Express 0 . 23 ¯ as a rational number in simplest form.

Assuming x to be 0 . 23 ¯ x = 0 . 23 ¯ . . . ( 1 ) Multiplying both sides by 100 , we get 100 x = 23 . 23 ¯ . . . ( 2 ) Subtracting 1 from 2 , we get 100 x − x = 23 . 23 ¯ − 0 . 23 ¯ 99 x = 23 x = 23 99 Thus, simplest form of 0 . 23 ¯ as a rational number is 23 99 .

Show that 2 3 is irrational.

Assuming 2 3 to be a rational number. ∴ 2 3 = p q where p and q are some integers and q ≠ 0 , also HCF ( p , q ) = 1 . . . ( 1 ) ⇒ 2 q = 3 p ⇒ ( 2 q ) 2 = ( 3 p ) 2 ⇒ 2 q 2 = 9 p 2 p 2 is divisible by 2 p is divisible by 2 . . . ( 2 ) Let p = 2 m , where m is some integer. ⇒∴ 2 q = 3 p ⇒ 2 q = 3 ( 2 m ) ⇒ ( 2 q ) 2 = 3 ( 2 m ) 2 ⇒ 2 q 2 = 4 9 p 2 ⇒ q 2 = 2 9 p 2 ⇒ q 2 is divisible by 2 q is divisible by 2 . . . ( 3 ) From 2 and 3 , 2 is a common factor of both p and q , which contradicts 1 . Hence, our assumption is wrong. Thus, 2 3 is irrational.

Write a rational number between 3 and 2 .

Since 3 = 1 . 732 … . So, we may take 1 . 8 as the required rational number between 3 and 2 . Thus, the required rational number is 1 . 8 .

E M B I B E

Secondary School Mathematics for Class 10>Number Systems>Chapter 1 - Real Numbers>Exercise 1E

R S Aggarwal and Veena Aggarwal Solutions for Exercise 5: Exercise 1E

Author:R S Aggarwal & Veena Aggarwal

R S Aggarwal Mathematics Solutions for Exercise - R S Aggarwal and Veena Aggarwal Solutions for Exercise 5: Exercise 1E

Attempt the free practice questions from Exercise 5: Exercise 1E with hints and solutions to strengthen your understanding. Secondary School Mathematics for Class 10 solutions are prepared by Experienced Embibe Experts.

Questions from R S Aggarwal and Veena Aggarwal Solutions for Exercise 5: Exercise 1E with Hints & Solutions

EASY

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 1 - Real Numbers>Exercise 1E>Q 5

If $a$ and $b$ are two prime numbers then find LCM $(a, b)$

EASY

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 1 - Real Numbers>Exercise 1E>Q 14

If two irrational numbers are $4 + \sqrt{5}$ and $m - \sqrt{n}$ , where $m, n \in N$ , and their sum is $8$ i.e., a rational number, then find the value of $m + n$ .