Okieinexile's Blog

By Bobby Neal Winters

In real life, I teach mathematics.  “Mathematics has been very good to me.” (Those of you who catch the reference, are welcome to hear that line in the voice of Chico Escuela.) As a part of one of my courses, I teach some of the mathematics of calendars.

The class starts at 8:00am, a time on the clock that some students believe is an imaginary number.  The students in the class are docile, which is a gentle way of saying “semi-comatose.”  I ask them questions, but that is something like throwing a rubber ball into a black hole: Nothing bounces back.

One calendar-related question I ask is this: “By looking at the date, how do you know whether a year is a leap year?”

They know the answer–or I should say an answer–but their mouths aren’t necessarily able to enunciate it at that (to them) tender time of the morning.

All of you are shouting in your minds: If it’s divisible by 4.  And I say shouting, because you know the answer and you like to be right for once in math class.

But–and this is why people HATE math teachers–this is only partially correct.

The full answer is this: If it’s divisible by 4, unless it’s divisible by 100, but then yes if it’s divisible by 400.

If you got here, you’ve read that sentence at least twice and have given up.  Here’s the thing.  The divisible by 4 method works 397 times out of 400.  The only time you have to be careful is at the turn of the century, so 1700, 1800, and 1900 were not leap years even though they were divisible by 4 because they were divisible by 100. However, the most recent turn of the century in the year 2000 was a leap year because it was divisible by 400.

I don’t remember hearing anything about this because, well, why would we?  We didn’t have to break the divisible by 4 rule.  Quite frankly, this was a blessing because given all of the other issues with the Millenium and the Y-2K bug, leaving out a leap day in a year divisible by 4 would’ve been viewed as a conspiracy.

So why is this an issue? Why do we have leap years?

Well, you know, but I am going to tell you again because this is how we remember things.  The year is NOT 365 days long.  It is about 365 and one-fourth days long, and that is why we add a day every four years.  We accumulate a fourth of a day error every year and are a full day short after four years. This means you are planting your potatoes a day too early.

Putting an extra day in will catch us up and keep the seasons right.

But look at the third sentence of the previous paragraph.  It has the word “about” in it.  The year is “about” 365 and one-fourth days long. What is that about covering up? Well, a decimal.  The year is 365.2422 days long, and that is rounded off. 

When we put in an extra day every 4 years, that is actually 0.0078 days too much.  That is 8 one-thousanths of a day.  That is pretty small. It’s only about 18 minutes.

Over a long lifetime, say 100 years, it would only put the calendar off by about 3 quarters of a day.  You are only planting your potatoes a little late, it’ll be fine.

This is what managers refer to as somebody else’s problem.

The thing is, this is a calendar. We don’t change calendars. They kept on using it.  After 1500 years, they were almost 12 days off. If you followed this calendar to plant your crops, you would be two weeks late.  You will be hungry, two weeks longer than you need to be.

As I said above, if you go 100 years adding an extra day every 4 years, you will be 0.78 days too long.  If you leave out that day every hundred years, you will be 0.22 days too short, which is a relatively small improvement.  If you stick with this system for 400 years, you will be almost a day short, so why not just go ahead and have a leap year then, adding an extra day.  

If you adopt this system, you will be off by 0.12 of a day–that is less than 2 hours–every 400 years. 

It works out that this calendar will be off by 3 days after 10 thousand years.  That is somebody else’s problem.

Bobby Winters grew up near Harden City, Oklahoma.  He teaches mathematics and computer science, does woodworking, and blogs at okieinexile.com.