Question

proove that root 3 is ratonal no.

Answer 1

let √3 be an rational no.
√3 =a/b where a and b are co-primes.
3=a²/b²
3b²=a²
3 divides a²
3 divides a
3 is a factor of a
a=3c
3b²=(3c²)
3b²=9c²
b²=3c²
3 divides b²
3 divides b
3 is a factor of b
3 is a common factor of a and b
hence √3 is an irrational no.

Answer 2

let us assume to the contrary , that root 3 is rational

therefore,root 3=r/s=a/b

where a and b are integers and are co-prime.

squaring both sides

root 3 the whole square =(a/b)the whole square

3=a sq./b sq.

a sq. =3b sq.

therefore 3 divides a sq

therefore 3 divides a------------(i)

let a/3=c/1

a=3c

squaring both sides

a sq = (3c) the whole sq. =3 sq. c sq. =9c sq.but a sq. = 3b sq.

substituting above

therefore 3b sq. = 9c sq.

b sq. =9c sq./3

b sq. = 3c sq.

therefore 3 divides b sq.

by theorem,

                  3 divides b--------------(ii)

(i.e.,) 3 is the facter of both a and b{from (i) and (ii)}

this contradicting to the original assumption that a and b are co-prime numbers

this contradiction is wrong due to our incorrect assumption that root 3 is rational

therefore ,we can conclude that root 3 is irrational