Complex Numbers and the Plane

I'm writing this article assuming you know what complex numbers are. If not, I'm sure there's a well-written article on this great website describing them.

The complex plane, or the Argand plane as I was taught, is a visual representation of complex numbers. Imagine taking the complex number $a+bi$ and graphing it on the Cartesian plane as $(a,b)$. This is precisely what the Argand plane is: instead of an $x$-axis, there is a $\Re(z)$ axis (the real part of $z$, or $a$ if $z=a+bi$) and an $\Im(z)$ axis (the imaginary part of $z$, or $b$ if $z=a+bi$). The Argand plane essentially just turns $a+bi$ into the coordinate $(a,b)$ on the Cartesian plane.

There are many benefits of doing this.

First, every complex number can be written either as $a+bi$ or $re^{i\theta}$, which is similar to how the Cartesian plane can be written in rectangular or polar form.

Second, complex numbers have the same basic operations as vectors do. Imagine $z=a+bi$ as the vector $\vec{z}=\left<a,b\right>$. Adding complex numbers performs the same function as adding vectors: if we take $z=u+v$, we get the same point as $\vec{z}=\vec{u}+\vec{v}$.

Third, multiplication of complex numbers performs the function of homothety about the origin. This is very important when you learn how to complex bash. This is made possible by the convenient polar form of complex numbers.

Question: Now we've seen both imaginary numbers and of course, the real numbers. But, what happens if we put them together?

Well, let's start with an example. Say we have a number $z,$ and let's say that:

$z=$ $5$$+$$3i$

Whoa, what happened there? Well, we have a real number plus an imaginary number. But that doesn't make sense?!

$z$ $is$ $complex$ $number.$

It has a real part and an imaginary part.

Here are some problems to practice basics of complex numbers before you go into detail: http://cdn.kutasoftware.com/Worksheets/Alg2/Operations with Complex Numbers.pdf

Have fun!