MATH 110: Course Materials - Real Numbers

COLLEGE ALGEBRA MATH 110		Southwestern Adventist University
Distance Education	Lawrence E. Turner, Jr., Ph.D.

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Decimal Representation of Rational Numbers

All Rational Numbers may be written as a fraction reduced to lowest terms; indeed, this is inherent in the defining property of a Rational Number:

A Rational Number is a number that may be written as the ratio of two integers.

Actually, a more precise definition is:

A Rational Number is a number that may be written as an Integer divided by a Natural Number.

This second definition does away with any possibility of division by zero as well as having to deal with negative denominators; however, the first one is the one more commonly stated.

Decimal Representation

The decimal representation of any Rational Number is either a:

terminating decimal, or
repeating decimal

To see that this must be true, consider the process of long division whereby we can generate the decimal expansion of a fraction by explicitly dividing the denomintor (divisor) into the numerator (dividend).

At each step of the process, we obtain a remainder. This remainder is in the range:

0 ≤ remainder < divisor

If the remainder is zero, then the process terminates; that is, we get a terminating decimal.

Since there are at most divisor−1 other possible remainders (1, 2, 3, ..., divisor−1), the decimal expansion must repeat with a period not larger than this value.

As examples:

5/8 becomes:
division problem

5/7 becomes:
division problem

The conversion from any terminating decimal expansion back to the fraction form is not hard: write the number as an integer (with the decimal point just to the right of the last non-zero digit) and divide by the appropriate power of 10.

As an example:

r = 1.23456 = 123456/100000

Of course, the fraction can be reduced to lowest terms as: r = 3858/3125.

For repeating decimals, the trick is to first multiply the number by a power of 10 to "shift" the decimal expansion by one period, then subtract the original number. This should clear the repeating decimal fraction.

As an example:

r = 2.345634563456...

The period of the repeating decimal is 4 digits (consisting of the group 3456) and by multiplying by 10⁴ = 10000 we should be able to "shift" the decimal expansion by 4 positions.

Therefore, 10000r = 23456.34563456...

Subtracting r we get: 9999r = 23456.34563456... − 2.34563456... = 23454.

Dividing both sides by 9999, we get the resultant fractional representation for the original value: r = 23454/9999 = 2606/1111.