# How many complex numbers satisfy $z\bar{z}=1$?

How many complex number satisfy $$z\bar{z}=1$$

Edit:

How about $$zz^*=1$$

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Write $z=a+bi$ and what equation in $a,b$ do you get? – Thomas Andrews Nov 5 '12 at 14:25
What do you mean by $zz*=1$= – Fredrik Meyer Nov 5 '12 at 14:38
I guess $z^*$ is physics notation for the complex conjugate. – GEdgar Nov 5 '12 at 14:55

## 2 Answers

Infinitely many. If $z=x+iy$, $z\overline{z}=(x+iy)(x-iy)=x^{2}+y^{2}=|z|^{2}$, so you're asking how many $z$ satisfy $|z|=1$, which is every $z$ on the unit circle.

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As many complex numbers as there are on the unit circle in the complex plane.

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Second spite downvote I have gotten in two days. Hopefully this isn't a common thing... – rschwieb Nov 5 '12 at 16:40
Upvoted to spite them! Also, this answer is useful... – The Chaz 2.0 Nov 12 '12 at 1:13