# What exactly is a number?

We've just been learning about complex numbers in class, and I don't really see why they're called numbers.

Originally, a number used to be a means of counting (natural numbers).

Then we extend these numbers to instances of owing other people money for instance (integers).

After that, we consider fractions when we need to split things like 2 pizzas between three people.

We then (skipping the algebraic numbers for the purposes of practicality), we use the real numbers to describe any length; e.g. the length of a diagonal of a unit square.

But this is when our original definition of a number fails to make sense- when we consider complex numbers, which brings me to my main question: what is a rigorous definition of 'number'?

Wikipedia claims that "A number is a mathematical object used to count, label, and measure", but this definition fails to make sense after we extend $\mathbb{R}$.

Does anyone know of any definition of number that can be generalised to complex numbers as well (and even higher order number systems like the quaternions)?

• Possible duplicate of What is a number? Jul 12 '14 at 19:44
• I don't know, but they're usefull Jul 12 '14 at 19:45
• "...consider fractions when we need to split things like 2 pizzas between two people..."? Is that an introduction to fractions or the fact that life is inherently unfair? ;) Jul 12 '14 at 19:59
• I don't need fractions to split two pizzas between two people: I get one, you get the other. Now, splitting one pizza between two people ... Jul 12 '14 at 20:03
• @Théophile I don't need fractions to split $n$ pizzas between $m$ people, I get $n$, the rest of them get none. ;-) Jul 12 '14 at 20:09

A basic question is, what would be the purpose of such a definition? Would it clarify anything if we came up with a definition that, say, included quaternions and not matrices or analytic functions?

Most of the usages of the term "number" are due to historical choices that have lived on. I'd be interested in seeing things that were called "numbers" initially, but are not called "numbers" now, I suppose, but any definition that applies is just a hack to justify choices at the boundaries, I think.

As mentioned, I haven't seen quaternions called "numbers." We say $1+i$ is a "complex number," but we just say $1+i+j+k$ is a "quaternion." At least in my experience.

Algebraic numbers

In "number theory," we often deal with "algebraic extensions" of the rational numbers. For example, $\mathbb Q(\sqrt{2})$ is the set of numbers of the form $a+b\sqrt 2, a,b\in\mathbb Q$. These can be seen as a subset of $\mathbb R$, but they actually exist more abstractly - for example, algebraically, we don't know whether $\sqrt{2}<0$ or $\sqrt{2}>0$ - the number exists as an algebraic object entirely - an object which, when squared, equals $2$.

The same thing happens with $\mathbb Q(\sqrt{-1})$. It would be strange to call $\sqrt{2}$ a "number" and not call $\sqrt{-1}$ a "number" in this context. Mathematicians call the two fields "algebraic number fields."

For example, $\mathbb Q(\sqrt{2})$ can be seen as isomorphic to a subset of $\mathbb R$, but it is also isomorphic to a (different) subset of $\mathbb C$.

Complex Numbers

There are also ways to see, inside the 'real numbers,' that the complex numbers sort of have to exist. My favorite way: If you look at the radius of convergence of the Taylor series of $f(x)=\frac{1}{x^2-x}$ at $x=a$, you get the radius of convergence is $\min(|a|,|a-1|)$. That is, the zeros of the denominator "block" the Taylor series. If you look at the Taylor series of $g(x)=\frac{1}{1+x^2}$ at a real number $x=a$, you get that the radius of convergence is $\sqrt{1+a^2}$. There is something (a zero of $1+x^2$?) "blocking" the Taylor series of $g(x)$ that looks like it is exactly a distance $1$ from $0$ in a direction perpendicular to the real line.

Complex numbers also are necessary for breaking down real matrices into component parts. Well, not "necessary," but the representation of matrices in, say, Jordan canonical form, becomes quite a bit more complicated without complex numbers. So complex numbers, oddly, make matrices seem more regular (or, if you prefer, hide the complexity.)

Also, complex numbers are really necessary for understanding quantum theory in physics. Everything you think you intuit about the universe, in terms of "measurements" being real numbers, starts to fall apart at the quantum level. The universe is far stranger than it seems.

$p$-adic Numbers

$p$-adic numbers are probably called "numbers" just because their construction is essentially the "same" as the construction of the reals, only using a different metric on $\mathbb Q$, and because they can be used to answer questions about the natural numbers.

Ordinals, Cardinals

Ordinal and cardinal numbers represent a different type of extension of the natural numbers.

I think of "ordinals" as being like the results of a race with no ties. Every runner has a result "ordinal" and any non-empty subset of the runners has a "winner."

