Help with finalising my IA on complex numbers

[Saguaro]

Hello, I have written my IA on complex numbers in which I included the "Phantom Graphs" and DeMoivre's theorem. We have had our first draft and my teacher told me what I thought she would, that a) the ideas don't connect and b) its not long enough. I have a couple weeks to complete it and I am panicking as I have no idea what to do with it now. I would like to add some information about the applications of hyperbolic functions of complex numbers and something about Euler's formula, but I don't know how to do it. I know I can't just write out proofs for all this as I might as well have copied that from a book. And so what do I do with the hyperbolic functions? 

[kw0573]

You are HL, yes?
If so, there is no point explaining DeMoivre's theorem or Euler's formula because they are in syllabus. You would demonstrate as much knowledge just by using them correctly in problem solving. Because there is limit to what teachers can say to a first draft, you should bring it to a fellow student (possibly in Year 2) to look at it from student's perspective. It's discouraged to send them on IBS in case plagiarism checker picks it up.

I like the inclusion of phantom graphs. The content, for the most part, is relevant and good. The problem lies in the organization. Think of how you are taught to structure essays. You cannot just have arguments disjointed in the paper, you have to have a central idea, and explain your reasoning systematically. In that sense, your IA should have an overall goal.

Because you are not solving a large problem, you need to first find the overall argument to your exploration then find some way to group ideas together to support that claim. It could be showing why complex numbers are useful. Then, you can have a section for visualizing complex numbers, and another one for applying them. Or you can organize by connections to other parts of syllabus (algebra, functions, trig). You should go beyond proofs, and show some interesting problems that can be made easier with complex numbers or a representation. 

I suggest you look through HL past papers on complex numbers and trig. They have questions on using complex numbers to find exact values to some trig expression. It's somewhat rare to see complex numbers with hyperbolic functions because for example cosh(x) = 1/2 (e^x + e^(-x)), if x is complex, cosh (e^(iθ)) = 1/2(e^(e^(iθ)) - e^(e^(-iθ))). For the differential equations, the differential equation d²y / dx² + ky = 0 has circular trig solutions if k > 0, and hyperbolic solutions if k < 0, similar to the discriminant of a quadratic. Something like this could be worth while to discuss, except for that differential equations are too advanced for a year 1 student. The past papers may be helpful to provide some starting points. 

Edited April 8, 2018 by kw0573