complex numbers- how do I prove the following statement?

Question

given: $$ \left | z_{1}\right |=\left | z_{2}\right |=...=\left | z_{n}\right |=1 $$

How do I prove:

$$ (1+\frac{z_{2}}{z_{1}})(1+\frac{z_{3}}{z_{2}})*...*(1+\frac{z_{n}}{z_{n-1}})(1+\frac{z_{1}}{z_{n}}) \in \mathbb{R} $$

?

Duplicate of http://math.stackexchange.com/q/1184048/215011 – grand_chat Apr 19 '15 at 14:35 — grand_chat, Apr 19 '15 at 14:35

score 0 · Answer 1 · edited Apr 19 '15 at 14:36

0

You need to consider of the angle from the origin of each complex number $1 + \frac{z_{k+1}}{z_k}$. That's $\frac{1}{2}\arg \frac{z_{k+1}}{z_k} = \frac{1}{2}(\arg z_{k+1} - \arg z_k)$. Add these up and see they cancel out to give $0$.

edited Apr 19 '15 at 14:36

meta_warrior

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answered Apr 19 '15 at 14:33

wlad

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The "1" in the expression will affect the argument.. – meta_warrior Apr 19 '15 at 14:44
That's why it's multiplied by 1/2 – wlad Apr 19 '15 at 14:45

complex numbers- how do I prove the following statement?

1 Answers1