Express the following complex number in polar form

Question

I've just began studying complex numbers for the first time and there are no solutions to the book I'm following. Are my answers correct for the following introductory questions? Thanks.

Q. Express the following in the form $re^{i\theta}$.

i. $i^3$

Ans: $e^{\frac{\pi i}{2}} $.

ii. $1-i$

Ans: $\sqrt{2} e^{\frac{7\pi i}{4}} $.

iii. $\sqrt{2}(1+i)$

Ans: $2e^{\frac{\pi i}{4}}$.

Answer 1

Yes you are correct on the bottom two.

For the the top one, remember $i^3=-i$

Be careful in questions where they restrict the values of $\theta$. For example if $-\pi <\theta \leq\pi$, your second would be $\sqrt2e^{\frac{-i\pi}{4}}$ instead.

But yeah, just follow the steps I suspect you have been.

If your number is $x+iy$ we have:

$$r=\sqrt{x^2+y^2}$$ $$\theta=\arctan(\frac yx)$$ With the second, double check you got the right angle, and add or subtract $\pi$ to get the correct angle if need be.

Answer 2

For (i), recall that $i = e^{i\pi/2}$. Then $i^3 = (e^{i\pi/2})^3 = e^{3\pi i/2} = e^{-\pi i/2}$.

The other answers are correct.