In mathematics, there seem to be a lot of different types of numbers. What exactly are:

  • Real numbers
  • Integers
  • Rational numbers
  • Decimals
  • Complex numbers
  • Natural numbers
  • Cardinals
  • Ordinals

And as workmad3 points out, some more advanced types of numbers (I'd never heard of)

  • Hyper-reals
  • Quaternions
  • Imaginary numbers

Are there any other types of classifications of a number I missed?

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    $\begingroup$ You've missed out many types of number: complex numbers, imaginary numbers, integers, rationals, hyper-reals, quaternions, etc. Good question though :) $\endgroup$ – workmad3 Jul 20 '10 at 19:29
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    $\begingroup$ As a general comment, the book Foundations of Analysis by Landau covers the basic constructions up to the complex numbers in full detail (albeit without much motivation). $\endgroup$ – Akhil Mathew Jul 20 '10 at 23:17
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    $\begingroup$ what about surreal numbers? Seriously, I would split this question in several parts. $\endgroup$ – mau Jul 29 '10 at 19:45
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    $\begingroup$ @Alistair: Digging up a very old question to replace the informal word "category" by a formally more correct phrase seems a bit unnecessary to me. $\endgroup$ – TMM May 10 '13 at 18:16
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    $\begingroup$ I don't understand the point of this question. Consult wikipedia or any other book which introduces number systems. Why do we write this up here once again ... $\endgroup$ – Martin Brandenburg May 10 '13 at 20:21

10 Answers 10

  • Natural numbers
    The "counting" numbers. (That is, all integers, that are one or greater).

  • Whole numbers
    The Natural numbers, and zero.

  • Integers
    The Whole numbers, and the negatives of the Natural numbers.

  • Rational numbers
    Any number that may be expressed by any integer A divided by any integer B, where B is not zero.

  • Irrational numbers
    Any number that cannot be expressed as a rational number, but is not imaginary. All irrational numbers have an infinite decimal representation.

  • Real numbers
    All of the Rational and Irrational numbers.

  • Imaginary numbers
    All Real numbers, multiplied by the square root of negative one. Imaginary numbers are signified by the letter i.

  • Complex numbers
    Numbers composed of the sum of a Real and an Imaginary number. This includes all Real and all Imaginary numbers.

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    $\begingroup$ Your definitions of rationals, irrationals, and reals are interdependent, of course. You're relying on the intuitive understanding of reals it seems, which is fair enough at this level... $\endgroup$ – Noldorin Jul 20 '10 at 19:38
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    $\begingroup$ Some definitions of natural numbers include 0. $\endgroup$ – Isaac Jul 20 '10 at 19:42
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    $\begingroup$ Your definition of irrational numbers/real numbers is circular. $\endgroup$ – user126 Jul 21 '10 at 23:20
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    $\begingroup$ @Harry: you still don't get it, do you? This isn't MO, you need to lower the level or people won't understand! $\endgroup$ – Vivi Jul 22 '10 at 1:13
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    $\begingroup$ I was hoping the answers (and the site in general) wouldn't descend into wikipedia style chin stroking but be accessible to dumbies like me. $\endgroup$ – Chris S Jul 22 '10 at 9:01

The natural numbers can be defined by Peano's Axioms (sometimes called the Peano Postulates):

  1. Zero is a number.
  2. If n is a number, the successor of n is a number.
  3. zero is not the successor of a number.
  4. Two numbers of which the successors are equal are themselves equal.
  5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

(This definition includes 0 in the natural numbers; altering rules 1, 3, and 5 to refer to one instead of zero excludes 0 from the natural numbers. Whether or not 0 is a natural number varies in various texts.)

The whole numbers are the natural numbers with the additive identity element called 0.

The integers are the whole numbers and their additive inverses.

The rational numbers are numbers that can be expressed as a ratio of an integer to a non-zero integer.

The real numbers are the set of numbers that are limits of Cauchy sequences of rational numbers.

The irrational numbers are the real numbers that are not rational numbers.

The complex numbers are the numbers that can be expressed as a + b * i where a and b are real numbers and i behaves like a real number under addition/multiplication/distribution/etc., with the added rule that i2 = -1.

