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 the unique positive square root of the positive operator "A""A", and "U" is defined by . If "A" is not invertible, then it still has a polar composition in which "P" is defined in the same way (and is unique). The unitary operator "U" is not unique. Rather it is possible to determine a "natural" unitary operator as follows: "AP" is a unitary operator from the range of "A" to itself, which can be extended by the identity on the kernel of "A". The resulting unitary operator "U" then yields the polar decomposition of "A". By Choi's result, a