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Today's date, March 14th, is considered Pi Day, since written in the North American (and a few other places') convention of putting the month before the day, 3/14 can be read as the three most significant digit of

$\pi =3.1415926535897932384626433832795...$

I personally disagree with this choice, mainly for two reasons:

1. it depends on the decimal representation of $\pi$; in hexadecimal, we have $\pi =3.243F...$ so that the correct day would be March 2nd (hexadecimal 24 is decimal 36, and no month is that long);

2. it depends on the (IMO barbaric) convention of putting the month before the day in numerical date representations, which is far from being international (day/month/year being much more common) or standard (the ISO standard goes for year-month-day).

While the second point is debatable (I'm not aware of ISO standard recommendations for dates without years, which might be used as a resolution), the first point is easily fixed by going for a fractional representation of $\pi$ instead; and a well-known fractional approximation of $\pi$ is given by $\frac{22}{7}$, that dates as far back as Archimedes at least. Of course, 22/7 is not really possible in the month/day representation, but it makes perfect sense as July 22nd.

Mathematically speaking, July 22nd is preferable to March 14th as Pi Day also because the relative error introduced by approximating $\pi$ as $\frac{22}{7}$ is 0.04%, while the $3.14$ truncation has a relative error of 0.05%, so July 22nd is a better approximation to $\pi$ than March 14th.

## The best Pi Day

Arguably, in the year 2015, the “American way” has another benefit: writing the year in the short (two-digit) form, we get an even better approximation of $\pi$ as $3.1415$, and by further appending the time we have an actual instant in which $\pi$ is presented exactly.

The argument fails miserably when taking into account that the time to be considered (9:26:53) would need to be padded by a 0 before “appending” it to the date, and there's always the decimal representation issue, and the fact that the time is sexagesimal …

If the year is to be considered, better dates can be chosen, with the following argument. Consider the continued fraction expansion of $\pi$:

$\left[3,7,15,1,292,...\right]$

Side note: the afore­men­tio­ned $\frac{22}{7}$ is exactly the continued fraction $\left[3,7\right]$.

Taking only the first three terms, corresponding to the date 3/7/15, we get $\left[3,7,15\right]=\frac{333}{106}$, which is accurate to 0.0026% (better than the 0.0029% of $3.1415$, even). So, March 7th 2015 (resp. July 3th 2015, depending on date notation) are both better approximations than March 14th (resp. June 22nd) for $\pi$ this year.

But as it happens, we can do better: if we take the first four terms of the continued fraction, we get $\left[3,7,15,1\right]=\left[3,7,16\right]=\frac{355}{113}$ which is an excellent approximation to $\pi$, with an error of less than one in ten million (a hint to this is the following huge 292 number).

The best date-fractional approximation of $\pi$ will happen next year, on March 7th in North America, Belize and whichever other country prefers month before day, and on July 3rd in the rest of the world.

## Going beyond $\pi$

As a mathematician, one gets to think: we stop at $\pi$ day, aside from the obvious pi/pie pun? There are so many other interesting numbers to look into!

### Tau Day and the $\tau$ manifesto

For example, there are people that believe that $\pi$ is wrong, and $\tau =2\pi$ should be considered the fundamental constant of the circle, and

$\tau =6.283185307179586476925286766559...$

The proposed Tau Day (in the manifesto linked above) is thus on June 28th, following the North American tradition. We obviously disagree, and would rather look for a fractional date choice. We thus go and look at the continued fraction representation of $\tau$,

$\left[6,3,1,1,7,2,146,...\right]$

And taking the first three terms we get $\left[6,3,1\right]=\left[6,4\right]=\frac{25}{4}$ which approximates $\tau$ to 0.5% (May 25th). Sadly, in this case the fractional approximation is worse than the truncation (0.05%), due to the fact that the next continued fraction term $\left[6,3,1,1\right]=\left[6,3,2\right]=\frac{44}{7}$, which approximates $\tau$ to 0.04%, is not a good date (although it was on 6/3/2). Is this a hint that $\pi$ is, in fact, better than $\tau$?

### $e$, $\varphi$, what else?

Similarly, we can go looking for the best date for Napier's number $e$, for which we can choose $\frac{19}{7}$, accurate to 0.15% and dating to July 19th. For the golden ratio $\Phi$ the best candidate is $\frac{11}{9}$, with an error of 0.21% and dating to either September 11th (oops), or November 9th, depending on convention.

(It should be mentioned that the continued fraction representation of $e$ and $\varphi$ is not interesting enough to give us something useful for the year.)

For anybody interested in exploring rational approximating dates, I've also cooked up a quick'n'dirty Ruby script that does the finding for you.

Have fun.