Question

Prove that the following are irrational:

(i) $fraction numerator 1 over denominator square root of 2 end fraction$

Solution

(i) $fraction numerator 1 over denominator square root of 2 end fraction$
Let us assume, to the contrary, that $fraction numerator 1 over denominator square root of 2 end fraction$ is rational.
So, we can find coprime integers a and b (≠ 0) such that
$rightwards double arrow space space space space space space fraction numerator 1 over denominator square root of 2 end fraction equals straight a over straight b rightwards double arrow square root of 2 equals straight b over straight a$
Since, a and b are integers, $straight a over straight b$ si rational, and so $square root of 2$ is irrational
So, we conclude that $fraction numerator 1 over denominator square root of 2 end fraction$ is irrational.

Some More Questions From Real Numbers Chapter

For Daily Free Study Material Join wiredfaculty

Real Numbers

Prove that the following are irrational:

(i) $fraction numerator 1 over denominator square root of 2 end fraction$

Some More Questions From Real Numbers Chapter

Express each number as a product of its prime factors: (i) 140

Express each number as a product of its prime factors: (ii) 156

Express each number as a product of its prime factors: (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Express each number as a product of its prime factors: (iv) 5005

Express each number as a product of its prime factors: (v) 7429

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54

Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21

Find the LCM and HCF of the following integers by applying the prime factorisation method. (ii) 17, 23 and 29

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location

For Daily Free Study Material Join wiredfaculty

Real Numbers

Prove that the following are irrational:(i)

Some More Questions From Real Numbers Chapter

Express each number as a product of its prime factors: (i) 140

Express each number as a product of its prime factors: (ii) 156

Express each number as a product of its prime factors: (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Express each number as a product of its prime factors: (iv) 5005

Express each number as a product of its prime factors: (v) 7429

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54

Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21

Find the LCM and HCF of the following integers by applying the prime factorisation method. (ii) 17, 23 and 29

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Prove that the following are irrational:

(i) $fraction numerator 1 over denominator square root of 2 end fraction$