + b i {\displaystyle a+bi} is equal to a − b i . {\displaystyle a-bi.} The complex conjugate of z {\displaystyle z} is often denoted as z ¯ . {\displaystyle {\overline {z}}.}
In polar form, 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 reference
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