General Number Sets

I am not an expert, merely a student. Apologies if I've not delved into octonions as much as you would have wanted to.

There are many numbers in this world, but many aren't alike to each other. 

e is quite different to 7, for example - e is a transcendental number with a never-ending decimal expansion, whereas 7 is an integer which has no complex decimal expansion. Compare them with i however - e and 7 are real, whereas i is imaginary. But they're all numbers at the end of the day, part of the incomprehensible language called maths.

How to classify these numbers? Put them into sets, such as:

the natural numbers, N - these are the whole numbers starting from 0 or 1 (depending who you ask - should you ask the ISO, it's 0), going all the way up to infinity. Why natural? Well, they're the most obvious numbers, which can be used for counting or tallying a set. Three sheep, seven penguins, ninety-four readers. I assume this is why 0's status is vague, as you can't really count 0 items in a set (if you do, it looks like this: {}). The only operations you can apply with natural numbers are addition and multiplication, along with the latter's inverse - division (since you're only dealing with positive whole numbers), and with this come two identities - the additive identity, 0 (where if you add 0 to a number, there's no change), and the multiplicative identity, 1 (similar but for multiplication).

There are two primary uses for natural numbers - ordinal numbers, for ordering items (the third sheep; the seventh penguin); and cardinal numbers, for counting. 0 is notably not an ordinal number, as what could the 0th item be?

Going up a logical level, you get the integers Z - these numbers include all natural numbers, but also incorporating whole negative numbers. Whereas the range of natural numbers was {x: x ≽ 0}, the range of integers is {x: -∞ ≼ x ≼ ∞}. At this point, you can also subtract numbers as negative numbers exist.

With natural numbers and integers, there are certain properties which can be used. Alongside the additive and multiplicative identities, you also have these three cases, which will remain constant up to and including the complex numbers:

• The commutative property (as in, x + y = y + x). You can shuffle the numbers around and the output is the same (though not in division).
• The distributive property (as in, x(y + z) = xy + xz). You can expand brackets as such and the output is the same.
• The associative property (as in, x + (y + z) = (x + y) + z. You can shuffle the brackets around and the output is the same).
  

The next highest number set are the rational numbers Q, where numbers can be expressed as the ratio between two integers. This is where fractions come in - 1/3 is rational, and 7/1 is rational as well. However, the key issue is that this ratio has to be between integers. So √2 isn't rational, as it cannot be expressed as such. Instead, it's irrational. 

After this come the real numbers R. These are all the numbers mentioned so far, including ones such as e and √2. Things get a bit more chaotic after this - it's like entering a storm that's waging outside your house. 

Because it's time to mention complex numbers C. At this point, numbers are of the form a + bi, featuring a real component a and an imaginary component bi. I've written about i before - to put it simply, any number bi = √-b². Complex numbers are the first number set so far in which a value in that set consists of two components, which allows you to plot them on an Argand diagram and get a polar co-ordinate (of the form a,bi).

So far, all the properties I mentioned earlier still apply. That is, until you get hit by the exceptionally harsh gale that are quaternions H (designated as such after William Rowan Hamilton, who first considered them). At this point, the commutative property no longer applies, and you have four components - a + bi + cj + dk, and the bolded values are part of the general equation i² + j² + k² = ijk = -1. This comes about due to six different identities, at which point you can no longer simply rearrange the components of a multiplicative function. 

Octonions O are like an avalanche compared to quaternions. You're up to eight components at this point, and the associative property no longer applies. Indeed, the general formula is an extension of the general quaternion formula, though this time with eight variables instead of four, and there are plenty of e's involved (as in a + b e₀ + c e₁ +...+ h e₇). This blogpost isn't about quaternions or octonions, however, and I'm a bit uncertain as to whether or not I'd like to get into more detail on them in the future. I might, however.

And this, Honourable Reader, is the end. Past octonions, the storm is so chaotic that only the real numbers up to and including the octonions are normed division algebras - as in, division is possible. Should you be brave and try to get out the eye of the storm, sedenions S await, what with their sixteen components and this being effectively 5D maths, and as you go further and further out, more properties are lost and the outcomes are unusual - but I'm not that courageous.