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 the original estimate and its square. Furthermore, ("x" + "c") ≈ "x" + 2"xc" when "c" is close to 0, because the tangent line to the graph of "x" + 2"xc" + "c" at "c" = 0, as a function of "c" alone, is "y" = 2"xc" + "x". Thus, small adjustments to "x" can be planned out by setting 2"xc" to "a", or "c" = "a"/(2"x"). The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopher Heron of Alexandria, who first described it.