Question

3. The following real numbers have decimal expansions as given below. In each case
decide whether they are rational or not. If they are rational, and of the form
you say about the prime factors of q?
(1) 43.123456789
(ii) 0.120120012000120000...
(1) 43.123456789​

Answer 1

Answer: 2^n × 5^m form

Step-by-step explanation: i ) Let x = p/q be a rational number , such

that the prime factorisation of q is of the

form 2ⁿ × 5^m , where n and m are

non - negative integers . Then x has

a decimal expansion which terminates.

ii ) The number which is non - terminating and

non - repeating is called an

irrational number.

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i ) x = 43.123456789

is a rational .

x = 43123456789/( 1000000000 )

= 43123456789/( 10^9 )

= 43123456789/( 2 × 5 )^9

= 43123456789/( 2^9 × 5^9 )

Here , q = 2^9 × 5^9 ( 2ⁿ × 5^m form )

43.123456789 is a terminating decimal.

ii ) 0.120120012000120000....

is non - terminating and non - repeating

decimal .

Therefore , it is an irrational number.

iii ) 43.123456789123456789....

is a non - terminating , repeating

decimal. So it is a rational number.

x = 43.123455789123456789....---( 1 )

10^9 x = 43123456789.123456789....--(2 )

subtracting ( 1 ) from ( 2 ) , we get

10^9 x = 43123456746

x = 43123456746/10^9

x = 43123456746/( 2 × 5 )^9

x = 43123456746/( 2^9 × 5^9 )

Therefore ,

q = 2^n × 5^m form

Given number is a rational .

I hope this helps you