Question

the following real number have decimal expansion as given below. in each case decide whether they are rational or not. if they are rational, and of the form p by q, what you say about the prime factors of q? 43.123456789​

Answer 1

Answer:

Hi ,

********************************************

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.

*******************************************

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.

: )