Question

show3) Show that
$2 + \sqrt{3}$
is an irrational number.

​

Answer 1

Step-by-step explanation:

We can prove it by contradictory method............

We assume that 2 + v3 is a rational number.

=> 2 + √3 = p/q, where p & q are integers, 'q' not =0.

=> √3 = (p/g) - 2

=> √3 = (p- 2q)....................1

=> here, LHS √3 is an irrational number.

But RHS is a rational number.. Reason- the difference of 2 integers is always an integer. So the numerator (p-2q) is an integer

& the denominator 'q' is an integer.&q' not = 0 This way, all conditions of a rational number are satisfied.

=> RHS (P-29)/q is a rational number.

But, LHS is an irrational

=> LHS of. (1) is not = RHS.

=> Our assumption, that 2 + v3 is a rational number is incorrect.

therefore, 2+√3 is an irrational number.