# Is there another type of number more advanced than complex numbers? [duplicate]

I am a student and I was introduced to complex numbers about a year ago. I am curious to know whether there is another type of number system more advanced than complex numbers.

So the way I was introduced to complex numbers was by being told the history of numbers. First, there were integers, which were used to count things (such as ten fingers, two eyes, and so on.) Then we came up with decimals that came in handy for example when we say "the glass is half full." Next, we used negative numbers. That was handy in for example in accounting to show debt. Then we came up with imaginary numbers, which are used in, for example, quantum mechanics. So my question is are there any other types of number systems?

Surely, there must be infinite types of numbers. This is because I imagine the real numbers to take up a dimension (similar to the x-axis) and imaginary numbers to take another dimension (similar to the y-dimension.) As a result, I am thinking there must be an infinite type of number because we can add as many dimensions as we like. Am I correct to assume this?

• Look up quaternions, octonions, Cayley algebras. – Deepak May 2 at 18:15
• You should take a look at the "$p$-adic number system" (en.wikipedia.org/wiki/P-adic_analysis) where $p$ is prime (in particular the 2-adic system) developed in the XXth century with many interesting/paradoxical features. – Jean Marie May 2 at 18:27

There are many kinds of numbers we can obviously say are as "advanced", or more advanced. The Cayley-Dickson construction lets you double the dimension as often as you like, e.g. the already mentioned quaternions are a $$4$$-dimensional generalization of complex numbers.
There are also some which aren't comparable to complex numbers as "more advanced". For example, $$\Bbb C_p$$ "combines" $$\Bbb Q_p$$ and $$\Bbb C$$, so how does $$\Bbb Q_p$$ compare to $$\Bbb C$$?