What are complex numbers, and how do you represent them on an Argand diagram?

Complex numbers are ones that have a real and imaginary component and are graphed per the diagram below.

Explanation:

In the world of mathematics, what we're told is not possible a more basic class suddenly becomes vitally important in a more advanced class (I still remember long ago when learning to subtract that I was taught you couldn't take away more than you already had - that 3 minus 2 was fine but 2 minus 3 was a complete no no... and then suddenly in a more advanced class we're dealing with the very important topic of negative numbers.)

And that is the case here when dealing with complex numbers.

Let's first talk about what they are. Remember hearing about how it made no sense whatsoever to have the expression $\sqrt{- 1}$? How there is no number that, when squared, would ever result in $- 1$? Well... it turns out that nature considers the $\sqrt{- 1}$ important because it pops up in all sorts of applications (electrical engineering for one) and if we don't include this mathematical oddity into our equations, we end up getting wrong results.

And it's not like we only get $\sqrt{- 1}$. We get $\sqrt{- 2}$ and $\sqrt{- \frac{1}{2}}$ and $\sqrt{- \pi}$. To deal with these numbers, we factor out the $\sqrt{- 1}$ and call that $i$ - and anything that includes an $i$ term is called an imaginary number.

And so we work with numbers that have 2 components - the real component (ex. $2 , \frac{1}{2} , \pi$) and the imaginary component and express them added together, like $2 + \frac{5}{3} i$ and $\pi - i$ and the like -that is called a complex number.

To graph a complex number, we need to devise a grid that will allow for the real component and the imaginary component and an Argand diagram allows us to do that. An Argand diagram is one that has the real component on the x-axis and the imaginary component on the y-axis that looks like this: