Pi Day in Wolfram|Alpha
In my blog post last month, I wrote about Valentine’s Day in Wolfram|Alpha. Strangely, we received a number of comments indicating that the computational power of Wolfram|Alpha was not always sufficient to melt the hearts of some non-mathematically inclined sweethearts of the world. But not to fear; I have decided to persist undeterred in spite of that disappointing and surprising news, now that we’re on the verge of another holiday (and a more inherently mathematical one).
The holiday in question is Pi Day. As with a large number of other holidays, simply typing its name (in this case, “pi day”) into Wolfram|Alpha gives you basic calendrical information about it:
Now, because Wolfram|Alpha users are both intelligent and discriminating, all of you have I’m sure already noticed that when the digits in the date 3/14 (March 14 in the United States style for dates—a bit more about this later) are run together with a decimal place between, the result is 3.14. And that that decimal expansion is connected with a certain famous mathematical constant given by the ratio of the circumference of a circle to its diameter. And that little fact explains why Pi Day is celebrated on the 14th of each March.
So far so good. Now let’s do a little further exploration. You may note that while typing 3/14 directly into Wolfram|Alpha assumes by default that you were referring to the rational number “three-fourteenths” (which is indeed a reasonable assumption), clicking the handy “Use as a date instead” link gives you information about that date (for the current year).
As you can see and won’t be surprised to learn, Wolfram|Alpha is kind enough to inform you about the occurrence of Pi Day for this particular calendar date.
So that’s a little fun, but you’re probably asking what other explorations you can do here, especially since you might correctly guess that Wolfram|Alpha knows a little something about pi. Let’s start by just typing pi into Wolfram|Alpha (for those of you with the Wolfram|Alpha App, you can have fun just typing the character in directly). This gives the following result:
For starters, that’s quite a screenful (and I even truncated it in the screen shot above so that it wouldn’t take up the whole page). So it’s clear you can compute many things about pi in Wolfram|Alpha. In fact, even the large number of properties shown above only scratch the surface of Wolfram|Alpha’s computational universe for pi (if you’ll pardon my mixed metaphors). You can also compute the first hundred digits (or first thousand digits for that matter), symbolic forms for the (non-simple) continued fraction representation, get a list of some formulas for pi, find some background information about the transcendence of pi, learn a little about the person who proved the transcendence of pi, and so on and so forth. If you have the time and inclination, certainly feel free to explore further on your own.
After discovering a bit about Pi Day, discerning users might also wonder about other mathematics-related holidays. If you’re one of those people, your wonderings are well-founded since there is indeed at least one more—and it’s even based on pi as well. As a hint, consider the convergents of pi.
However, since Wolfram|Alpha doesn’t (yet) provide information on continued fraction convergents when asked (see MathWorld for more information about these), a further hint is in order. Note that in the output illustrated above, the (simple) continued fraction of pi is given as [3; 7, 15, ...]. What that notation means is that ? = 3+1/(7+1/(15+1/…)). Taking only the first two terms then gives the approximation 3+1/7. And typing that into Wolfram|Alpha gives this:
Since you now know enough to be dangerous (if you didn’t already), I won’t completely spoil the fun, but if you need another little hint, look at the bottom output pod and note that in Europe and other countries, the preferred date format is day/month instead of the United States month/day (which explains why Wolfram|Alpha is considerate enough when assuming something like 10/12 as a date to allow you to assume both day/month and month/day).
That basically wraps up what I wanted to say here, except for noting that are actually a few other “pi holidays” floating around. However, they are sufficiently obscure that even Wolfram|Alpha doesn’t bother telling you about them (at least not yet). But you can of course still compute them if you like. For example, you would be correct if you wondered whether “new year’s eve + 314 days” is sometimes considered to be a pi day worthy of note. In closing, let’s just be grateful that no one has yet proposed celebration of “Euler-Mascheroni Constant Day” or “Littlewood-Salem-Izumi Constant Day.” :)