## not every 4 years a leap year?

Posted: December 4, 2012 in Mathematics
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I have just found out that not every four years there is a leap year. It maybe true for my lifetime and probably yours, but it may not be true for your kids or grandkids.

Before I go any further, I have to define what a leap year is. A leap year is when there is a Feb 29th in the particular year. For those years that do not have Feb 29th are called common years. The reason that a leap year has an extra day (366 days) is that the planet Earth rotates around the sun in approximately 365.242374 days. So that’s about 365.25 days. And if you multiply this number by 4, you will get an extra day every four years.

$365.25 \times 4 = (365+0.25) \times 4 = 365\times 4+1$

So this is how we were taught in elementary school that whenever the year is a multiple of 4, it is a leap year. For example, year 2000, 2004, 2008, 2012, etc all have the leap day Feb 29th. But if you look further beyond the calendar, you will find that there is no Feb 29th in the year 2100. You can check it in your electronic devices like laptop or iPhone. Certainly 2100 is a multiple of 4, and if you keep adding a 4 in the sequence 2000, 2004, 2008, …, etc; 2100 is in the list. So what the hell is happening?

I looked that up in wikipedia, this is what I found out in pseudocode:

if year modulo 400 is 0 then is_leap_year else if year modulo 100 is 0 then not_leap_year else if year modulo 4 is 0 then is_leap_year else not_leap_year 

In simple English, usually every 4 years is a leap year except for those years that are a multiple of 100 but not a multiple of 400. For example, 2100, 2200, and 2300 are multiple of 100, so those are considered common years. The year 2000 is a multiple of 400, so it is still considered a leap year and so as year 2400. If you can live that long, you will witness 7 consecutive common years: 2097, 2098, 2099, 2100, 2101, 2102, 2103.

Certainly I will be dead by then. But maybe our kids or grandkids can live through the year 2100.