Question

Show that √5 is an irrational numbers .​

Answer 1

Answer:

Let us assume the opposite, i.e., √5 is a rational number. Hence, √5 can be written as in the form ab where a and b(b≠0) are co-prime (no common factor other than 1 ). By theorem: if p is a prime number and p divides a2, then p divides a, where a in a positive number. ... This means that 5 is a common factor of a and b.

Answer 2

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.

Hence proved