Question

prove that ✓2 is an irrational number ??






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Answer 1

Answer:

To prove that √2 is an irrational number, we will use the contradiction method. ⇒ p2 is an even number that divides q2. Therefore, p is an even number that divides q. ... This leads to the contradiction that root 2 is a rational number in the form of p/q with p and q both co-prime numbers and q ≠ 0.

Step-by-step explanation:

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Answer 2

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Proof that root 2 is an irrational number.

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 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.