Question

Prove that root 5 is irrational.
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Answer 1

To prove that √5 is irrational number

Let us assume that √5 is rational

Then √5 = \frac{a}{b}

b

a

(a and b are co primes, with only 1 common factor and b≠0)

⇒ √5 = \frac{a}{b}

b

a

(cross multiply)

⇒ a = √5b

⇒ a² = 5b² -------> α

⇒ 5/a²

(by theorem if p divides q then p can also divide q²)

⇒ 5/a ----> 1

⇒ a = 5c

(squaring on both sides)

⇒ a² = 25c² ----> β

From equations α and β

⇒ 5b² = 25c²

⇒ b² = 5c²

⇒ 5/b²

(again by theorem)

⇒ 5/b-------> 2

we know that a and b are co-primes having only 1 common factor but from 1 and 2 we can that it is wrong.

This contradiction arises because we assumed that √5 is a rational number

∴ our assumption is wrong

∴ √5 is irrational number

Answer 2

let √5 be a rational number.

√5 = p/q

Squaring Both sides

(√5)² = (p/q)²

5 = p²/q²

5q² = p²

Here,

5 is a factor of

AND 5 is also the factor of p²....(¡)

let p be 5

5q² = p²

5q² = (5)²

5q² = 25

q² = 5

here,

we see that

5 is a factor of q

AND 5 is also a factor of q².....(¡¡)

FROM (¡) AND (¡¡) WE see that

5 is a common factor of p and q.

So, Our assumption is wrong.

√5 is an irrational number.

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