Cardinal numbers are like a pile of beans, and determining whether two piles of beans have the same amount in them.

Ordinals are by far the most weird, because even addition of ordinals is non-commutative.

In this case, then, ordinals and cardinals are "measurements" of something.

Non-standard real number definitions

There are also lots of variations of the real numbers that we call "numbers," basically because they are a variant of the real numbers.

Conclusion: Exclusions

The hardest part of coming up with a definition for "number" is to exclude: Why don't we call matrices, or functions, or other similar things "numbers?" Things we see as primarily functions are not seen as "numbers," but it is hard to exclude them with anything rigorous. Indeed, one way to see the complex numbers is as a sub-ring of the ring of real $2\times 2$ matrices, and one reason we need complex numbers is that they are great at representing the operation of rotation - that is why we see them come in studying real matrices.

Zero-divisors are often a sign that a thing isn't a number, but we do have $g$-adic numbers with $g$ not prime, which is a ring with zero divisors. (Usually, $g$-adic numbers are not actually used anywhere, since they are just products of rings of $p$-adic numbers...)

Does anybody refer to the elements of ring $\mathbb Z/n\mathbb Z$ as "numbers?" Not in my experience.

I also haven't seen finite field elements referred to as "numbers."

So, no, the entire history of mathematics has not ascribed a single logical meaning to the word "number," so that we can distinguish what is and isn't a number. As noted in comments, "Cayley numbers" is another name for the octonions, but there are zero occurrences in Google NGram of the singular phrase, "Cayley number." So octonions are numbers, but a single octonion is not a "number?" That's just the world we live in. Number, being the most basic idea in mathematics, gets generalized in a lot of interesting ways, not all consistent, and not the same way over time.

Recall, the ancients didn't define $0$ as a "number."

Q: How many beans do you have?

A: I don't have beans.

(I suspect this failure was due to the confusion between cardinals and ordinals - we count finite cardinals by arbitrarily sorting and then computing the ordinal of the last element, but that fails when counting an empty collection...)

• Quaternions have been referred to as hypercomplex numbers. Jul 12 '14 at 23:49
• Quaternions, perhaps, are not often called numbers. But Cayley numbers are. Jul 12 '14 at 23:55
• In fact, the ancients did not consider $1$ to be a number either. (Euclid's definition: A number is a multitude of units.) Jul 13 '14 at 7:04
• When in ancient times it was realized that incommensurable quantities exist (like $1:\sqrt{2}$), they did not consider these quantities to be "numbers" (greek (singular) arithmos). Instead, they began "calculating" with geometric objects such as line segments (since this was more general than "numbers" (i.e. $\mathbb{Q}$)). Only later was the concept of number/arithmos generalized to include irrational quantities. Jul 14 '14 at 10:36
• (Hyperbolic numbers are the numbers that appear on Fox News, right? :) ) Jul 14 '14 at 18:25

There is no concrete meaning to the word number. If you don't think about it, then number has no "concrete meaning", and if you ask around people in the street what is a number, they are likely to come up with either example or unclear definitions.

Number is a mathematical notion which represents quantity. And as all quantities go, numbers have some rudimentary arithmetical structures. This means that anything which can be used to measure some sort of quantity is a number. This goes from natural numbers, to rational numbers, to real, complex, ordinal and cardinal numbers.

Each of those measures a mathematical quantity. Note that I said "mathematical", because we may be interested in measuring quantities which have no representation in the physical world. For example, how many elements are in an infinite set. That is the role of cardinal numbers, that is the quantity they measure.

So any system which has rudimentary notions of addition and/or multiplication can be called "numbers". We don't have to have a physical interpretation for the term "number". This includes the complex numbers, the quaternions, octonions and many, many other systems.

• I'd be happy to hear counterarguments, because it seems that this point was made by others as well, and yet... I'm somehow the only one to receive a downvote. Jul 13 '14 at 11:22
• "So any system which has rudimentary notions of addition and/or multiplication can be called "numbers"." So matrices are numbers? And functions too? Jul 21 '14 at 5:57
• @Martin: Shocking. I know. Jul 21 '14 at 7:26
• @Martin: Note, by the way that the complex numbers has a matricial representation. So some matrices are numbers. Why not all of them, then? And if matrices are linear transformations, this means that some functions are numbers too. Jul 21 '14 at 7:29
• @Martin That depends on your PoV on functions. For example, $\mathbb Q[\pi] \simeq \mathbb Q[x]$, which means that we can regard at least some transcendent numbers as functions, in this particular case on the affine line. According to the period conjecture, periods determine a generic point of the corresponding period torsor, so can be meaningfully identified with functions on it. On the other hand you have the numbers-as-functions principle which associates to any integer a function on $\mathrm{Spec} \mathbb {Z}$. Dec 28 '17 at 1:32

