376 Chapter 4 Decimals
WHYOnce we detect a repeating pattern of remainders, the division process can
stop.
Solution means.We can use three dots to show that a repeating pattern of 5 and 4 appears in the
quotient:and therefore,Or, we can use an overbar to indicate the repeating part (in this case, 54), and write
the decimal equivalent in more compact form:and therefore,6
11
0.54
6
11
0.54
6
11
0.545454...
6
11
0.545454...
Write a decimal point and five additional zeros to the right of 6.It is apparent that 6 and 5 will continue to reappear as remainders.
Therefore, 5 and 4 will continue to reappear in the quotient. Since the
repeating pattern is now clear, we can stop the division process.0.54545
(^11) 6.00000
55
50
44
60
55
50
44
60
55
5
116 6 ^11
The repeating part of the decimal equivalent of some fractions is quite long. Here are
some examples:A block of three digits repeats.A block of four digits repeats.A block of six digits repeats.Every fraction can be written as either a terminating decimal or a repeating
decimal. For this reason, the set of fractions (rational numbers) form a subset of the
set of decimals called the set of real numbers.The set of real numbers corresponds to
all points on a number line.
Not all decimals are terminating or repeating decimals. For example,does not terminate, and it has no repeating block of digits. This decimal cannot be
written as a fraction with an integer numerator and a nonzero integer denominator.
Thus, it is not a rational number. It is an example from the set of irrational numbers.0.2020020002...
6
7
0.857142
13
101
0.1287
9
37
0.243
3 Round repeating decimals.
When a fraction is written in decimal form, the result is either a terminating or a
repeating decimal. Repeating decimals are often rounded to a specified place value.