More details at http://www.welchlabs.com/blog Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space.
In part 5 we begin to see what makes imaginary numbers so unique, by investigating their special relationship with real numbers. It's interesting that mathematicians regularly made use of \(\sqrt{-1}\) for the 200 years after Bombelli's death without recognizing its deeper connection to the real numbers. I think this says a lot about how difficult it is to wrap your head around imaginary numbers. If math geniuses like Euler missed this, it certainly isn't obvious.
This episode concludes with a concept we'll dig into more next time: the complex plane. This is the plane formed by adding the imaginary axis to the real axis. The complex plane is more useful than ordinary planes because of the special relationship between its axes - this is what we'll explore next time.
Enjoy!