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 to converge. In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) "T" on a complex Hilbert space, "B" is a square root of "T" if "T" = "B* B", where "B*" denotes the Hermitian adjoint of "B". According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator "T" such that "T" is itself positive and ("T") = "T". The operator "T" is the unique non-negative square root of "T". A bounded non-negative operator on a complex Hilbert space is self adjoint by definition. So "T" = ("T")* "T".