What is a real number (also rational, decimal, integer, natural, cardinal, ordinal...)?
The natural numbers can be defined by Peano's Axioms (sometimes called the Peano Postulates):
- Zero is a number.
- If n is a number, the successor of n is a number.
- zero is not the successor of a number.
- Two numbers of which the successors are equal are themselves equal.
- (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.
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.
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 letteri.Complex numbers
Numbers composed of the sum of a Real and an Imaginary number. This includes all Real and all Imaginary numbers.