The imaginary numbers are sometimes defined to be the "pure imaginary" numbers--complex numbers for which the "real part" a = 0, sometimes with the added restriction that b is not zero--and are sometimes defined to be the non-real complex numbers.

The algebraic numbers are numbers that are solutions to polynomial equations with integer coefficients.

The transcendental numbers are complex numbers (sometimes limited to real numbers) that are not algebraic.

  • $\begingroup$ So now I have to go and read about cauchy sequences in order to understand real numbers? And additive identity for whole numbers? Sorry if this appears snarky + rude, but this is what I see a lot of with maths books/sites trying to explain concepts in plain english, but also trying to impress piers with their knowledge $\endgroup$ – Chris S Jul 22 '10 at 8:57
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    $\begingroup$ @Chris: I wrote this answer after two others that provided perfectly reasonable plainer-language answers that were slightly less technically precise--that is, the goal of this answer was precision and technical correctness despite losing some simplicity of explanation. Roughly speaking, Cauchy sequences in this context are are sequences that converge at all. Additive identity is an early algebra concept, so not that fancy. $\endgroup$ – Isaac Jul 22 '10 at 13:30
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    $\begingroup$ @Chris: Some things that are easy to understand informally are hard to define formally. The real numbers are a classic example of this fact. If you can find a more elementary definition of the real numbers as an extension of the rationals, I will be very surprised. We can give another definition of the real numbers (the unique metric-complete archimedean field), but this definition, while fine, doesn't actually prove that such a structure exists. It is for this reason that we give constructions of the real numbers. $\endgroup$ – user126 Jul 29 '10 at 12:53

I think you were being a little too hard on Isaac. The truth is that the real numbers are a sophisticated mathematical construction and that any explanation of what they "are" which pretends otherwise is a convenient fiction. Mathematicians need these kind of sophisticated constructions because they are what is required for rigorous proofs. Before people explicitly constructed the real numbers and used them to define and prove things about other concepts, it was never totally clear what was true or what was false, and everybody was very confused.

For example, Cantor proved that the number of points in the plane is the same as the number of points on a line. Many people thought that this was impossible before he did it; they had an intuition that you couldn't possibly "fit" the plane into the line. More generally, people were pretty sure you couldn't fit $\mathbb{R}^n$ into $\mathbb{R}^m$ if $n$ was greater than $m$. It wasn't until quite a bit later that mathematicians formalized and proved a rigorous mathematical statement which justified this intuition called invariance of domain, which says you can't do this in a continuous way. One of the many mathematical constructions you need to even state this theorem is the construction of the real numbers. (Another is a formal definition of what "continuous" means, but one thing at a time.)

So, what are the real numbers? They are a formal way to fill "holes" in the rational numbers, which is necessary for all sorts of things. The most basic thing they are necessary for is doing geometry. You probably know that the square root of 2 is irrational. What this means is that it is impossible to think about the diagonal of a square as being the same kind of object as the sides of a square while only using rational numbers. But you can rotate a diagonal, and it looks just like the side of a square, only a bit longer. So you'd like a number system in which you can sensibly talk about any number you can construct geometrically. You'd also like to be able to talk about rotation! You can't do that with just rational numbers, either.

So how do you fill in enough holes to do geometry? Dedekind came up with a very clever way to do this. It starts by observing that a rational number $q$ is completely determined by the set of rational numbers greater than it and the set of rational numbers less than it. For example, 1/2 is completely determined by the fact that it's always between 1/2 + 1/n and 1/2 - 1/n. (For the initiated, this is a special case of the Yoneda lemma.) But there are "numbers," such as the square root of 2, which aren't rational, and yet have the property that we can always tell what rational numbers are greater than it and what rational numbers are less than it. For the square root of 2, these are precisely the fractions p/q such that 2q^2 < p^2 and such that 2q^2 > p^2, respectively. Dedekind's brilliant idea was the following:

Define a real number to be a partition of the rational numbers!

In Dedekind's construction, the square root of 2 quite literally is the set of rational numbers that are greater than it, and the set of rational numbers that are less than it. You can define all the usual arithmetic operations on these "numbers," called Dedekind cuts, and prove all the wonderful theorems you'll find in a standard book on real analysis. In particular, the property that guarantees that all the holes are filled is called completeness.

