Prove that root 3 irrational.

1.	Prove that root 3 irrational.
Answer» Let root 3 is a rational no.Therefore root 3 = P/q (where q# 0)Squaring both side 3 =p^2/ q^23q^2 = p^2Because p^2 divided by 3Therefore P also divided by 3 Now, P = 3mSquaring both side (p)^2 = (3m)^2 3q^2 = 9m^2q^2 =9/3 =3q^2= 3m^2Because q^2 divided by 3Therefore q also divided by 3Therefore p and q have at least 3 as a common factor.So our assumption root 3 is a rational no. is incorrect. We have assume √3 is rational Therefore √3 =p/q and p and q are co primes p not equal to 0 Therefore p = √3q Squaring on both sides p^2 = 3q^2 Theorem : let p be a prime number ,it divides a^2 than p divides a, where a is a positive integer 3 divides q ^2 .3 divides q Let p = 3x (3x)^2 =3q ^2 9x^2 = 3 q ^2 3x^2 = q ^2 Theorem : let p be a prime number ,it p divides a^2 than p divides a where a is a positive integer 3 divides q^2 3 divides q That mean p and q has 2 common factorWhich is a contradicts the fact a and b Therefore √3 is irrational Root 3 is a rational numberWhere a and b are positive integer and has on common number(Root 3) =a÷b 3a=b3 divodes aPut a= 3m in eq 13.b2 =9m b2= 3m 3divides bSo root3 has common numberWhich is contradicationSo root3 is an irrational number

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