**By Nicholas McLean, University of Adelaide**

In my project I was working with complex numbers and seeing how certain complex numbers behaved under certain functions. I then assigned each complex numbers to one of two sets depending on how they behaved and looked at when every complex number belonged to one of the two sets only. In this post I will attempt to outline what complex numbers are and why they exist and why they are useful.

There is a theorem in mathematics called the fundamental theorem of algebra which states that every quadratic polynomial has exactly two roots (or solutions), when accounting for multiplicity. Remember from high school algebra this means that an equation of the form ax^{2}+bx+c=0 has exactly two solutions for x. These solutions can be found using the infamous quadratic formula. Take for example y=x^{2}-1. This looks like the picture shown in figure 1. As can be seen in the figure this quadratic (taking y = 0) clearly has roots of 1 and -1 as that is where it crosses the x-axis. However, what about the quadratic seen in figure 2, y=x^{2}+1? That does not cross the x-axis but according to the fundamental theorem of algebra it must have two roots. So where are its two roots?

It seems our picture here is incomplete in some sense as it cannot capture where these roots lie. To grasp at this concept we must extend our notion of numbers to something called the complex numbers. We all know about the real numbers. They include numbers like 0,1,15,53,π,e,-100,1/2,-1/7, etcetera. The complex number are exactly that but with a new number added called “i”. Now “i” is defined as the square root of -1. For an intuitive explanation as to why this is, if we ask ourselves the question what number squares to -1 (in other words what number when multiplied by itself gives -1) we don’t have such a number, hence it is reasonable to define such a thing. We call a number of the form a+bi a complex number where a and b are real numbers.

Complex numbers now provide an answer to the question we asked before, ‘where the roots of the quadratic, y = x^{2} +1, lie?’. Letting y = 0 because we are looking for where the quadratic crosses the x-axis, we see that x^{2}=-1 will give the roots. Since we defined “i” as the square root of -1, i and –i are our solutions.

This means that no matter what quadratic is given we can always find two roots if we use complex numbers.