Question

Prove that 4-√3 is irrational number

Answer 1

Proof by contradiction ,

Let us consider 4 -√3 is a rational number, so it can be expressed as p/q where p, q are co primes , integers.

4 - √3 = p/q

4 - p/q = √3

4q - p/q = √3

Here, q & p are integers, so 4q - p / q is rational, but we already know that √3 is irrational.

So this contradiction has arrived because of our faulty assumption that 4 - √3 is rational. Hence it is irrational.

Answer 2

Thanks for your question !

     Let we assume that 4- √3 is a rational number. But,

     We know that, rational nunbers can be written in p/q form. Hence, the 4- √3 should be equal to p/q as we assumed it as rational number.

     Therefore, 4 - √3 = p / q

                         4 - p/q = √3

        On L.H.S., the 4 - p/q are rational numbers but on R.H.S., √3 is an irrational number.

        Hence, this is an contradiction. Therefore, 4 - √3 is an irrational number.

Proved. Done.