Question

how to prove 2+root3 is irrational number

Answer 1

Hi friend,

Let 2+√3 is a rational number.
A rational number can be written in the form of p/q.
2+√3=p/q
√3=p/q-2
√3=(p-2q)/q
p,q are integers then (p-2q)/q is a rational number.
But this contradicts the fact that √3 is an irrational number.
So,our supposition is false.
Therefore,2+√3 is an irrational number.
Hence proved.

Hope it helps

Answer 2

Hey there !

Let us assume that :-
2 + √3 is a rational number.

Let , 2 + √3   = r , where "r" is a rational number

Squaring both sides ,

[2 + √3 ]² = r²

2² + 2 x 2 x √3 + [√3]²  = r²

4 + 4√3 + 3 = r²

7 + 4√3 = r²

4√3 = r² - 7

√3 = r² - 7÷ 4

So , 
we see that LHS is purely irrational.
But , on the other side , RHS is rational.

This contradicts the fact that 2+√3 is rational.
Hence , our assumption was wrong.

Hence ,
2+√3 is a irrational no: