What is a real number (also rational, decimal, integer, natural, cardinal, ordinal...)?

Solution 1:

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.

Solution 2:

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.

Solution 3:

• 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.

Solution 4:

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

Decimals

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.

Cardinals

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.

Ordinals

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.

Solution 5:

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.