Last week we looked at some questions that arose leading up to the year 2000, triggered by the 20th anniversary of that event. Now we’ll start from a different question about that year, to look at the story of leap years.

## Was 2000 a leap year, or not?

Here is the question from 1999:

Year 2000Why isn't the year 2000 a leap year?Since every year is 365.26 days long, wouldn't every 100 years be a double leap year? Does it have to do with some weird mathematical process? Thanks.

There are a couple misunderstandings here to be untangled!

Doctor Rick answered:

Hi, Ben. I don't know where you got your information, but2000 _will_ be a leap year. The years 1900 and 1800 and 1700 werenotleap years, but 1600 was, and 2000 will be.

That is, 2000 was special, not because it was **not** a leap year, but because, unlike other century years, it **was**!

Here is an interesting Web site with information about the Gregorian calendar (which we use now) and its leap-year rule, which I quote: The Julian and the Gregorian Calendars, by Peter Meyer http://www.magnet.ch/serendipity/hermetic/cal_stud/cal_art.htm "In the Gregorian Calendar a year is a leap year if either (i) it is divisible by 4 but not by 100 or (ii) it is divisible by 400. In other words, a year which is divisible by 4 is a leap year unless it is divisible by 100 but not by 400 (in which case it is not a leap year). Thus the years 1600 and 2000 are leap years, but 1700, 1800, 1900 and 2100 are not."

I now have to access the link via archive.org.

You can think of the rule this way:

- Start with a basic calendar in which every year has 365 days:

no leap years - Change every year divisible by 4 to a leap year:

leap years 1600, 1604, 1608, 1612, 1616, …, 1696, 1700, 1704, … - Now change every year divisible by 100 back to a normal year:

leap years 0000, 1604, 1608, 1612, 1616, …, 1696, 0000, 1704, … - Finally, change every year divisible by 400 to a leap year again:

leap years**1600**, 1604, 1608, 1612, 1616, …, 1696, 0000, 1704, …

So 2000 was the first application of this last rule since the Gregorian calendar began; but that means that my generation missed its only chance to see a multiple-of-4 year that wasn’t a leap year. It’s sort of an “Oops, nothing special here, folks!” sort of rule.

This Web site also differs with you about the length of a year, stating: "The mean solar year during the last 2000 years is365.242 days(to three decimal places)." It is because this figure is slightly _less_ than 365 1/4 days (not greater, as you stated) that it is necessary to _omit_ occasional leap days (rather than add any). To be precise, 3 days are omitted every 400 years. The average length of a calendar year is thus 365 1/4 - 3/400 = 365.25 - 0.0075 = 365.2425, which matches the astronomical figure given above pretty well.

So rule 2 above would raise the length of a year to 365.25 (by adding 1/4 = 0.25 day per year), and rule 3 reduces that by 1/100 = 0.01 day, to 365.24. Then rule 4 raises it by 1/400 = 0.0025 day, to 365.2425 days. To match the current length of a year, it should be 365.2424, so this is good enough for now. (See below for more!)

If the year were 365.26 days long, your calculation would be correct, we would need to insert an extra leap day every 100 years. The calculations I've described don't seem too weird to me, but the matter of defining exactly what a year is turns out to be pretty complicated, as you will see from this Web site.

Be sure to read the whole linked page if you are interested!

### Different ways to measure a year

Ben replied:

Thanks, I am skilled in Math and I didn't really mean weird, sorry 'bout that. I always thought years were 365.26 so it's good to know that they are 365.242. Thanks again.

Doctor Rick explained:

Hi again, Ben. I took no offense at the word "weird"; in fact I wanted to acknowledge thatthe definitions of year and day do get pretty complex and even weird- though they make sense when you get to know them. I did a quick Web search to verify my hunch about your figure for the length of a year. One site I found has a long list of definitions of various astronomical periods: PREDICTABLE PERIODIC EVENTS (Jan Curtis, Alaska Climate Research Ctr.) http://climate.gi.alaska.edu/Curtis/astro1.html

Again we have to link to archive.org.

