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 formulas that give up to four square roots, if the matrix has any roots. If is a diagonal "n" × "n" matrix, one can obtain a square root by taking a diagonal matrix , where each element along the diagonal is a square root of the corresponding element of . If the diagonal elements of "D" are real and non-negative, and the square roots are taken with non-negative sign, the matrix will be the principal root of . If a matrix is idempotent, one of its square roots is the matrix itself. An "n" × "n" matrix is diagonalizable if