prove that square root 2 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/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.
