Question

Show that (5 - 2`sqrt(3)`) is irrational.

Answer 1

Let x = 5 - 2`sqrt(3)` be a rational number.
x = 5 - 2`sqrt(3)`
⇒` x^2 = (5 - 2sqrt(3))2`
⇒ `x^2 = 5^2 + (2sqrt(3))^2 – 2(5) (2 sqrt(3) )`
⇒ x2 = 25 + 12 – 20√3
⇒ `x^2 – 37 = – 20sqrt(3)`

⇒`( 37− x^2)/20 =sqrt(3)`
Since x is a rational number, `x^2` is also a rational number.
⇒ 37 -` x^2` is a rational number
⇒ `(37− x^2)/20` is a rational number
⇒`sqrt(3)` is a rational number
But `sqrt(3)`  is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, (5 - 2`sqrt(3)` ) is an irrational number.