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 , then the Jordan decomposition into generalized eigenspaces of can be deduced from the partial fraction expansion of . The corresponding projections onto the generalized eigenspaces are given by real polynomials in . On each eigenspace, has the form as above. The power series expression for the square root on the eigenspace show that the principal square root of has the form "q"("A") where "q"("t") is a polynomial with real coefficients. Another way to find the square root of an "n" × "n" matrix "A" is the Denman–Beavers square root iteration. Let "Y" = "A" and "Z" = "I", where