Question

prove that √5 is irrational

Answer 1

Heya!

Here is yr answer....


Let us assume √5 is rational.....

=> √5 = a/b (a, b are co-primes)

=> b√5 = a

by squaring on both sides....

=> 5b² = a²

=> a² = 5b²

Here, 5 divides a²

Therefore, 5 also divides 'a' --------(1)

Let a=5c

by sub. a=5c in a²=5b²

=> (5c)² = 5b²

=> 25c² = 5b²

=> 5c² = b²

=> b² = 5c²

Here, 5 divides b²

Therefore, 5 also divides 'b' ----------(2)


From (1) & (2)

we can conclude that, a, b are not co-primes

So, our assumption is false..

Hence, √5 is irrational.


Hope it helps....

Answer 2

HI

Proof:

Let us assume to the contrary that √5 is rational

Therefore, √5 = a/b , where a and b are coprime integers and b ≠ 0

Squaring on both sides,

5 = a²/b²

5b² = a² ......(1)

Since, a² is divisible by 5,

Therefore, a is divisible by 5

Let a = 5c and substitute in eq(1)

5b² = (5c)²

5b² = 25c²

b² = 5c²

Since, b² is divisible by 5,

Therefore, b is divisible by 5

➡️ a and b have at least one common factor i.e. 5

This contradicts the fact that a and b are coprime.

➡️ This contradiction has arisen due to our incorrect assumption that √5 is rational.

Therefore, we conclude that √5 is irrational.

Hence Proved !

Hope it proved to be beneficial....