Rational Numbers and Their Decimal Expansions

We must understand the terms Terminating and Non-terminating Recurring Decimals to learn this concept. Rational Numbers are of form `p/q` where q is not equal to 0, then the expansion is either a terminating decimal or a non-terminating recurring decimal. Terminating decimals are like 1.2, 1.3, 6.2, 6.3, etc. And Non-terminating recurring decimals are like 1.323232323232, 1.632632632632, etc.

1)Theorem: Let x be a rational number whose decimal expansion terminates. Then x can be

expressed in the form p, q where p and q are coprime, and the prime

factorisation of q is of the form `2^n5^m` where n, m are non-negative integers.

Example: Rational Number 1.2 can be written as `p/q=12/10`, i.e. `6/5,` here the factor for q is 5. Rational Number 1.07 can be written as `p/q=107/100`, here the factor for q is 2×5×2×5

2)Theorem: Let x=`p/q` be a rational number, such that prime factorisation of q is of form

`2^n5^m` where n, m are non-negative integers. Then x has a decimal expansion

which terminates.

Explanation: This is exactly the opposite of the previous theorem. If x=p/q=107/100 is a rational number, then here, x=1.07 is terminating.

3)Theorem: Let x=p/q be a rational number, such that the prime factorisation of q is not of

the form 2^n5^m where n, m are non-negative integers. Then, x has a decimal

expansion which is non-terminating recurring.

Explanation: Here, `x=p/q`, where p&q are coprime but in this case, q is not to the power of `2^n5^m`. Then x will always be Non-terminating Recurring. For example 3/14=0.214285714285