# Math Help - Find the smallest integer

1. ## Find the smallest integer

Find the smallest integer n for which (1+i/1-i)^n=1

2. Hello,
Originally Posted by varunnayudu
Find the smallest integer n for which (1+i/1-i)^n=1
$1+i=\sqrt{2} e^{i \frac{\pi}{4}}$
$1-i=\sqrt{2} e^{-i\frac{\pi}{4}}$

So $\frac{1+i}{1-i}=e^{i \frac{\pi}{2}}=i$

Hence $\left(\frac{1+i}{1-i}\right)^n=i^n$
n=0
or n=4
etc...

3. ## Where to find a complete explanation for this

Thanks for ur reply but please can u tell me from where have u gottten this from, i need it as a reference. If u could please recommend me a book from which i can refer this from.

4. Hmm I don't know any reference in English.

I used this formula :
$a+ib=\sqrt{a^2+b^2} \cdot e^{i \arctan \frac ab}$

then use the properties of i :
$i^2=-1$
then $i^4=1$
etc..

http://en.wikipedia.org/wiki/Complex...the_polar_form