Question

prove that
$\sqrt{3}$
is a irrational number

Answer 1

LET US ASSUME TO THE CONTRARY THAT ROOT3 IS RATIONAL. THEN THERE EXIST CO PRIME POSITIVE INTEGERS a AND b

root 3 =a/b
root 3 b =a
sq . both sides
root 3 b sq =a sq ( eq 1)

therefore 3 divides a sq

hence 3 divides a

so we can write a =3c for some integer c
substitute a=3c
3b sq =3c sq
3b sq = 9 c sq
b sq = 9 / 3 c sq

b sq = 3csq
3 divides b sq and so 3 divides b

therefore a and b have atleast 3 as a common factor
so root 3 is rational

this contradiction the fact that a and b have no common factor other than 1 . So our assumption is not correct

therefore root 3 is irrational.


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