Question

Show that 5+ root2 is not a rational number

Answer 1

Assume to reach the contradiction that  5 + √2  is rational.

Let  5 + √2 = x,  where x is rational.

    x = 5 + √2

⇒  x² = (5 + √2)²

⇒  x² = 25 + 2 + 2 · 5√2

⇒  x² = 27 + 10√2

⇒  x² - 27 = 10√2

⇒  (x² - 27) / 10 = √2

Here makes a contradiction!

In  (x² - 27) / 10 = √2,  at LHS, since x is rational, then so will be  (x² - 27) / 10,  but the RHS,  √2,  is not rational.  Here it actually seemed  √2  being written in  p/q  form. Is it possible?!

Hence proved by contradiction that  5 + √2  is not a rational number!