## Question Prove that √7 is an irrational number.​

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10 months 2021-07-07T15:51:52+00:00 2 Answers 0 views 0

Given √7

To prove: √7 is an irrational number.

Proof:

Let us assume that √7 is a rational number.

So it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

√7 = p/q

Here p and q are coprime numbers and q ≠ 0

Solving

√7 = p/q

On squaring both the side we get,

=> 7 = (p/q)2

=> 7q2 = p2……………………………..(1)

p2/7 = q2

So 7 divides p and p and p and q are multiple of 7.

⇒ p = 7m

⇒ p² = 49m² ………………………………..(2)

From equations (1) and (2), we get,

7q² = 49m²

⇒ q² = 7m²

⇒ q² is a multiple of 7

⇒ q is a multiple of 7

Hence, p,q have a common factor 7. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√7 is an irrational number.

2. To prove : is irrational.

Proof :

Assume that is rational.     7 and b² is a factor of a², which means 7 is also a factor of a. Substitute Eq. 2 in Eq. 1   7 and c² are factors of b². Which means 7 is a factor of b too.

7 is a common factor of a and b. But they were co prime numbers. This is a contradiction. The contradiction was arisen due to wrong assumption.

Hence proved, is irrational.