February 2024
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News
Pi Day – The International Day of Mathematics – In 2019, UNESCO declared March 14th as the International Day of Mathematics. This was in recognition that 3.14 is a close approximation of pi ~ 3.14159265... However, July 22nd, which is known in some circles as Pi Approximation Day, comes from the fraction 22/7 ~ 3.142857... which is a closer approximation to pi than 3.14.
If you do celebrate on March 14th, be cautious doing so with a pie. The Greek letter pi is pronounced by Greeks like the second syllable of hippie, which is close to the way the word pea is pronounced. I realize partying with pies is a lot more fun than partying with peas, but I thought you should know.
A Big Leap for George Washington
A good story helps a child’s engagement with a math topic. Today’s story concerns how animals count when they have a different number of fingers than people do.
Poor George Washington! Imagine being told when you are twenty that you’ve been celebrating your birthday on the wrong day. We’ll get back to George Washington after a while, but first I want to tell the story of how George Washington’s situation was the result of an unusual combination of contributors – a Roman emperor, a Pope, and a reluctant Church of England. Along the way we’ll talk about different types of years, the Age of Aquarius, the Precession of the Equinoxes, and we’ll even talk about continued fractions and better ways to do leap years and approximate pi. This will be quite a journey
Three Kinds of Years
How long is a year? There is interesting geometry contained in this question.
It is tempting to think of a year as the time it takes for the earth to return to a point where it lines up with the stars in the same way. This is called the sidereal year. However, this is not the year we want to use for our calendars. While the sidereal year is a good measure of a year from an astronomical standpoint, it doesn’t correspond well enough to the timing of the seasons on Earth. What we need is a year that keeps the seasons in their normal places on the calendar. If we used the sidereal year for our calendar, the coldest and warmest times of the year would, over the centuries, slowly drift to different months.
To keep the seasons where they are, we use the tropical year for our calendar. The vernal equinox is the moment in March when the sun, as viewed from the center of the earth, is directly above the equator. The tropical year is the length of time between vernal equinoxes. The length of the tropical year is on average about 365.2421897 days long. This length is approximately 20.409552 minutes shorter than the sidereal year. This difference of 20 minutes is not large for one year, but as we shall see, it can add up.
The lengths of the sidereal and tropical year can vary minutely from year to year, and it is also subject to various very technical definitions. The lengths also drift over periods of centuries. So you may see slightly different numbers should you look them up. I have chosen to ignore all of those subtleties for this note.
A third kind of year is the anomalistic year. The earth’s orbit is an ellipse that has the sun as one of the two foci. The perihelion is the point of the earth’s orbit when it is closest to the sun. In recent years the perihelion happens around January 3rd. The anomalistic year is the time between visits to the perihelion. This year is about 5 minutes longer than the sidereal year. The earth’s orbit is so close to being circular that a change of when perihelion occurs in the calendar does not have much effect on our day-to-day lives, so it is mostly ignored. One small effect it does have is that the length of a day varies slightly during the year, but it is only by a few seconds.
It all started with Julius Caesar
No matter how you try, there are no simple ways to deal with the period of time the moon takes to orbit the earth and the period of time the earth takes to orbit the sun. Since antiquity, many civilizations have come up with various calendars, some of which work better than others.
In 46 BCE, Julius Caesar added a leap year every four years. This was in recognition of their thinking a year was 365 ¼ days long. Thus the Julian calendar was born. This corrected the rather obvious problem that a year is not 365 days long. The Julian calendar is still used by the Eastern Orthodox Church and several other places and organizations.
Strangely, the Greeks knew a century before the time of Julius Caesar that, due to Hipparchus’ work on the precession of the equinoxes, the tropical year is shorter than 365 1/4 days. Even though they had very poor time instruments in those days, Hipparchus (a Greek astronomer, geographer and mathematician) was able to use old records of when equinoxes occurred. Over a span of 108,478 days, 297 vernal equinoxes occurred. This gave an average of 365.24579 days, which is a little high, but is amazingly accurate given the instruments and mathematics of the time. Julius Caesar ignored this, and it would be 1600 years before this oversight was officially addressed.
The Age of Aquarius
Over the years, the 20.409552 minute difference between sidereal and tropical years adds up. If you live a very, very long time (keep reading to get a sense of how long!) and each year you look up at the night sky at the time of the vernal equinox, you will notice that the Zodiac constellation overhead changes very slowly over time. There are 12 Zodiac constellations, and 1/12th of the length of the year is approximately 365.25 x 24 x 60 / 12 = 43830 minutes. If you divide this by 20.409552 minutes, you will find that it takes about 2147.5 years to shift from one Zodiac sign to the next. That is, it takes about 2147.5 years to precess about 30 degrees in the night sky. This is called the precession of the equinoxes, and it is the basis of the song The Age of Aquarius in the musical Hair.
