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 matrix is an "n" × "n" complex matrix "ρ" that is positive semidefinite with trace 1. If "ρ" can be expressed as where formula_36 and ∑ "p" = 1, the set is said to be an ensemble that describes the mixed state "ρ". Notice {"v"} is not required to be orthogonal. Different ensembles describing the state "ρ" are related by unitary operators, via the square roots of "ρ". For instance, suppose The trace 1 condition means Let and "v" be the normalized "a". We see that gives the mixed state "ρ". In the Unscented Kalman Filter (UKF), the square root of