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Prove that √2 is an irrational …

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Prove that √2 is an irrational number?

Posted by Mycharla. Pushpa Kumari 4 years, 5 months ago

CBSE > Class 10 > Mathematics

2 answers

Himanshu Raj 4 years, 5 months ago

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 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.

Parth Gopani 4 years, 5 months ago

Any root of imperfect square is an 'irrational number'. As '2' is not a perfect square(i.e.4,9,16....) so root of it is surely irrational.

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