Finding the Smallest n for Real z^n in Complex Numbers

[nayfie]

I have solved part a, I just have no idea how to go about doing part b. If anybody could point me in the right direction, that would be greatly appreciated!

Homework Statement

a. Express z = \frac{1 + \sqrt{3}i}{-2 -2i} in the form rcis\theta

b. What is the smallest positive integer n such that z^{n} is a real number? Find z^{n} for this particular n.

Homework Equations

z^{n} = r^{n}cis(n\theta)

\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}}cis(\theta_{1} - \theta_{2})

The Attempt at a Solution

Part A

z_{1} = 1 + \sqrt{3}i

r = |z_{1}| = \sqrt{4} = 2

\theta = arccos(\frac{1}{2}) = \frac{\pi}{3}

z_{1} = 2cis(\frac{\pi}{3})

z_{2} = -2 -2i

r = |z_{2}| = \sqrt{8} = 2\sqrt{2}

\theta = arccos(\frac{-2}{2\sqrt{2}}) = \frac{3\pi}{4}

z_{2} = 2\sqrt{2}cis(\frac{3\pi}{4})

\frac{z_{1}}{z_{2}} = \frac{2}{2\sqrt{2}}cis(\frac{\pi}{3} - \frac{3\pi}{4}) = \frac{1}{\sqrt{2}}cis(\frac{-5\pi}{12})

Part B

No idea :(

 

[chiro]

I have solved part a, I just have no idea how to go about doing part b. If anybody could point me in the right direction, that would be greatly appreciated!

Homework Statement

a. Express z = \frac{1 + \sqrt{3}i}{-2 -2i} in the form rcis\theta

b. What is the smallest positive integer n such that z^{n} is a real number? Find z^{n} for this particular n.

Homework Equations

z^{n} = r^{n}cis(n\theta)

\frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}}cis(\theta_{1} - \theta_{2})

The Attempt at a Solution

Part A

z_{1} = 1 + \sqrt{3}i

r = |z_{1}| = \sqrt{4} = 2

\theta = arccos(\frac{1}{2}) = \frac{\pi}{3}

z_{1} = 2cis(\frac{\pi}{3})

z_{2} = -2 -2i

r = |z_{2}| = \sqrt{8} = 2\sqrt{2}

\theta = arccos(\frac{-2}{2\sqrt{2}}) = \frac{3\pi}{4}

z_{2} = 2\sqrt{2}cis(\frac{3\pi}{4})

\frac{z_{1}}{z_{2}} = \frac{2}{2\sqrt{2}}cis(\frac{\pi}{3} - \frac{3\pi}{4}) = \frac{1}{\sqrt{2}}cis(\frac{-5\pi}{12})

Part B

No idea :(

Hint: What property does a real number have (what is its imaginary component equal to)?

Based on an argand diagram (a two dimensional graph with real and imaginary part), what is the argument (angle) equal to?

 

[nayfie]

Oh right. So they want me to find the smallest value of n such that arg(z) = k\pi (so that the imaginary part = 0)?

 

[nayfie]

Okay it turns out that's the way to solve the question.

Thank you for the help!