Page 50 - ICSE Math 7

However, if the dividend is not completely divisible and the remainder is non-zero, no matter how long
the division is, then the decimal is known as non-terminating decimal. For example, 23 = 3.2857…
is a non-terminating decimal. 7

Recurring decimals

Sometimes in a non-terminating decimal, a digit or a group of digits repeat themselves. Such a decimal
2
is known as recurring decimal. For example, = 0.666… is a recurring decimal. Recurring decimals
3 2
can be represented by using a bar or dots over the repeating digits. So, = 0.666… can be represented
3
.
as 0.6 or 0.6. If more than two digits are repeated, then we put a dot over the first and last repeating
. .
digits only. For example, 7.12342342… can be written as 7.1234 or 7.1234.
Example 15: Find whether the following are terminating or non-terminating decimals.

(a) 17 (b) 11 (c) 23 (d) 0.5608
15 5 0.7 0.8

Solution: (a) 17 = 1.1333… (b) 11 = 2.2
15 5
So, 17 is a non-terminating So, 11 is a terminating decimal.
15 5
decimal.

(c) 23 = 230 = 32.8571… (d) 0.5608 = 5.608 = 0.701
0.7 7 0.8 8
So, 23 is a non-terminating So, 0.5608 is a terminating decimal.
0.7 0.8
decimal.

Conversion of recurring decimals into fractions

Let’s learn conversion of recurring decimals into fractions by the following examples.

Example 16: Convert the following decimals into fractions.
. .
(a) 0.85 (b) 1.3134

Solution: (a) Let x = 0.85 = 0.8555…(1)
\ 10x = 8.555…(2) (Multiply by 10 since only 1 digit is repeated)
Subtracting (1) from (2)
⇒ 10x – x = 8.555… – 0.855…

⇒ 9x = 7.7
7.7 77 77
⇒ x = = =
9 9 × 10 90
77
So, 0.85 =
90
. .
(b) Let x = 1.3134 = 1.3134134…(1)
\ 1,000x = 1313.4134…(2) (Multiply by 1,000 since 3 digits are repeated)

Subtracting (1) from (2)
⇒ 1,000x – x = 1313.4134… – 1.3134…

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