A proof that the square root of s for s not a perfect square is simply normal to base 2

Since E. Borel proved in 1909 that almost all real numbers with respect to Lebesgue measure are normal to all bases, an open problem has been whether simple irrationals like square root of 2 are normal to any base. We show that each number of the form square root of s for s not a perfect square is simply normal to base 2, that is, the averages of the first n digits of its dyadic expansion converge to 1/2.