1.1: Basic Axioms for Z

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Since number theory is concerned with properties of the integers, we begin by setting up some notation and reviewing some basic properties of the integers that will be needed later:

\[\begin{aligned} \mathbb{N} &=\{1,2,3,\cdots\} \quad \text{(the natural numbers or positive integers)} \\ \mathbb{Z} &=\{\cdots,-3,-2,-1,0,1,2,3,\cdots\} \quad\text{(the integers)} \\ \mathbb{Q} &=\left\{ \frac nm \mid n,m\in\mathbb{Z}\text{ and }m\neq 0\right\} \quad \text{(the rational numbers)} \\ \mathbb{R} &=\text{the real numbers}\end{aligned}\] Note that \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\). I assume a knowledge of the basic rules of high school algebra which apply to \(\mathbb{R}\) and therefore to \(\mathbb{N}\), \(\mathbb{Z}\) and \(\mathbb{Q}\). By this I mean things like \(ab=ba\) and \(ab+ac=a(b+c)\). I will not list all of these properties here. However, below I list some particularly important properties of \(\mathbb{Z}\) that will be needed. I call them axioms since we will not prove them in this course.

Some Basic Axioms for \(\mathbb{Z}\)

If \(a\), \(b\in\mathbb{Z}\), then \(a+b\), \(a-b\) and \(ab\in\mathbb{Z}\). (\(\mathbb{Z}\) is closed under addition, subtraction and multiplication.)
If \(a\in\mathbb{Z}\) then there is no \(x\in\mathbb{Z}\) such that \(a<x<a+1\).
If \(a\), \(b\in\mathbb{Z}\) and \(ab=1\), then either \(a=b=1\) or \(a=b=-1\).
Laws of Exponents: For \(n\), \(m\) in \(\mathbb N\) and \(a\), \(b\) in \(\mathbb{R}\) we have
1. \(\left( a^n \right)^ m=a^{nm}\)
2. \((ab)^n=a^nb^n\)
3. \(a^na^m=a^{n+m}\).
  These rules hold for all \(n,m \in \mathbb{Z}\) if \(a\) and \(b\) are not zero.
Properties of Inequalities: For \(a\), \(b\), \(c\) in \(\mathbb{R}\) the following hold:
1. (Transitivity) If \(a<b\) and \(b<c\), then \(a<c\).
2. If \(a<b\) then \(a+c<b+c\).
3. If \(a<b\) and \(0<c\) then \(ac<bc\).
4. If \(a < b\) and \(c < 0\) then \(bc < ac\).
5. (Trichotomy) Given \(a\) and \(b\), one and only one of the following holds: \[a=b , \quad a<b , \quad b<a.\nonumber\]
The Well-Ordering Property for \(\mathbb N\): Every non-empty subset of \(\mathbb{N}\) contains a least element.
The Principle of Mathematical Induction: Let \(P(n)\) be a statement concerning the integer variable \(n\). Let \(n_0\) be any fixed integer. \(P(n)\) is true for all integers \(n \ge n_0\) if one can establish both of the following statements:
1. \(P(n)\) is true if \(n=n_0\).
2. Whenever \(P(n)\) is true for \(n_0\le n\le k\) then \(P(n)\) is true for \(n=k+1\).

We use the usual conventions:

\(a\leq b \text{ means } a<b \text{ or } a=b\),
\(a>b \text{ means } b<a\), and
\(a\geq b \text{ means } b\leq a\).

Important Convention

Since in this course we will be almost exclusively concerned with integers we shall assume from now on (unless otherwise stated) that all lower case roman letters \(a,b,\dots,z\) are integers.