Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Terminating and Repeating Decimals

Table 6.7 illustrates an important relationship between the fraction and decimal representations of equivalent numbers: Every fraction p/q is equivalent to a terminating or repeating decimal. An alternative notation for each number, called vinculum notation, is also given.

· In every terminating decimal there is a digit, to the right of which, every subsequent digit is zero. For example, the decimal representations of 1/1, 1/2, 1/4, 1/5, 1/8, and 1/10 all terminate in an infinite string of zeros.

· In every repeating decimal there is a digit, to the right of which, a non-zero digit or finite string of digits repeats itself an infinite number of times. For example, the decimal representations of 1/3 and 1/6 each repeat a single digit an infinite number of times. By contrast, the decimal representation of 1/7 repeats the string 142857 an infinite number of times.

Fraction	Decimal Notation	Vinculum Notation
1/1	1.0000000000000000 …
1/2	0.5000000000000000 …
1/3	0.3333333333333333 …
1/4	0.2500000000000000 …
1/5	0.2000000000000000 …
1/6	0.1666666666666666 …
1/7	0.1428571428571428 …
1/8	0.1250000000000000 …
1/9	0.1111111111111111 …
1/10	0.1000000000000000 …

Table 6.7: Repeating and Terminating Decimals

Given any rational number p/q, it is a simple matter to write the number as a decimal. One merely performs the division to a sufficient number of decimal places to reveal whether the result is a repeating or terminating decimal. The inverse process, converting a repeating or terminating decimal number to an equivalent rational number, is more complicated. Table 6.8 demonstrates three such conversions. Note the differences as well as the similarities from one demonstration to the next.

In general, two multipliers (both powers of ten) are used to generate a system of equations based on a given repeating decimal. This system of equations is then solved to obtain a fraction equivalent to the repeating decimal. Choosing those multipliers is the most challenging aspect of the entire procedure. The following procedure outlines the principles used in selecting the multipliers:

· Let x be a repeating decimal, written as , in which the variables a, b, c, d, and e represent digits in the decimal number.

· The notation is used to indicate that the digits c, d, and e repeat endlessly.

· One multiplier repositions the decimal point immediately to the right of the first repetition of cde. In this case, this involves a shift of 5 positions. Consequently, the multiplier is 10⁵, or 10,000.

· The other multiplier repositions the decimal point immediately to the left of the first repetition of cde. This involves a shift of 2 positions, so the multiplier is 10², or 100.

· Using these multipliers, the following equations are obtained: and . Notice that the decimal parts of both equations are identical.

· When the second equation is subtracted from the first, the decimal portions cancel out, leaving

As you review the conversions in Table 6.8, think about how this general strategy was adapted to deal with the particular conditions given in each conversion.

Let x = 0.777 … = · Multiplying both sides of this equation by 10 yields 10x = 7.777 · Subtracting x = 0.777 from 10x = 7.777 yields 9x = 7.000 · Dividing both sides of this equation by 9 yields x = 7/9 In summary, 10 x = 7.777 - x = 0.777 9x = 7.000 So x = 7/9

Let x = 0.7575 … = · Multiplying both sides of this equation by 100 yields 100x = 75.7575 · Subtracting x = 0.7575 from 100x = 75.7575 yields 99x = 75.000 · Dividing both sides of this equation by 99 yields x = 75/99 In summary, 100x = 75.7575 - x = 0.7575 99x = 75.000 So x = 75/99

Let x = 0.37575 … = · Multiplying both sides of this equation by 1000 yields 1000x = 375.7575 · Multiplying x = 0.37575 by 10 yields 10x = 3.7575 · Subtracting 10x = 3.7575 from 1000x = 375.7575 yields 990x = 372.000 · Dividing both sides of this equation by 990 yields x = 372/990 In summary, 1000x = 375.7575 … - 10x = 3.7575 … 990x = 372.000 … So x = 372/990

Table 6.8: Decimal to Fraction Conversions

Example 6.3

Find a fraction equivalent to 0.03232 …

Solution 6.3

x = 0.03232

10x = .3232

1000x = 32.3232

990x = 32.000

x = 32/990

Example 6.4

Find a fraction equivalent to 0.102323

Solution 6.4

x = 0.102323

100x = 10.2323

10000x = 1023.2323

9900x = 1013.000

x = 1013/9900