Question

0.12345,Given real number is expressed in decimal form. Find whether they are rational or not. If rational, express them in the form p/q. Comment on factors of q.

Answer 1

Hi ,

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Let x be a rational number whose

decimal expansion terminates.

Then we can express x in the form

of p/q , where p and q are coprime,

and the prime factorisation of q is

of the form 2^n5^m , where n and m

are non - negative integers.

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x = 0.12345

x = 12345/100000

x = ( 5 × 2469 )/( 5 × 20000 )

x = 2469/20000 = p/q

q = 20000 = 2^5 × 5⁴

Therefore ,

q is of the form 2^n5^m ,

where n = 5 , m = 4

x = 0.12345 = 2469/20000

= 2469/(2^5×5⁴)

Is a terminating decimal.

I hope this helps you.

: ,)

Answer 2

REAL NUMBERS :

a number which is either rational or irrational is called a real number.

RATIONAL NUMBERS :

A number that can be expressed as p/q , where p,q are Integers and q≠0 is called a rational number.

IRRATIONAL NUMBERS :

A number that cannot be expressed as p/q , where p,q are Integers and q≠0 is called a irrational number.

TERMINATING DECIMAL EXPANSION :

The number which terminates after a finite number of steps in the process of division is said to be terminating decimal expansion.

Given : 0.12345

ANSWER :

0.12345 is terminating ,so it is a RATIONAL NUMBER.

0.12345 = 12345/100000

= 2469 × 5/ 2000×5 = 2469/20000 = 2469/2^5 × 5⁴

[q is of the form 2^n5^m]

0.12345 =2469/20000

Hence, 0.12345 is a RATIONAL NUMBER.

HOPE THIS ANSWER WILL HELP YOU...