Figured I might as well add something about the complex numbers. The story here is beautiful, and if you're really interested you should check out Tristan Needham's Visual Complex Analysis. Some people say that the point of the complex numbers is to let you solve polynomials, but this is really selling them short. The complex numbers are an inherently geometric construction, and should be understood as such. Their geometry and topology just happens to be responsible for the fact that you can solve polynomials with them, but it's also responsible for much more.

Here is a quick sketch. Now that you've got the real numbers on your hands, you can rigorously talk about plane geometry. In plane geometry, an important notion is that of similarity. Informally, two figures are similar if they have the same shape. More formally, two figures are similar if you can rotate, translate, and scale one figure so that it matches up with the other. So similarity is all about a certain collection of transformations of the points in the plane. It was Klein who first realized that the important features of different flavors of "geometry" are captured in what kind of transformations are allowed. So to do geometry the modern way we should focus our attention on these transformations, which form a group.

To make this easier, let's ignore the translations for now. We'll pick an origin for our plane, and we'll only allow rotations and scalings about this origin. Rotations and scalings have the property that they are both linear transformations; this means that if you know what the transformation does to two points $u, v$, you also know what it does to the vector sum $u + v$. In particular, a linear transformation is determined by what it does to the point $(1, 0)$ and to the point $(0, 1)$.

However, rotations and scalings satisfy an extra property: they are, in fact, determined by what they do to the point $(1, 0)$. This is because $(0, 1)$ can be obtained from $(1, 0)$ by a rotation by 90 degrees, and rotations and scalings commute with each other: if you rotate x degrees then y degrees, that's the same as rotating y degrees then x degrees, which is the same as rotating x+y degrees. Similarly, if you rotate x degrees then scale by 2, that's the same as scaling by 2, then rotating x degrees. So if you know what a rotation-and-scaling does to $(1, 0)$, you just rotate that vector by 90 degrees, and you know what it did to $(0, 1)$.

So to every rotation-and-scaling, we can assign two real numbers: the coordinates of the image of the point $(1, 0)$. In general, a rotation by $\theta$ angles followed by a scaling by $r$ sends $(1, 0)$ to $(r \cos \theta, r \sin \theta)$. A different transformation, say a rotation by $\phi$ angles followed by a scaling by $s$, sends $(1, 0)$ to $(s \cos \phi, s \sin \phi)$. And their composition sends $(1, 0)$ to $(rs \cos (\theta + \phi), rs \sin (\theta + \phi))$. In other words, composition of rotations-and-scalings defines a multiplication law on pairs of real numbers. What is this law, exactly? Well, by the angle addition formulas, it's

$$(a, b) * (c, d) = (ac - bd, ad + bc).$$

And this is precisely the rule for multiplication in the complex numbers, where $(a, b)$ corresponds to $a + bi$. You get the rule for addition by observing that not only can you compose two rotations-and-scalings, you can also add their results.

Together, the real numbers and the complex numbers provide a foundation for much of modern mathematics and physics. For example, the complex numbers turn out (for reasons which are still not well understood) to be fundamental in the description of quantum mechanics.

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    $\begingroup$ I disagree with the opening paragraph. The real numbers only appear as a "sophisticated mathematical construction" when you decide to go into the mathematical construction business. We've had good synthetic versions of their theory for some time -- e.g. the complete ordered field axioms, Hilbert's axioms for Euclidean geometry. $\endgroup$ – Hurkyl Mar 3 '12 at 9:07
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    $\begingroup$ @Hurkyl: If you think the definition of the reals as the unique complete ordered field (up to isomorphism) would be a good answer to this question, why not write it up and submit it? I like the synthetic perspective as well, and it would be nice to see it represented here. $\endgroup$ – Vectornaut Jul 20 '14 at 17:54
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    $\begingroup$ @Hurkyl: I can't help but point out, however, that the definition of a complete ordered field can look pretty imposing when you're seeing it for the first time. Moreover, if you're given a complete ordered field, and you want to get your hands on some interesting elements of it, I think you'll end up going through a construction process much like the one Qiaochu describes. So, if I wanted to be snarky, I might characterize the synthetic definition of the reals as a sophisticated mathematical non-construction. $\endgroup$ – Vectornaut Jul 20 '14 at 17:54

Real numbers

Real numbers are any numbers you can locate (even approximately) on an infinite number line. This is a theoretical number line with infinite "resolution" that extends infinitely in both positive and negative directions.

