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.

• There's an entire area of mathematics called complex analysis,
– anon
May 13 '15 at 18:12
• Your teacher makes a strange claim. There is in fact an incredible number of interesting things you can say about $\mathbb C$. The complex numbers are deep and wide and full of surprises. I think you should write the report and show your teacher he/she is wrong. May 13 '15 at 18:12
• The equation $e^{\pi i}=-1$ is considered one of the coolest equations in math, it relates four fundamental numbers, $e$, $\pi$, $i$, and $1$. When this was first discovered it made a big impact. May 13 '15 at 18:14
• You can also try reporting on the fundamental theorem of algebra. May 13 '15 at 18:17
• Have a look at Gaussian integers, the subset of complex numbers whose real and complex parts are whole numbers. May 13 '15 at 18:17

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

• I am honestly trying to give you a bunch of keywords that you can follow up on. If you want me to expand on anything in particular, I would be happy to do so.
– Joel
May 13 '15 at 18:27
• Haha, thank you very much. This is really helpful! May 13 '15 at 18:30
• You can easly write an entire report solely on complex integration May 13 '15 at 18:45
• I thought it was the cubic equations that were troubling people and eventually led to the acceptance of complex numbers. These equations might have real solutions but require complex numbers to solve yhem algebraically. May 14 '15 at 4:19
• You are probably right about the cubic equations. I was simplifying things a bit for this answer. The fact is that complex numbers first arose in the study of roots of polynomials @andreysokolov
– Joel
May 14 '15 at 14:34

You'll probably find some inspiration in Needham's Visual Complex Analysis.