website/content/posts/numbers.md

draft	title
true	Numbers aren't real and can hurt you

Numbers aren't real and can hurt you

Okay so like the question of 'What is a number' should be an easy enough question to answer, but of course it isn't cause when is anything ever simple on this godforsaken plane of existence.

Naturals

Let's start with the caveman bits, Natural Numbers (\mathbb{N}) (god I really hope latex renders here). This is the stuff every 3 year old knows, 1,2,3,\ldots. Seems basic enough because it is.

Just fucking kidding is 0 a natural number? who knows, I sure as hell don't.

Also, like what exactly is 0 or 1?

I mean if we want to use the Zermelo-Fraenkel definition, 0 = \{\}, 1 = \{0\}, 2 = \{0,1\}, \dots ad absurdum

Cardinals

Naturals

Okay so baby steps, now that we have some semblance of a place to start, we can move onto the cardinals. They're called that since they're used to measure the cardinality of a set (also known as it's size). these are generally the same as the Naturals since like, the hell other kind of unnatural number could you use to count the number of something?

Aleph and Beth

Cardinals Sigh you just had to ask didn't you

Note: it was my rhetorical question, but throughout this work I will be blaming the consequences of these on you, the reader. I do not apologize

\aleph_0 is the first of these, it's the cardinality of \mathbb{N}, of the number of natural numbers. You see that zero there? That means there's more of them

Integers

Naturals Snapping back to normalcy for a second, we have the integers. These are just the natural numbers and their evil twins, negative numbers (they're evil because they keep showing up in my budget and then I get sad).

Rationals

Integers ...actually this one is also pretty normal, take an integer, divide it by an integer that's not zero. Congratulations, you have a rational number

Reals

Rationals

[!TODO] figure out how to construct this from the rationals

Complex Numbers

Reals Okay so you know how the Reals start with you starting at zero and going left or right? Turns out we can make up a new number that lets us go up and down as well. What is this cursed number we pulled out of our asses? i, or \sqrt(-1) Well actually it doesn't really go up and down, it's more like you're spinning a number around 0 (let's call this the origin for clarity). For example, if you rotate 1 90° around the origin, you get i. If you only go 45°, you get \sqrt2+i\sqrt2. All three of these values have the same distance from the origin (1) but they are at different angles. We have two ways of representing complex numbers: a+bi and re^{\theta i}. The first tells you the position of the point in terms of the real and imaginary units. The second tells you the position in terms of the distance and angle from the origin

Vectors

Matrices

Split-Complex

Complex Numbers Remember how I said we could use i to rotate numbers around a circle (technically e^{\theta i}). What if you could rotate things in the opposite of a circle You can! We defined i earlier as the square root of -1, if we take the square root of 1 instead we get j

[!QUOTE] I'm an electrical engineer and we use j for the square root of -1

Shut up.

[!QUOTE] isn't that just 1?

No. because I said so. That's not even a joke or anything that's actually how it's defined

Dual

Quaternions

Octonions and Sedenions

RGB

Color Charge

Surreal Numbers

Algebraic Structures

p-adics

Conclusion

Q: What the fuck is a number? A: A miserable pile of math