In my thinking, a number is simply an object which satisfies some set of algebraic rules. In particular, these rules are typically constructed to allow an equation a solution. Most interesting, these solutions were unavailable in the less abstract version of the number, whereas, with the extended concept of number the solutions exist. This seems to be the theme in the numbers which have been of interest in my studies. For example:

1. $$x+3=0$$ has no solution in $$\mathbb{N}$$ yet $$x = -3 \in \mathbb{Z}$$ is the solution.
2. $$x^2-2=0$$ has no solution in $$\mathbb{Q}$$ yet $$x = \sqrt{2} \in \mathbb{R}$$ is the solution.
3. $$x^2+1=0$$ has no solution in $$\mathbb{R}$$ yet $$x = i \in \mathbb{C}$$ is the solution.

However, even this theme is not enough, a new type of number can also just be something which allows some new algebraic rule. For example,

1. $$2+j$$ is a hyperbolic number. Here $$j^2 = 1$$ and we have some rather unusual relations such as $$(1+j)(1-j) = 1+j-j-j^2 = 0$$. The hyperbolic numbers are not even an integral domain, they've zero divisors.
2. $$1+\epsilon$$ is a dual number or null number as I tend to call them. In particular $$\epsilon^2=0$$ so these capture something much like the idea of an infinitesimal.

This list goes on and on. Indeed, the elements of an associative algebra over $$\mathbb{R}$$ are classically called numbers. There is a vast literature which catalogs a myriad of such systems from around the dawn of the 20-th century. In fact, the study of various forms of hypercomplex analysis is still an active field to this day. Up to this point, the objects I mention are all finite dimensional as vector spaces over $$\mathbb{R}$$. In the 1970's physicists began throwing around super numbers. These numbers were infinite dimensional extensions of the null numbers I mentioned above. They gave values for which commuting and anticommuting variables could take. Such variables could be used to frame the classical theory of fields for fermions and bosons. In my view, the supernumbers of interest were those built from infinitely many Grassmann generators. Since the numbers themselves were built from infinite sums, an underlying norm is given and mathematically the problem becomes one of Banach spaces.

My point? A number is not just for counting.

• The hyperbolic number example is missing its '^2' superscript after the expansion of terms, but a single character edit isn't possible ;-) Jul 13 '14 at 22:21
• @PhilipOakley hey, thanks! Fixed. Jul 14 '14 at 0:34

"If it looks like a duck, and quacks like a duck, it is a duck... or something that is so close that is good enough"

In Mathematics you don't really work with objects, you work with the properties. So, we define an object (for example, a vector) in a certain context (linear algebra) as something that has a series of properties (in the example, you can add them, multiply by a scalar, create a base...), and that is what we use.

In my first year at university, my Algebra professor spent a couple of months talking about vectors and matrices. One day, he wrote down the properties of a vector that we had been using (roughly listed above), and showed that they were also applicable to polynomials: you can add two polynomials, 0 is a polynomial, you can multiply by scalars... In short, everything that he had said about vectors automatically applied to polynomials, with no extra work required! It also allowed for expansions: in linear algebra a matrix can be seen as an object that inputs a vector and outputs a transformed vector; and for each (square) matrix, there are particular vectors that have nice properties. Well, we can then define operators on polynomials pretty much the same way: something that eats a polynomial and spits a polynomial (for example, the derivative).

I was quite impressed by this, and it improved my intuition of mathematics.

So, coming back to your question, a number is something that behaves more or less like a number (or something that you accept as a number).

And quoting another professor from that year: "don't think, calculate!". It is good to think about the Phylosophy of all this, but don't let it get in the way!

