# The Rational Numbers

• John Stillwell
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

The natural numbers 0, 1, 2, ... are the numbers used for counting. They are generated from 0 by the successor operation +1 (add one). In other words, the set ℕ = {0, 1, 2, ...} of natural numbers is the closure of the set {0} under successor, that is, the intersection of all sets S such that 0 ∈ S and n + 1 ∈ S when nS. This definition yields several logically equivalent properties of ℕ known as induction. The most commonly used versions of induction are:
1. I.

If 0 ∈ S, and if n +1 ∈ S when nS, then ℕ ⊆ S.

2. II.

If 0 ∈ S, and if n + l ∈ S when 0, 1, ..., nS, then ℕ ⊆ S.

3. III.

If T ⊆ ℕ is nonempty then T has a least member (that is, an nT such that the closure of {n} under successor includes all of T).

## Keywords

Natural Number Prime Divisor Rational Solution Algebraic Integer Congruence Class
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.