Prove√5 is a rational number

We have to prove that √5 is an irrational number It can be proved using the contradiction method Assuming √5 as a rational number, i.e., can be written in the form a/b where a and b are integers with no common factors other than 1 and b is not equal to zero. √5/1 = a/b √5b = a By squaring on both sides (5b)2 = (a)2 b2 = a2/5 …. (1) It means that 5 divides a2. It means that it also divides a a/5 = c a = 5c By squaring on both sides a2 = 25c2 Substituting the value of a2 in equation (1) 5b2 = 25c2 b2 = 5c2 b2/5 = c2 As b2 is divisible by 5, b is also divisible by 5 a and b have a common factor as 5 It contradicts the fact that a and b are coprime This has arisen due to the incorrect assumption as √5 is a rational number. Therefore, √5 is irrational.

We have to prove that √5 is an irrational number It can be proved using the contradiction method Assuming √5 as a rational number, i.e., can be written in the form a/b where a and b are integers with no common factors other than 1 and b is not equal to zero. √5/1 = a/b √5b = a By squaring on both sides (5b)2 = (a)2 b2 = a2/5 …. (1) It means that 5 divides a2. It means that it also divides a a/5 = c a = 5c By squaring on both sides a2 = 25c2 Substituting the value of a2 in equation (1) 5b2 = 25c2 b2 = 5c2 b2/5 = c2 As b2 is divisible by 5, b is also divisible by 5 a and b have a common factor as 5 It contradicts the fact that a and b are coprime This has arisen due to the incorrect assumption as √5 is a rational number. Therefore, √5 is irrational.