# Thread: Find the complex numbers

1. ## Find the complex numbers

Find the complex numbers $\displaystyle z_1, z_2, z_3$, so that $\displaystyle |z_1|=|z_2|=|z_3|=1$ and $\displaystyle z_1+z_2+z_3=z_1\cdot z_2\cdot z_3=1$.

Comment>
I know that $\displaystyle z_1, z_2, z_3$ are on circle with r=1.
And know $\displaystyle x_1+x_2+x_3=1$.

How to combine it for product?

Thank you.

2. ## Re: Find the complex numbers

Since arguments of complex numbers are added under multiplication of these numbers, $\displaystyle z_1z_2z_3=1$ means that $\displaystyle \arg z_1+\arg z_2+\arg z_3$ is a multiple of $\displaystyle 2\pi$.

3. ## Re: Find the complex numbers

Originally Posted by feferon11
Find the complex numbers $\displaystyle z_1, z_2, z_3$, so that $\displaystyle |z_1|=|z_2|=|z_3|=1$ and $\displaystyle z_1+z_2+z_3=z_1\cdot z_2\cdot z_3=1$.

Comment>
I know that $\displaystyle z_1, z_2, z_3$ are on circle with r=1.
And know $\displaystyle x_1+x_2+x_3=1$.

How to combine it for product?

Thank you.