I think I have a good grasp of complex numbers and their applications. Quick Google searches show quarternions and octonions. I figured beyond complex numbers would be needed for higher dimensions, but why are the next more complex numbers seemingly 4 and 8 dimensional rather than 3, 4, 5, etc. TIA
In: Mathematics
There are two main pathways I want to go here: algebraic closure and on what numbers *are* in a more metamathematical sense. I’ll start with a simplified version of algebraic closure, hop into metamathematics, and then tie up some loose ends on algebraic closure. Finally, after that, we can answer your question.
So, we often invent new number types because one of our operations “break” the system. For example, consider natural numbers and addition. Can you add two natural numbers and not get a natural number back? Nope! So we call the naturals “closed” under addition. But what about subtraction? Well, 5 – 10 is not a natural number, so… we broke something. But what if we extended to include negatives and make the integers? Now we are good! Still good with multiplication, too. But… not division. So now we need to make rational numbers. Ok, ok, good. But where the heck does pi go??? This allows us to extend to the classic real numbers. Ok, good. Now let’s play with polynomials and roots. Things are looking fine, until…. x^2 + 1 = 0. Ok, not a problem, just another extension into the complex numbers. And this is where we are. We have extended so much, but can we keep going like we have in the past? Surprisingly, no! In terms of finding roots of polynomials and our standard operations, the complex numbers are the end of our journey. And that point of containing all numbers possible in polynomial roots is called algebraic closure. You might be familiar with this as the fundamental theorem of algebra.
But you read about other number types, so where does that lead us? Well, it leads us into metamathematics and abstract algebra. One thing that mathematicians love is the generalization and abstraction of structures and ideas. For example, a lot of things *act like* numbers but aren’t. You are familiar with function composition and matrices, yes? Well, think of the function f(x) = x and trying to use it in function composition. It makes no difference, right? Now, consider a square matrix of all 0s except for a line of 1s on the diagonal from the top left to bottom right. Using that with matrix multiplication makes no difference at all. Doesn’t that kind of remind you of adding 0 or multiplying by 1? We call these “identities,” and just like the function and matrix listed, we can generalize them far beyond our “normal” mathematical structures. We can generalize other things, too. Who says the symbol “+” has to mean “add these two things”? What if it is just a function? Instead of *a*+*b* think of it as +(*a*,*b*). And as a function, we can redefine it. But if we use the same symbol, we ought to keep in the spirit of similar properties, so things like negatives and commutativity and associativity and a “0” would be good to hang on to, even if everything looks different. This idea of “keep the core structure but remake the core rules” is a topic called “abstract algebra,” and it is a beautiful subject.
When we make something in abstract algebra, we want it to have some general properties to the ideas can be used all over. The relevant structure here is called a “field.” It has two operations “+” and “×” that we use on whatever we are playing with. To keep in the spirit of our normal math, fields have the properties of commutativity, associativity, identities, distribution, and inverses (“opposites” for our operations). And we want to use them in polynomials due to their frequent use in math in general. Since polynomials have exponents, we want to take roots. When we have a field which is closed under everything, including roots, we are said to be algebraically closed, like the complex numbers.
Now, you will notice that some of these choices were arbitrary. Why two operations instead of one or three or twenty? Why did we decide to keep those properties? Matrices are not commutative under multiplication, so that seems questionable to include. What makes polynomials special? I, for one, think logarithms are pretty awesome. And that brings us to our broader abstract algebra: these choices *are* arbitrary, and we do not need to keep them. Throw them out, add new ones, have fun with it!
And that is what leads us to things like quaternions, octonions, and other things like p-adic numbers or a bajillion other number types. With quaternions, we lose commutativity: *ab* is not the same as *ba*. Octonions lose associativity. However, their structures come from different types of extensions with different uses. P-adic numbers come because we redefined distance but kept some core properties.
However, these come from different abstractions used for different reasons. Under the operations you work with, complex numbers are the end of the road, but take a different road and more numbers can come about as easily as simply making a new definition which does not lead to contradictions. And that is where they come from. The types of extensions lead to the dimension doubling but do not necessarily need to be the only ones. We just don’t have many mainstream uses for 3D numbers that cannot be defined in other ways, but get niche enough and you can likely find something.
As for uses, mostly things built on logical structures. So computer science, modelling, physics, etc. can all have fun with our more alien number types.
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