Question

Show that 9 - 4√7 is an irrational number. Given that √7 is an irrational number.​

Answer 1

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9th

Maths

Number Systems

Irrational Numbers

Prove that √(7) is an irrat...

MATHS

Prove that

7

is an irrational number.

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ANSWER

Let us assume that

7

is rational. Then, there exist co-prime positive integers a and b such that

7

=

b

a

⟹a=b

7

Squaring on both sides, we get

a

2

=7b

2

Therefore, a

2

is divisible by 7 and hence, a is also divisible by7

so, we can write a=7p, for some integer p.

Substituting for a, we get 49p

2

=7b

2

⟹b

2

=7p

2

.

This means, b

2

is also divisible by 7 and so, b is also divisible by 7.

Therefore, a and b have at least one common factor, i.e., 7.

But, this contradicts the fact that a and b are co-prime.

Thus, our supposition is wrong.

Hence,

7 is irrational.