Can We Discover a Number Set More General Than Reals with Similar Properties?

[trees and plants]

Hello there.We know that we have sets of numbers like the real numbers, complex numbers, quaternions, octonions.Could we find a set of numbers more general than that of real numbers that has basic properties of the real numbers like commutativity, order, addition, multiplication, division and can be used to solve problems? I know that complex numbers were discovered after attempts of solving polynomial equations, but perhaps algebraic geometry could find other kinds of numbers? Thank you.

 

[member 587159]

I believe you are looking for this:

https://en.wikipedia.org/wiki/Ordered_field

Note that the complex numbers are not an ordered field.

Maybe also useful:

https://en.wikipedia.org/wiki/P-adic_number

 

[mfb]

You can show that everything beyond complex numbers must lose properties of a field. Quaternions are not commutative any more, octonions are not associative any more, sedenions have zero divisors.
I'm not sure where exactly the proof is but if you click around starting from Hurwitz's theorem you probably find it.

 

[Stephen Tashi]

Could we find a set of numbers more general than that of real numbers that has basic properties of the real numbers like commutativity, order, addition, multiplication, division and can be used to solve problems?

Unless you say specifically what you mean by "basic properties of the real numbers", this is not a specific question. For example, in your list of basic properties of the real numbers, you omitted associativity.

And it isn't clear what you mean by "order". For example, a finite field ( often introduced to students in a concrete way as "clock arithmetic"https://en.wikipedia.org/wiki/Modular_arithmetic ) shares many properties with those of the real numbers. However, although one can "order" the elements of a finite field , the property " If $a < b$ and $c > 0$ then $ca < cb$" may not hold. Does the failure of this property disqualify a finite field from being, in your words, "a set of numbers more general than that of real numbers"?

If you study the technicalities needed to understand @mfb 's recommendation of Hurwitz's Theorem https://en.wikipedia.org/wiki/Hurwitz's_theorem_(composition_algebras) or the Frobenius Theorem https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras) you'll get an idea of the details you need to fill-in to make your question specific.