# Roots of unity

1. Jun 1, 2014

### MissP.25_5

Hello everyone.

How to find the 4th root of -4? I know it's just plugging in the number into the formula but how since n=4, how can we calculate that without calculator? And how to draw it? Here I attached what I have done so far.

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2. Jun 1, 2014

### ehild

How you draw z=-4? What is the angle of -4? It is not 0 as you wrote.

ehild

3. Jun 1, 2014

### Fightfish

It would help to express -4 in complex exponential form.

4. Jun 1, 2014

### SteamKing

Staff Emeritus

z = -4 + i0, or (-4, 0)

θ = arctan (0/-4) = π

Even though z = -4, draw it on the complex plane properly.

5. Jun 1, 2014

### MissP.25_5

Oh yes, that was a careless mistake. Ok, so now I have the values:
r=4
n=4
θ=∏

But how do I find 4√4 ?

6. Jun 1, 2014

### HallsofIvy

Staff Emeritus
You know that $$x^4= (x^2)^2$$, right? So $$\sqrt[4]{4}= \sqrt{\sqrt{4}}$$. What is the square root of 4? What is the square root of that?

7. Jun 1, 2014

### SteamKing

Staff Emeritus
Use this information and apply Euler's formula.

z = r e$^{i θ}$

z$^{1/n}$ = r$^{1/n}$ e$^{i kθ / n}$, $0\leq k \lt n$

See:

http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Roots.aspx

8. Jun 1, 2014

### MissP.25_5

If I compute z = r e$^{i θ}$, wouldn't that bring us back to the start? Because that is -4.
Could you elaborate please? I don't really get it.

9. Jun 1, 2014

### MissP.25_5

Thanks! I never thought of that. But what if we were to find the 5th root? I don't think this method can be applied.

10. Jun 1, 2014

### SteamKing

Staff Emeritus
That's your problem in a nutshell. You are working with complex numbers and you don't understand what is going on.

I wrote these two formulas as a reminder of

1. how to express any number in exponential form, using Euler's formula, and

2. how to find the n nth roots of said number.

Your original problem was to find the 4 fourth roots of -4, or in other words, solve the equation

$z^{4}+4 = 0$ or

$z^{4}= -4$

Let's say the solutions to this equation are the complex numbers

$ω_{1}, ω_{2}, ω_{3}$, and $ω_{4}$

By writing -4 in the form $z = r e^{i θ}$, where z = -4,
we can use the second formula from the quote,

z$^{1/n}$ = r$^{1/n}$ e$^{i kθ / n}$

to calculate the numerical values of ω

I really recommend that you study the article linked below very carefully:

http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Roots.aspx

For a better visual representation of the cyclic nature of such roots:

http://mathworld.wolfram.com/RootofUnity.html