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I've been working with this identity but I never gave it much thought. Why is $ |z|^2 = z z^* $ ? Is this a definition or is there a formal proof?

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    $\begingroup$ what is the $z^*$ ? $\endgroup$ – Mher May 9 '14 at 16:20
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    $\begingroup$ Why do you aks this? Do you know what's the "usual" representation of a complex number as $\;z=x+iy\;,\;\;x,y\in\Bbb R\;$ ? Do you know what's the conjugate of a complex number? Then de the multiplication $\;z\overline z\;$ and that's all! $\endgroup$ – DonAntonio May 9 '14 at 16:21
  • $\begingroup$ @Mher Safaryan It's the complex conjugate. $\endgroup$ – hb20007 May 9 '14 at 16:24
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    $\begingroup$ @DonAntonio I'm asking because I had no idea it's this simple. The identity was presented to me baldly along with a sloppy explanation to complex numbers back when I took my intro class. Unfortunately, not all educators are able to present mathematics in an accessible, intuitive way. That's a real skill. $\endgroup$ – hb20007 May 9 '14 at 16:25
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I take it that $z^*$ means the conjugate of $z$, then it follows from nothing more than algebra:

$$zz^* = (a+bi) \cdot (a-bi) = a^2 - abi + abi + b^2 = a^2 + b^2 = |z|^2$$

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Let $z=x+iy$, for $x,y \in \mathbb{R}$. Then $zz^*=(x+iy)(x-iy)=x^2+y^2=|z|^2$.

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There is no formal proof: it's a definition.

Looking at $z=x+yi$ and doing $$ zz^*=(x+yi)(x-yi)=x^2+y^2 $$ shows that, when we interpret a complex number as a point in the Argand-Gauss plane, $|z|$ represents the distance of the point from the origin.

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