Quick, how many days are there in March? Thirty? Thirty-one? Definitely not thirty-two. Probably. At least some of you are reciting a poem right now. Or counting on your knuckles. It's okay, we've all been there. Let he who has never been surprised by March 31st cast the first stone.
Because we use it every day, we don't often stop to think about how weird our calendar is. In particular, we talk about a month as though it were a well-behaved unit. But it's pretty strange, no? Most of the time it's 31 days long. Much of the time it's 30 days long. Occasionally it's 28 days long and, even more rarely, 29.
For deeply ingrained historical reasons we have seven-day weeks, and for deeply ingrained gravitational reasons we have 365-day* years. Because 7 is not a factor of 365, it's tricky to come up with an intermediate unit of time, even though it would be very convenient to have one. Hence our months are four weeks long...more or less. Well, more, but by differing amounts. Confusing.
Not everyone is satisfied with the status quo. Over the years there have been many proposals to improve the Western Calendar, with different methods of reconciling the mathematical dissonance among weeks, months, and years. One extreme example is the Ordinal Calendar, which simply numbers the days in a year from 1 to 365, ignoring months altogether. The International Fixed Calendar solves the problem by making each month 28 days long (precisely four weeks), and adding a 13th month called Sol between June and July. Because 13 × 28 = 364, an extra day --- not belonging to any month --- is added before the first of every year.
All of the proposed calendars have their pros and cons. The Ordinal Calendar, for example, makes it trivial to locate an individual date within a year, or to calculate the time between dates. But it's awkward to describe particular stretches of time (think, for instance, about how you would have to describe your summer vacation). The International Fixed calendar is what's called a perennial calendar (the same date falls on the same day of the week in each year), which simplifies long-term planning. It also makes monthly statistics more meaningful, since a month is a bona fide unit of time. These two facts make the calendar particularly attractive to businesses (Eastman Kodak even used it as their official calendar for over six decades!). But because 13 is prime, it's tricky to divide the year into semesters, trimesters, or quarters, all of which are simple (and useful) in the current calendar.
Regardless of which reform calendar you might want to consider, you need some way to convert dates among the different representations, and that gives us a reason to talk about functions. Because any rule that maps a date in one calendar to a date in another must be able to do so unambiguously (each date on the original calendar has to correspond to only one date in the new one), the rule meets the definition of a function. In fact, it's a 1-to-1 function, since each date in the first calendar must be mapped to exactly one date in the second, and vice versa.
We also have a reason to talk about function composition. You may notice that it's really difficult to convert directly from the Western Calendar to the International Fixed Calendar, for example. But it's simpler to convert from Western to Ordinal, and it's really easy to convert from Ordinal to International Fixed. If we name the function that converts from Western to Ordinal f, and the function that converts from Ordinal to International Fixed g, and if x is a date in the Western Calendar, we can convert x to an International Fixed date by evaluating g(f(x)). Cool.
Things stay function-y if we look at rules for converting dates within a given year into days of the week, also. Every date gets mapped to a single day of the week (March 7th of any year can't be both a Friday and a Tuesday), so that's a perfectly nice function. But, unlike our previous cases, the inverse is no longer a function, since each day of the week would be mapped to multiple dates (52, to be precise...maybe 53). Even so, still cool.
Teachers, want to have this conversation with your students? Check out our lesson materials here!
* We're going to ignore leap years here, which add their own brand of weirdness to the discussion that gets handled differently in different calendars. In the International Fixed Calendar, for example, Leap Day is added to the calendar in the summer. Like the day before Jan 1st, it doesn't belong to any month.