# Thread: complex numbers in a fourth degree equation

1. ## complex numbers in a fourth degree equation

Problem:
Solve the equation $\displaystyle z^4=4+4i$ and graph the solutions in an argand diagram.

I'm leaning towards a solution where I use:

$\displaystyle |z|(cos(0)+isin(0))$
$\displaystyle z^4=r^4(\cos(4v)+i\sin(4v))$

But it keeps messing up. Help is heavily appriciated.

2. Originally Posted by Greenb
Problem:
Solve the equation $\displaystyle z^4=4+4i$ and graph the solutions in an argand diagram.

I'm leaning towards a solution where I use:

$\displaystyle |z|(cos(0)+isin(0))$
$\displaystyle z^4=r^4(\cos(4v)+i\sin(4v))$

But it keeps messing up. Help is heavily appriciated.
$\displaystyle 4 + 4i = 4 \sqrt{2} \text{cis} \, \left( \frac{\pi}{4} + 2 n \pi\right)$ where n is an integer.

Compare this with $\displaystyle z^4 = r^4 \text{cis} \, (4 \theta)$ to get the value of $\displaystyle r$ and the values of $\displaystyle \theta$ necesary to write down the four distinct solutions for $\displaystyle z = r \text{cis} \, \theta$.

3. Ah you got $\displaystyle \frac{\pi}{4}$ from $\displaystyle \tan(\theta)=\frac{b}{a}$

4. Originally Posted by Greenb
I can follow you all the way except how you get the angle $\displaystyle \frac{\pi}{4}$
Draw an argand diagram showing 4 + 4i ...... It's in the first quadrant. And $\displaystyle \text{Arg} (4 + 4i) = \tan^{-1} \frac{4}{4} = \tan^{-1} 1$ ....

5. Somehow I keep getting reversed solutions..

Mine are;
$\displaystyle z_1=1.51+0.30i$
$\displaystyle z_2=-0.30+1.51i$
$\displaystyle z_3=-1.51-0.30i$
$\displaystyle z_4=0.30-1.51i$

$\displaystyle z_1=0.30+1.51i$
$\displaystyle z_2=-1.51+0.30i$
$\displaystyle z_3=-0.30-1.51i$
$\displaystyle z_4=1.51-0.30i$

Is the equation wrong?

$\displaystyle \sqrt[4]{\sqrt{2}*4}(\cos(\frac{\pi+n*2\pi}{4})+i\sin(\fra c{\pi+n*2\pi}{4}))$
where n=0,1,2,3

6. It must be the angle seeing as the answer is right only reversed, in my opinion. Maybe I'm substituting it wrong?

7. Originally Posted by Greenb
Somehow I keep getting reversed solutions..

Mine are;
$\displaystyle z_1=1.51+0.30i$
$\displaystyle z_2=-0.30+1.51i$
$\displaystyle z_3=-1.51-0.30i$
$\displaystyle z_4=0.30-1.51i$

$\displaystyle z_1=0.30+1.51i$
$\displaystyle z_2=-1.51+0.30i$
$\displaystyle z_3=-0.30-1.51i$
$\displaystyle z_4=1.51-0.30i$

Is the equation wrong?

$\displaystyle \sqrt[4]{\sqrt{2}*4}(\cos(\frac{\pi+n*2\pi}{4})+i\sin(\fra c{\pi+n*2\pi}{4}))$
where n=0,1,2,3
It's not hard to see that solutions in the answer key are wrong. Simply raise them to the power of 4. I get 4 - 4i as the result .....

Your solutions are the correct fourth roots of 4 + 4i.

By the way you have a typo in you 'equation', it should be $\displaystyle \sqrt[4]{\sqrt{2}*4}\left(\cos \left(\frac{\frac{\pi}{4}+n*2\pi}{4}\right)+i\sin\ left(\frac{\frac{\pi}{4}+n*2\pi}{4}\right)\right)$

8. Darn! Your right, but seriously this has been bugging me all day.... Thanks for your persistence with me hehe