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 pairs: If "R" is a square root of "M", then –"R" is also a square root of "M", since (–"R")(–"R") = (–1)(–1)("RR") = "R" = "M". Let where "A", "B", "C", and "D" may be real or complex numbers. Furthermore, let "τ = A + D" be the trace of "M", and "δ = AD – BC" be its determinant. Let "s" be such that "s" = "δ", and "t" be such that "t" = "τ" + 2"s". That is, Then, if "t" ≠ 0, a square root of "M" is Indeed, the square of "R" is Note that "R"