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What are numbers?

Introduction

Informally:

Construction of the Number System

Formally (following the mainstream in math) the numbers are constructed from scratch out of the axioms of Zermelo Fraenkel set theory (a.k.a. ZF set theory) [Enderton77, Henle86, Hrbacek84]. The only things that can be derived from the axioms are sets with the empty set at the bottom of the hierarchy. This will mean that any number is a set (it is the only thing you can derive from the axioms). It doesn't mean that you always have to use set notation when you use numbers: just introduce the numerals as an abbreviation of the formal counterparts.

The construction starts with N and algebraically speaking, N with its operations and order is quite a weak structure. In the following constructions the structures will be strengthen one step at the time: Z will be an integral domain, Q will be a field, for the field R the order will be made complete, and field C will be made algebraically complete.

Before we start, first some notational stuff:

Although the previous notations and the constructions that follow are the de facto standard ones, there are different definitions possible.

Construction of N

Informally n = { 0,...,n - 1 } (thus 0 = { } , 1 = { 0 } , 2 = { 0,1 } , 3 = { 0,1,2 } ). We will refer to the elements of N by giving them a subscript _n. The relation <_n on N is defined as: a_n <_n b_n iff a_n in b_n. We can define +_n as follows: Define *_n as:

Construction of Z

We define an equivalence relation on N x N: (a_n,b_n) ==_z(c_n,d_n) iff a_n +_n d_n = c_n +_n b_n. Note that ==_z ``simulates'' a subtraction in N . Z = { [(a_n,b_n)]_z | a_n, b_n in N } . We will refer to the elements of Z by giving them a subscript _z. The elements of N can be embedded as follows: embed_n : N --> Z such that embed_n(a_n) = [(a_n,0_n)]_z. Furthermore we can define:

Construction of Q

We define an equivalence relation on Z x (Z { 0_z }): (a_z,b_z) ==_q (c_z,d_z) iff a_z *_z d_z = c_z *_z b_z. Note that ==_q ``simulates'' a division in Z . Q = { [(a_z,b_z)]_q | a_z in Z and b_z in Z { 0_z } } . We will refer to the elements of Q by giving them a subscript _q. The elements of Z can be embedded as follows: embed_z : Z --> Q such that embed_z(a_z) = [(a_z,1_z)]_q. Furthermore we can define:

Construction of R

The construction of R is different (and more awkward to understand) because we must ensure that the cardinality of R is greater than that of Q .
Set c is a Dedekind cut iff

You can think of a cut as taking a pair of scissors and cutting Q in two parts such that one part contains all the small numbers and the other part contains all large numbers. If the part with the small numbers was cut in such a way that it doesn't have a largest element, it is called a Dedekind cut. R = { c | c is a Dedekind cut } . We will refer to the elements of R by giving them a subscript _r. The elements of Q can be embedded as follows: embed_q : Q --> R such that embed_q(a_q) = { b_q | b_q <_q a_q } . Furthermore we can define:

There exists an alternative definition of R using Cauchy sequences: a Cauchy sequence is a s : N --> Q such that s(i_n) +_q((-1)_q *_q s(j_n)) can be made arbitrary near to 0_q for all sufficiently large i_n and j_n. We will define an equivalence relation ==_r on the set of Cauchy sequences as: r ==_r s iff r(m_n) +_q((-1)_q *_q s(m_n)) can be made arbitrary close to 0_q for all sufficiently large m_n. R = { [s]_r | s is a Cauchy sequence } . Note that this definition is close to ``decimal'' expansions.

Construction of C

C = R x R. We will refer to the elements of C by giving them a subscript _c. The elements of R can be embedded as follows: embed_r : R --> C such that embed_r(a_r) = (a_r,0_r). Furthermore we can define:

There exists an elegant alternative definition using ideals. To be a bit sloppy: C = R [x]/< (x *_r x) +_r 1_r > , i.e. C is the resulting quotient ring of factoring ideal < (x *_r x) +_r 1_r > out of the ring R [x] of polynomials over R . The sloppy part is that we need to define concepts like quotient ring, ideal, and ring of polynomials. Note that this definition is close to working with i^2 = -1: (x *_r x) +_r 1_r = 0_r can be rewritten as (x *_r x) = (-1)_r.

Rounding things up

At this moment we don't have that N is a subset of Z , Z of Q , etc. But we can get the inclusions if we look at the embedded copies of N , Z , etc. Let

For these sets we have N' subseteq Z' subseteq Q' subseteq R' subseteq C. Furthermore these sets have all the properties that the ``informal'' numbers have.

What's next?

Well, for some of the more alien parts of math we can extend this standard number system with some exotic types of numbers. To name a few:

Cardinals and ordinals are commonly used in math. Most mortals won't encounter (let alone use) hyperreals, quaternions, and octonions.

References

J.H. Conway. On Numbers and Games, L.M.S. Monographs, vol. 6. Academic Press, 1976.

H.B. Enderton. Elements of Set Theory. Academic Press, 1977.

G.M. Dixon. Division Algebras; Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Kluwer Academic, 1994.

J.M. Henle. An Outline of Set Theory. Springer Verlag, 1986.

K. Hrbacek and T. Jech. Introduction to Set Theory. M. Dekker Inc., 1984.

L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.

This subsection of the FAQ is Copyright (c) 1994, 1995 Hans de Vreught. Send comments and or corrections relating to this part to J.P.M.deVreught@cs.tudelft.nl


next up previous contents
Next: Number Theory Up: Fundamentals Previous: Algebraic structures


Alex Lopez-Ortiz
Fri Feb 20 21:45:30 EST 1998