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 Square root of a matrix In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. Matrix is said to be a square root of if the matrix product is equal to . In general, a matrix can have several square roots. For example, the matrix formula_1 has square roots formula_2 and formula_3, as well as their additive inverses. Another example is the 2×2 identity matrix formula_4 which has infinitely many symmetric rational square roots given by where formula_6 is any Pythagorean triple—that is, any set of positive integers such that formula_7. However,