Prove that √3is irrational

1.	Prove that √3is irrational
Answer» Let us assume to the contrary that √3 is a rational number.It can be expressed in the form of p/qwhere p and q are co-primes and q≠ 0.⇒ √3 = p/q⇒ 3 = p2/q2\xa0(Squaring on both the sides)⇒ 3q2\xa0= p2………………………………..(1)It means that 3 divides p2\xa0and also 3 divides p because each factor should appear two times for the square to exist.So we have p = 3rwhere r is some integer.⇒ p2\xa0= 9r2………………………………..(2)from equation (1) and (2)⇒ 3q2\xa0= 9r2⇒ q2\xa0= 3r2Where q2\xa0is multiply of 3 and also q is multiple of 3.Then p, q have a common factor of 3. This runs contrary to their being co-primes. Consequently, p / q is not a rational number. This demonstrates that √3 is an irrational number.

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