Decimal Representation Of Rational Numbers

Real Numbers of Class 10

Theorem: Let x = p/q be a rational number such that q ≠ 0 and prime factorization of q is of the form 2ⁿ x 5^m where m, n are non-negative integers then x has a decimal representation which terminates.

e.g. 0.275 = 275/10³ = 5² x 11/2³ x 5³= 11/2³ x 5 = 11/40

Theorem: Let x = p/q be a rational number such that q ≠ 0 and prime factorization of q is not of the form 2^m x 5ⁿ, where m, n are non-negative integers, then x has a decimal expansion which is non-terminating repeating.

e.g. 5/3 = 1.66666....

Rational number	Form of prime factorisation of the denominator	Decimal expansion of rational number
x = p/q, where p and q are coprime and q ≠ 0	q =2^m x 5ⁿ where n and m are non-negative integers	terminating
x = p/q, where p and q are coprime and q ≠ 0	q ≠ 2^m x 5ⁿ where n and m are non-negative integers	non-terminating