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 Square root of a 2 by 2 matrix A square root of a 2×2 matrix "M" is another 2 by 2 matrix "R" such that "M" = "R", where "R" stands for the matrix product of "R" with itself. In general there can be no, two, four or even an infinitude of square root matrices. In many cases such a matrix "R" can be obtained by an explicit formula. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root. Square roots of a matrix of any dimension come in