- Decimal Representation of Rational Numbers in Terms of Terminating Or Non-terminating Recurring Decimals
To learn this conept we need to understand the terms Terminating and Non-terminating Recurring Decimals. Rational Numbers are of form `p/q` where q is not equal to 0 then the expansion is either terminating decimal or 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
Expalnation: This is exactly the oppsite of the previous theorem. If x=p/q=107/100 is a rational number, then here x=1.07 which 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