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What years are leap years? What day of the week will it be, a year from today? How do I find the day of the week for any date? How do I find a calendar for any year? How common are Friday the 13ths?
What years are leap years?
It takes the earth about 365.2422 days to go around the sun, but a normal calendar year is only 365 days. The extra fraction of a day adds up: circling the sun four times takes 1460.9688 days, but four calendar years would only be 1460 days. That .9688 is almost a whole day, so every four years we add an extra day to our calendar, February 29. We call that year leap year. To make things easier, leap years are always divisible by four: 2004 and 2008 will both be leap years.
For hundreds of years, people used a calendar called the Julian calendar that followed this rule, adding a leap year every four years. However, because .9688 isn't exactly a whole day, the Julian calendar slowly began to disagree with the real seasons. In 1582, Pope Gregory fixed this problem by ordering everyone to use a new set of rules. These rules are named the Gregorian calendar, after him. They work like this:
People in English-speaking countries didn't start using the Gregorian calendar until 1752. Some countries, such as Iran, still use other systems.
From the Dr. Math archives:
From the Web:
What day of the week will it be, a year from today?
Suppose that today is February 13, 2053. The day of the week is a Thursday. One week (seven days) from today, on February 20, 2053, it will be Thursday again. After two weeks (fourteen days), it will be Thursday once more. We need to figure out how many weeks and days will have passed after a year. 2053 is not a leap year, so 365 days will pass between February 13, 2053 and February 13, 2054. Let's divide by 7 to find out how many weeks that is: 365 / 7 = 52, remainder 1, or fifty-two weeks with one day left over. Fifty-two weeks after February 13, 2053 is yet another Thursday, so fifty-two weeks and one day later must be a Friday.
We can use the same method for any date, but we have to be careful. Because some years are leap years, a year from today might be 366 days and not 365. For instance, there are 366 days between Saturday, November 20, 2055 and November 20, 2056, because 2056 is a leap year and February 29, 2056 lies between the two Novembers.
How do I find the day of the week for any date?
There are two popular formulas that you can use to find the day of the week for a given date. You should be careful when you use these formulas, though, because they only work for the Gregorian calendar. (People in English-speaking countries used a different calendar before September 14, 1752.)
The following formula is named Zeller's Rule after a Reverend Zeller. [x] means the greatest integer that is smaller than or equal to x. You can find this number by just dropping everything after the decimal point. For example, [3.79] is 3. Here's the formula:
f = k + [(13*m-1)/5] + D + [D/4] + [C/4] - 2*C.
Now let's substitute our example numbers into the formula.
f = k + [(13*m-1)/5] + D + [D/4] + [C/4] - 2*C
Once we have found f, we divide it by 7 and take the remainder. Note that if the result for f is negative, care must be taken in calculating the proper remainder. Suppose f = -17. When we divide by 7, we have to follow the same rules as for the greatest integer function; namely we find the greatest multiple of 7 less than -17, so the remainder will be positive (or zero). -21 is the greatest multiple of 7 less than -17, so the remainder is 4 since -21 + 4 = -17. Alternatively, we can say that -7 goes into -17 twice, making -14 and leaving a remainder of -3, then add 7 since the remainder is negative, so -3 + 7 is again a remainder of 4.
A remainder of 0 corresponds to Sunday, 1 means Monday, etc. For our example, 100 / 7 = 14, remainder 2, so January 29, 2064 will be a Tuesday.
The Key Value Method
This method uses codes for different months and years to speed up the calculation of the day of the week. You might even be able to memorize the codes. We'll use December 16, 2482 as an example.