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3.2 Efficient infinite series. ... 3.7 Complex functions. 3.8 Continued fractions. 3.9 Iterative algorithms. 3.10 Asymptotics. 3.11 Hypergeometric ... formulas of the ...
Rational numbers have two continued fractions; the version in this list is the shorter one. ... 1, 1, 1, 2 2, 3 2, 13 2, 129 2, 25298 2, ... 3.00000 00000 [3; ] Pi
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
Two equations are used so that one can check they both give the same result; it is helpful if the equations used to cross-check the result reuse some of the arctangent arguments (note the reuse of 57 and 239 above), so that the process can be simplified by only computing them once, but not all of them, in order to preserve their independence.
In the top figure the fraction 1/9000 in Excel is displayed. Although this number has a decimal representation that is an infinite string of ones, Excel displays only the leading 15 figures. In the second line, the number one is added to the fraction, and again Excel displays only 15 figures.
An easy mnemonic helps memorize this fraction by writing down each of the first three odd numbers twice: 1 1 3 3 5 5, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits: 1 1 3 分之(fēn zhī) 3 5 5. (In Eastern Asia, fractions are read by stating the denominator first ...
((x),(y) = {239, 13 2} is a solution to the Pell equation x 2 − 2 y 2 = −1.) Formulae of this kind are known as Machin-like formulae . Machin's particular formula was used well into the computer era for calculating record numbers of digits of π , [ 39 ] but more recently other similar formulae have been used as well.
Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10 −5). Its simple continued fraction is periodic and given by: