## Complex Vectors

The complex number i or j is a pseudo-unit vector. It’s purpose is to enable vector operations on scalars. Put another way, it’s an adapter so you can plug scalars into vector operations.

To elaborate, start with Aristotle. He came up with an idea of four causes: material, formal, efficient, and final. These causes are effective descriptions of what something is. Here is a fairly good description.

Think for a moment of a phone charger, USB to wall plug. When you think of what that is, you don’t think of transistors and bridge rectifiers, transformers, and coils. You describe it by what it does and how you use it. This is Aristotle’s formal cause. The thing is an adapter so you can plug your phone into the wall. The material cause, the circuitry and plastic, is honestly mostly irrelevant and makes you take a long detour around a short mental trip. Think of a phone adapter. What is it? It’s an adapter.

That’s what i is and how it should be considered. The mathematical definition, sqrt(-1) is the material cause. That’s what makes it up. But to almost everyone who uses it, what i is is an adapter. It lets anyone do vector math on scalars.

What’s a scalar? What’s a vector?

Numbers come in two basic groups: scalars and tensors. A scalar is a number by itself. A tensor is any concatenation of more than one scalar together. Vectors are a particular type of tensor, a bunch of scalars put together, in a certain order. For your curiosity, they’re organized with one index in particular order. Which particular order doesn’t make them vectors or not; they just need an order.

Let’s think sports. The score of one team is a scalar. Yesterday, 9/24, JMU scored 32 points against Appalachian State (American College Football). The 32 is a scalar. It’s a number by itself.

Clearly, there’s some utility in that single number. The JMU coaches know if they’re doing well or not. The ASU coaches know if their defense is doing well or not. It’s a high number, so it was probably an exciting game. The number is useful but exists by itself. It is a scalar.

But it doesn’t tell me who won the game. For that, I need another number: the other team’s score. ASU got 28, so the final score was 32-28. 32-38 is a vector. It’s got two scalars, 32 and 28, put together in a certain order.

What order? JMU/ASU. Could you flip it around? Easily. But you need to be careful so the numbers retain their meaning.

Two numbers together tell you something that one number doesn’t: who won the game. What’s more, the vector 32-28 tells you something that would be difficult to grab with the numbers unordered. It’s trivially obvious that you can do something with the scores as a vector that is difficult or impossible with the scores as independent scalars, and the punchline here is pretty simple. Vectors are super useful in a bunch of circumstances.

Give me one step of trust: vectors are super useful in many ways. Velocities, scores, electrical currents, and a million other things are more usefully addressed as vectors.

Vectors are so useful that people wanted to do vector math on scalars. Which, if you think about it, doesn’t really make a lot of sense, but it turns out it’s also super useful. So a group of enterprising mathematicians figured out a way to do it.

Complex numbers let you do vector math on scalars. It’s what ‘i’ does. It’s what ‘i’ is. It’s the thing, like your phone charger, you plug your phone into one end, stick the other end into your wall outlet, and they work together. Complex formulations let you do vector math on scalars.

And seriously, that’s it. Don’t worry about the sqrt(-1). It’s as irrelevant to most functions as the bridge rectifier inside your charger is to whether or not it charges your phone. Yeah, it’s there, and yeah, it needs to be there, but if you’re just plugging in your phone, it’s not worth a lot of time. It’s certainly not worth aggravation and frustration.

Complex numbers are just stuck with two really terrible names: imaginary or complex numbers. They’re neither moreso than any other numbers.