One neat property about real numbers is that they are orderable -- that is, given any two real numbers, you can tell which one is "higher" and which one is "lower" than the other.

Real numbers are closed under multiplication, addition, and subtraction. That is, if you perform any of these operations on two real numbers, their result will always be real as well. They are almost closed under division, except for the whole divide-by-zero issue.

Not real numbers:

  • infinity

  • the square root of -1

  • 1/0


There isn't a rigorous definition of "decimals", because depending on where you use it, you'll get different definitions.

In the elementary sense, it means any number that has a "decimal part"; or a part after the radix (decimal point, etc.).

In a more advance sense, it means any number written in Base 10.

Natural numbers

Natural numbers are often also called "counting numbers", because they are the numbers you count with. (0,) 1, 2, 3, 4, etc.

There is some disagreement in the mathematics community over whether or not 0 is a natural number.


In linguistics, this means the natural "numbers" themselves (1, 2, 3, etc.) But you probably don't want to know about linguistics.

In Set Theory, two sets have the same cardinality if each element could be paired up with an element of the other set.

{1,2,3} and {4,5,6} share the same cardinality because you can pair up 1&4, 2&5, 3&6.


In linguistics, this means 1st, 2nd, 3rd, etc. But you probably don't want to know about linguistics.

In Set Theory, an ordinal is a well-ordered set.

  • $\begingroup$ The definition of "ordinal" is incorrect. -1 $\endgroup$ – user126 Jul 21 '10 at 23:13
  • $\begingroup$ @Henry Can you help me correct it? $\endgroup$ – Justin L. Jul 22 '10 at 0:24
  • $\begingroup$ Look it up on wikipedia. An ordinal is a certain type of pure set. $\endgroup$ – user126 Jul 22 '10 at 1:31
  • $\begingroup$ an ordinal is a well-ordered set. $\endgroup$ – mau Jul 29 '10 at 19:47
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    $\begingroup$ Well, an ordinal is more like an equivalence class of well-ordered sets under order-preserving bijections. The actual construction (as a pure set or something else) doesn't really matter. Ordinals measure the "order type" of the set just like cardinals measure the size. As an example, the ordered set $\lbrace 1,2,\dotsc,\omega\rbrace$ is the same cardinality as $\lbrace 1,2,\dotsc\rbrace$, but its order type is strictly greater. $\endgroup$ – Paul VanKoughnett Oct 12 '10 at 21:09

Going closer to the foundations of mathematics than most ever need or are comfortable with, we can also write down the classic model of $\mathbb{N_0}$ in ZF. This is of course going to be very bird's eye view and rather devoid of details.

"There exists a set, $I$, such that $\emptyset \in I$ and $x \in I \Rightarrow x \cup \{x\} \in I$." This is the axiom of infinity. We've the chosen the letter "I" to represent it since it is infinite and we can do induction over $I$ (thus sets satisfying this axiom are also called "inductive sets").

Then we construct the ordinals starting with $0 = \emptyset$, $1 = 0 \cup \{0\} = \{\emptyset, \{\emptyset\}\}, 2 = 1 \cup \{1\} \ldots$ Let us not discuss limit ordinals or why "the ordinals" is not a set - it is all rather complicated. Instead let me point out that from the existense of $I$ it's possible to show the existense of the smallest infinite ordinal, $\omega$, the first ordinal with infitely many preceding ordinals. Then we define $\mathbb{N}_0$ to be $\omega$ along with this "simple" definition of sum: $x+0 = x$, $x + 1 = x \cup \{x\}$ and $x + (y \cup \{y\}) = x+y \cup \{x+y\}$.

The next couple of steps are easy if you're familiar with equivalence classes.

