i = √(-1) or i² = -1

This number is imaginary. It exists on an axis perpendicular to real numbers.

But that’s absolute JARGON and confusion

What the heck is “perpendicular to real numbers”?

Think of it like such.

Take a number line.

Copy it but put i next to every number. Rotate it 90° and line it up to the original line so that the 0 is in the same spot as the 0i (which… equals 0). 
 

What you have done just there is created the “Complex Plane” (or apparently also the “Argand Plane”)

Now I know this looks like a graph, but IT ISN’T BANISH THAT NOTION FOREVER TO THE NETHER REGIONS OF HECK.

This is just a very fancy number line. There is no “x” and no “y”, just real and imaginary numbers.

Pick a point on this plane. Say the point is like (3, 4i). The point is actually just the number 3 + 4i (if you want a generic number, it’s often just referred to as (a + bi). Which looks scary, but you just need to treat it like every other number. And by that I mean that your eyes need to glaze over when you look at it and forget what it means and just do the calculations.

Let’s think about the “value” of the number, though. The best way to really get that idea is to use what is called the “absolute value”. This is defined as the distance between the number and zero. For real numbers, you just make the number positive.

But how do you calculate the distance on a fancy number line?

This, my friends, is where my good friend Pythagorean comes in!

You draw a straight line on the fancy number line from 0 to your number. (3, 4i in our case).

The length of this line is the absolute value. To calculate it, we need a triangle. The sides are weird and I definitely need to draw up a visual aid. I’ll try to explain here. The longest side, called the hypotenuse, is the straight line you just drew. Next, draw a line straight up/down (depending if the point is below/above the real number line) from the point you chose, stopping at the real number line. Finally, draw a straight line from the end of THAT line to 0.

*phew*

Alright. We want to calculate the length of the longest side—the absolute value. A smart person that might have been called Pythagorean but also prolly not figured out that the length of this side is dependent on the other sides.

He said!

This length = √(Side1² + Side2²)

Funnily enough, the lengths of side one and two are just the coefficients of the real and imaginary parts of your number. (Side1 = 3 and Side2 = 4, for example, when we have (3, 4i))

And, doing some terrible math, we find that this length, and therefore the absolute value is equal to 5.

Yeah!

…what does that means…

5 is how big the number is. That. Yeah!

NEXT.

Exponents with i in them are complicated (math pun?). Here’s the formula -> 

e^(a+bi) = e^a(cos(b)+ i sin(b)).

I think. Imma fact check that when I like… actually have internet. (I did fact check it, and I was NOT CORRECT. It’s fixed now)

The reason that that is true is a bit out of the scope of this blog, but! I’ll give an overview.

That is fun.

yep

imaginary numbers.

If you didn’t take anything away from this, at least remember (this is the secret to all math):

your eyes need to glaze over when you look at it and forget what it means and just do the calculations.

Those people that say it should be intuitive are lying.