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Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then

$$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$

Hint: Note that $|a|^2 = a\overline a$

I have been stuck on this problem for a while and cant seem to find somewhere to start.

Edit: The question linked is a different question

Edit: I have been able to prove the |w|=1 case, but when proving the z case I take these steps and cant seem to figure out where to go next. $|1-\overline{w}z| = |z\overline{z} - \overline{w}z| = |z||\overline{z} - \overline{w}| = |\overline{z} - \overline{w}|$

Brandon Klein
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    Hint: assume $|z|=1$, so you can write $z=e^{i\theta}$ for some $\theta\in \mathbb R $. – Dimitris Apr 23 '15 at 22:54
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    Somebody down-voted this question. Why? ${}\qquad{}$ – Michael Hardy Apr 23 '15 at 23:11
  • @MichaelHardy Maybe because a little research would find existing answers on this very site, not to mention elsewhere. –  Apr 23 '15 at 23:22
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    "Maybe." We shouldn't have to take "maybe" for an answer. Whoever down-voted it should explain their objections here in the comments. ${}\qquad{}$ – Michael Hardy Apr 23 '15 at 23:35
  • @MichaelHardy I was not the first to downvote, but I have just done so now for the reason of lack of shown research effort. This is in line with the tooltip that appears on hovering over the downvote arrow. – apnorton Apr 24 '15 at 00:35

2 Answers2

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We can suppose that $|w| = 1$ (the problem is equivalent if $|z| = 1$). $$|1 - \overline{w}z| = \left|1 - \frac zw\right| = \frac1{|w|} |w - z| = |w - z| $$

Alonso Delfín
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Kevlar
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  • I get the w direction pretty well and most of the z direction, but I have gotten to a point where I am getting $|\overline{z}-\overline{w}|$ and don't know how to simplify this to $|w-z|$. Is there a theorem or anything that could explain if this is possible? – Brandon Klein Apr 24 '15 at 23:35
  • The absolute value of a complex number doesn't change if you take the opposite or the conjugate so $|\overline{z}-\overline{w}| = |z-w| = |w-z|$. – Kevlar Apr 25 '15 at 21:14
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Ok so we know that $z$ and $w$ are distinct and so $$\left|\frac{w-z}{1-\bar{w}z}\right|\neq 0,$$ right? So try proof by contradiction and remember how norms work with division: $$\left|\frac ab\right|=|a||b|$$. Note that the only positive real number that squares to $1$ is $1$.

theGrolarBear
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