Mark.g Switch

Ever wondered why a year has 365 or 366 years? Why is it every 4 years there is 1 day added to February to make it 366 years?

If 1 day equals one turn of the earth, and 1 month equals 1 orbit of the moon, then 1 year is one full orbit of earth around the sun. But why is it that every 4 years the orbit 'LEAPS' 1 day? It does not make sense!

At 15+years old, I have thought about this problem and came to one inevitable explanation that I did not bother verifying because I knew I was right. I knew that 1 year does not have 365days or 366days BUT more accurately 365.25 days. Hence, every a quarter day adds up to 1 day every 4 years. Since then I did not bother checking my explanation because I knew that I was right - BUT today when I read about it online I have not only found out that I was right but I learn something MORE! I was very close - 1 year has actually 365.2422 days!

Now you may think that the argument ends here and that the 3rd n 4th decimal places could be neglected - guess what? WRONG! WRONG! WRONG! WRONG!

According to the Gregorian Calendar 1 year equals to 365.2422 days and the decimal places have a very key importance at maintaining timekeeping accuracy.

Every one knows that every 4 years we will have a leap year of 366 days - but how many of you people knows that there are restrictions to this rule!

According to TimeandDAte.com:

In the Gregorian calendar, the calendar used by most modern countries, the following three criteria determine which years will be leap years:

1. Every year that is divisible by four is a leap year;
2. of those years, if it can be divided by 100, it is NOT a leap year, unless
3. the year is divisible by 400. Then it is a leap year.

What does this means? Well it means that 1800, 1900, 2100, 2200, etc IS NOT a leap year because it is divisible by 100 and not by 400 although they are divisible by 4. This also means that the year 2000 which occurred only recently is a leap year because it is divisible by 4, 100 and 400.

Why is this so I wondered? And within 30 minutes I manage to make a quick calculation and verify the reason why.

Note that 365.2422 days makes a year and not 365.25 days. Which the latter would coincidentally result that every 4 years we can add EXACTLY 1 day into the year because 4 x 0.25 = 1 day. But obviously this is not the case. We have 0.2422 extra days and not 0.25 days - therefore 4 x 0.2422 = 0.9688 days which is NOT exactly 1 day and after four years we are (1 - 0.9688 = 0.0312) days leading ahead the actual planetary orbit. Therefore we need to 'recalibrate' our calendar such that it compensate for this gaining of 0.0312 days every four years. How do we do that? We wait until this 0.0312 days accumulate to 1 day and since we are gaining 0.0312 days every 4 calendar year we have to MINUS it from our calendar again to keep it accurate to our planetary orbit. How do we do this? The same way! So how many of 0.0312 days do we need to have so that it equals to 1 day? We need 1/0.0312 = 32.0513 of it and if we take the closest 2 digit it is 32 times. And since 0.0312days is added in 4 years then 32 times of it means 32 x 4 = 128years. So every 128years we need to MINUS 1 day. BUT of course we do not like to deal with 128 because it would mean that we have to COUNT in 128 year cycle which is horribly confusing. Imagine if that 1800 is the beginning then the year (1800+128 = 1928) we have to minus 1 day, and again at (1928+128=2056) and again at (2056+128) and so forth. Hence to simplify things we round it to 100 years. And every 100 years we have to minus 1 day which falls on 1800 1900 2000 2100 (MUCH EASIER). However, cutting the remaining 28years means we lose accuracy again! Therefore we have to recompensate this again! Let's calculate this. In 100 years there are 25 leap year cycle and there are 32 leap year cycle in 128 years - we also concluded that in every leap year cycle we GAIN 0.0312 days. In 128 years (32 leap year cycle) has a gain of 0.0312 x 32 = 0.9984 days and in 100 years (25 leap year cycle) has a gain of only 0.0312 x 25 = 0.7800 days. So if we MINUS 1 day in 100 years it would mean that we are now LAGGING BEHIND the planetary orbit - and by how much? By 1-0.78 = 0.22days! So every 100 years we are going to LAG 0.2200 days because we MINUS 1 whole day when the calendar has only gained 0.7800 days! This explains the reason why we do not consider the years 1800 1900 2100 2200 2300 as leap years! But the inaccuracy of 0.22days ALSO presents incompatibility with the planetary orbits! So how we fix this? We recalibrate our calendar once again using the same technique! Now that we are lagging 0.22days EVERY 100 years we need to ADD IT UP again when this 0.22days over 100 years accumulate to 1 day. Mathematically again, 1/0.22 = 4.545454 times and since 0.22days is lost every 100 years and it needed 4.5454 times of 0.22days to gain 1 day therefore we need 4.5454 x 100 = 454 years to loss 1 day such that we can add it back into the calendar! But of course we do not want to count in a cycle of 454 years again therefore we use a cycle of 400 years. Of course 400 years only equals to only 4 times of 0.22days lost every 100 years which only totals up to 0.88days. And if we add 1 whole day to the year it means that we are going to be LEADING by 1 - 0.88 = 0.12 days! And that is the reason why we do have leap year in 1600, 2000, 2400, 2800, etc which are divisible by 400. Of course this 0.12days presents inaccuracy as well and if we were to continue we need to minus 1 day again in every 3333.3333 years (Calculation: 1/0.12=8.333, 8.333 x 400 = 3333.333 years) but we do not want to count 3333.333 years cycle so we use 3000 years. Therefore in every 3000 year cycle we need to MINUS 1 day. But of course this year cycle already gets to the range where it would be longer than our lifespan of existence probably.

The end. I hope you could follow the math up to this point.