Additional Math Pages & Resources

Friday, March 25, 2011

Four Hundred

Yesterday was my 400th posting in this series of articles about how we adults can use math we learned in elementary school. You could read one of my previous posts about the calories required to heat 400 ml of water, or go here to for more on the number 400 and its ramifications.

Today I have decided to explore how we use the number 400 to adjust our Gregorian calendar.

We divide 400 into the calendar year date at the beginning of a century, to see if that year will be a leap year or not. You don't follow me? Let me explain a little bit further.

If the solar year was exactly 365 days, we wouldn't need leap years. But it isn't 365 days. It's longer.

A year is almost 365.25 days (365 days + 6 hours) long, so every 4 years we add one day to our calendar. This resolves the issue of the year being 1/4 day (6 hours) longer than 365 days.

Sadly, a year is not quite 365.25 days either. It's a bit less. About 11 minutes less. So we make another adjustment.

Years evenly divisible by 100 are not considered leap years, unless they are also evenly divisible by 400. The years 1600 and 2000 were leap years, but 1700, 1800 and 1900 were not leap years, nor will 2100 be a leap year. This extra-fine adjustment process resolves the issue of the year being 11 minutes shorter than 365 days, 6 hours.

The average number of days per year by this formula is

365 + 1/4 day − 1/100 day + 1/400 day = 365.2425 days.

Did you get that?

365.0000 + .2500 - .0100 + .0025 = 365.2425

You might ask, How can we turn this decimal number (.2425) into our familiar time units of hours, minutes and seconds?

.2425 days = X hours, Y minutes and Z seconds

Let's get the number of seconds in a day (24 x 60 x 60 = 86,400)

.2425 x 86400 = 20952. Our year is 365 days and 20952 seconds.

Now you are likely to say, But I want hours and minutes and seconds!

I have to warn you that the following process is relatively easy to follow if you do it longhand, but confusing when you use a calculator - because we want integer remainders in second, NOT decimal remainders of various time periods. Here's what I mean:


Divide 20952 by 3600 (seconds in an hour) to learn the number of hours

20952 ÷ 3600 = 5 hours with a remainder of 2952 seconds

Divide 2952 by 60 (seconds in a minute) to learn the number of minutes

2952 ÷ 60 = 49 minutes with a remainder of 12 seconds

Thus the year (as adjusted by the Gregorian calendar) is 365 days, 5 hours, 49 minutes, and 12 seconds long.

I did a bit of research and the most accurate measurements of the movement of the earth say the year 2000 was 365 days, 5 hours, 48 minutes, 45.19 seconds (and slowing). You might ask me,
 
What will we have to do to adjust for this 26.81 second discrepancy between actual and theoretical calendar years?

I'll reply, I've given you the pattern, so you can figure it out.

I'm taking a couple days off to go on a short vacation.