Page 64 - ICSE Math 7

Decimal representation of rational numbers

Every rational number can be converted into an equivalent decimal form either by using long
division method or writing an equivalent rational number whose denominator is some power of
10 only.
Decimal numbers are of two types—terminating and non-terminating decimal numbers.

Terminating decimals are those which have a finite number of digits in the decimal part
(i.e., right of the decimal point).
7
Let’s convert into decimals.
8

8 7.0 0.875
–0 Try This
70 7 = 7 = 7 × 5 3 = 875 = 0.875
–64 or 8 2 3 2 × 5 3 10 3 Convert the following rational
3
60 numbers into decimals.
5
4
9
–56 (a) 11 (b) 7 (c) 9
40
–40
0

Point to remember

A rational number can be represented by a terminating decimal iff the prime factorization of its
denominator (in the simplest form) contains no primes other than 2 or 5.
Non-terminating decimals are those whose decimal representation has infinite
number of digits in the decimal part. In fact, non-terminating decimals are also 11 2.0 0.18
of two types, i.e., non-terminating recurring (or repeating) and non-terminating –0
non-recurring decimals. 20
2 –11
For example, can be written as 0.18.
11 90
At this point if the division is continued the quotient will emerge as 0.181818 ... –88
because the remainder 2 is equal to the original dividend and the division process 2
will keep on repeating.

2 · ·
∴ = 0.181818 ... = 0.18 or 0.18 (the bar or dot above 18 means 18 is repeating)
11
On the other hand decimals like 1.732050807 ... neither terminate nor have any repeating
part. In fact these decimals are not equal to any rational number. So they are called irrational
numbers.
The decimal 1.732050807 ... represents an irrational number √ 3.

p
A rational number where q ≠ 0 is represented either by a terminating decimal or a non-terminating
q
1
repeating decimal. For example, 3 = 1.625, = 0.142857.
10 7

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