• He probably gave you that "don't think, calculate" quote because you were doing linear algebra ; in most parts of mathematics, the opposite is true : don't calculate, think! Because calculating will give you an answer to a question, but more often than not, thinking will give you understanding to an answer you would have gotten by calculating but not understood. I loved the answer though, +1 Nov 21 '15 at 6:52
• @PatrickDaSilva he was actually teaching calculus, and his most repeated phrase was to "never start a calculation without knowing the result". The "don't think" part was applied to everything else but maths (philosophical interpretations, likelihoods of different exercises popping up in the exam, etc), meaning to keep us from spending more time than necessary. Nov 21 '15 at 22:15
• "In Mathematics you don't really work with objects, you work with the properties." Nov 29 '20 at 0:55
• @aderchox sorry, I don't understand, was part of the comment lost in the aether, perhaps? Nov 29 '20 at 16:06
• @Davidmh No, I really enjoyed that part of your answer, so I wanted to re-write it once again in the comments section. Sorry for not being clear on that. Thanks. Nov 29 '20 at 16:09

These questions are addressed in mathematical philosophy (e.g. see Russell's Introduction to Mathematical Philosophy). The very short answer to your question is classes.

(I'll have to elaborate some other time unless someone beats me to it)

• This is actually the best answer here, though if you hadn't heard it before, you really would have no way to know. Jul 13 '14 at 0:23
• FWIW, the second edition of Russell's book is at Project Gutenberg (typeset in LaTeX). Jul 14 '14 at 10:43

This is a deep and philosophical question. You are right, kids start by learning to count things like 3 apples and 7 cars but quickly build up to the real numbers, which we can make sense of by money and distances. When you start adding things like the imaginary numbers, it gets less clear.

My revelation came when I learned some electricity and magnetism. There, the complex numbers are used to describe alternating current. I was blown away by the thought that this number, $i$, which I did not really consider a serious concept at the time, to have a physical interpretation. So I had to ask myself the same question you are now.

To us, a number is just an abstract idea, that we use to make sense of physical or mathematical ideas. In general, a number is no different from an element of a group or ring. If you don't know what those are, do not worry about it. But they are just as abstract as the idea of a number. An example of a group describing something would be a knob that you turn on a fan. It might have zero, one, two, and three. When you turn past three you get back to zero. This is the group $\mathbb{Z}/4\mathbb{Z}$, which means that 4 is the same as zero.

I have heard, but do not have a reliable source, that some cultures used to have different numbers for different objects. To describe two cows, they would use a different word than they would to describe two sheep, for example.

I suppose the point is, that a number is just a way to describe something. There is no reason to think that the number line is the whole story. Perhaps some people just wish it was, because that would simplify things, so they do not talk about these other descriptions that are much like numbers.

• Here's a reliable source: it happens in English. Pair, twins, couple, mates, duo all mean "two of" slightly different things. For groups of greater size, we have group, cluster, herd, hive, flock, and so on, including my two favorites, clowder (of cats) and murder (of crows). Jul 12 '14 at 22:21
• In electronics, we usse the imaginary unit for separating a sine wave from a cosine wave (or as a way to distinguish a 90° delay) Jul 14 '14 at 17:26
• @ryanreich Yeh but the words become more vague the bigger the number being described. Mostly after two really. Aug 2 '14 at 13:40

Actually, the "practical" examples of numbers start to break down at $\mathbb{R}$. Yes, the algebraic numbers have interpretations in geometry (though these are already more abstract than owing someone a dollar or splitting a pizza), but there are many numbers in $\mathbb{R}$ that are not algebraic and in fact aren't really describable in any ordinary way.

On the other hand, numbers in $\mathbb{C}$ can be interpreted as points in a plane. And by extending $\mathbb{R}$ to $\mathbb{C}$, we ensure that all polynomials in $x$ have a factorization into factors that are linear in $x$; equivalently, a polynomial of degree $n$ has $n$ roots. So just as real numbers let us solve problems that rationals do not, complex numbers let us solve problems that reals do not.

• Great point. We have a very poor feel for most of the real numbers, and mostly tend to forget that. Jul 14 '14 at 3:16

Does anyone know of any definition of number that can be generalised to complex numbers as well (and even higher order number systems like the quaternions)?

There is a general consensus that no such definition could possibly exist. "Number" is inherently a fluffy concept. However, a good guideline is: if the analogy with any of the usual numbers systems $\mathbb{Z},\mathbb{Q}, \mathbb{R},\mathbb{C}$ is fruitful and deserves to be emphasized, you are free to call it a number.

Examples:

1. The quaternions
2. The ordinal numbers
3. The cardinal numbers
4. The surreal numbers / surcomplex numbers
5. The isomorphism types of totally ordered sets
6. The isomorphism types of partially ordered sets

Bertrand Russell in the early 20th century defined "number" as

anything which is the number of some class

and "the number of a class" as

the class of all those classes that are similar to it

where similarity of classes is defined by the existence of a bijection between them.

(Paraphrased from Introduction to Mathematical Philosophy.)

