The decimal expansion of some real numbers are given below. for each number, state whether it is rational or irrational of the number is rational,then assuming it to be of the form p/q, what can be said about the prime factors of its denominator? give reasons to justify your answer
(a) 34.98
(b) 23.353355333555…
(c) 9.8734
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