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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?

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Your answer for $w^\ast$ might be marked as being (technically) wrong. They specified polar form with $0 \le \theta<2\pi$, so for $w^\ast$ this angle would be $8\pi/5$. –  André Nicolas Jun 9 '11 at 16:10
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1 Answer

up vote 1 down vote accepted

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?

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So it's plotting a circle that's been translated one up the y-axis? Also I think $w^5$ = 1? But why does it take 5 to get all the way around? Why not 2 or 3? Maybe because $w$ is the 5th root of something? –  Lerp Jun 9 '11 at 15:38
    
Yes, because $w$ is a 5th root of unity (i.e. 1) –  Chris Taylor Jun 9 '11 at 15:43
    
Thanks, you've been a great help! –  Lerp Jun 9 '11 at 15:52
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