Question

Prove that `14 - 2sqrt(3)` is an irrational number, given that `sqrt(3)` is irrational.

Answer 1

Let us assume that `14 - 2sqrt(3)` is a rational number.

If it is rational, it can be represented as x, where x is rational.

`14 - 2sqrt(3) = x`

Rearrange the equation to isolate the irrational term:

`14 - x = 2sqrt(3)`

`(14 - x)/2 = sqrt(3)`

Now, evaluate both sides:

On the Left Hand Side (L.H.S.):

Since 14 and 2 are rational integers and we assumed x is rational, the difference and division of rational numbers is also rational.

Thus, `(14 -x)/2` is a rational number.

On the Right Hand Side (R.H.S.):

We are given that `sqrt(3)` is an irrational number.

This leads to a contradiction: Rational = Irrational.

Our initial assumption that `14 - 2sqrt(3)` is rational is incorrect.

Therefore, `14 - 2sqrt(3)` is an irrational number.

Hence, `14 - 2sqrt(3)` is proved to be irrational.