Real Numbers

A real number is any number which can be represented by a point on a number line. The numbers 3.5, −0.003, 2/3, π, and √2 are all real numbers.

The real numbers include the rational numbers, which are those which can be expressed as the ratio of two integers, and the irrational numbers, which cannot. (In the list above, all the numbers except pi and the square root of 2 are rational.)

It is thought that the first real number to be identified as irrational was discovered by the Pythagoreans in the sixth century B.C. Prior to this discovery, people believed that every number could be expressed as the ratio of two natural numbers (negative numbers had not been discovered yet). The Pythagoreans were able to show, however, that the hypotenuse of an isosceles right triangle could not be measured exactly by any scale, no matter how fine, which would exactly measure the legs.

To see what this meant, imagine a number line with an isosceles right triangle drawn upon it, as in Figure 1. Imagine that the legs are one unit long.

The Pythagoreans were able to show that no matter how finely each unit was subdivided (uniformly), point P would fall somewhere inside one of those subdivisions. Even if there were a million, a billion, a billion and one, or any other number of uniform subdivisions, point P would be missed by every one of them. It would fall inside a subdivision, not at an end. Point P represents a real number because it is a definite point on the number line, but it does not represent any rational number a/b.

Point P is not the only irrational point. The square root of any prime number is irrational. So is the cube root, or any other root. In fact, by using infinite decimals to represent the real numbers, the mathematician Cantor was able to show that the number of real numbers is uncountable. An infinite set of numbers is "countable" if there is some way of listing them that allows one to reach any particular one of them by reading far enough down the list. The set of natural numbers is countable because the ordinary counting process will, if it is continued long enough, bring one to any particular number in the set. In the case of the irrational numbers, however, there are so many of them that every conceivable listing of them will leave at least one of them out.

The real numbers have many familiar subsets which are countable. These include the natural numbers, the integers, the rational numbers, and the algebraic numbers (algebraic numbers are those which can be roots of polynomial equations with integral coefficients). The real numbers also include numbers which are "none of the above." These are the transcendental numbers, and they are uncountable. Pi is one.

Except for rare instances such as √2 ÷ √8 , computations can be done only with rational numbers. When one wants to use an irrational number such as π, √3 , or e in a computation, one must replace it with a rational approximation such as 22/7, 1.73205, or 2.718. The result is never exact. However, one can always come as close to the exact real-number answer as one wishes. If Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group.

the approximation 3.14 for π does not come close enough for the purpose, then 3.142, 3.1416, or 3.14159 can be used. Each gives a closer approximation.