# NCERT Mathematics Class 9 Exemplar Ch 1 Number Systems Part 4

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Exercise 1.2

1. Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

Answer: Yes. Let x = 21, y = be a rational number. Now x + y = 21 + = 21 + 1.4142 ... = 22.4142 ... Which is non-terminating and non-recurring. Hence x + y is irrational.

2. Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.

Answer: No. 0× = 0 which is not irrational.

3. State whether the following statements are true or false? Justify your answer.

(i) is a rational number.

(ii) There are infinitely many integers between any two integers.

(iii) Number of rational numbers between 15 and 18 is finite.

(iv) There are numbers which cannot be written in the form p q , q ≠ 0 , p, q both are integers.

(v) The square of an irrational number is always rational.

(vi) is not a rational number as and are not integers.

(vii) is written in the form , and so it is a rational number.

Answer: (i) False. Although is of the form but here p, i.e., is not an integer.

(ii) False. Between 2 and 3, there is no integer.

(iii) False, because between any two rational numbers we can find infinitely many rational numbers.

(iv) True. is of the form but p and q here are not integers.

(v) False, as = which is not a rational number. ANSWERS

(vi) False, because = = 2 which is a rational number.

(vii) False, because = = which is p, i.e., 5 is not an integer.

4. Classify the following numbers as rational or irrational with justification:

(i)

(ii) 3

(iii)

(iv)

(v)

(vi)

(vii) 0.5918

(viii)

(ix)

(x)

(ii) 3 = 9 , which is the product of a rational and an irrational number and so an irrational number.

(iii) = ,which is the quotient of a rational and an irrational number and so an irrational number

(iv) , which is a rational number.

(v) Irrational, , which is the quotient of a rational and an irrational.

(vi) = , which is a rational number.

(vii) Rational, as decimal expansion is terminating.

(viii) , which is a rational number.

(ix) Rational, as decimal expansion is non-terminating recurring.

(x) Irrational, as decimal expansion is non-terminating non-recurring.