NCERT Class 10 Maths Solutions Chapter - 1 Real Numbers, Ex-1.4

Ex - 1.4

Question 1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:



Solution


3125 = 5^₅
The denominator is of the form 5^m.

8 = 2^₃
The denominator is of the form 2^m. 

455 = 5 x 7 x 13
Since the denominator is not in the form 2^_m x 5^_n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating. 

1600 = 2^₆  5^₂
The denominator is of the form 2^_m x 5^_n.

343 = 7^₃
Since the denominator is not in the form 2^_m x 5^_n, and it has 7 as its factor, 

The denominator is of the form 2^_m x 5^_n.

Hence, the decimal expansion of   is terminating.

(vii)

Since the denominator is not of the form 2^m  5ⁿ, and it also has 7 as its

The denominator is of the form 5^_n. 
10 = 2 x 5
The denominator is of the form 2^_m x 5^_n.

30 = 2 x 3 x 5
Since the denominator is not of the form 2^m  5ⁿ, and it also has 3 as its factors,

Concept Insight: The  concept used in this problem is that

The decimal expansion of rational number  where p and q are coprime numbers,

terminates if and only if the prime factorisation of q is of the form 2ⁿ5^m, where n and m are non negative integers. Do not forget that 0 is also a non negative integer so n or m can take value 0.
Generally mistake is committed in identifying terminating decimals  when either of the two prime numbers  2 or 5 is appearing in the prime factorisation.

Question 2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Solution

Concept Insight: This is  based on performing the long division and expressing the rational number in the decimal form learnt in lower classes.

Question 3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , what can you say about the prime factor of q?

Solution

(i)    43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form   and q is of the form 2^m x 5ⁿ, 

i.e., the prime factors of q will be either 2 or 5 or both. 

(ii)    0.120120012000120000...
The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form and q is not of the form 2^_m x 5^_n  i.e., the prime factors of q will also have a 

factor other than 2 or 5.


Concept Insight: The  concept used in this problem is that, 

If the decimal expansion of rational number  , [where p and q are coprime numbers] terminates, then  prime factorization of q is of the form 2ⁿ5^m, where n and m are non negative integers.