Square root of 3 billion digits. The fraction (…) for the square root of three can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than (approximately ). The rounded value of 1.732 is correct to within 0.01% of the actual value. Archimedes reported , accurate to (six decimal places) and (four decimal places), respectively. It can be expressed as the continued fraction , expanded on the right. It can also be expressed by generalized continued fractions such as which is evaluated at every second term. The following nested square expressions converge to
Square root of 2 if is a monic polynomial with integer coefficients, then any rational root of is necessarily an integer. Applying this to the polynomial , it follows that is either an integer or irrational. Because is not an integer (2 is not a perfect square), must therefore be irrational. This proof can be generalized to show that any root of any natural number which is not the square of a natural number is irrational. See quadratic irrational or infinite descent for a proof that the square root of any non-square natural number is irrational. One proof of the number's irrationality is the