$\mathbb{Z} = \mathbb{N_0}\times\mathbb{N_0}/\sim$, where $(n,m) \sim (n',m')$ iff $(n+m') = (m+n')$. E.g. the integer -2 is the equivalence class consisting of $\{(0,2),(1,3),(2,4),\ldots\}$. Then you show that there is a natural inclusion $\mathbb{N_0} \hookrightarrow \mathbb{Z}$ so it makes sense to write things like $0 \in \mathbb{Z}$ even though we technically defined 0 to be $\emptyset$ and $\emptyset \notin \mathbb{Z}$.

$\mathbb{Q} = \mathbb{Z}\times\mathbb{Z}\setminus\{0\}/\sim$, where $(a,b) \sim (c,d)$ iff $ad=bc$. Again we have natural inclusions $\mathbb{N}_0 \hookrightarrow \mathbb{Q}$ and $\mathbb{Z} \hookrightarrow \mathbb{Q}$ and each equivalence class is one rational number, i.e. a fraction. E.g. $\tfrac{2}{3}$ is the set $\{(2,3),(4,6),(6,9),\ldots\}$.

When we want the real numbers it gets hard. Typically in a first course in analysis you would give the students the reals in the form of an extra axiom: There exists a field, $\mathbb{R}$, together with an order ("inequality sign") $\leq$ such that: 1) $\mathbb{R}$ is an ordered field, and 2) Every subset of $\mathbb{R}$ that has an upper bound has a least upper bound.

Why don't you start from scratch? Well... first of all there's the issue of time, but also that of mathematical maturity. Any construction of $\mathbb{R}$ is going to require quite a bit of machinery. Some algebra, some topology and being able to manipulate things like "sequences of equivalence classes of sequences of rational numbers" without getting lost. Even formulating the notion of uniqueness of the real numbers can be a bit tricky. We want not just that any two fields satisfying 1) and 2) to be isomorphic (both as fields and w.r.t. order), but also that the isomorphism should be unique.

So what different constructions are there? Quite a lot actually. My 3 personal favourites are:

A) Dedekind cuts. Based on the "obvious" observation that a real number is uniquely determined by which rational numbers it is smaller than and which it is larger than. Pros: Easy to prove that this construction has the least upper bound property. Difficulties: It has to be a field, so we need to define multiplication and division. How do you divide subsets of rational numbers? Especially when the divisor contains $0$?

B) Equivalence classes of rational cauchy sequences. Based on the observation that the only way a rational cauchy sequence can not converge is if the limit is somehow "not there". So we just set the limit to be the sequence itself! This is of course nonsense, but it can be made precise. Pros: How easy it is to show it's a field: The set of rational cauchy sequences under coordinate-wise addition and multiplication is a ring and the rational sequences that converge to $0$ are a maximal ideal! Difficulties: Defining the order (note that $(\tfrac{1}{n})$ belongs to the same equivalence class as $0$) and then showing that the order is in fact total. Keeping track of sequences of equivalence classes of rational cauchy sequences when you're trying to prove that your field is complete can be a bit tricky.

C) "Almost-homomorphisms" of $\mathbb{Z}$. Based on the observation (at least since we got pixel-based displays) that a real number $\alpha$ is in some sense the same as choosing a rule for how to draw a line with slope $\alpha$ on an infinite ($\mathbb{Z}\times\mathbb{Z}$) computer display. Pros and cons? Not sure actually. It could be a fun little project to work out the details I guess.

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    $\begingroup$ Thinking back on this, it's become clear to me that I should probably clarify what the point of all this is. Explicit constructions of $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ serve mostly to show that existence and uniqueness (up to isomorphism) of "the counting numbers", "the integers", "the rationals" and "the reals" is possible in and consistent with ZF. That's why when we show e.g. the [en.wikipedia.org/wiki/… theorem), we just use that $\mathbb{R}^n$ is complete rather than any specific set representation. $\endgroup$ – kahen Oct 13 '10 at 13:17
  • $\begingroup$ So apparently I'm not allowed to edit that comment any more, and the link is unfixable. So here's a naked link to the wikipedia article on the Bolzano-Weierstraß theorem: en.wikipedia.org/wiki/Bolzano-Weierstrass_theorem $\endgroup$ – kahen Oct 13 '10 at 13:24

The natural numbers are defined by the Peano axioms, as in the answer of Isaac.

