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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

π=3.1415926535897932384626433832795...

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

  1. it depends on the decimal representation of π; in hexadecimal, we have π=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 π instead; and a well-known fractional approximation of π is given by 227, 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 π as 227 is 0.04%, while the 3.14 truncation has a relative error of 0.05%, so July 22nd is a better approximation to π 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 π as 3.1415, and by further appending the time we have an actual instant in which π 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 π:

[3,7,15,1,292,...]

Side note: the afore­men­tio­ned 227 is exactly the continued fraction [3,7].

Taking only the first three terms, corresponding to the date 3/7/15, we get [3,7,15]=333106, 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 π this year.

But as it happens, we can do better: if we take the first four terms of the continued fraction, we get [3,7,15,1]=[3,7,16]=355113 which is an excellent approximation to π, 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 π 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 π

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

Tau Day and the τ manifesto

For example, there are people that believe that π is wrong, and τ=2π should be considered the fundamental constant of the circle, and

τ=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 τ,

[6,3,1,1,7,2,146,...]

And taking the first three terms we get [6,3,1]=[6,4]=254 which approximates τ 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 [6,3,1,1]=[6,3,2]=447, which approximates τ to 0.04%, is not a good date (although it was on 6/3/2). Is this a hint that π is, in fact, better than τ?

e, ϕ, what else?

Similarly, we can go looking for the best date for Napier's number e, for which we can choose 197, accurate to 0.15% and dating to July 19th. For the golden ratio Φ the best candidate is 119, 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 ϕ 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.