Question

Prove that
$\sqrt{5}$
is irrational ​

Answer 1

Let us assume that √5 is a rational number.

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

⇒ √5 = p/q

On squaring both the sides we get,

⇒5 = p²/q²

⇒5q² = p² —————–(i)

p²/5 = q²

So 5 divides p

p is a multiple of 5

⇒ p = 5m

⇒ p² = 25m² ————-(ii)

From equations (i) and (ii), we get,

5q² = 25m²

⇒ q² = 5m²

⇒ q² is a multiple of 5

⇒ q is a multiple of 5

Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√5 is an irrational number.

Answer 2

Answer:

no it is not

Step-by-step explanation:

=

q

p

Squaring on both the sides we get,

5 = \frac{ {p}^{2} }{ {q}^{2} }5=

q

2

p

2

5 {q}^{2} = {p}^{2}5q

2

=p

2

p = 5kp=5k

p is divisible by 5

k is positive integer

Again squaring on both the sides

{p}^{2} = 25 {k}^{2}p

2

=25k

2

{q}^{2} = \frac{ {25k}^{2} }{5}q

2

=

5

25k

2

{q}^{2} = 5 {k}^{2}q

2

=5k

2

q is divisible by 5

We can say that p and q have a common factor which is 5. p and q are co- prime.

\sqrt{5} \: is \: an \: irrational \: number

5

isanirrationalnumber

I hope it will help u mate