Question

show that root 2 is an irrational​

Answer 1

Answer:

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Answer 2

Answer:

Given √2

To prove:

√2 is an irrational number.

Proof:

Let us assume that √2 is a rational number.

So it can be expressed in the form p/q where p, q are co-prime integers and q≠0

√2 = p/q

Here,

p and q are coprime numbers and q ≠ 0

Solving:

√2 = p/q

On squaring both the side we get,

⇒ 2 = (p/q)²

⇒ 2q² = p² ... (1)

p²/2 = q²

So,

2 divides p and q is a multiple of 2.

⇒ p = 2m

⇒ p² = 4m² ... (2)

From equations (1) and (2), we get,

2q² = 4m²

⇒ q² = 2m²

⇒ q² is a multiple of 2

⇒ q is a multiple of 2

Hence, p, q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√2 is an irrational number.

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