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 "UB". Indeed, take "B" = "T" to be the unique non-negative square root of "T". If "T" is strictly positive, then "B" is invertible, and so is unitary: If "T" is non-negative without being strictly positive, then the inverse of "B" cannot be defined, but the Moore–Penrose pseudoinverse "B" can be. In that case, the operator is a partial isometry, that is, a unitary operator from the range of "T" to itself. This can then be extended to a unitary operator "U" on the whole space by setting it equal to the identity on the kernel of "T". More generally,