Question

Prove that $space 3 plus 2 square root of 5$ is irrational.

Solution

Let us assume, to the contrary that

space 3 plus 2 square root of 5

is rational.

That is, we can find coprime a and b (b ≠ 0)
$such space that space space space space space space space space space space space 3 plus 2 square root of 5 equals straight a over straight b$
$Therefore comma space space straight a over straight b space equals space 3 space plus 2 square root of 5 space$
$rightwards double arrow space space space space space space fraction numerator straight a minus 3 straight b over denominator straight b end fraction equals 2 square root of 5 rightwards double arrow space space space space space space space space fraction numerator straight a minus 3 straight b over denominator 2 straight b end fraction equals square root of 5 rightwards double arrow fraction numerator straight a over denominator 2 straight b end fraction minus 3 over 2$
Since a and b are integers, we get $fraction numerator a over denominator 2 b end fraction minus 3 over 2$ is rational, and so $square root of 5$ is rational.
But this contradicts the fact that $square root of 5$ is irrational.

This contradiction has arisen because of our incorrect assumption that $space space space 3 plus 2 square root of 5$ is rational

So, we conclude that $space space 3 plus 2 square root of 5$ is irrational.

Some More Questions From Real Numbers Chapter

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

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Real Numbers

Prove that $space 3 plus 2 square root of 5$ is irrational.

Some More Questions From Real Numbers Chapter

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

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

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For Daily Free Study Material Join wiredfaculty

Real Numbers

Prove that is irrational.

Some More Questions From Real Numbers Chapter

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

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

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Prove that $space 3 plus 2 square root of 5$ is irrational.