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Rational Numbers |
The simplest numbers are the natural numbers, 1, 2, 3, .... The natural numbers are also called the whole numbers, positive integers, or positive rational integers. The natural numbers are closed with respect to addition and multiplication—that is, the sum and product of two natural numbers are always natural numbers. Because the quotient (the result of dividing) of two natural numbers, however, is not always a natural number, it is convenient to introduce the positive fractions to represent the quotient of any two natural numbers. The natural number n is identified with the fraction n/1. Furthermore, because the difference of two positive fractions is not always a positive fraction, it is expeditious to introduce the negative fractions (including the negative integers) and the number zero (0). The positive and negative integers and fractions, and the number 0, comprise the rational number system.
The sum, difference, product, or quotient of two rational numbers is always a rational number. Division of any number by zero, however, is not allowed (see Zero). It can be shown that every rational number can be represented as a repeating or periodic decimal—that is, as a number in the decimal notation, which after a certain point consists of the infinite repetition of a finite block of digits. Conversely, every repeating decimal represents a rational number. Thus, 617/50 = 12.34000 ..., and 2317/990 = 2.34040 .... The first expression is usually written as 12.34, omitting the infinite repetition of the block consisting of the single digit 0. The second expression is frequently written as
to indicate that the block of two digits, 4 and 0, is repeated infinitely. The first type of expression, where the infinitely repeating block consists of the digit 0, is called a finite or terminating decimal, and the second type of expression is called a repeating decimal.
