Question

Prove that V2 is an irrational number​

Answer 1

Step-by-step explanation:

To prove that the square root of 2 is irrational is to first assume that its negation is true. Therefore, we assume that the opposite is true, that is, the square root of 2 is rational. ... Since we assume that 2 is rational, we must describe or express 2 as a rational number

Answer 2

Solution:-

Let us assume on contrary that √2 is rational number then there exists positive integer such that

→ √2 = a/b ( a & b are co-prime and b≠0)

Squaring both side we get

→ (√2)² = (a/b)²

→ 2 = a²/b²

→ 2b² = a²

→ b² = a²/2

here a² is divisible by 2

therefore a is also divisible by 2

a/2 ---------------(i)

Now, let a = 2c for positive integer c

Squaring both side we get

→ a² = 2c²

→ 2b² = 4c². ( a² = 2b²)

→ b² = 4c²/2

→ b² = 2c²

→ c² = b²/2

It means that b² is divisible by 2

therefore, b is also divisible by 2

b/2----------------(ii)

From equation (i) and (ii) we observe that a and have at least two common factor so our assumption is wrong that a & b are co - prime

Therefore, √2 is an irrational number .

Hence Proved!!