Types of numbers - is there a fundamental type?

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I'm not sure exactly how to frame this question, but will have a go.

In the way one could say that positive integer numbers are a subset of all real numbers, positive or negative and fractional numbers; real numbers themselves could be described as being a subset of complex numbers because they consist of both a real and an imaginary part. Is there a type of number that embraces all other types of number? A fundamental type of number for want of a better way of expressing it?

I think you are asking for the highest level of abstraction in our number system, is that correct? Reading your question reminds me of object oriented programming. You are searching for the super class of all numbers?

Who knows. I would wonder if when finding that all ecompassing set of numbers you could then identify a larger set.....

That would just be "all numbers." I don't think there is a special name for it. We give names to subgroups of numbers because it is useful to distinguish between them, but I don't see any purpose for having a special designation that just means "all numbers," so I doubt it exists.

When I was a student (many years ago) I thought I knew what "all numbers" meant until I was introduced to the concept of complex numbers and realised that what I thought of as "all numbers" was really a subset of complex numbers. I'm wondering if complex numbers themselves are only a subset of yet another level of abstraction and if so is there no limit to the levels of abstractions or is there a fundamental (logical) limit? I seem to remember ruveyn mentioning once before an even higher abstraction regarding numbers but can't remember what that was.

If you're serious, read up on cardinality regarding the various sets of numbers.

There must be a limit. Abstractions are human inventions, and humans could not invent an infinite number of anything. Off the top of my head, I can think of a higher abstraction than complex numbers. Analogous to an ordered pair of real numbers (x, y) there could be an ordered pair of complex numbers ( (a,b), (c,d) ) where a and c are the real parts, and b and d are the imaginary parts. There could also be an ordered triple, quadruplet, etc., up to infinity.

There are hypercomplex numbers. There are hyper-reals. There are surreal numbers

What makes a number what it is is the set of algebraic laws it follows.

Different algebraic axioms define different sorts of numbers.

Historically numbers got their start by answering the questions: how many and how much.

ruveyn

So would this mean that in principle there is no ultimate fundamental type of number? Could future mathematicians define new and currently unthought of algebraic axioms?

I think I'm grasping what you are saying. When I was first introduced to physics I was introduced to scalar quantities where the numbers represented not only how many (speed, momentum) but directionality too.

So an algebraic axiom would be a set of rules for manipulating numbers that refer to different countable "things" other than how many and how much (or what direction).

Eventually numbers were put to work answering the questions where and in what direction. This is how vectors and tensors originated.

ruveyn

No.

Excellent! This is really helpful to my understanding of numbers, thank you.

In classical physics all equations can be broken down into no more than five 'dimensions': Mass, Length, Time, Temperature and Electrical Current. But the numbers associated with these all fall into the categories of how much, where, when and in what direction.

So are numbers used to quantify anything other than what has already been mentioned: How much, how many, where, when and what direction? Anything else that is distinct from these?

I don't think mathematicians have ever formally defined what it means for some object to be a "number". All number systems use the natural numbers (i.e. 0, 1, 2, 3, ...) as basic existential building blocks and then go on to define new kinds of arithmetic. There isn't an all inclusive "number type" because once you go beyond a certain point you start losing useful properties.

The natural numbers have the nice property of being well-ordered, i.e. any set of natural numbers has a unique least element. Once you add negative numbers you already lose this property. Once you get to the complex numbers you lose any meaningful ordering.

Negative numbers are nice because they allow you to define a unique additive inverse for every integer. Rational numbers allow you to define a unique multiplicative inverse for every nonzero number. Real numbers extend the rational numbers creating a system that can be used to describe continuous variables for things like calculus and geometry, but you lose countability.

You can also take another route and define transfinite ordinals that take the concept of "well-ordered" to its logical limit, and transfinite cardinals that take the concept of equinumerosity to its logical limit. These concepts of "number" diverge a bit from the usual types of numbers we deal with in everyday life as they don't have such nice or meaningful arithmetic properties. You can define a way to add and multiply transfinite ordinals and cardinals but the meaning isn't exactly intuitive. You can't even say a+b=b+a.

A vector is not a truly a "number", it is a collection of quantities which when combined have both a magnitude and direction. A Tensor however, is a function; which is not the same thing mathematically as a number.

The problem is in mathematics we have multiple-inheritance. The natural numbers can be thought of as a subclass of real numbers, or a subclass of ordinals, yet neither the class of ordinals nor the class of real numbers are subclasses of the other. The hierarchy branches in the upward as well as the downward direction. There is no strict upward chain of abstraction even though the development from natural numbers to complex numbers gives that impression.

The best ultra-abstract definition I can think of is that all full number systems require the development of peano aritmetic at some place in their logical construction. The induction principle is foundational to all number systems. In the ZFC system that most mathematicians refer to as their axiomatic foundation, the induction principle is included in the axiom of infinity.

But complex numbers are semantically 2 dimensional vectors. In any case it makes more sense to define numbers in conjunction with a set of arithmetic operators and rules of operation. The constructive approach is only used to formally verify the existence of such sets or classes. I mean, in ZFC all numbers are technically sets, but the sets themselves tell you absolutely nothing without understanding the arithmetic that's been defined in the process of constructing said sets. Add to that there are multiple ways to define numbers which may semantically refer to different sets yet have the same algebraic structure via some isomorphism.