The decimal expansion of some real numbers are given below. for each number, state whether it is rational or irrational of the num

1 thought on “The decimal expansion of some real numbers are given below. for each number, state whether it is rational or irrational of the num”

1. Answer:

   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

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