Question

prove that root 3 is irrational number​

Answer 1

Answer:

we have to prove that √3 is irrational

let us assume √3 is rational

then

√3 can written in the form of a/b

√3=a/b

√3b=a

S.O.B.S

squaring on both sides

(√3b)²=a²

3b²=a²

b²=a²/3

hence

3 divides a²

theorem: if p is a prime number and p divides a²

then, where 'a' is a positive number

then, 3 shall also divide 'a'. __________(1)

hence we can say

a/3=c

a=3c

now we know that

3b²=a²

substituting a=3c

3b²=(3c)²

3b²=9c²

b²=9c²/3

b²=3c²

b²/3=c²

hence

3 divides b²

and by using the same theorem

so,

3 divides also b ____________(2)

by (1)&(2)

3 divides both a & b

hence

3 is a factor of a & b

so,

a & b have factor 3

therefore

hence

a & b are not co-prime

assuming it's wrong

by

contradiction

√3 is irrational