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 Conversely, it is trivially true that every operator of the form "B* B" is non-negative. Therefore, an operator "T" is non-negative if and only if "T" = "B* B" for some "B" (equivalently, "T" = "CC*" for some "C"). The Cholesky factorization provides another particular example of square root, which should not be confused with the unique non-negative square root. If "T" is a non-negative operator on a finite-dimensional Hilbert space, then all square roots of "T" are related by unitary transformations. More precisely, if "T" = "A*A" = "B*B", then there exists a unitary "U" such that "A" =