The Complex Numbers!

The complex numbers are one of the coolest fields out there. They might not feed your stomach, but hopefully they will be able to feed your mind! This field is super cool because its existence is strange to those of us who saw them in school. This is that number group that we saw and said “why do we care about imaginary numbers? If it’s imaginary, then why should we care?” However, complex numbers are interesting as a field since they can be proven in multiple ways.

First, just as a heads up, calling them imaginary is a bit disingenuous. As we will see, these aren’t things we made up for no reason. As such, “imaginary” is just the term that really stuck with them and thus that’s what the second part is called now. In reality, complex numbers are what can be defined as an extension field. What is an extension field? An extension field can be defined in the following way: Let F be a field and f(x) a non-constant polynomial in F[x]. Then there is an extension field K of F that contains a root of f(x). In layman’s terms, if there is a field (let’s say the real numbers) and there is a non-constant polynomial in the field (let’s say x^2+1=0), then there is an extension field K (say the complex numbers) that contains the root of that polynomial. This is the complex numbers.

However, what if we wanted to prove that the complex numbers are a field the good old fashioned way? By proving all 12 axioms and see if they do indeed work? Now, I know that you’ve probably never thought that, but since you’re this far into the website, why don’t we find out. Below this is a video on the Complex Numbers and the proof of them being a field. However, before watching, I challenge you! You know all the axioms. Try to prove some of them that you might be interested in!



This video gives the official proof on why the complex numbers are a field!

For one more application, how else could we represent the complex numbers? I ask this because of the chains of the idea that these are “imaginary” numbers. As such, what if I told you that the complex numbers actually represent a type of matrix? The fact that we can represent these numbers in a form that is probably more familiar to us will hopefully be extremely interesting to you as it was for me. Watch the video below to see some cool transformations of the Complex Numbers!



A video on a Complex Number Matrix!