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 numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by "i" (sometimes "j", especially in the context of electricity where ""i"" traditionally represents electric current) and called the imaginary unit, which is "defined" such that . Using this notation, we can think of "i" as the square root of −1, but notice that we also have and so −"i" is also a square root of −1. By convention, the principal square root of −1 is "i", or more generally, if "x" is