Question

Prove root 5 is irrational, using co-prime a and b.

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Answer 1

Let us assume that √5 is a rational number. This implies that √5 can be expressed in the form a/b where b ≠ 0 and a & b are co-primes.

⇒ √5 = a/b

Squaring on both sides;

⇒ (√5)² = (a/b)²

⇒ 5 = a²/b²

⇒ 5b² = a²

⇔ 5 divides a².

∴ 5 divides "a" as well. → 1

We know that; If p is a prime number and divides q², then p divides 'q' as well where 'q' is a positive integer

Let us take a = 5c for any positive integer c.

⇒ 5b² = a² (From previous equation)

⇒ 5b² = (5c)²

⇒ 5b² = 25c²

⇒ b² = 5c²

⇔ 5 divides b².

∴ 5 divides "b" as well. → 2

From 1 & 2 we can say that both a & b have factors other than 1 & themselves. This contradicts the fact that they are co-primes. This is due to the incorrect assumption that √5 is a rational number.

∴ √5 is an irrational number.

Answer 2

Given:

√5

We need to prove that √5 is irrational

Proof:

Let us assume that √5 is a rational number.

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

⇒√5=p/q

On squaring both the sides we get,

⇒5=a²/b²

⇒5b²=a² —————–(i)

a²/5= b²

So 5 divides a

a is a multiple of 5

⇒a=5m

⇒a²=25m² ————-(ii)

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

5b²=25m²

⇒b²=5m²

⇒b² is a multiple of 5

⇒b is a multiple of 5

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

√5 is an irrational number