3.5 The Complex Conjugate
Definition 3.9 (Conjugate). Let \(z=x+iy\). The conjugate of \(z\), denoted \(\overline {z}\), is \[\overline {z}=x-iy,\] so the conjugate is obtained by changing the sign of the imaginary part. For example, if \(z=7+13i\) then \(\overline {z}=7-13i\).
Note 3.10. The conjugate matters because of one identity: \[z\overline {z}=(x+iy)(x-iy)=x^{2}+y^{2}=|z|^{2},\] which is always real and never negative. Multiplying a complex number by its conjugate is the standard way of turning something complex into something real, and it is what makes division possible.
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