WITHOUT ACTUAL DIVISION, CHECK WHICH OF THE FOLLOWING RATIONAL NUMBERS CAN BE EXPRESSED AS NON-TERMINATING REPEATING DECIMALS.

 

To understand the decimal expansion of rational numbers, let us start by taking a few examples of rational numbers.
The numbers etc. are rational numbers.
Let us consider the decimal expansions of a few rational numbers.
(a) 
(b) 
(c) 
(d) 
(e) 
We notice that in examples (a) and (e), the decimal expansion is terminating (there is a finite number of digits after the decimal point), whereas in examples (b)(c), and (d), the digits after the decimal point are repetitive (same digit or set of digits occur again and again) and non-terminating (there is an infinite number of digits after the decimal point).
Hence, we can say that the decimal expansion of a rational number can be of two types:
(i) Terminating
(ii) Non-terminating and repetitive
Let us now consider a few examples.
Example 1:
Write the decimal expansion of and find if it is terminating or non-terminating and repetitive.
Solution:
We perform the long division of 1237 by 25.
Hence, the decimal expansion of is 49.48, which is terminating.
Example 2:
Write the decimal expansion of and find if it is terminating or non-terminating and repetitive.
Solution:
We perform the long division of 2358 by 27.
We can see that the remainder 9 is obtained again and again. Hence, the decimal expansion of is non-terminating and repetitive.
In the previously discussed examples, we carried out the long division method in order to check the decimal expansion of rational numbers. Now, we will do this without carrying out the actual long division method.
Let us start by taking a few rational numbers in the decimal form.
(a)
(b)
0.275
On prime factorizing the numerator and the denominator, we obtain
Can you see a pattern in the two examples?
We notice that the given examples are rational numbers with terminating decimal expansions. When they are written in theform, where p and q are co-prime
(the HCF of p and q is 1), the denominator, when written in the form of prime factors, has 2 or 5 or both.
The above observation brings us to the given theorem.
If x is a rational number with terminating decimal expansion, then it can be expressed in the form, where p and q are co-prime (the HCF of p and q is 1) and the prime factorisation of q is of the form 2n5m, where n and m are non-negative integers.
Contrary to this, if the prime factorisation of q is not of the form 2n5m, where n and m are non-negative integers, then the decimal expansion is a
non-terminating one.
Let us see a few examples that will help verify this theorem.
(a) 
(b) 
(c) 
(d) 
Note that in examples (b) and (d), each of the denominators is composed only of the prime factors 2 and 5, because of which, the decimal expansion is terminating. However, in examples (a)and (c), each of the denominators has at least one prime factor other than 2 and 5 in their prime factorisation, because of which, the decimal expansion is non-terminating and repetitive.
To summarize the above results, we can say that:
Let be any rational number.
  1. If the prime factorization of q is of the form 2m5n, where m and n are non-negative integers, then x has a terminating decimal expansion.
  1. If the prime factorisation of q is not of the form 2m5n, where m and n are non-negative integers, then x has a non-terminating and repetitive decimal expansion.
Let us solve a few more examples to understand this concept better.
Example 3:
Without carrying out the actual division, find if the following rational numbers have a terminating or a non-terminating decimal expansion.
(a) 
(b) 
Solution:
(a)
As the denominator can be written in the form 2n5m, where n = 6and = 2 are non-negative integers, the given rational number has a terminating decimal expansion.
(b)
As denominator cannot be written in the form 2n5m, where n and m are non-negative integers, the given rational number has a non-terminating decimal expansion.
Example 4:
Without carrying out the actual division, find if the expression  has a terminating or a non-terminating decimal expansion.
Solution:
 = 
As the denominator can be written in the form 2n5m, where n = 7and = 0 are non-negative integers, the given rational number has a terminating decimal expansion.
Hence, 5.5859375 is the decimal expansion of the given rational number.

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