Question

Prove that v2 is irrational
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Answer 1

Answer:

Step-by-step explanation:

Let us assume that √2 is rational

√2=a\b where a and b are co prime

b√2=a (by cross multiplication)

Now, squaring both sides

2b2=a2

a=2c

By substituting we get 2b2=4c2

Hence, our assumption is wrong.

Hence √2 is irrational

Answer 2

Let us assume to the contrary that √2 is rational. Then, there exist positive integers a and b such that,

√2 = a/b (where, a and b, are co-prime, i. e, HCF = 1)

Squaring both the sides

=> (√2)² = (a/b)²

=> 2 = a²/b²

=> 2b² = a² ......(1)

=> 2 divides a²

=> 2 divides a

=> a = 2c for some integer c

=> a² = 4c²

=> 2b² = 4c² ....( from 1)

=> b² = 2c²

=> 2 divides b²

=> 2 divides b

Therefore, we can observe that 2 is a common factor of a and b. But this contradicts the fact that a and b are co-prime. This means that our supposition is wrong.

Hence, √2 is irrational number.