Number systems

1. Introduction

This web page discusses perhaps the most obviously basic part of analysis, or of mathematics, that is the numbers we work with. The numbers are usually classified as being whole numbers (integers), fractions (rationals), real numbers, or complex numbers. However there is more to this than just saying what each kind of number is. We should also describe the fundamental properties (or axioms) satisfied by these numbers so that we are sure our work relies on only these and no other implicit or hidden property.

You should be reasonably familiar with the usual number systems of mathematics, the natural numbers, the integers, the rationals, the reals, and the complex numbers. This web page introduces these in a fairly informal way to agree some notation and terminology for the way these main number systems used in these pages. These systems will be subject to a lot more discussion as our work gets more advanced and the methods we have of describing them get more detailed.

2. The natural numbers

The natural numbers are the usual counting numbers, $0 , 1 , 2 , 3 , 4 , …$. We shall use the symbol $ℕ$ for the set of all such numbers, so $ℕ = { 0 , 1 , 2 , 3 , … }$. Equally important for these numbers are certain operations that apply to them. For instance, for each natural number $n$ there is a next number $n + 1$. Also, there are important operations of addition and multiplication on natural numbers, $n + m$ and $n · m$, and two natural numbers can be compared to see which is the greater via the $<$ or $⩽$ relation.

One complication is that there are two competing conventions for $ℕ$: whether to include $0$ or not. I am firmly in the first camp, and for me $ℕ = { 0 , 1 , 2 , 3 , … }$. I am aware you may be reading material written by, or studying with, another mathematician who prefers the other convention. Where possible I will write text, examples, exercises, in such a way that it doesn't matter which $ℕ$ you are thinking of. In the few occasions when it does matter, I will try to remind you which my $ℕ$ is.

Apart from these arithmetic operations, the most important feature of the natural numbers is induction. This is the main method of proof for anything to do with natural numbers and is implicit in any mathematics where the elipsis $…$ appears.

Principle of Induction.

Suppose a statement $S ( k )$ is true for some number $k$; suppose also that whenever $S ( n )$ is true then $s ( n +1 )$ also holds. Then $S ( n )$ holds for all $n ⩾ k$.

You will have seen many examples of this principle being used to good effect, and will see many more examples in these notes. Usually (but not always) the base case $S ( k )$ is the case of the smallest natural number $k = 0$ itself.

An important and useful variation of induction is what I like to call total induction which says that in proving the induction step, $s ( n +1 )$, it is permissible to assume that $S ( m )$ folds for all $k ⩽ m ⩽ n$. In principle (though not always in practice) this makes the proof of the induction step easier at no extra cost.

Theorem.

Suppose a statement $S ( k )$ is true for some number $k$; suppose also that whenever $S ( m )$ is true for all $m = k , k +1 , … , n$ then $s ( n +1 )$ also holds. Then $S ( n )$ holds for all $n ⩾ k$.

A variation of this is the least number principle (also called smallest counterexample) as follows.

Theorem.

Suppose a statement $S ( n )$ is false for some number $n ∈ ℕ$. Then there is a least $n ∈ ℕ$ such that $S ( n )$ is false, i.e., a number $n ∈ ℕ$ such that $S ( n )$ is false and $S ( k )$ is true for $k = 0 , 1 , 2 , … , n - 1$.

3. The integers

The next important ingredent is subtraction. By adding to the set of natural numbers the negatives of these numbers we get the integers. An integer is a whole number, which may be positive or negative. The set of integers is denoted $ℤ$.

The set of integers has a very close connection with the natural numbers. Indeed the set of natural numbers is the set of integers that are greater than or equal to zero. The negative numbers makes arithmetic in the integers much more convenient, however, and $( ℤ , + )$ is an abelian (or commutative) group with identity $0$ and inverses $- x$ for each $x ∈ ℤ$. If we take away the number zero we also get an abelian semigroup for multiplication, $( ℤ - { 0 } , · )$, i.e., the associativity and commutativity axioms hold for multiplication, and there is a multiplicative identity $1$. Furthermore, the distributivity law, $( y + z ) = x ⁢ y + x ⁢ z$ also holds. This means that $( ℤ , + , · )$ is a ring. It is a particularly nice ring with an order relation $<$ and other nice properties, and you will most likely have spent some time studying the properties of numbers in $ℤ$ and their congruence relations via the idea of one number being a factor of another.

4. The rationals

The next stage is to extend our number system still further to add multiplicative inverses $1 n$ for each non-zero integer $n$. We then close our resulting collection of numbers under addition and multiplication. The result is the set $ℚ$ of all rational numbers, or fractions. A rational number will be written as $p q$ where $p$ and $q$ are integers, and $q ≠ 0$. Note that this is true for all rational numbers, and (unlike in primary school) we do not care whether the fraction is "proper" or not. So $p 1$ is a perfectly good rational, and this explains why set of the rational numbers includes the set of integers.

The main complication is that a rational number will not have a unique representation as $p q$. There will be many such. For example, if $p q$ represents a certain number then so do $- p - q$ and $2 ⁢ p 2 ⁢ q$ represent the same number.

The operations of addition and multiplication have to be extended to the new system. That's OK, and it is done as you learnt in primary school, essentially by putting the result over a common denominator. big advantage of the rationals over the integers is that in the rationals you can also divide by any nonzero number since $( p q ) / ( r s ) = p ⁢ s q ⁢ r$.

The disadvantage of the rationals is that there are still lots of operations one cannot do. The most obvious is taking square roots. You may have come across the well known theorem attributed to Euclid that says that $2$ is not a rational number. There are plenty of other equations that might reasonably have roots, but do not have roots in the rationals. Going down this line might lead you to the field of algebraic numbers. However, studying this is best done as part of a course in algebra, rather than analysis.

Analysis takes a more radical and much more powerful approach of closing the field of rationals under the process of taking limits. This is what the field of real numbers $ℝ$ is, and this is what these web pages are really about.