• That's just defining something like cardinal number, really, while the questioner wants to know about all the things we call "number." It's not clear how Russell's definition even handles ordinal numbers... Jul 13 '14 at 1:11
• The book Logicomix is a comic book about Bertrand Russell's struggle to answer this question. I would recommend it to those of us who find "Introduction to Mathematical Philosophy" a bit hard to digest. Jul 14 '14 at 16:34

The question is controversial in philosophy of mathematics. Some answers:

(A) Numbers are sets. According to Bertrand Russell:

• The number 0 is the empty set.
• The number 1 is the set of all single-membered sets.
• For natural number n, the successor of n is the set of all sets that are equinumerous with {0, 1, ... n}, where two sets are equinumerous if there is a one-to-one mapping between them.

(See Russell's Introduction to Mathematical Philosophy.) Similar views are taken by Frege (Foundations of Arithmetic) and Cantor (Contributions to the Founding of the Theory of Transfinite Numbers). A more recent view is that 0 is {}, 1 is {0}, 2 is {0,1}, and so on. People who take this sort of view then try to construct other numbers out of sets. E.g.:

• The number 2/3 is the set of ordered pairs {<2,3>, <4,6>, <6,9>, ...}
• A real number is a set of sequences of rational numbers with a certain convergence property.

These views enable you to derive the familiar truths of arithmetic from set theory. However, it is not very natural to suppose that when I say "My gas tank is 2/3 full" I am talking about this sort of object. (For a hilarious critique, see http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf.)

(B) Natural numbers are properties.

The best way to explain what property one is referring to is to give examples. Thus, my hands are two (they together have the property of twoness); the Earth and the Moon are two; green and blue are two colors; etc. The number 2 is what all those examples have in common. This is defended in Byeong-Uk Yi's excellent "Is Two a Property?" (https://tspace.library.utoronto.ca/bitstream/1807/25255/1/Is%20two%20a%20property.pdf)

To address your question about complex numbers: if you accept the set-theoretic account of other numbers, then complex numbers are no problem. You can identify a complex number with an ordered pair of real numbers. This is not very different from, e.g., the construction for rational numbers.

If you take the property view (as I do), complex "numbers" are not numbers in the same sense as natural numbers. They might still be ordered pairs of numbers. (Whether we want to extend the term "number" to include them is then a semantic question.)

Note incidentally that other presently-accepted numbers used to not be accepted. 0 wasn't always a number; it began life as a mere placeholder (something like a punctuation mark used to indicate that a position is unoccupied). The concept "number" had to be extended to include zero. Negative numbers weren't always accepted either, since it was considered absurd to speak of having less than nothing. Transfinite "numbers" are another late addition, which is motivated by the set-theoretic account of numbers. So there is precedent for extending the concept.

• If you take the property view, which kinds of "numbers" (negative, rational, real, etc.) are legitimately numbers, other than the natural numbers? Jul 13 '14 at 3:28
• Complicated question. Short answer: they're all different kinds of properties/relations, but "number" might be a broad enough term to cover all of them. Real numbers (including fractions) are relationships between magnitudes. A negative number is just a number that is used to measure something that is 'opposite' to something else. A complex number is an ordered pair of real numbers (ala Hamilton).
– Owl
Jul 14 '14 at 8:24

This is a book by kirillov ,titled what are numbers?,it could help you http://www.math.upenn.edu/~kirillov/MATH480-S08/WN1.pdf http://www.math.upenn.edu/~kirillov/MATH480-S08/WN2.pdf

This doesn't exactly answer your question, but here are some characteristics all numbers I know of satisfy.(I know of counting numbers, integers, rational numbers, real numbers, complex numbers, quaternions and octonions).

Number 1: All the counting numbers seem to be equipped with two binary operations which receive the name of addition and multiplication.

Number 2: They are an extension of the counting numbers, in the sense that there is a function from the counting numbers to all the other number systems such that $f(a)+f(b)=f(a+b)$ and $f(a)\times f(b)=f(a\times b)$ for $a$ and $b$ counting numbers.

Unfortunately I'm still young inexperienced and those were the only ones I could think of.

