A report about complex numbers I was told to make a report for mathematics, and I could choose my own subject. I chose complex numbers, because I really think they are interesting. However, my teacher says that there isn't a lot of information to find about complex numbers.
Now I have found some things to write about complex numbers:


*

*What are complex numbers?

*History of complex numbers

*Application of complex numbers 

*Elementary operations
The problem is, I can't think of anything else to write about. I was maybe thinking about Euler's formula $e^{ix}=\cos x + i \sin x$, and then Euler's identity. But this is not enough. I was hoping if you have any ideas for this report.
 A: Complex numbers are an extension of the real numbers. Every complex number, $z$, can be expressed as $$z=x+iy$$ where $x$ and $y$ are real numbers and $i$ is a number for which $i^2 = -1$. In particular, the complex numbers can be viewed as a plane.
The imaginary number is a root of the equation $x^2+1=0$. We know that this equation cannot have any real roots, because when we graph it, it does not intersect the $x$-axis. Therefore, the solution to $x^2+1=0$ cannot be "real", and so it earned the infamous qualification "imaginary".
Imaginary numbers first appeared in the work of Hero of Alexandria. And there they appeared as the square root of a negative number. By then it was known  that any real number squared must be positive, and Hero of Alexandria flipped the sign of the number insider the square root without mention.
It wasn't until much later before complex numbers appeared again. This time it was in the context of solving cubic equations by Cardano (who lived roughly between 1500 and 1570). However, he still considered complex numbers extraneous and not proper mathematics.
It wasn't until the time of Euler before complex numbers became more accepted, in equations like $$e^{i\pi}+1=0.$$ More serious consideration of complex numbers began in the 1800s.
The subsequent development of complex numbers in the context of calculus occurred mostly in the 1800s. The goal was to resolve questions like the Prime Number Theorem. It also is used to answer questions in Physics, Approximation Theory, and has many other applications.
A quick list of important topics in complex analysis are the following:
Liouville's Theorem
Cauchy's Theorem
Picard's Theorem
Riemann Hypothesis
Riemann Sphere
Mobius Transformations (See: https://www.youtube.com/watch?v=JX3VmDgiFnY )
Entire Functions
Analytic Functions
Euler's Formula $e^{i\theta} = \cos(\theta) + i \sin(\theta)$.
Analytic Number Theory (Prime Number Theorem, Dirichlet Series, Dirichlet's Theorem on Arithmetic Progressions, etc.)
Honestly, the field of Complex Analysis is a huge area of study. This is just a taste of what goes on there.

You might find the following book useful as primary source material:
"An imaginary tale: The story of $\sqrt{-1}$" by Paul Nahin
A: You'll probably find some inspiration in Needham's Visual Complex Analysis.
