Question

Classify the following decimal fractions as terminating and non-terminating recurring decimals.

(1) 0.777...
(2) 0.777
(3) 4.7182
(4) $4.\overline{7182}$
(5) 9.165165
(6) $9.\overline{165}$
(7) 0.52888
(8) $0.52\overline{8}$
(9) $72.\overline{136}$

Answer 1

(1) 0.777….
In the decimal fraction 0.777…. the digit ‘7’ is repeated after the decimal point, 7 is the recurring digit so 0.777…. is in non-terminating and recurring decimal form.

(2) 0.777
In the decimal fraction 0.777, the digit ‘7’ is repeated up to three decimal places but after that no recurrence is observed. So, 0.777 is in terminating, decimal form.

(3) 4.7182
In the decimal fraction 4.7182, there are no digits after 2, i.e. no recurrence is observed. So, 4.7182 is in terminating decimal form.

(4) $4.\overline{7182}$
In the decimal fraction $4.\overline{7182}$, the term 7182 is repeated after the decimal point, i.e. 7182 is the recurring term. So, $4.\overline{7182}$ is in non-terminating and repeating decimal form.

(5) 9.165165
In the decimal fraction 9.165165, there are no digits after the last 5,i.e. no recurrence is observed. So, 9.165165 is in terminating decimal form.

(6) $9.\overline{165}$
In the decimal fraction $9.\overline{165}$, the term 165 is repeated after the decimal point,i.e. 165 is the recurring term. So, $9.\overline{165}$ is in non-terminating and repeating decimal form.

(7) 0.52888
In the decimal fraction 0.52888, there are no digits after the last 8,i.e. no recurrence is observed. So, 0.52888 is in terminating decimal form.

(8) $0.52\overline{8}$
In the decimal fraction $0.52\overline{8}$, the digit 8 is repeated after the decimal point,i.e. 8 is the recurring term. So, $0.52\overline{8}$ is in non-terminating and repeating decimal form.

(9) $72.\overline{136}$
In the decimal fraction $72.\overline{136}$, the term 136 is repeated after the decimal point,i.e. 136 is the recurring term. So, $72.\overline{136}$ is in non-terminating and repeating decimal form.

So all the decimal fractions can be classified as:

Terminating	Non-terminating
0.777	0.777...
4.7182	$4.\overline{7182}$
9.165165	$9.\overline{165}$
0.52888	$0.52\overline{8}$
	$72.\overline{136}$