You can also view the natural numbers as the cardinalities of finite sets, which implies that zero is a natural number.

Now the other number domains arise because mathematicians want to give values for certain operations which otherwise are only defined partially. Alternatively, they want to give solutions to equations of certain forms, which cannot always be solved in the smaller domain.

  • integers arise from subtraction (solutions of a + x = b)

  • rational numbers arise from division (solutions of a * x = b)

  • real numbers arise from taking limits or upper bounds

  • complex numbers arise from taking roots (solutions of polynomial equations)

  • cardinal numbers arise as the sizes of sets. They can be constructed as isomorphism classes of sets, e.g. the number one would then be the class of all sets containing one element. They are useful for talking about the different sizes of infinite sets, which can get extremely large. Some set theorists compete over who can define the largest cardinal.

  • ordinal numbers arise as the sizes of well-orderings. They can be constructed as isomorphism classes of well-orderings, e.g. the number two would then be the class of all well-orderings containing two elements (with the first smaller than the second in the ordering). Interestingly, the ordinals are themselves well-ordered in a single well-ordering, so they come in a well-defined sequence, and an ordinal number can be viewed as describing its position in this sequence.

  • $\begingroup$ Or you can view the natural numbers as the cardinalities of finite nonempty sets. Really, there is no good reason to prefer one over the other. The complex numbers also do not arise from taking roots of polynomials with rational coefficients (which is what you seem to be implying). $\endgroup$ – Qiaochu Yuan Jul 28 '10 at 7:34
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    $\begingroup$ I changed the order so now complex numbers come after real numbers, as usual. W.r.t. zero or the empty set, I find it strange to exclude them now that we fully accept their existence. In my experience as a computer scientist it is better to always at least consider the case 0 and only exclude it where necessary. $\endgroup$ – starblue Jul 29 '10 at 19:33

You might as well start at the 'lowest':

  • Integers are all the whole numbers,
  • Natural numbers (N) are the set of positive integers, {1, 2, ...} (0 optional),
  • Rational numbers (Q) are any number that can be represented a/b with a and b being Integers (|b| < 0.
  • Real numbers are all the Rational numbers and all the others.

  • Cardinality is the number of elements in a set.

  • An Ordinal is a well ordered set.
  • $\begingroup$ An ordinal is a well-ordered set. $\endgroup$ – Qiaochu Yuan Jul 28 '10 at 7:35

Ordinals aren't simply sets well-ordered by the membership relation, but they're also transitive i.e. such a relation is transitive on them ( meaning that $y$ is transitive iff $ x \in y \Rightarrow x \subset y$ )


You tossed out a ton of names associated with number systems, and I thought you might like to know this:

With the exception of the natural numbers, ordinals and cardinals, all of the things in your post are examples of rings. Rings are a type of algebraic object that gather up most of the important properties of number systems.

By "important properties" I'm thinking of addition, subtraction, multiplication and distributivity. It turns out we can do quite well without division and commutativity :)

Rather than thinking of number systems as a menagerie of names like the one in your post, this might help you get a handle on most of them all at once. (Actually after the next paragraph we'll be able to pick the natural numbers back up.)

What are the technicalities keeping the three exceptions I mentioned from being rings? For one thing, the natural numbers, cardinals and ordinals don't really have any appropriate notion of subtraction which is part of the definition of rings. Secondly, all rings are supposed to be made up of a set of points, and while the natural numbers are a set of points, the cardinals and ordinals are not.

The natural numbers actually form a semiring, which is basically a ring that might not have subtraction. In this sense they are just a number system that is not quite as nice as a ring.

The cardinals and ordinals have addition and multiplication operations too which might qualify them to be semirings if they weren't so darn big (being a set is part of the definition of a semiring, too). But if you were willing to accept a semiring-that-is-a-proper-class then you would have a type of object which encompasses all the examples you gave.


There are many types of numbers, though the natural numbers, the integers, the rational, the decimal, the real, and the complex form a nice self-complete expository whole. Hopefully, the following block(s) of text aren't too poorly formatted.

1) The set of natural numbers consists of numbers with which we count {0,1,2,3,...}. As noted in some of the other answers, some people think that 0 is not a natural number (see one of my desktop backgrounds). Whether or not it is, it's a matter of taste.

