Rational Number

A rational number is one that can be expressed as the ratio of two integers such as 3/4 (the ration of 3 to 4) or -5:10 (the ration of -5 to 10). Among the infinitely many rational numbers are 1.345, 1⁷⁄₈, 0, -75, √25, √0.125, and 1. These numbers are rational because they can be expressed as 1345:1000, 15:8, 0:1, -75:1, 5:1, 1:2, and 1:1 respectively. The numbers π, √2 , i, and √5 are not rational because none of them can be written as the ratio of two integers. Thus any integer, any common fraction, any mixed number, any finite decimal, or any repeating decimal is rational. A rational number that is the ratio of a to b is usually written as the fraction a/b.

Rational numbers are needed because there are many quantities or measures which natural numbers or integers alone will not adequately describe. Measurement of quantities, whether length, mass, or time, is the most common situation. Rational numbers are needed, for example, if a farmer produces and wants to sell part of a bushel of wheat or a workman needs part of a pound of copper.

The reason that rational numbers have this flexibility is that they are two-part numbers with one part available for designating the size of the increments and the other for counting them. When a rational number is written as a fraction, these two parts are clearly apparent, and are given the names "denominator " and "numerator " which specify these roles. In rational numbers such as 7 or 1.02, the second part is missing or obscure, but it is readily supplied or brought to light. As an integer, 7 needs no second part; as a rational number it does, and the second part is supplied by the obvious relationship 7 7/1. In the case of 1.02, it is the decimal point which designates the second part, in this case 100. Because the only information the decimal point has to offer is its position, the numbers it can designate are limited to powers of 10: 1, 10, 100, etc. For that reason, there are many rational numbers which decimal fractions cannot represent, 1/3 for example.

Rational numbers have two kinds of arithmetic, the arithmetic of decimals and the arithmetic of common fractions. The arithmetic of decimals is built with the arithmetic of integers and the rules for locating the decimal point. In multiplying 1.92 by 0.57, integral arithmetic yields 10944, and the decimal point rules convert it to 1.0944.

Common fraction arithmetic is considerably more complex and is governed by the familiar rules

If one looks closely at these rules, one sees that each rule converts rational-number arithmetic into integer arithmetic. None of the rules, however, ties the value of a rational number to the value of the integers that make it up. For this the rule (a/b)b = a, b ≠ 0 is needed. It says, for example, that two 1/2s make 1, or twenty 3/20s make 3.

The rule would also say that zero 5/0s make 5, if zero were not excluded as a denominator. It is to avoid such absurdities that zero denominators are ruled out.

Between any two rational numbers there is another rational number. For instance, between 1/3 and 1/2 is the number 5/12. Between 5/12 and 1/2 is the number 11/24, and so on. If one plots the rational numbers on a number line, there are no gaps; they appear to fill it up.

But they do not. In the fifth century B.C. followers of the Greek mathematician Pythagoras discovered that the diagonal of a square one unit on a side was irrational, that no segment, no matter how small, which measured the side would also measure the diagonal. So, no matter how many rational points are plotted on a number line, none of them will ever land on √2 , or on any of the countless other irrational numbers.

Irrational numbers show up in a variety of formulas. The circumference of a circle is π times its diameter. The longer leg of a 30°-60°-90° triangle is √3 times its shorter leg. If one needs to compute the exact length of either of these, the task is hopeless. If one uses a number which is close to π or close to √3 , one can obtain a length which is also close. Such a number would have to be rational, however, because it is with rational numbers only that we have computational procedures. For π one can use 22/7, 3.14, 3.14159, or an even closer approximation.

More than 4,000 years ago the Babylonians coped with the need for numbers that would measure fractional or continuously variable quantities. They did this by extending their system for representing natural numbers, which was already in place. Theirs was a base-60 system, and the extension they made was similar to the one we currently use with our decimal system. Numbers to the left of what would be a "sexagesimal point" had place value and represented successive units, 60s, 3600s, and so on. Numbers smaller than 1 were placed to the right of the imaginary sexagesimal point and represented 60ths, 3600ths, and so on. Their system had two deficiencies that make it hard for contemporary archaeologists to interpret what they wrote (and probably made it hard for the Babylonians themselves). They had no zero to act as a place holder and they had no symbol to act as a sexagesimal point. All this had to be figured out from the context in which the number was used. Nevertheless, they had an approximation for √2 which was correct to four decimal places, and approximations for other irrational numbers as well. In fact, their system was so good that vestiges of it are to be seen today. We still break hours down sexagesimally, and the degree measure of angles as well.

The Egyptians, who lived in a later period, also found a way to represent fractional values. Theirs was not a place-value system, so the Babylonian method did not suggest itself. Instead they created unit fractions. They did not do it with a ratio, such as 1/4, however. Their symbolism was analogous to writing the unit fraction as 4^-1 or 7^-1. For that reason, what we would write as 2/5 had to be written as a sum of unit fractions, typically 3^-1 + 15^-1. Clearly their system was much more awkward that of the Babylonians.

The study of rational numbers really flowered under the Greeks. Pythagoras, Eudoxus, Euclid, and many others worked extensively with ratios. Their work was limited, however, by the fact that it was almost entirely geometric. Numbers were represented by line segments; ratios by pairs of segments. The Greek astronomer Ptolemy, who lived in the second century, found it better to turn to the sexagesimal system of the Babylonians (but not their clumsy cuneiform characters) in making his extensive astronomical calculations.