# NCERT Solutions for Class 9 Math | Exercise 1.5

###### CBSE Chapter 1 | Number Systems | Rationalizing Irrational Numbers

Chapter 1 of CBSE NCERT Class 9 Math covers number system. Concepts covered in exercise 1.5 include rational numbers, rationalizing irrational numbers by multiplying with the conjugate of the denominator to simplify the expressions.

Solutions to Question 1 and Question 3 are given in this page. For the other questions, click on the solution links given adjacent to the questions.

### NCERT Solutions for Class 9 Math | Exercise 1.5

1. Question 1

Classify the following numbers as rational or irrational:
i) 2 - √5
ii) (3 + √23) - √23
iii) $$frac{2√7}{7√7}$ iv) $\frac{1}{√2}$ v) 2π #### Answer i) 2 - √5 is irrational as it contains the irrational number √5 in it. √5 cannot be expressed in the form $\frac{p}{q}$ where p and q are integers and q ≠ 0. ii)$3 + √23) - √23 = 3 + √23 - √23 = 3 is a natural number.
All natural numbers are rational numbers. 3 can be expressed in the form $$frac{p}{q}$. iii) $\frac{2√7}{7√7}$ = $\frac{2}{7}$ which is rational as it can be expressed in the form $\frac{p}{q}$. Property used for$iv) and (v): The product or quotient of a rational number and an irrational number is an irrational number.
iv) $$frac{1}{√2}$ contains the irrational number √2. So, $\frac{1}{√2}$ is irrational. It cannot be expressed in the form $\frac{p}{q}$ where p and q are integers and q ≠ 0. v) 2π is irrational because π is irrational. 2. Question 2 Simplify each of the following expressions:$i) (3 + √3) (2 + √2)
(ii) (3 + √3) (3 - √3)
(iii) (√5 + √2)2
(iv) (√5 - √2)(√5 + √2)

3. Question 3

Recall, π is defined as the ratio of the circumference (c) of a circle to its diameter (d). That is, π = $$frac{c}{d}$. This seems to contradict the fact that π is irrational. How can you resolve this contradiction? #### Answer There is no contradiction. The measure of c will be irrational if d is rational or the measure of d will be irrational if c is rational or both c and d will be irrational. Essentially, at least one of c or d will be irrational. Because at least one of c or d is not an integer, though it is of the form $\frac{c}{d}$, π is not a rational number. 4. Question 5 Rationalise the denominators of the following:$i) $$frac{1}{√7}$$ii) $$frac{1}{√7 - √6}$$iii) $$frac{1}{√5 + √2}$$iv) $\frac{1}{√7 - 2}$

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