Question

Prove that 1 + √7 is an irrational number.

↬Need Required Answer ​

Answer 1

Answer:

Answer:

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

Answer 2

Let us assume that √7 is a rational

number.

So it t can be expressed in the form$\frac{p}{q}$

where p,q are co-prime integers and q*0

$\sqrt{7} = \frac{p}{q}$

Here and q are coprime numbers and q

#0

Solving

$\sqrt{7} = \frac{p}{q}$

On squaring both the side we get,

$⇒7 = { \frac{p}{q} }^{2}$

$⇒7 {q}^{2} = {p}^{2}$

$⇒ p \frac{2}{7} = {q}^{2}$

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

$⇒ p = 7m$

${p}^{2} = 49 {m}^{2}$

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

$7 {q}^{2} = 49 {m}^{2}$