Question

prove that root 2is not a rational number​

Answer 1

Answer:

Euclid's proof starts with the assumption that √2 is equal to a rational number p/q. From this equation, we know p² must be even (since it is 2 multiplied by some number). Since p² is an even number, it can be inferred that p is also an even number. ... Hence √2 is not a rational number.

Answer 2

Answer:

√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 p 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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