# Thread: [SOLVED] Complex No.s

1. ## [SOLVED] Complex No.s

How exactly do I do these questions:

1. z = 1 + $\displaystyle \sqrt{3}$ . Find the smallest positive integer n for which $\displaystyle z^n$ is real and evaluate $\displaystyle z^n$ for this value of n. Show that there is no integral value of n for which $\displaystyle z^n$ is imaginary.

2. Find the modulus of $\displaystyle \frac{7-i}{3-4i}$. Evaluate $\displaystyle tan[\arctan\frac{4}{3} - \arctan\frac{1}{7}].$Hence, find the principal argument of $\displaystyle \frac{7-i}{3-4i}$ in terms of $\displaystyle \pi$.

2. Oh w8, found the solutions ...

SORRY!

3. Originally Posted by xwrathbringerx
1. z = 1 + $\displaystyle \sqrt{3}$ . Find the smallest positive integer n for which $\displaystyle z^n$ is real and evaluate $\displaystyle z^n$ for this value of n. Show that there is no integral value of n for which $\displaystyle z^n$ is imaginary.
I am sure that you mean $\displaystyle z = 1 + \sqrt 3 \color{red}~i$. In that case $\displaystyle \arg (z) = \frac{\pi }{3}$.
What is the smallest $\displaystyle n$ such that $\displaystyle n\left(\frac{\pi }{3}\right)=k\pi$, that is a multiple of $\displaystyle \pi$.

Is the any $\displaystyle n$ such that $\displaystyle n\left(\frac{\pi }{3}\right)=k\left(\frac{\pi }{2}\right)?$