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The Mystery of 355/113

Today's date is a good excuse to memorize a few more digits of 3.1415926535897932384626433832795....

And yet decimal approximations to pi are an artifact of our ten-fingered anatomy. Fractional approximations to pi are more satisfying, and they promise to teach us something more universal about pi.

We all know that 22/7 is a very good approximation to pi. 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:

chart?cht=tx&chl=\pi\hspace{7}\approx\hspace{7}\frac{22}{7}-\frac{1}{791}\hspace{7}=\hspace{7}\frac{355}{113}.

Remembering 355/113

The fraction 355/113 is incredibly close to pi, within a third of a millionth of the exact value. 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. For example, use any scientific calculator to compute cos(355) in radians. The oddball result is due to the freakish closeness of 355/113 to pi.

A cute mnemonic makes it easy for our base-10 species to remember this useful fraction. Write down the first three odd numbers twice: 1 1 3 3 5 5. Then divide the decimal number represented by the last 3 digits by the decimal number given by the first three digits.

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.

The Unusualness of 355/113

In a 2000 lecture on rational approximations of pi, Fritz Beukers defines the "quality" of a rational approximation p/q as a number M such that

chart?cht=tx&chl=\left|{\pi-\frac{p}{q}}\right|=\frac{1}{q^M}

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.

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. But it is rare to find M larger than 2. The best two approximations for pi we have seen are:

chart?cht=tx&chl=\left|{\pi-\frac{22}{7}}\right|=\frac{1}{7^{3.429...}}   and   chart?cht=tx&chl=\left|{\pi-\frac{355}{113}}\right|=\frac{1}{113^{3.201...}}

These approximations both have quality M > 3, which is unusually good. 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.

Other Good Rational Approximations for Pi

The fraction 355/113 overestimates pi by less than chart?cht=tx&chl=\frac{1}{3748629}.

It is not easy to approximate pi as economically as 355/113, but you can certainly try. 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:

chart?cht=tx&chl=\frac{355}{113}-\frac{1}{3748629}\hspace{7}=\hspace{7}\frac{1330763182}{423595077}

chart?cht=tx&chl=0\hspace{7}%3C\hspace{7}\pi-\frac{1330763182}{423595077}\hspace{7}%3C\hspace{7}\frac{1}{151648960887729}

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. By subtracting a couple fractions you are near the 10-15 limits of IEEE 754 double-precision arithmetic. This new fraction is an excellent and economical way to approximate pi as the difference between two fractions.

On the other hand, this new fraction is not as remarkable as 355/113:

chart?cht=tx&chl=\left|{\pi-\frac{1330763182}{423595077}}\right|=\frac{1}{423595077^{1.64379...}}

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.

Looking for 355/113 in Wallis's Rational Expression

Close coincidences in math are usually a hint of something deeper. 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.

There are several beautifully computable and well-known sequences of rational numbers that converge to pi. In 1655, shortly before the dawn of calculus, English mathematician John Wallis painstakingly computed an extended rational expression for the area of a circle to come up with one of the most memorable rational expressions for pi:

chart?cht=tx&chl=\pi=2\quad\times\quad\left(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdot\quad\cdot\quad\cdot\quad\cdot\right)

The partial expansions of this infinite fraction are:

chart?cht=tx&chl=4,\qquad\frac{8}{3},\qquad\frac{32}{9},\qquad\frac{128}{45},\qquad\frac{256}{75},\qquad\frac{512}{175},\qquad\frac{4096}{1225},\qquad\frac{32768}{11025},\qquad\cdots

This series converges very slowly despite having huge denominators, and our efficient fraction 355/113 is nowhere in sight.

Looking for 355/113 in Taylor Series

While developing modern calculus, Gregory and Leibniz systematicaly computed what we call the Taylor expansion for arctan. That formula leads to another pretty rational expansion for pi:

chart?cht=tx&chl=\pi=4\quad\times\quad\left(\frac{1}{1}-\frac{1}{3}%2B\frac{1}{5}-\frac{1}{7}%2B\frac{1}{9}-\cdots\right)

Partial expansions give us the converging sequence of rationals:

chart?cht=tx&chl=4,\qquad\frac{8}{3},\qquad\frac{52}{15},\qquad\frac{304}{105},\qquad\frac{1052}{315},\qquad\cdots

Unfortunately this sequence again converges slowly, and 355/113 does not appear.

Looking for 355/113 in Gauss's continued fraction

Gauss developed a clever generalized continued fraction for arctan that gives us a faster-converging expression:

chart?cht=tx&chl=\pi=4\quad\div\quad\left({1%2B\frac{1\cdot1}{3%2B\frac{2\cdot2}{5%2B\frac{3\cdot3}{7%2B\frac{4\cdot4}{9%2B\ddots}}}}}\right)

This expands to a sequence of rationals that converges much more quickly than the previous examples:

chart?cht=tx&chl=4,\qquad\frac{16}{5},\qquad\frac{22}{7},\qquad\frac{179}{57},\qquad\frac{952}{303},\qquad\cdots

Ah, hah! Here we can see that Gauss's expansion of arctan has justified the rational approximation 22/7, which appears as the third approximation in the sequence. This seems to be a side-effect of the efficiency of Gauss's sequence of rational numbers, which gets much closer to pi for a denominator of any given size.

But 355/113 is still beyond the reach of this sequence. It does not appear here.

Looking for 355/113 in Lange's Sequence

In 1999 Lange derived another elegant expression for pi based on a continued fraction for arcsin:

chart?cht=tx&chl=\pi={3%2B\frac{1\cdot1}{6%2B\frac{3\cdot3}{6%2B\frac{5\cdot5}{6%2B\frac{7\cdot7}{6%2B\ddots}}}}}

The sequence of partial fractions begins:

chart?cht=tx&chl=3,\qquad\frac{22}{7},\qquad\frac{160}{51},\qquad\frac{1462}{465},\qquad\frac{2307}{735},\qquad\cdots

This is another fast-converging sequence that includes 22/7. But again, 355/113 does not appear.

The quest is not over; Mathematicians are still hunting for faster-converging sequences of rational pi approximations. Beukers's lecture on this topic is worth a read. But 355/113 does not appear naturally in any these sequences that can be used to derive pi.

Where Does 355/113 Come From?

So, then, where does 355/113 come from? Is its nearness to pi a mere coincidence? A mathematical accident? A freak of nature?

It may be.

Or it might be a hint that there exists some as-yet undiscovered sequence of rationals that converge much faster towards pi, for which the highly precise 355/113 is just one remarkable number among many.

Posted by David at March 14, 2010 04:03 AM



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