When I first learned about imaginary numbers, I kept seeing the fact that an imaginary number is a number that when squared, it results in a negative number. I thought that was just a feature of imaginary numbers, but turns out that is just the actual definition. The $i$ is called an imaginary unit, so any number multiplied by the imaginary unit $i$ gives a negative number So, $i^2$ is -1, meaning $\sqrt(-1) = i$. You can still perform all of the basic operations on imaginary numbers as you can on real numbers. To solve for an exponential imaginary unit, such as $i^5$, you break it down into $i^2$, since we know the definition of that. So we would have $i^2\cdot i^2\cdot i$ which is $-1\cdot-1\cdot i$ which is $i$. $\sqrt(-x) = i\sqrt(x)$.

Imaginary numbers help us fill in the gaps for equations that don’t have real number solutions. Oddly enough, it’s the same reason that negative numbers were created. They needed an explanation for what could be made of equations such as 1-2. I was reading some notes from an Australian mathematician that I found very interesting. “Math is not a science, it does not represent reality, it is a system which merely works”. It exists for convenience. I love that about it. It’s reasoning for why things are the way that they are. The only reason imaginary numbers sound so abstract is because of the name and the fact that they are the opposite of real numbers. I used to think they were so confusing I wouldn’t be able to understand it very well, but they’re actually quite simple.