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 is related to the property of silver ratios. if the square root symbol is interpreted suitably for the complex numbers and . for square roots and only one minus sign. Similar in appearance but with a finite number of terms, appears in various trigonometric constants: It is not known whether is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two. The identity , along with the infinite product representations for the sine and cosine, leads to products such as and or equivalently,