Question

Prove that 2+/3 is a irrational number

Answer 1

Let 2 + root 3 is rational number.

2 + root 3 is rational nd 2 is rational .

(2 + root 3 ) - 2 .....( Difference of two rational number is rational) .

So root 3 is rational .

This contradicts the fact that root 3 is irrational .

The contraction arises by assuming that (2 + root 3) is rational .

Hence , 2 + root 3 is irrational .

Answer 2

First We have to assume that 2 +√3 is a rational number
So their exist a rational number p/q such that p/q have no common Factors
2+√3=p/q
√3=(p-2q)/q
Here both Denominator and numera are interger say they are equal to a r.no m/n where m and n have no common factors
√3=m/n
√3n=m 
Squaring on BS
3n²=m²
n²=m²/3
If 3 divides m² 3 also divides m hence we can say m=3c (c is some integer)
3n²=9c²
n²/3=c
Here 3 divides n² so it also divides n 
But 3 also divides m 
So 3 is the common factor of m,n
CONTRADICTION!
This arises due to assumption that 2+ √3 is rational
So, 2+√3 is Irrational