## Tuesday, February 28, 2012

### Leapin' Lizards - Leap Year is Mathematical!

We live by and teach our students that there are exactly 365 days in a year. In reality, the earth turns approximately 365 and a quarter times (six extra hours) on its axis by the time it has completed a full year's orbit around the sun, which means that every so often the calendar has to catch up.  Since those six extra hours add up to 24 hours over the course of four years (4 × 6 = 24), our calendar includes a leap year every fourth year.  (It's similar to receiving a free ice cream cone after getting your frequent buyers card stamped the fourth time.)  That is the reason the month of February has 29 days instead of 28 for a total of 366 days in the year. This year of 2012 is a leap year.

But why is the word "leap" used?  Believe it or not, it has to do with patterns. Typically, a calendar date that is on, say, a Monday one year will fall on a Tuesday the next year; then Wednesday the year after that, and so on. However every fourth year, thanks to the extra day in February, we "leap" over Thursday and that same calendar date lands on a Friday instead.  (For example, in 2010, Christmas was on a Sunday, but because 2012 is a leap year, this year, Christmas will be on Tuesday, not Monday.)

Believe it or not, there is a mathematical formula for figuring out leap years. (Don’t you love it?)  It goes like this: A leap year is any year whose date is exactly divisible by four except for those years that are divisible by 100, not 400. (No, I didn’t make this up!) So years that are evenly divided by 100 are not leap years; however, if the years are also evenly divisible by 400, they are leap years.

For example, 1600 and 2000 were leap years, but 1700, 1800, and 1900 were not.  In the same way, 2100, 2200, 2300, 2500, 2600, 2700, 2900, and 3000 will not be leap years, but 2400 and 2800 will be. Therefore, in a period of two thousand years, we will have 485 leap years. By this rule, the average number of days per year will be 365 + 1/4 − 1/100 + 1/400 = 365.2425, which is 365 days, 5 hours, 49 minutes, and 12 seconds.

So why does this formula have to be so difficult?  Because, in reality, the exact number of days in a solar year is slightly less than 365.25 (365.242374, to be exact), so the algorithm is designed so that a leap year is omitted every so often to account for underestimating the length of the earth's orbit.

Unfortunately, there's an exception to the "divide by 4" rule.  (You knew there would be).  For some time, astronomers have been able to more precisely estimate the earth's orbit. In reality, that number is roughly 365.2422 days, or 365 days, 5 hours, 48 minutes and 46 seconds, just a smidgen under the 365.25 days previously discussed.  By comparing the numbers, we see that the number above is off by 26 seconds. To make up for this, a rule states there can only be 97 leap years over the span of 400 years, not 100 as you may think. [Source: U.S. Navy Astronomical Center] One way to remember the rule is this:  Years that occur at the turn of centuries such as 1900 and 2000 must be evenly divisible by 400. This is why 1900 wasn't a leap year but the year 2000 was.
Here is something fascinating for those whose birthday falls on February 29th.  Over the course of their lives, these people will enjoy 75% fewer birthdays than the rest of us.  Does that also mean they are 75% younger, too?  So if your birthday is Leap Year Day, Happy, Happy Birthday to You!

1. Fascinating. I had no idea that we skipped leap years at the turn of most centuries. In fact, I found it so strange that I had to look it up for myself. Thanks for teaching me something new today!

-Emily

2. Thank you so much for this wonderful post!

3. Thank- you so much for sharing this wealth of information! I found it very fascinating too :)

Color Me Kinder

4. Love this! I shared this with my readers today. Great post and thanks for sharing!

Michelle