I will quote the relevant sections: "Earth'sTropical year365.24219 Days "Interval for Earth to return to same equinox. This explains why leap years exist. Leap years also occur only in years when centuries are evenly divisible by four (e.g., 1600, 2000, 2400, etc.). The Gregorian calendar therefore is equal to 365 days 5 hours 49 minutes 12 seconds. "Earth'sSidereal year365.25636 Days "Interval for Earth to return to same fixed star. "Earth'sAnomalistic year365.25964 Days "Interval for Earth to orbit the Sun as measured from its closest point (perihelion) to its return back. This period is slightly less than five minutes longer than the sidereal year because the position of the perihelion point moves along the Earth's orbit by about 1.1 minutes of arc yearly. During this current epoch, the Earth is closest the Sun just after the new year. It will take about 12,500 years for this date to advance six months." In other words, the length of a year depends on whether you are measuring the time for the earth to return to the same place in orbit relative to the stars (sidereal year), or relative to the direction of the earth's tilt (tropical year), or relative to the perihelion of the earth's orbit (anomalistic year).The figure relevant to the calendar is the tropical year, because it relates to the seasons. The figure you know is correct, but it's one of the other kinds of "year."

The tropical year is relevant because the purpose of the calendar is to keep the equinoxes (and therefore the seasons) aligned with the calendar. (The distance from the earth to the sun has only a small effect on the seasons; so at most the change in the perihelion will only make the northern hemisphere summer a little warmer. The bigger effect is that, because the earth moves faster in orbit near perihelion, northern hemisphere winter is currently shorter than southern hemisphere winter, and that will reverse in 12,500 years!)

## The reason for leap years

Now let’s back up to a 1998 question about leap years in general, for a good summary:

Why Do We Have Leap Year? Dear Dr. Math, Why do we need to have one extra day each 4 years? Thanks, Pat

Doctor Rob answered:

This is an astronomical question, but I think I know the answer. In short, the reason is to preserve the alignment of dates on the calendar with the seasons of the year. As the Earth revolves around the Sun, it rotates on its axis. When it has made exactly one orbit around the Sun, it has made 366.2422 rotations on its axis. One of those rotations is accounted for by its revolving about the Sun. (Think of a planet like Mercury for which one side always faces the Sun. After one revolution, it has made one rotation, but the Sun has never set on one side of Mercury, and never risen on the other.) That means that 365.2422 days have elapsed. An ordinary year contains 365 days, not 365.2422 days. Since .2422 is about 1/4, every four years we have fallen behind by almost a full day. If we didn't do anything about this, after 700 years we would have Summer in January and Winter in July! As a result, we insert an extra day, 29 February, to make a Leap Year. This arrangement results in what is called theJulian Calendar, supposedly invented by Julius Caesar (more likely just decreed by him). The average year is 365.25 days under this calendar.

This covers my rules 1 and 2.

If you thought the mention of 366.2422 was wrong, and still don’t get it after it was explained, see here:

One Circle Revolving Around Another

Now we need rules 3 and 4:

Of course .2422 is not exactly 1/4, so we will be drifting a little, even with Leap Years. As a result,every year divisible by 100 is declared *not* to be a leap year. 1900 was not a leap year under this calendar. That means that the average year is 365.24 days, still a little off. To be even more accurate, every year divisible by 400 is declared to be a leap year, after all! Thus 2000 will be a leap year. This system is called theGregorian calendar, since it was established by order of Pope Gregory in 1582. This was only adopted in English-speaking countries in 1752, however, to be made retroactive. In the Gregorian calendar, the average year is 365.2425, which isoff only 3 days every 10000 years. No doubt someone will make more rules to fix even that slight deviation sometime in the future. If you think this is complicated, you should see how the date of Easter is calculated!

Now, how about a Rule 5 to handle that extra little drift?

## Will the Gregorian calendar last forever?

Here’s a question about the future of leap years, from 2004:

Will Zeller's Rule Work Indefinitely? I showed an equation involving Zeller's Rule to a college teacher and he told methe equation may not work for very distant years. He said that it may be impossible to write an equation relating the day of the week to a given year; month; and day of month because the exact value relating the two may haveirrational numbersinvolved. His statement came as a shock to me; I thought that since the use of the rounding down function is used throughout "Zeller's Rule," the equation would work indefinitely. Who's right? Is the equation sure to work in 4561? Indefinitely?

Zeller’s Rule is a formula for finding the day of the week, which encapsulates the rules of the Gregorian calendar. We’ll be examining Zeller’s Rule next week; the question here is really about the calendar itself, not the formula. I replied to this intriguing question:

Hi, Hunter. Zeller's formula exactly corresponds to the Gregorian calendar, andwill work as long as that calendar is used. It is true that, over a very long time, that calendar would need further adjustment, just as the Julian calendar did; but until a new calendar is defined, you can't say the formula is wrong! The point is thata calendar is not a measurement of reality, but a legal concept established by law, and therefore it remains valid as long as, but ONLY as long as, the law is in effect. So the formula you havedoes not refer to the astronomically measured length of a year, and does not depend on physical reality for its accuracy; on the other hand, it depends on the whim of governments, so no one can really say how long it will remain valid!

