We shall assume you know the basics of numbers—addition, multiplication and so on. Whole numbers are also called integers. We refer to a number as a multiple of another number if it equals the product of that number with an integer; for example the numbers 6, 12, 18, 36 are some of the multiples of 6, as are −6, −12, −24  (negative multiples) and even 0 (0 = 6 times 0).

It is natural to think of the numbers we use in terms of sets. The set N of all positive integers or natural numbers is the first number system we encounter;  the natural numbers are used to count things. If one adds zero, to account for the possibility of there being nothing to count, and negatives for subtraction, the result is the set Z of integers. We write

                    N = {1,2,3,...}

                        Z = {...−3,−2,−1,0,1,2,3,...}

The use of a string of dots (an ellipsis) means that the set continues without end. Such sets are called infinite (as opposed to finite sets like {0,1,3}). We also write {1,2,...,20} to mean the set of all positive integers from 1 to 20. When no confusion arises, the ellipsis means “continue in the obvious way.”

The set Q of rational numbers consists of the ratios p/q, where p and q are integers and q ≠0. In other words,

                       Q = {p/q : p ∈ Z,q ∈ Z,q ≠ 0}.

Each rational number has infinitely many representations as a ratio. For example,

                                   1/2 = 2/4 = 3/6 = ...

An alternative definition is that Q is the set of all numbers with a repeating or terminating decimal expansion. Examples are

                      1/2 = 0.5

                  −12/5 = −2.4

             3/7 = 0.428571428571...

In the last example, the sequence 428571 repeats forever, and we denote this by writing

The denominator q of a rational number cannot be zero. In fact, division by zero is never possible. Some people—even, unfortunately, some teachers—think this is a made-up rule, but it is not. In fact, it follows from the definition of division. When we write x = p/q, we mean “x is the number which, when multiplied by q, gives p.”  So what would x = 2/0 mean? There is no number which, when multiplied by 0,  gives 2. Whenever you multiply any number x by 0, you get 0. You can never get 2.

How about x = 0/0? There are suitable numbers x, in fact every number will give 0 when multiplied by 0, but we wanted a single answer. So “x = 0/0” tells us nothing about x; it doesn’t specify any number.

The integers are all rational numbers, and in fact they are the rational numbers with numerator 1. For example, 5 = 5/1.

The positive integers are called “natural,” but there is no special name for the positive rational numbers. However, we have a notation for this set, Q+. In general,  a superscript + denotes the set of all positive members of the set in question.

The final number system we shall use is the set R of real numbers, which consists of all numbers which are decimal expansions, all numbers which represent lengths.

Not all real numbers are rational; one easy example is √2. In fact, if n is any natural number other than a perfect square (that is, n is not one of 1, 4, 9, 16, . . . ), then √n is not rational. Another important number which is not rational is π, the ratio of the circumference of a circle to its diameter.

Remember that every natural number is an integer; every integer is a rational number; every rational number is a real number. Do not fall into the common error of thinking that “rational number” excludes the integers, and so on. There are special words for such things. Rational numbers which are not integers are called proper fractions, and real numbers which are not rational are called irrationals.

This is not the end of number systems. For example, the set C of complex numbers is derived from the real numbers by including square roots of negative numbers, plus all the sums and products of the numbers. However, we will not encounter them in this book.