Question

Prove that
2 +
underroot3 is an irrational number.​

Answer 1

let us asume to the contrary that 2+√3 is rational

=>2+√3=a/b where a and b are integers and b≠0

=>√3=(a-2b) /b

(a-2b)/b is rational since a and b are integers

=>√3 is rational

=> √3= p/q 'where p and q are integers and q ≠0 and p and q are coprimes

squaring both sides

3=p^2/q^2

=> q^2=p^2/3

=>3 divides p

=> p= 3c for some integer c

=> q^2=(3c)^2/3

=> q^2=9c^2/3

=>q^2=3c^2

=>c^2=q^2/3

=> 3 divides q

=> 3 is a common factor of p and q

but this contradicts the fact that p and q are coprimes

hence our assumption is wrong

√3 is irrational

=>2+√3 is irrational

HENCE PROVED