There are several layers of reality here. The rules for the calendar, as we’ve seen, are intended to keep the calendar in sync with the solar system; and the formula is just an embodiment of those rules. The solar system gives us certain parameters that we can’t change, which accounts for the need for an approximation that, in its several rules, works something like a decimal approximation to an irrational number, each digit (or rule) getting us closer to what we need, and keeping it accurate to within a day for longer and longer time periods. But beneath this we have the fact that those parameters gradually change, so that we might eventually need to change the calendar because the year is no longer 365.2422 days long. That’s a separate issue I didn’t touch, because it’s about reality, not math.

A calendar is onlya way to fit a whole number of days into each year, while staying as close as possible to what is, as you were told, a theoretically irrational number of days per year astronomically (and also somewhat variable, and subject to errors in measurement). Thereforeno completely regular calendar can be exactly correct forever; but then, in a sense, no calendar is really exact anyway, since the whole point is to approximate the year with whole numbers. It's just that an extra day will have to be added or dropped eventually. What is surprising is that such a simple set of rules (and therefore a simple calculation) happens to be able to do such a good job of approximating the physical length of a year.

It’s easy to imagine a world in which we’d need a leap day, say, after 3 years, and then again after 4 years, and then an extra leap day after 17 years, or whatever! All the 4’s and 100’s make it impressively simple.

The second link below has been taken down (ironically, because “astronomy and astrophysics knowledge evolves”), so it’s good that I copied the part that mattered:

Our Calendar FAQ has links to several sites about calendars; here is one that explains the Gregorian calendar with links to other details, historical and astronomical: Gregorian Calendar http://scienceworld.wolfram.com/astronomy/GregorianCalendar.html This site summarizes how the rules were changed from the Julian to the Gregorian calendar, and mentions a proposed additional rule that would keep it accurate for 20,000 years: Calendars http://csep10.phys.utk.edu/astr161/lect/time/calendars.html However, the Julian year still differs from the true year of 365.242199 days by 11 minutes and 14 seconds each year, and over a period of 128 years even the Julian Calendar was in error by one day with respect to the seasons. By 1582 this error had accumulated to 10 days and Pope Gregory XIII ordered another reform: 10 days were dropped from the year 1582, so that October 4, 1582, was followed by October 15, 1582. In addition, to guard against further accumulation of error, in the new Gregorian Calendar it was decreed that century years not divisible by 400 were not to be considered leap years. Thus, 1600 was a leap year but 1700 was not. This made the average length of the year sufficiently close to the actual year that it would take 3322 years for the error to accumulate to 1 day.

The Julian calendar, with its average year of 365.25 days, was off by 0.007801 = 1/128 of a day, so it gained a day every 128 years. The Gregorian calendar averages 365.2425 days, and is therefore off by 0.000301 = 1/3322 of a day.

### Making it last

A further modification to the Gregorian Calendar has been suggested:years evenly divisible by 4000 are not leap years. This would reduce the error between the Gregorian Calendar Year and the true year to 1 day in 20,000 years. However, this last proposed change has not been officially adopted; there is plenty of time to consider it, since it would not have an effect until the year 4000. That is, the length of a year in the Julian calendar was 365 + 1/4 - 1/100 = 365.24 (off by 0.002199 days, or 1 day in 454 years) and in the Gregorian calendar is 365 + 1/4 - 1/100 + 1/400 = 365.2425 (off by 0.000301, or 1 day in 3322 years) while with the 4000 year rule it will be 365 + 1/4 - 1/100 + 1/400 - 1/4000 = 365.24225 (off by 0.000051, or 1 day in 19,607 years)

So there’s our Rule 5:

5. change every year divisible by 4000 back to a normal year:

leap years **4000**, 4004, 4008, 4012, 4016, …, 4096, 0000, 4104, …

For a reference to this “Herschel proposal”, see Wikipedia.

Given that such a simple addition (which has not been made only because it is not needed yet) would fix the Gregorian calendar so effectively, we can safely say that you can use Zeller's formula up to the year 4000. After that--if the change is actually made in law-- you can just add a term to the formula and keep it correct.

But this assumes the earth’s orbit has not changed much in that time; that’s for astrophysicists, not mathematicians, to discuss.

## A puzzle

We can close with this little puzzle, whose answer relates to leap years:

When Were They Born? Two people celebrate their birthdays on the same day this June. One of them is exactly 2555 days older than other. In what years were they born?

Doctor Ian provide a hint:

Hi Tina, Note that 2555 = 7 * 365. What this means is that there are no leap years between their birthdays. When is it possible to go seven years without a leap year?

You should be able to answer that now…