Think rationally

*plane

Imaginary numbers are typically introduced in a high school “algebra 2” course in my neck of the woods, like junior year unless you are accelerated or held back. I feel like it’s really common for them to not be taught well - the naming is something that occasionally trips up the teachers.

The teaching of them is something that really interests me - they’re the kind of thing that triggers the “when am I ever going to use this?”/rants about not learning how to do taxes. You can talk about the relevance to electronics, but DC electronics is already hard enough for most to comprehend.

I like to connect it to rotation. Show them the pattern of powers of i with physical movement - quarter turns.

Also tbf, rational numbers are just the lattice modded out by the equivalence relations (a, b) ~ (c, d) iff ad = bc, and the equivalence classes just happen to form an ordered field. If you show an undergrad math/science student this esoteric definition of rationals, the motivation of "2D numbers" makes a lot more sense. Of course, please don't show this to an elementary or middle schooler, a high schooler might be able to handle this if they're passionate about math.

Lateral numbers is my preferred term

Quaternions are useful, but no means necessary or intrinsic to 3d graphics.

They are useful for describing rotations in 3d space, but the same can be done with a matrix, or even Euler angles. It’s just a more calculations.

That said, basically all the major game engines use quaternions, so unless you are creating your own 3d renderer, you have to learn at least the basics about them.

Quaternions really shine at spherical interpolation (Slerp - yes thats the real terminology) between two rotations.

I can define zero as the absence of 1. But also 1 itself is a reference to the infinitely fractional 0.00...1 and I'd consider that a different number to the whole number 1

Oh yeah? Prove it

That would be 2. 3 would be {{},{{}},{{},{{}}}}.

Especially for solipists.

-1 / 0 can be defined to be a special new number, ⭐.

This system of numbers is actually mathematically consistent and has all the properties that a number system should have. But using simple algebra, you can prove that in this new number system, ⭐ = 0. And then you can also prove that all other numbers equal zero.

This number system is known as the zero ring, and it's the ring where the only number is 0 and the following operations are defined:

• 0 + 0 = 0
• 0 - 0 = 0
• 0 × 0 = 0
• 0 ÷ 0 = 0
• 0^0 = 0

All other symbols, including 1, 2, and ⭐, are just alternative labels for 0 in this number system. In other words, defining division by zero is essentially the equivalent of reality collapsing in mathematics.

So yes, you can validly define division by zero. You just get degenerate mathematics if you do. So be careful what you wish for.