I am stuck on a past exam question. I don't have a clue what it's on about and would appreciate any help.
In this question, $w$ denotes the complex number $cos{\frac{2}{5}\pi} + isin{\frac{2}{5}\pi}$
i) Express $w^2$, $w^3$ and $w^*$ in polar form, with arguments in the interval $0 \le \theta < 2\pi$
My working (I'm pretty sure this is right just it's part of the same question)
$$|w| = 1$$
$$arg(w) = \frac{2}{5}\pi$$
$$\therefore |w^2| = 1$$
$$arg(w^2) = \frac{2}{5}\pi + \frac{2}{5}\pi = \frac{4}{5}\pi$$
$$|w^3| = 1$$
$$arg(w^3) = \frac{2}{5}\pi + \frac{2}{5}\pi + \frac{2}{5}\pi = \frac{6}{5}\pi$$
$$\therefore w^2 = cos{\frac{4}{5}\pi} + isin{\frac{4}{5}\pi}, w^3 = cos{\frac{6}{5}\pi} + isin{\frac{6}{5}\pi}, w^* = cos{\frac{2}{5}\pi} – isin{\frac{2}{5}\pi}$$
ii) The points in an Argand Diagram which represent the number
$$ 1, 1 + w, 1 + w + w^2, 1 + w + w^2 + w^3, 1 + w + w^2 + w^3 + w^4 $$
are denoted by A, B, C, D, E respectively. Sketch the Argand diagram to show these points and join them in the order stated.
I could do this by brute force just bashing numbers into my calculator to find the coordinates but surely there has to be a better way? Is there a property of complex numbers when they're added together like that?
Best Answer
Think geometrically. Do you know about Euler's formula? It states that
$$e^{i\theta} = \cos\theta + i \sin\theta$$
By comparing, you can see that your number is $w = e^{2\pi i/5}$, which geometrically corresponds to the point in the argand diagram a distance 1 from the origin, with argument $2\pi/5$, which is 1/5 of the way round a circle.
Squaring to get $w^2$ leaves the modulus unchanged and doubles the argument, so the point $w^2$ is 2/5 of the way round the circle, and $w^3$ and $w^4$ are 3/5 and 4/5 of the way round the circle.
Try plotting the points $1, w, w^2, w^3, w^4$ on the argand diagram and see what you notice about them. Now can you start adding them up?