What is not a matter of taste are the defining properties of the natural numbers. In particular, there is a (binary) operation called +, which takes two numbers a and b and spits out a third number a+b, which is the sum of a and b. It satisfies the usual properties that you would expect from counting: it is commutative (the order of the summands doesn't matter, i.e. a+b=b+a) and associative (the order in which you add summands to each other doesn't matter, i.e. (a+b)+c=a+(b+c), so you can always write a+b+c for a specific number).

There is also an order relation < such that for any two different numbers a and b either a

The order plays nice with addition in that a < b implies a+c < b+c. We also have a cancellation property that if a < c, then there exists a number b such that a+b=c.

The order also satisfies one unobvious property called the well-ordering principle and states that if you take any collection of natural numbers, there is a smallest one, i.e. a number that is smaller than all the other numbers in the collection.

The well-ordering principle, together with cancellation, implies that there is a smallest number, and that any two consecutive numbers differ by the same number. From this it follows that either the smallest number is 0 (in the sense that 0 is the number such that 0+a=a+0=a) or that it is 1, where 1 is the common difference between any two consecutive numbers.

Choosing one or the other of the two possibilities defines the natural numbers uniquely as either {0,1,2,...} or {1,2,...}.

It is a fun exercise to show that the above axioms are equivalent to the axioms of Peano (note that the axioms of Peano state nothing about addition, so in fact BOTH {0,1,2,...} and {1,2,...} satisfy the axioms; how we define addition with respect to the smallest element is what specifies one of the two sets)

2) The set of integers, also known as whole numbers, is {..., -3, -2, -1, 0, 1, 2, 3, ...}. It arises as we try to "complete" the arithmetic of the integers in the following sense.

For any two numbers a < b we have a number b-a such that (b-a)+a=b, thus we have subtraction of a smaller number from a bigger number. We do not have such numbers b-a if a>b (i.e. there is no natural number which when you add a to you get b if a>b), but from experience we see that numbers such as ... -3, -2, -1 seem to be sensical to add and subtract as we try to solve equations and whatnot.

But while numbers such as b-a when a < b are defined, the same number could be represented in different ways. For example, 2 is both 5-3 and 8-6. So we need to answer the question of when b-a=d-c for a < b, c < d, and by algebra the answer is that they're equal whenever b+c=d+a. Hence, we can define all the integers, positive and negative, by taking them to be pairs of natural numbers (a,b) with the caveat that (a,b)=(c,d) whenever a+c=b+c.

For example, the integer -2 consists of (among others) the pairs (5,3), (6,8), (0,2). This is all seems like much formal ado about intuitive nothings, but that is only because we can represent any integer as either the pair (a,0), which we write simply as a, which we write as -a, and think of as (a,0) is the integer which when you add a to you get 0, or as (0,a), which we think of as the integer to which which you add 0 you get the natural number a. Hence what we actually do arithmetic with are the pairs {... (3,0), (2,0), (1,0), (0,0), (0,1), (0,2), (0,3),...}, but the fact that we only need to throw in a negative sign to make the arithmetic meaningful is precisely because there exists these special representative pairs.

Exercise: define the addition and order on the pairs of natural numbers for the above to work (we lose of course the property that any set of integers has a smallest integers [e.g. the set of negative integers], but the principle still holds for sets of positive integers).

3) The rational numbers (or fractions) are obtained by the exact same process of completion as above, except with respect to multiplication. the fraction a/b, where a and b are integers corresponds to the number which when multiplied by b gives you a. Note that a/b=c/d if and only if ad=bc, which mirrors the fact that a-b=c-d if and only if a+b=b+c.

4) The real numbers are best understood as coming from decimals (or decimal expansions), so what are decimals and where do they come from?

Rational numbers as fractions are hard to compare to one another. Is 7/5 bigger or smaller than 4/3? The answer is yes, becayse 7*3>3*5. In general if you have a/b and c/d, we have a/b < c/d, a/b=c=d or a/b>c/d if correspondingly adbc. But that's a tedious process, is there some way of writing them down so that it is clearer who's bigger than who? Better, can the process of writing them down be easier than the process of checking who's bigger than who by cross-multiplication?

