Consider the following quadratic equation
\[x^2 + 3 = 0\]
It seems pretty simple, but if you put this into the quadratic formula and type it into your calculator, it will give you an error. Up to now, we have simply disregarded these non-real solutions. Now, we want to make some progress on solving these kinds of equations. So, we define a new "imaginary" number
\[\sqrt{-1} = i\]
\(i\) is our "imaginary" number. So, if we solve the above equation
\[x = \frac{\pm \sqrt{-12}}{2} = \pm \frac{\sqrt{12}}{2}i\]
A complex number is one with both a real and imaginary part. They are written in the form \(a + bi\). Just like real numbers, complex numbers have some properties that we can explore
First, adding complex numbers
\[(x + ui) + (y + vi) = (x + y) + (u + v)i\]
For example,
\[2 + 3i + 5 - i = 7 + 2i\]
Let's try multiplying complex numbers
\[(x + ui)(y + vi)\]
\[xy + yui + xvi + uvi^2\]
\[xy - uv + (yu + xv)i\]
We define a new concept - the conjugate. If we have
\[z = a + bi\]
The its conjugate is
\[\bar{z} = a - bi\]
What is the motivation to do this? Try
\[z \bar{z} = (a + bi)(a - bi)\]
\[z \bar{z} = a^2 + abi - abi + b^2\]
\[z \bar{z} = = a^2 + b^2\]
This is sort of "extracting" the real terms.
We can also define an inverse of a complex number, for example
\[(1 + i)(i - i) = 2\]
\[(1 + i)^{-1} = \frac{1 - i}{2}\]