The best approximate fraction for π

The number π is irrational, but can be approximated well by some fractions. The best known is probably 22/7, which was already known to Archimedes. But which fraction is actually the “best” approximation fraction, and how can we assess this at all? An answer is given in the following video, which is a “mathematical meditation” on this question:

At one point in the video, it is shown that some numerators and denominators add up. More precisely, in the sequence of approximate fractions

\frac{3}{1} ; \frac{22}{7} ; \frac{333}{106} ; \frac{355}{113} ; \frac{103993}{33102} ; \frac{104348}{33215}

we observe that

\begin{align*} 22+333 &= 355 && && && && 355 &+& &103993 &&=& &104348 \\ 7+106 &= 113 & & && && && 113 &+& &33102 &&=& &33215 \end{align*}

These are the numerators and denominators of the fourth and sixth fractions, which can be calculated in this way. If we look into the continued fraction representation of $\pi$ , then we notice that in the fourth and sixth place in each case a $1$ occurs before the $+$ :

\pi=3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+\frac{1}{1+\ddots}}}}}}

In fact, we can state in advance that the sum relationship just seen really holds whenever a $1$ appears before the $+$ in the continued fraction representation. Let’s check this by adding a seventh member to the sequence of numbers (these are the OEIS sequences A002485 and A002486):

\frac{3}{1} ; \frac{22}{7} ; \frac{333}{106} ; \frac{355}{113} ; \frac{103993}{33102} ; \frac{104348}{33215}; \frac{208341}{66317}

We do the math, and indeed it is true:

\begin{align*} 103993 &&+&& 104348 &&=& &208341 \\ 33102 &&+&& 33215 &&=& &66317 \end{align*}

The numbers before the [catex]+[/catex] in the continued fraction representation also form a sequence. This is the sequence OEIS A001203:

q=(3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, ... )

We can see from it that the sum relationship holds at the 4th, 6th, 7th, 8th, 10th, 12th, 15th, and so on place is valid. In fact, there is a general rule that governs the relationship between the numerators in the sequence of approximate fractions. It reads:

\rm numerator_{\it i} =numerator_{\it (i-1)} \cdot \it q_i + \rm numerator_{\it (i-2)}

The $q_i$ are the elements of the sequence $q$ . Let’s check this on the third element:

333=22\cdot 15+3

is correct. The corresponding rule then also applies to the denominators:

\rm denominator_{\it i} =denominator_{\it (i-1)} \cdot \it q_i + \rm denominator_{\it (i-2)}

Again, we check this by using the third element as an example:

106=7\cdot 15+1

Right! 🙂 The validity of these regularities is best understood by subtracting adjacent members in the sequence of approximate fractions.

For more info and surfing suggestions on this topic, check out the following links:

Here are approximations for π on Wolfram’s Mathworld. However, many of the expressions shown on this page are not rational approximations.
Approximations in the Online Encyclopedia of Integer Sequences (OEIS).
A very comprehensive list of approximate fractions can be found on the Fractional Approximations of Pi page.
Proof that 355/113 is optimal on Math Stack Exchange.
The article about continued fraction expansions of $\pi$ on Wikipedia.
An article and discussion about “The Mystery of 355/113” by David Bau