• I can't recall quaternions and matrices called "numbers." We say $1+i$ is a complex number, we say $1+i+j+k$ is a quaternion, not a "quaternion number." Obviously, matrices have can be seen as extensions of the counting numbers, but matrices are never called numbers, as far as I've seen. I'd suggest that the notions of "numbers" usually includes associate and commutative $+,\times$ and distributive law. Jul 12 '14 at 20:54
• I never mention matrices in my answer though, well yes, if you don't consider quaternions or octonions you can include associativety and commutativity as characteristics, I thought quaternions were considered numbers, because I have heard of people talk about the quaterions as a number system. Jul 12 '14 at 21:06
• I was wrong about associative and commutative, since we also call ordinals "numbers," and ordinals aren't even commutative under addition... Jul 12 '14 at 21:18
• I failed to edit my first comment properly. My point about matrices was meant to be that your definition would include matrices. Jul 12 '14 at 21:20
• Oh, I see, but they would have to be over a a field that already satisfied those characteristics. Jul 12 '14 at 21:21

There isn't a rigorous definition of number.

It doesn't make sense to come up with a rigorous definition of number until there is some specific (and hopefully useful) concept you wish to formalize: and we already have formalizations of concepts like "real number" if that's the specific concept we wish to talk about.

As an aside, I have used complex numbers to label things, and to measure things, so calling complex numbers "numbers" certainly continues to make sense by the Wikipedia definition.

I've even used much more exotic things for the purposes that Wikipedia delineates: e.g. there are things I've labelled and measured by using Abelian groups. For example, homology can be used to measure various properties of topological spaces, and the values of such a measurement are Abelian groups. (not elements of Abelian groups, but the Abelian groups themselves)

• Hurkyl, did you defect from Physicsforums? :) Jul 13 '14 at 0:16

There are two schools of thought here:

1 - Intuitive quantity.
2 - Arithmetic quantity.


For me, I go with the Intuitive concept, which means: restrict the concept of numbers - into measuring quantity, either positive or negative. (Set of Real numbers)

Complex numbers are a problem because they measure more than quantity. They have some sort of "direction", but don't behave arithmetically like normal x,y vectors.

Instead, Complex numbers can be represented by a matrix:

 a -b
b  a


The Arithmetic concept suggests that numbers are whatever consistently obeys a certain, arbitrarily defined, arithmetic operators. Such numbers do not necessarily correspond to anything physical - namely, Quantity. Rather, they represent raw data.

Since these numbers are consistent with their corresponding operators, using them in equations is useful, because it remains consistent.

• Negative numbers also have a "direction", so why accept them as "intuitive" if having a direction is objectionable? For that matter, the "intuition" behind a definition of even the positive real numbers is esoteric. Jul 13 '14 at 3:20
• Negative numbers are not that intuitive. Complex numbers were introduced by Cardano before negative numbers were accepted. Jul 14 '14 at 14:50

Mathematics is really just a system of formal logic; you define the rules of a universe then use those rules, and only those rules, to build systems. Within that rule system, you can define what a number is, some operations you can do on numbers, and start proving results about what happens with those numbers. Eventually, you build up more complex systems, and start defining all of modern mathematics.

The current most common foundation for mathematics is considered to be Zermelo-Fraenkel ("ZF") set theory, generally augmented with the Axiom of Choice to make "ZFC" set theory. The subject is often called "mathematical philosophy", as another answer stated. Its modern formulation typically has only 9 axioms, and from just those 9 everything else derives.

As an example of how you can get the concept of numbers from just sets, we can derive a simple counting system just from the rules that there exists an empty set, you can construct a set as a collection of other sets, and finally the existence of an infinite set (which will become the set of natural numbers). Formally, the existence of the empty set is implied by several of the ZFC axioms, the ability to construct a set as a collection of other sets is from the Axiom Schema of Specification and Axiom of Power Set, and finally the existence of an infinite set is its own Axiom of Infinity.

Taken directly from Wikipedia, a simple formulation of the natural numbers arrives, starting with zero being the empty set. A successor function S(n) defined in set notation as S(n) = n + 1 = n ∪ {n}.

0 = ∅ = { }
1 = { 0 } = { { } }
2 = { 0, 1 } = { { }, { { } } }
3 = { 0, 1, 2 } = { { }, { { } }, { { }, { { } } } }


http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

There are a lot of good answers already. This isn't much of a deep, mathematical answer. It's just my thoughts on the matter. A "soft answer" to your "soft question", if you will :)

To me, a number basically is a constant that has notions equivalent to basic addition and multiplication over the natural numbers. By that definition, the above often-mentioned Matrices of which each element is a number would also be numbers.

Now I could stop right there. The rest is rather about Complex Numbers and an alternate origin of them, outside of defining the square root of -1 or finding roots to polynomials.

