Complex numbers

Very clueless on 7b, looked at the ms and they seem to have rotated the side by 90 degrees but not sure what they have done after that

I see now thanks, would it be possible to solve this using the roots of unity so transforming a unit square to the new square or would that approach take far too long?

I see now thanks, would it be possible to solve this using the roots of unity so transforming a unit square to the new square or would that approach take far too long?

Youd have both a rotation and a translation so yes its possible but its more complex.
The key thing with questions like these are to get used to seeing complex addition as translations and complex multplication as scaling (modulus) and rotation (arg).

You definitely can. In fact I'd encourage you to share your solution using roots of unity.
But that's an overkill. A geometric approach seems the easiest.

Thanks for the insight I'll attempt the approach I stated later.

Not quite familiar with these types of questions as they're not really on my exam spec but would it be possible if you could come up with a question covering these bases thanks

Using complex numbers/operations for geometry/geometric transformations is a fairly thing as it enables things like scaling and rotations to be combined in "simple" algebra. There are a reasonable number at
https://madasmaths.com/archive/maths_booklets/further_topics/various/complex_numbers_part_2.pdf
and Id imagine drfrostmaths will have a fair few. I just googled something like
complex operations geometry questions
Also
https://www.maths.ox.ac.uk/system/files/attachments/complex.pdf
popped up and seemed readable/advanced a level. I think thats a chapter of his book where there are exercises as well
https://www.google.co.uk/books/edition/Towards_Higher_Mathematics_A_Companion/34swDwAAQBAJ?hl=en&gbpv=1&printsec=frontcover

Can you upload the first part/whole question. It sounds like A,B,C,D are on a circle centred on the origin, otherwise the description won't make sense.

Yes, so you have four points A,B,C,D on a circle of radius 12 centered on the origin and these correspond to the 4, fourth roots of -12i (12e^(-i3pi/2) ... So the midpoints of each side correspond to
* For rectangular representation - find the mean of two adjacent vertices
* For polar / exponential representation - mean arg of two adjacent vertices, and mod is given by a simple triangular / area calculation.
They've gone down the 2nd route to keep everything in polar / exponential complex form.

Not sure what you mean as I found the mean and kept it in exponential form but I need to take it to the power of 4 to get w yet inside the bracket is composed of the two vertices. So I don't actually understand why geometrically the argument is the mean of the arguments of the two complex numbers inside the brackets. And isn't A B C D representing a square not a circle?

* A,B,C,D have the same mod (radius r) so you can imagine them on a circle, but yes, they are the vertices of a square which is centered on origin. The side midpoints will also lie on a circle with radius r/sqrt(2), but again they will define a square as the angle between adjacent points/vertices will be 90.
* Geometrically, the mean of the arguments, simply divides the angle between A and B into two. So if A was arg -30 and B arg 60, then the midpoint of AB would lie at arg 15, so bisecting the 90 degree angle between A and B.
Not too sure what you mean about the other points.

Its a bit hard to read your args for A,B,C,D, but it looks like there are no i's in the exponentials? Euler would be turning in his grave.

But yes to your first line. The midpoint is indeed the average of the two points. However, what is the second line ()^4? That is why you calculate a midpoint as
r e^(ix)
where you get expliclt values for r and x. Then taking the fourth power gives
r^4 e^(i4x)
which is easy to evaluate.

My bad my I's disappeared, but as you I have e^(3ipi/8)+ e^(7ipi/8) which I want to simplify to one exponential so I can apply de moivre's where I can just multiply by 4 to the argument however my issue is that how would I simply my exponentials to one exponential?