Pope Gregory and Controlling Easter
The Christian church does not care about Hair or The Age of Aquarius, but it does care about Easter. The date for Easter is calculated relative to the vernal equinox, and the Christian church was unhappy that by the late 1500’s the date of the vernal equinox was ten days off.
Pope Gregory XIII set about to fix this. In 1582, after much study and lots of discussion among scientists, a ten day adjustment was made to the calendar and a new system of leap years was adopted. To adjust the calendar that year, the date October 4, 1582 was followed by October 15, 1582. Imagine the confusion of losing ten days!
The new system of leap years was this: a leap year will occur every 4 years, except it will not be a leap year every 100 years, except it will be a leap year every 400 years. This is the system we use today in all places that use the Gregorian calendar. It assumes that the tropical year is 365.2425 days long, which is pretty close to 365.2421897 days. This made the year 2000 not only the marker of a 1000 year change, but also a 400 year exception in the leap year rule! It was quite a year for some of us!
George Washington and Why England was Late to the Party
At first thought, you would think that George Washington, being born in 1732, would be safe from this craziness. However, the Church of England was reluctant to adopt something put forth by a pope of the Catholic church. This reluctance translated to waiting almost two centuries, which is an impressive amount of procrastination. As a consequence, England and all its colonies did not adopt the new Gregorian calendar until September of 1752. That’s when, at the age of 20, George Washington watched his birthday change from February 11, 1732 to February 22, 1732. How confusing! When do you give him his birthday cake?
But Wait, There is a Better Way
Continued fractions offer us a better approach to go about all this. A continued fraction is something of the form
There is a great deal of mathematics concerning these fractions, but I’ll only touch on the topic very lightly. These are, in some sense, the best way to approximate an irrational number with a fraction. Here, “best” refers to using a small denominator.
An important idea to be comfortable with is that the reciprocal of the reciprocal is the original number. In symbols this looks like x = 1 / (1 / x). For example, 4 = 1 / 1/4.
Approximating Pi
Suppose we want to approximate pi = 3.14159265358 … with continued fractions. Start by writing pi = 3 + 0.14159265… . Next, look at the reciprocal of the fractional part,
pi = 3 + 0.14159265… = 3 + 1 / 7.0625133059… ~ 3 + 1/7
So pi is approximately 3 + 1/7 = 22/7 ~ 3.142857, which is a famous approximation to pi accurate to two decimal places. This is very accurate using a denominator as small as 7. We could of course approximate pi to two decimal places using 314/100, but that uses a much larger denominator.
If we go a bit further, and cheat a little, we can get an even better approximation. Continuing where we left off,
pi = 3 + 0.14159265… = 3 + 1 / 7.0625133059… = 3 + 1 / (7 + 1 / 15.996…)
If we use 16, instead of the 15 we are supposed to use, we have
pi ~ 3 + 1 / (7 + 1/16) = 355 / 113 ~ 3.141592920…
which is accurate to six decimal places! I also like it very much as an approximation because it is easy to remember – just take 113355 and cut it down the middle!
A Better System for Leap Years
People came up with doing a leap year every four years by taking 365.25 and doing a continued fraction representation of it (though I doubt they thought of it that way). Let’s use these ideas with our better knowledge of the length of a tropical year.
As mentioned earlier, a tropical year is approximately 365.2421897 days. If we start with 365.25 as the average length of a year using a leap year every four years, then we get an average error of 365.25 – 365.2421897 = 0.0078103 days per year. Let’s use the ideas of continued fractions on this error. 1 / 0.0078103 is approximately 128.036054953. This means that it would be better to have the leap year exception every 128 years instead of every 100 years.
365.2421897
= 365 + 0.25 - 0.0078103
= 365 + 1/4 - 1 / 128.036054953…
~ 365 + 1/4 - 1/128
~ 365.2421875
How good is this? Well, 365.2421897 - 365.2421875 = 0.0000022, and 1 / 0.0000022 is approximately 454545. We would only need to adjust this leap year rule once every 454545 years! That’s a lot better than having to fix it every 400 years.
In the interest of full disclosure, the drift and the variability in the length of the year will start to come into play for intervals that are this long. However, I thought those refinements could wait for another year, so to speak.
Wrap Up
I hope you’ve enjoyed this journey through history, religion, orbital mechanics, and rock musicals. I realize this is a bit advanced for Early Family Math, but I couldn’t resist covering it during February of a leap year.
If you have any questions or comments, please send them our way! We would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, we would love to talk with you about ways we can work together!
February 18, 2024
Chris Wright
Chris@EarlyFamilyMath.org
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