Eli5: What are complex numbers?

Question

What are complex numbers and their practical applications?

In: Mathematics

Answer 1

Just to add a quip,

Yes, “complex” and “imaginary” are *extremely unfortunate* names as people get caught up in the words and what they mean, but the names come from a sort of historic precedent and, frankly, a bit of mathematician humor.

An example of an imaginary number is “3 + 2i”. Don’t think “algebra”, don’t think “arithmetic”, just because you see the “i” there, and the addition – think “notation” or “convention” rather than terms to be solved. What you have here is a 2 on the normal number line you’re used to, and a 2 along the second axis, what we call the “imaginary” axis. As others have said, this is effectively a singular 2D number, you can plot this on an xy graph and the construct represents a single number, just like you can plot “7” on a number line which is just a 1D graph.

I mean look, someone had to invent the decimal dot at some point, and I’m sure the notation we take for granted today blew the minds of everyone who saw it the first time.

And there are types of numbers that extend in multiple dimensions. Another one is the quaternion, which is in the form of “a + b*i* + c*j* + d*k*”. Beyond the complex number, these extend into 4 dimensions.

Where do we use them? A lot of rotations in math. I’m sure there are other uses, but in video games, this is what I principally use them for. Imagine a 2D graph, xy, like we’ve all done in grade school. Imagine an arrow from the origin, [0,0], to [1,0]. Basically, the arrow is 1 unit long and is pointing to the 1 on the normal number line. Often in video games we want to rotate a thing, and our arrow here represents our thing. The important thing is if I give some degrees of rotation, the arrow has to remain 1 unit long, because it’s easy to skew the length as you compute the new direction the arrow points. Because in video games and other things, while the line indicates direction and that should be all that matters, that arrow length can cause our thing to stretch, which is typically not what you want. We call this the unit circle. Quaternions are used to represent rotations in 3D space, because you can rotate along all 3 axis, xyz, at the same time.