leap year calculation

Question

In order to find leap years, why must the year be indivisible by 100 and divisible by 400? I understand why it must be divisible by 4. Please explain the algorithm.

Answer 1

In general terms the algorithm for calculating a leap year is as follows...

A year will be a leap year if it is divisible by 4 but not by 100. If a year is divisible by 4 and by 100, it is not a leap year unless it is also divisible by 400.

Thus years such as 1996, 1992, 1988 and so on are leap years because they are divisible by 4 but not by 100. For century years, the 400 rule is important. Thus, century years 1900, 1800 and 1700 while all still divisible by 4 are also exactly divisible by 100. As they are not further divisible by 400, they are not leap years

Answer 2

There's an algorithm on wikipedia to determine leap years:

function isLeapYear (year):
    if ((year modulo 4 is 0) and (year modulo 100 is not 0))
    or (year modulo 400 is 0)
        then true
    else false

There's a lot of information about this topic on the wikipedia page about leap years, inclusive information about different calendars.

Answer 3

I'm sure Wikipedia can explain it better than I can, but it is basically to do with the fact that if you added an extra day every four years we'd get ahead of the sun as its time to orbit the sun is less than 365.25 days so we compensate for this by not adding leap days on years that are not divisible by 400 eg 1900.

Hope that helps

Answer 4

You really should try to google first.

Wikipedia has a explanation of leap years. The algorithm your describing is for the Proleptic Gregorian calendar.

More about the math around it can be found in the article Calendar Algorithms.

Answer 5

A quick Google for "calculating leap years" came up with this page, which describes the problem clearly and gives an (admitedly VB) implementation, as the first hit.

Answer 6

Leap years are arbitrary, and the system used to describe them is a man made construct. There is no why.

What I mean is there could have been a leap year every 28 years and we would have an extra week in those leap years ... but the powers that be decided to make it a day every 4 years to catch up.

It also has to do with the earth taking a pesky 365.25 days to go round the sun etc. Of course it isn't really 365.25 is it slightly less (365.242222...), so to correct for this discrepancy they decided drop the leap years that are divisible by 100.

Answer 7

If you're interested in the reasons for these rules, it's because the time it takes the earth to make exactly one orbit around the sun is a long imprecise decimal value. It's not exactly 365.25. It's slightly less than 365.25, so every 100 years, one leap day must be eliminated (365.25 - 0.01 = 365.24). But that's not exactly correct either. The value is slightly larger than 365.24. So only 3 out of 4 times will the 100 year rule apply (or in other words, add back in 1 day every 400 years; 365.25 - 0.01 + 0.0025 = 365.2425).

Answer 8

The length of a year is (more or less) 365.242196 days. So we have to subtract, more or less, a quarter of a day to make it fit :

365.242196 - 0.25 = 364.992196 (by adding 1 day in 4 years) : but oops, now it's too small!! lets add a hundreth of a day (by not adding that day once in a hundred year :-))

364.992196 + 0,01 = 365.002196 (oops, big too big, let's add that day anyway one time in about 400 years)

365.002196 - 1/400 = 364.999696

Almost there now, just play with leapseconds now and then, and you're set.

(Note : the reason no more corrections are applied after this step is because a year also CHANGES IN LENGTH!!, that's why leapseconds are the most flexible solution, see for examlple here)

That's why i guess

Answer 9

There are on average, roughly 365.2425 days in a year at the moment (the Earth is slowing down but let's ignore that for now).

The reason we have leap years every 4 years is because that gets us to 365.25 on average [(365+365+365+366) / 4 = 365.25, 1461 days in 4 years].

The reason we don't have leap years on the 100-multiples is to get us to 365.24 `[(1461 x 25 - 1) / 100 = 365.24, 36,524 days in 100 years.

Then the reason we once again have a leap year on 400-multiples is to get us to 365.2425 [(36,524 x 4 + 1) / 400 = 365.2425, 146,097 days in 400 years].

I believe there may be another rule at 3600-multiples but I've never coded for it (Y2K was one thing but planning for one and a half thousand years into the future is not necessary in my opinion - keep in mind I've been wrong before).

So, the rules are, in decreasing priority:

• multiple of 400 is a leap year.
• multiple of 100 is not a leap year.
• multiple of 4 is a leap year.
• anything else is not a leap year.