update nhmbers

content/posts/numbers.md

@@ -17,33 +17,50 @@ 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
 
-## Ordinals
+## Cardinals
 [Naturals](#Naturals)
 
-Okay so baby steps, now that we have some semblance of a place to start, we can move onto the **ordinals**. The ordinals are basically numbers used for putting things in order. How are they different from the naturals?
-
-Short answer: they aren't
-Long answer: ...for finite numbers
-
-## Cardinals
-
+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](#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](#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
-
-## Algebraics?
-
+[Integers](#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](#Rationals)
+> [!TODO] figure out how to construct this from the rationals 
 ## Complex Numbers
-
+[Reals](#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
 
-## 2D Hypercomplex
+## Split-Complex
+[Complex Numbers](#Complex%20Numbers)
+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