Number

Numbers are labels for quantity.

Naturals
The most important objects in mathematics are the Natural numbers, i.e. 1, 2, 3, etc. The first way to represent these objects with a series of tallymarks:



These objects are naturally ordered, and the successor to any Natural Number is clear. We can also easily compare any numbers, such as 𝍩𝍩𝍩𝍩𝍩𝍩𝍩𝍩𝍩 and 𝍩𝍩𝍩𝍩𝍩𝍩𝍩𝍩𝍩𝍩 by pairing marks up. If they pairing has none left over, they are the same say. If there are unpaired marks, then the one with more is bigger than the other.

They can be given letter names, such as n or m or x.

Integers
Recent incites from Physics over that last 100 years show that we may exist in a strange universe, where particles outnumber anti-particles, for some reason. Our number system might be revised, therefore, to include anti-tally-marks, what are traditionally called negative numbers.

As Matrices
As we start to imagine numbers which cannot be written as fractions, we must allow that there are multiple ways to represent such extensions. Because the number line is one-dimensional, a natural way to think about more numbers is to lay across the 1D number line another. This is a very reasonable way to imagine that i is half-way in between 1 and -1 (and so is -i).

A way to capture this conjectured two-dimensionality of extension numbers is to see numbers as vectors. Vectors are represented in vector-notation, but a collection of vectors is most easy to read as a matrix. For example, the number one can be seen as specifying some amount of x and y, an a matrix like this: $$ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$ This allows for a very easy depiction of i (the complex number) as $$ \begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix} $$ The split-complex number j can therefore be seen as $$ \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} $$ And an infinitesimal $$\epsilon$$ is viewed as $$ \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} $$

Ficticious Numbers
Since the late twentieth century, mathematicians have invented many new categories of numbers, none of which hold up under scrutiny. These include
 * Real number
 * Irrational number
 * Normal number
 * Computable number

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