State whether the given is irrational or not: π 2

A rational number is the one which can be represented in the form of p q where p and q are integers and q ≠ 0 Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p q where p and q are integers, q ≠ 0 . Therefore, π 2 is not an irrational number because its denominator is not equal to 0 .

State whether the given is irrational or not: 3 4 2 3

A rational number is the one which can be represented in the form of p q where p and q are integers and q ≠ 0 Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p q where p and q are integers, q ≠ 0 . 3 4 2 3 = 27 64 × 2 × 2 × 2 = 27 64 × 2 2 = 27 32 2 Therefore, 3 4 2 3 is an irrational number.

State whether the given is irrational or not: 18 × 8

A rational number is the one which can be represented in the form of p q where p and q are integers and q ≠ 0 Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p q where p and q are integers, q ≠ 0 . 18 × 8 = 3 × 3 × 2 × 2 × 2 × 2 = 3 × 2 × 2 × 2 = 12 Therefore, 18 × 8 is not an irrational number.

State whether the given is irrational or not: 7 - 1

A rational number is the one which can be represented in the form of p q where p and q are integers and q ≠ 0 Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p q where p and q are integers, q ≠ 0 . Therefore, 7 - 1 is an irrational number because it cannot be express as p q form.

State whether the given is irrational or not: 3 + 2 3 - 2

A rational number is the one which can be represented in the form of p q where p and q are integers and q ≠ 0 Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p q where p and q are integers, q ≠ 0 . Now, 3 + 2 3 - 2 = 3 2 - 2 2 = 9 - 2 = 7 Therefore, 3 + 2 3 - 2 is not an irrational .

State whether the given is irrational or not: 5 + 3 4 - 2

A rational number is the one which can be represented in the form of p q where p and q are integers and q ≠ 0 Irrational numbers are real numbers that cannot be represented as a simple fraction. These cannot be expressed in the form of ratio, such as p q where p and q are integers, q ≠ 0 . Now, 5 + 3 4 - 2 = 5 4 - 2 + 3 4 - 2 = 20 - 5 2 + 4 3 - 6 Therefore, 5 + 3 4 - 2 is an irrational .

Represent 10 and 20 on the number line.

Construction for representation of 10 on number line. Mark point A at 3 units from a fixed point O . Draw a line measuring 1 unit from point A such that A B ⊥ O A at point A join points O and A Then, O B = 10 Now, treating A as a part of number line , draw an arc with centre O and radius O B intersecting the number line at point E . Proof: Using Pythagoras theorem, In right triangle O A B . O B = O A 2 + A B 2 = 3 2 + 1 2 = 9 + 1 = 10 Now, O B = O E = 10 The point E represents 10 on the number line. We are given to represent the number 20 on the number line. We can write 20 as 20 = 16 + 4 . So, 20 = ( 4 ) 2 + 2 2 . To locate 11 on the number line, we first need to construct a length of 10 . Steps: a) Mark O at 0 and A at 4 on the number line, and then draw AB of 2 units perpendicular to O A . Then, by Pythagoras Theorem, we get, O B 2 = O A 2 + A B 2 = 4 2 + 2 2 = 16 + 4 = 20 ⇒ O B = 20 . b) Using a compass with centre at O and radius as O B , draw an arc intersecting the number line at point P . Thus, P corresponds to the number 20 . Therefore, O P is the required length 20 .

Prove that 3 - 5 is an irrational number.

Let us assume that 3 - 5 is a rational number. Now, let 3 - 5 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 5 = 3 - a b . Since a , b are integers, then 3 - a b is also a rational number. ⇒ 5 is a rational number. But, we know that, 5 is an irrational number. This shows our initial assumption was incorrect. Therefore, 3 - 5 is an irrational number.

E M B I B E

ICSE MATHEMATICS CLASS 9>Pure Arithmetic>Chapter 1 - Rational and Irrational Numbers>EXERCISE 1B

B Nirmala Shastry Solutions for Exercise 2: EXERCISE 1B

Author:B Nirmala Shastry

B Nirmala Shastry Mathematics Solutions for Exercise - B Nirmala Shastry Solutions for Exercise 2: EXERCISE 1B

Attempt the practice questions from Exercise 2: EXERCISE 1B with hints and solutions to strengthen your understanding. ICSE MATHEMATICS CLASS 9 solutions are prepared by Experienced Embibe Experts.