Decimal Representation Of Rational Numbers
Theorem:
Let be a rational number such that and prime factorization of q is of the form where m, n are non-negative integers then x has a decimal representation which terminates.
For example :
Theorem:
Let be a rational number such that and prime factorization of q is not of the form , where m, n are non-negative integers, then x has a decimal expansion which is non-terminating repeating.
For example :
Solved Examples Based on Decimal Representation Of Rational Numbers
Question:
Without actually calculating, state whether the following rational numbers have a terminating or non-terminating repeating decimal expansion.
(i) (ii) (iii)
Hint: If the denominator is of the form for some non negative integer m and n, then rational number has terminating decimal otherwise non terminating.
Solution:
(i)
Since which is not of the form .
\ It has non terminating decimal representation.
(ii)
Since q = which is of the form .
\ It has a terminating decimal representation.
(iii)
Since is not of the form . It has a non-terminating decimal representation.
Question:
What can you say about the prime factorization of the denominators of the following rationales:
(i) 36.12345 (ii)
Solution:
(i) Since 36.12345 has terminating decimal expansion. So, its denominator is of the form where m, n are non-negative integers.
(ii) Since has non terminating repeating decimal expansion. So, its denominator has factors other than 2 or 5.