What are the real counting, natural, integer, rational, irrational, real, trascendental, complex, and algebraic numbers?

This is a made up explanation of the kinds of numbers.

Part I: naturals and integers

Counting numbers: ℕ

A long time ago humans begun trading. Due to all the ongoing trade it became necessary to keep track of what – and more importantly how much, they had already traded. This way they could know how much merchandize was stil left to trade. And so humans invented numbers for this purpose.

For the numbers it’s not important what is specifically being traded; this is one of the important characteristics of numbers. If it is goats, cows, loaves of bread, vases of flour, enslaved people; it makes no difference from the standpoint of the numbers. The fact that all traded merchandize can by measured by some kind of unit is the only really important thing.

For example, imagine a farmer who had 1 pig, 1 sheep, and 3 cows and imagine this farmer wanted to keep track of how many animals the had. So this farmer would have to count all the 1 + 1 + 3 = 5 animals.

A very importnat attribute of this kind of numbers is the fact that they are ordered, as the counting numbers are made by counting, they have a natural implied order: one, then two, then five; but before five, there is three and four, and so on…

These are simplest kind of number: the natural (or counting) numbers. The set (or collection) of all these numbers – the naturals (for short), is denoted with the special symbol “ℕ”. This set is infinite because it has no end, there exists no number bigger than all the other numbers; in other words there exists a next number for all numbers.

One of the simplest ways to construct the naturals is by sucession. There’s some thing, any thing, and there’s a succesor for each thing. There is 1 of something, and it has a succesor: 2 of something, then 3, 4, and so on.

But the true power of numbers (including other kinds of numbers), lies in the ability to perform operations on them. By calculating – performing operations absractly on the numbers – we can anticipate results before we actually do anything real; such as trading merchandize.

Operations on numbers

The simplest kind of numbers in use by humans are associated with the simplest operation on numbers: addition; quite simply an extension of counting.
Recall that the natural counting numbers are built on succesion; and succesion is equivalent to add one. In this sense addition is simply repeated sucession.

If there are 7 things and we add 2 (i.e 7 + 2), this transaltes into: start with 7 and add one twice. Take the succesor of the succesor of 7; which is 9. This basic scenario is equivalent to asking “what do we get if we start with 7 units and add 2 more?”.

This is a rather simple example, but we can also ask more complex questions. For example the opposite to the previous one: “what do we need to add to 7 in order to get 9?” i.e. 7 + 𝐱 = 9. So far so good, the answer is 2.
But what if we swap the seven and the nine? “what do we need to add to 9 in order to get 7?” i.e. 9 + 𝐱 = 7.

If we were being rigorous this would be a problem; and since this is mathematics we are being rigiorous, therefore this is a problem. Luckily it is a solved problem.

The issue is that there isn’t a natural counting number such that we can add it to 9 and get out 7. We constructed the naturals by sucession. It is not possible to advance your way from 9 back to 7.

The solution we need are negative numbers. The natural numbers together with the negative numbers are called the integers (the integers for short) and they have the special symbol “ℤ”.