Question

prove that ✓2 is an irrational number ??





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Answer 1

Answer:

Explanation: To prove that √2 is an irrational number, we will use the contradiction method. ⇒ p2 is an even number that divides q2. ... Since p and q both are even numbers with 2 as a common multiple which means that p and q are not co-prime numbers as their HCF is 2.

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Answer 2

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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

Explanation :-

√2 = p/q

On squaring both the sides we get,

=>2 = p/q2

=> 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.