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 a 2 by 2 matrix may have complex entries even if "M" is a real matrix; this will be the case, in particular, if the determinant "δ" is negative. When a matrix can be expressed as real multiple of the exponent of a matrix, formula_5 then the square root is formula_6 In this case the square root is real, and can be interpreted as the square root of a type of complex number. If "M" is an idempotent matrix, meaning that "MM" = "M", then if it is not the identity matrix its determinant is zero, and its trace equals its rank which (excluding the