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 2 by 2 matrix use the simplified formula where "a" = ±√"A" and "d" = ±√"D"; which, for the various sign choices, gives four, two, or one distinct matrices, if none of, only one of, or both "A" and "D" are zero, respectively. Because it has duplicate eigenvalues, the 2×2 identity matrix formula_10 has infinitely many symmetric rational square roots given by where is any Pythagorean triple—that is, any set of positive integers such that formula_12 In addition, any non-integer, irrational, or complex values of "r, s, t" satisfying formula_13 give square root matrices. The identity matrix also has infinitely many non-symmetric square roots.