Since I learned about Geometric Algebra, I like to think of Complex Numbers, Quaternions and higher-dimensional notions as just subsets of Geometric Algebras. I think that name is reserved for something else already, but calling them "Geometric Numbers" would be fitting in my mind. - Geometric in the sense that those values have clear geometric interpretations, and numbers in the sense that they are constants that have a straight forward definition of multiplication and addition.

Let's, for instance, take a 2D Geometric Algebra: You'll have your base elements:

$$1, \; x, \; y$$

and they behave like this:

$$1 \cdot 1 = 1 \\ 1 \cdot x = x \cdot 1 = x \\ 1 \cdot y = y \cdot 1 = y \\ x \cdot x = y \cdot y = 1 \\ x \cdot y = -x \cdot y := I$$ and by just applying those rules, you'll find that $$I \cdot I = x \cdot y \cdot x \cdot y = \; | \left(x \cdot y = - y \cdot x \right)\\ = -y \cdot x \cdot x \cdot y = -y \cdot \left(x \cdot x \right) \cdot y = -y \cdot 1 \cdot y = \\ = -y \cdot y = -1$$

Thus you get the following multiplication table:

$$\begin{matrix} \textbf{1} & \textbf{x} & \textbf{y} & \textbf{I} \\ \textbf{x} & 1 & I & y \\ \textbf{y} & -I & 1 & -x \\ \textbf{I} & -y & x & -1 \end{matrix}$$

Notice two things here:

First of all, if you think of $x$ and $y$ as being two orthogonal directions in a plane, $I$ will rotate them counterclockwise. Alternatively you can think of a "sandwitching operator" like $x \cdot y \cdot x = -y$ - this kind of operation will mirror what ever is inside the "sandwitch" along the axis of the outside, the "bread", if you will. This holds true in higher dimensions too. You'll end up with a larger structure though. If you try this with three dimensions, say, $x$, $y$ and $z$ (naming, of course, is arbitrary), you'll end up with what is essentially the quaternions as part of the algebra.

And second, if you take just the $1$ and the $I$ together, you end up with precisely the complex numbers. Notice how $I^2=-1$.

If you define a generic such number as $a_R \cdot 1 + a_x \cdot x + a_y \cdot y + a_I \cdot I$, and multiply two such numbers by the above multiplication rules in just the same way as you'd multiply complex numbers, you basically have three different parts: The "scalar part" $a_R$ (for Complex numbers, this is the real part), the "vector part" $a_x x + a_y y$ (this behaves precisely like you'd hope it to, if you want to deal with geometry in a plane) and the so-called bivector (or 2-vector) part $a_I I$ (the imaginary part of Complex numbers, here consisting of the two vectors x and y, hence the name), which acts like a rotator of sorts.

Note that the 2D case is a little special because the bivector simultaneously is the so-called pseudo-scalar. For 3D you can find, in a very similar fashion, that you'll have

$$1\\ x, \; y, \; z \\ Ix = yz = xI, \; Iy = zx =yI, \; Iz = xy = zI \\ x \cdot y \cdot z = I$$

Consisting of 3 orthogonal bivectors (the three primary planes of the $\mathbb{R}^3$) and the trivector (or 3-vector) $I$.

And if you play around with those algebras, you'll very quickly find ways to express the scalar product and the vector product in straight forward ways. The capabilities of mirroring or rotating are also kept and so is the meaning of $1$ and $I$.

There are several neat extensions to this but the bottom line is that complex numbers appear everywhere (I gave one example, most gave others) and even if you don't like my above definition of what a number is (I'd not be surprised if somebody is quick to object), the way you can deal with them exactly as if they were reals, and the beautiful consequences, the introduction of complex numbers has, makes it more than justified to call them numbers.

I just want to share my own experience with similar questions. I believe the correct answer to this question is really important in letting one see and enjoy the power of Mathematics without having curtains blocking its light.

More than anything, I myself may have asked such a question from myself when thinking about dimensions greater than 3. For instance, the $$R^n$$ space. It seemed very complicated. I knew what would an $$R^3$$ space look like(for a long time, we have believed we live in one), but I used to ask myself how would $$R^n$$ be useful if it's not imaginable in real world to see what things would look like in nD (e.g, in 5D). But after some while, I learned some Algebra and realized that if I measured 5 things and put those 5 output numbers in a collection, that would be in 5D as well!

But wait, those were totally different "things"! The "geometrical" 1D 2D 3D spaces, and the dimensions in Algebra. So how would we be able to "ultimately" define what a dimension is?

The answer is, we will never be able to. A dimension can be many things depending on the context. But then how would we be able to know what we're talking about when we talk about "dimensions"? The answer is simple, we will pay attention to the context, and the context will "describe" the thing we're talking about.

