Question

1.Prove that √2 is not a rational number​

Answer 1

Answer:

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Answer 2

♣ Question :-

Prove that √2 is not a rational number

♦ Given :- √2

♦ To prove: - √2 is an irrational number.

♦ Proof :---

We will solve this question by contrary method .

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 co-prime numbers and q ≠ 0

Solving :-

• ⇒√2 = p/q

On squaring both the side we get,

• ⇒√2 = (p/q)²
• ⇒ 2 q² = p² ------------› Equation 1
• ⇒ (p²/2) = q²

So 2 divides p and q is a multiple of 2.

• ⇒ p = 2m²
• ⇒ p² = 4m² ------------› Equation 2

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

• ⇒ 2q² = 4m²
• ⇒ q² = 2m²

q² is a multiple of 2 So ,

• ⇒ 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

Hence √2 is an irrational number.

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