# 2 questions about complex numbers

• Apr 4th 2011, 11:15 PM
Lafexlos
Q1) Prove that for any two complex numbers $\displaystyle z_{1}$ and $\displaystyle z_{2}$ , $\displaystyle |\frac{z_{1}}{z_{2}}| = \frac{|z_{1}|}{|z_{2}|}$. Is it true to write Arg$\displaystyle (z_{1}z_{2})=$ Arg$\displaystyle (z_{1})$$\displaystyle +$ Arg$\displaystyle (z_{2})$. Support your ideas geometrically.

Q2) Show that $\displaystyle z^{\frac{1}{n}}$ has $\displaystyle n$ distinct values for natural number $\displaystyle n$. What can you say about non-natural values of $\displaystyle n$? Why?

For question 1, i know that, if one the argument is negative, that equation isn't true because of essential argument thing but i don't know how to explain it at exam paper and how to support it geometrically.
For question 2, i didn't understand a thing. so don't know where to begin solving.
• Apr 4th 2011, 11:19 PM
Prove It
Q1. Start by writing $\displaystyle \displaystyle z_1 = x_1 + y_1i$ and $\displaystyle \displaystyle z_2 = x_2 + y_2i$.

Evaluate $\displaystyle \displaystyle \left|\frac{z_1}{z_2}\right|$ and $\displaystyle \displaystyle \frac{|z_1|}{|z_2|}$. Are they equal?

For the second part of that question, do you know how to work with the exponential polar form of complex numbers?
• Apr 4th 2011, 11:31 PM
Lafexlos
Quote:

Originally Posted by Prove It
Q1. Start by writing $\displaystyle \displaystyle z_1 = x_1 + y_1i$ and $\displaystyle \displaystyle z_2 = x_2 + y_2i$.

Evaluate $\displaystyle \displaystyle \left|\frac{z_1}{z_2}\right|$ and $\displaystyle \displaystyle \frac{|z_1|}{|z_2|}$. Are they equal?

Thanks for it. i'll try. :)

Quote:

Originally Posted by Prove It
For the second part of that question, do you know how to work with the exponential polar form of complex numbers?

Edit: i know how to work with those $\displaystyle e^{z}=e^x(cos{y}+isin{y})$ for $\displaystyle z=x+yi$ and $\displaystyle e^{i\theta}=cos{\theta}+i sin{\theta}$ about exponential thing. hope, you meant those. :)