PSEB Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4 Guide
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:
(i)Let x = 13/3125 ………..(1)
Compare (1) with x = p/q
Here p = 13 and q = 3125
Prime factors of q = 3125 = 5 × 5 × 5 × 5 × 5 = 55 × 20
which are of the form 2n × 5m here n = 0, m = 5
which are non negative integers.
∴ x = 13/3125 have a terminating decimal expansion.
Question 2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Solution:
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 f, what can you say about the prime factors of q?
(i) 43.123456789
(ii) O.120120012000120000……
(iii) 4.3.123456789
Solution:
(i) Let x= 43.123456789 ……….. (1)
It is clear from the number that x is rational number.
Now remove the decimal from the number
Where p = 43123456789 and q = 109
Now, Prime factors of q = 100 = (2 × 5)9
⇒ Prime factors of q are 29 × 59
(ii) Let x = 0.120120012000120000
It is clear from the number that it is an irrational number.
(iii) Let x = 43.123456789 …. (1)
It is clear that the given number is a rational number because it is non-terminating and repeating decimal.
To show that (i) is of the form p/q
Multiply (1) with 109 on both sides,
109 x = 43123456789.123456789 …………….(2)
Subtract (1) from (2), we get:
