Question

Complex numbers help. *question below* Will give medal.

Answer 1

must have gone really below the screen

Answer 2

So this question came on my Math exam today, and unfortunately, I wasn't able to understand the question, so I wasn't able to answer it. Can someone walk me through this, please? It's question 5 :)

Answer 3

i'll explain this to u in morning :)

Answer 4

:)

Answer 5

I'd suggest converting/writing both numbers in polar form. The operations asked for in (a) are much simpler in polar form.

Answer 6

@ikram002p , have you forgotten me? xD

Answer 7

\[u=2i=2(\cos0+i\sin0)\]
\[w=\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\]
For (a), you need to find the \(x+iy\) form of \(w\), which makes use of the equations
\[\begin{cases}x^2+y^2=r^2&\text{where }r=1\text{ in this case}\\
\tan\theta=\dfrac{y}{x}&\text{where }\theta=\dfrac{2\pi}{3}\end{cases}\]
You also need to find \(uw\) and \(\dfrac{u}{w}\), which is where my suggestion for converting to polar comes in.

Given two complex numbers in polar coordinates, \(\large z_1=re^{i\theta}\) and \(\large z_2=se^{i\phi}\), you get the product \(\large z_1z_2=rse^{i(\theta+\phi)}\) (so that it's a matter of multiplying the moduli and adding the arguments), and the quotient \(\large \dfrac{z_1}{z_2}=\dfrac{r}{s}e^{i(\theta-\phi)}\) (so you just divide the moduli and subtract the arguments). Then convert back to the rectangular \(x+iy\) form.

Answer 8

The Argand diagram I'll leave to you. That's fairly simple to do once you find the rectangular form.

Part (c) is interesting. Lots of ways you can go about it. I'd suggest showing that the angle between any two of the numbers \(w\), \(uw\), and \(\dfrac{u}{w}\) is 60 degrees, or \(\dfrac{\pi}{3}\) radians.

Answer 9

I'm still confused with part (a). Do I need to solve those equations simultaneously? @SithsAndGiggles

Answer 10

Yes
\[\tan\frac{2\pi}{3}=\frac{\sqrt3}{-1}=\frac{y}{x}~~\implies~~y=-\sqrt3 x\]
Then solve for \(x,y\):
\[r^2=1=x^2+(-\sqrt3 x)^2=4x^2~~\implies~~x=\frac{1}{2}~~\implies~~y=-\frac{\sqrt3}{2}\]
Hence
\[w=\frac{1}{2}-i\frac{\sqrt3}{2}\]