11/7/2022 0 Comments Weird ways to calculate pi![]() ![]() So let us take a look at some computable sequences of rational numbers that converge to pi to see if 355/113 appears on some explainable path to pi. Looking for 355/113 in Wallis's Rational ExpressionĬlose coincidences in math are usually a hint of something deeper. While M > 1 tells us that the precision is better than a random denominator would give us, M is still much lower than the quality of 355/113. On the other hand, this new fraction is not as remarkable as 355/113: This new fraction is an excellent and economical way to approximate pi as the difference between two fractions. ![]() By subtracting a couple fractions you are near the 10 -15 limits of IEEE 754 double-precision arithmetic. In other words, subtracting 1/3748629 from 355/113 will provide an approximation to pi that is well within 10 -14 of the true value. If you have a good enough memory to remember the number 3748629, then you might find it handy to know that the following difference is even closer to pi: It is not easy to approximate pi as economically as 355/113, but you can certainly try. The fraction 355/113 overestimates pi by less than. Other Good Rational Approximations for Pi Both these fractions provide an approximation that have a precision with about triple what you would have any right to expect for their small denominators. These approximations both have quality M > 3, which is unusually good. The best two approximations for pi we have seen are: It is no surprise when we find a fraction that approximates pi with M around 1: for any q there will be a p/ q within 1/ q of any value. In words, the Beuker's quality M is the ratio between the number of digits of precision by the number of digits of the denominator. "quality" of a rational approximation p/ q as a number M such that In a 2000 lecture on rational approximations of pi, Fritz Beukers defines the The mystery is: why is this fraction so close to pi? The deeper you look, the more unique and unexplained 355/113 appears to be. ![]() Then divide the decimal number represented by the last 3 digits by the decimal number given by the first three digits. Write down the first three odd numbers twice: 1 1 3 3 5 5. The oddball result is due to the freakish closeness of 355/113 to pi.Ī cute mnemonic makes it easy for our base-10 species to remember this useful fraction. For example, use any scientific calculator to compute cos(355) in radians. This level of accuracy is far beyond its rights as a fraction with such a small denominator, and it causes various oddities elsewhere in math. The fraction 355/113 is incredibly close to pi, within a third of a millionth of the exact value. But this well-known fraction is is actually 1/791 larger than a slightly less-well-known but much more mysterious rational approximation for pi: We all know that 22/7 is a very good approximation to pi. Fractional approximations to pi are more satisfying, and they promise to teach us something more universal about pi. Today's date is a good excuse to memorize a few more digits of 3.1415926 53589793 23846264 33832795.Īnd yet decimal approximations to pi are an artifact of our ten-fingered anatomy. ![]()
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