The answer is yes, and we only need consider the usual way in which we write natural numbers. We can express the number 1729 compactly based on the distributive properties of addition and exponentiation: 1729=1000+700+20+9=10^3+7*10^2+2*10+9, where the digits 1,2,3,4,5,6,7,8,9,10 are the first ten successors of zero, and zero is denoted by 0.

So we can write rational numbers with denominator 10^n as a.b=a+b/10^n with a and b are natural numbers and b is smaller than 10^n. For example, 1729/100 can be written as 17.29 where 17 and 29 are natural numbers with 29<100.

Now, for any rational number c/d there exists a rational number with denominator closest 10^n that's closest to and smaller than c/d (this is intuitively obvious and not that hard to formalize). Then we can say that c/d ~ a.b where a+b/10^n is that rational number.

For example, the closest number with denominator 10 to 1/3 is 3/10=.3, with denominator 100 is 33/100=.33, and so on. Then that the infinite string 0.3333... equals 1/3 means that the closest rational number less than or equal to 1/3 with denominator 10^n is 333...3/10^n where you have n 3's.

From this, one can prove that rational numbers correspond have periodic expansions, i.e. that the string a.b has a repeating sequence of digits from some point onward.

(even better, we can compute decimal expansions by the usual process of long division by just adding a decimal point, and we can efficiently compare numbers by looking at the expansion until the first point of difference).

5) Real numbers come from taking arbitrary infinite sequences of digits as decimals.

From above, we know that decimals actually encode sequences of rational numbers with denominators 1,10, 10^2, 10^3, ...

When should two such sequences be equivalent? Consider the standard question of whether .999...=1.

The decimal expansion of 1 is just 1.000... because the sequence of rational numbers with denominators 10, 10^2, 10^3, ... has to be a sequence of rational numbers less than or equal to 1.

If we drop the requirement that the numbers be less than or equal to 1, keep the requirement that the sequence of rational numbers (approximations really) is non-decreasing, and say allow the rational number to be either the closest number less than 1 or 1 itself, then we obtain the other expansion 1=.999...

If you had a measuring instrument that was accurate to n decimal places, then upon measuring .999... and 1.000... you would always get .999...9 with n-2 9s.

So it turns out that with finite precision measuring, you can't tell the difference between certain decimal expansions, so we might as way say those two decimal expansions are equal. This is the process of completion by Cauchy sequences put into words.

And those expansions then are the real numbers, and they have addition, multiplication, subtraction and division and ordering just like the rational numbers, except that they're complete in that every sequence of real numbers converges.

6) Complex numbers It turns out that every polynomial with real coefficients of degree >2 has a square root. -1 does not have a a real square root, so we wish to throw a root of -1 in to algebraically complete the real numbers, and thus get the complex numbers. The algebraic construction is best done by explaining concepts such as rings and ideals and quotients and homomorphisms, which is too much machinery.

Instead, consider Euclidean geometry. It is a(n advanced) fact that you can coordinize it with real numbers, i.e. you can represent points by pairs of real numbers (a,b), lines by the solutions of the two variable equation ax+by=c, lengths given by square root of ((a-c)^2+(b-d)^2), etc.

Adding pairs (a,b) and (c,d) gives you (a,b)+(c,d) so addition of pairs corresponds to translations. Another way of interpreting a pair of (a,b) is by interpreting it in polar form (r, theta) where r is the distance of (a,b) form the origin and theta is the angle from the x-axis to (a,b). Then we can think of (a,b)~(r,theta) as a dilation and rotation with center (0,0) and sending (1,0) to (a,b), i.e. a dilation by a factor of r and a rotation by an angle of theta. But this then defines a multiplication of vectors, which turns out to distribute over addition. Clearly we have inverse rotations/dilations and inverse translations, giving us subtraction and division. Finally, we can identify the real numbers with pairs (r,0) which are only a dilation without any rotation.

These pairs are then our complex numbers and you can deduce all their properties from the above description. Note that the vector (0,1) multiplied by (0,1) gives (-1,0) since it corresponds to a rotation of 90 followed by a rotation of 90, which is a rotation of 180 of (1,0) which equals (-1,0). See Qiaouchu's answer for more details.


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