So to sum it up, in Mathematics, we don't really care about all-encompassing definitions, neither do we care about imaginations, because both of these would strive for concreteness, while Maths is abstract. The things we truly care about in Maths are: "Descriptions". And those turn out to be very useful.

Currently there is no accepted definition of a number, simply because no one has proposed one.

I'll propose one.

A number is an element of a number space. A number space is a subsemiring of a field.

• The concept of a number should be a relatively simple abstraction. Complicated abstractions should build on simpler abstractions.
• Note the similarity between vectors and vector spaces; a vector is an element of a vector space. Note how the analogous discussion for "What exactly is a vector?" would be equally confusing without the concept of a vector space.
• Whatever the accepted definition will be in the future, a number should be defined as an element of some structure (i.e. a set of elements + operations on them). It does not make sense to look at specific sets such as the empty set, the set $$\{\{\}, \{\{\}\}\}$$, or a specific matrix, and ask whether they are numbers. Rather, as with vectors, it is how a set of elements interact with each other under the operations of that structure.
• Some things have already been named numbers, such as dual numbers, ordinal numbers, and Clifford numbers. An accepted definition of a number does not necessarily need to include them, since these can just be badly named; perhaps because there was no accepted definition of a number at the time. Number spaces exclude these examples in general.
• When I say a set, I also allow set theories other than ZFC: e.g NBG which contains proper classes (1-level set contains 0-level sets) or other set-theories where there is an infinite hierarchy of sets (i-level set contains sets with level less than i).
• Every subsemiring of a field is commutative.

### Examples of number spaces

These are number spaces:

### Non-examples of number spaces

The following are not in general number spaces. By that I mean that there is at least one instance which is not a number space.

• Extended real numbers
• These may sound like they should be considered numbers, because they often seem kind of maybe useful. However, I instead view extended real numbers as a failed attempt at capturing hyperreals. There is subtlety when dealing with infinities. If you are using anything else than the topological properties of infinity or denoting by infinity some special case, then usually with extended real numbers you are just left confused (probably because inf - inf occurs). This is similar to denoting every infinite ordinal with the same symbol.
• Ordinal numbers
• Excluded because of non-commutative multiplication. When I think of ordinals, I think of transfinite induction and transfinite recursion. The things you do with ordinals are not like what you do with numbers.
• Commutative monoid
• Although a number space is an additive commutative monoid, an additive commutative monoid may not come with a multiplication operation. A vector space is in general excluded.
• Semiring
• Although a number space is a semiring, the multiplication in a semiring may not be commutative. A set of matrices which form a semiring is in general excluded.
• Commutative semiring
• Although a number space is a commutative semiring, a commutative semiring may not be a subsemiring of a field. A polynomial ring is a commutative semiring, but is in general excluded.
• Semifield
• Although a number space is a semifield, it may not be possible to extend a semifield to a field. Tropical semiring and ratios of positive-coefficient polynomials are excluded.
• Geometric algebra
• A geometric algebra is in general excluded, because of non-commutativity and zero divisors; either excludes it from being a subsemiring of a field. Note that geometric algebras may contain many number spaces as subsemirings (e.g. real numbers and complex numbers).
• Also known as Clifford algebra.
• Clifford numbers are a synonym for a geometric algebra.
• Dual numbers are a synonym for a certain geometric algebra; excluded because of zero divisors.
• Quaternions are a synonym for a certain geometric algebra; excluded because of zero divisors and non-commutativity.
• Octonions and other Cayley-Dickson constructions starting from quaternions are excluded because of lacking commutativity, and even associativity.
• Spinors, rotors, and versors can all be formulated in geometric algebras. Excluded in general because of non-commutativity.

### Software abstractions

In deciding what the definition of a number should be, it may actually be a good approach to think about what is useful as a software abstraction. Many of the mathematical structures can be translated as abstractions/concepts in software. There your aim is to build your abstractions so that a single algorithm will work with the largest possible set of types. Hence, the concept of a number should eventually emerge there as an abstraction in libraries dealing with numbers and geometry. The point here is that in this way the number concept emerges naturally on what you actually want to do with numbers as a way to minimize code-duplication, rather than just from what you would like them to be.

### Identity element

The concept of a number space could be slightly generalized: instead of a subsemiring, require to be a subsemirng (without identity element 1). Then the subring of even integers would be a number space too. On the other hand, this seems somewhat arbitrary (for example, the odd integers together with zero is not a number space). Hence, I chose not to do this generalization (at least for now).