CTK Exchange
Front Page
Movie shortcuts
Personal info
Awards
Reciprocal links
Terms of use
Privacy Policy

Interactive Activities

Cut The Knot!
MSET99 Talk
Games & Puzzles
Arithmetic/Algebra
Geometry
Probability
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
My Logo
Math Poll
Other Math sit's
Guest book
News sit's

Recommend this site

Manifesto: what CTK is about |Store| Search CTK Buying a book is a commitment to learning Table of content Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

CTK Exchange

Subject: "Leap Year"     Previous Topic | Next Topic
Printer-friendly copy     Email this topic to a friend    
Conferences The CTK Exchange High school Topic #176
Reading Topic #176
saideep
Member since Jun-1-02
Jun-03-02, 03:19 PM (EST)
Click to EMail saideep Click to send private message to saideep Click to view user profileClick to add this user to your buddy list  
"Leap Year"
 
   Why the years 1900, 1800, 1700 etc. are not considered to be leap years?
T SAIDEEP


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top

  Subject     Author     Message Date     ID  
  RE: Leap Year alexbadmin Jun-03-02 1
  RE: Leap Year Vladimir Aug-11-03 2
     RE: Leap Year Graham C Aug-20-03 3
         RE: Leap Year Vladimir Aug-20-03 4
             RE: Leap Year Graham C Aug-25-03 6
                 RE: Leap Year Vladimir Aug-26-03 7
                     RE: Leap Year bluediamond Aug-26-03 8
                         RE: Leap Year Vladimir Aug-27-03 9
     RE: Leap Year Vladimir Aug-23-03 5

Conferences | Forums | Topics | Previous Topic | Next Topic
alexbadmin
Charter Member
1051 posts
Jun-03-02, 03:25 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
1. "RE: Leap Year"
In response to message #0
 
   Check this

https://es.rice.edu/ES/humsoc/Galileo/Things/gregorian_calendar.html

Search the page for the word "leap".


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
Vladimir
Member since Jun-22-03
Aug-11-03, 00:36 AM (EST)
Click to EMail Vladimir Click to send private message to Vladimir Click to view user profileClick to add this user to your buddy list  
2. "RE: Leap Year"
In response to message #0
 
   LAST EDITED ON Aug-20-03 AT 05:17 PM (EST)
 
The solar year equals to 365.2422 days - not an integer. Using 1 leap year of 366 days for every 3 years of 365 days gives you the average length of the calendar year

(3365 + 366)/4 = 365.25 days

(pretty close), which appeared good enough to Julius Ceasar, who introduced this calendar in the ancient Roman empire Therefore, it is called the Julian calendar. At the time, the the spring equinox was on March 25. A few centuries later - in the year 325, the Julian calendar was adopted by the Christian Church. By this time, the spring equinox was on March 21. However, in everal centuries the Julian calendar got considerably out of step with the Sun:

(365.25 - 365.24220) = 0.0078 days/year
(365.25 - 365.24220)100 = 0.78 days/century
(365.25 - 365.24220)1000 = 7.8 days/millenium

By the end of the 16th century (1582), the difference was (1582 - 325)0.00788 10 days. The spring equinox, which is supposed to occur on or very close to every March 21, was occuring 10 days earlier, on March 11. An accurate solar calendar is crucial for agriculture - farmers have to know when to sow. To put the calendar back in step with the Sun, Pope Gregory 23th skipped 10 days and shortened the calendar year by not making leap years the century years not divisible by 400 (including 1700, 1800, 1900, but not 2000). Therefore, it is called the Gregorian calendar. The average calendar year is now

(365.25 * 400 - 3)/400 = 365.25 - 3/400 = 365.2425 days

and the Gregorian calendar loses

(365.2425 - 365.24220)1000 = 0.3 days/millenium

on the solar millenium. Of course, we could still improve the calendar by not making leap years the millenia years divisible by 4000. Such improved calendar would have the average calendar year

(365.2425 * 4000 - 1)/4000 = 365.2425 - 1/4000 = 365.24225 days

and it would lose only

(365.24225 - 365.24220)1000 = 0.05 days/millenium
(365.24225 - 365.24220)20000 = 1 day/20 millenia

on the solar millenium resp. 20 solar millenia.

By the end of the 16th century, the Christian Church had been split by the Reformation, and Protestants of those times would rather do the exact opposite of what the Pope said. Countries with predominant Protestant denominations and countries with predominant Ortodox Church accepted the Gregorian calendar at various later times, for example the United States after the Declaration of Independence. Russia was the last - she accepted the Gregorian calendar shortly after the Bolshevic revolution in 1917. By this time, the Julian calendar lost another 3 days, for a total of 13 days, which had to be skipped.

The only thing that remains to explain is the name "leap year". Since 365 = 1 mod 7, we have 52 weeks plus 1 day in a normal year. The days of week march forward by 1 day for the same dates in 2 subsequent normal years. For example, if July 4 falls on Sunday in a normal year, it will fall on Monday in the next normal year. In a leap year we have 52 weeks plus 2 days. Again, if July 4 falls on Sunday in a normal year, the day of week leaps over Monday if the next year has 366 days and July 4 will be on Tuesday.


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
Graham C
Member since Feb-5-03
Aug-20-03, 05:11 PM (EST)
Click to EMail Graham%20C Click to send private message to Graham%20C Click to view user profileClick to add this user to your buddy list  
3. "RE: Leap Year"
In response to message #2
 
   The Western calendar is still not as accurate as the Persian one.


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
Vladimir
Member since Jun-22-03
Aug-20-03, 08:20 PM (EST)
Click to EMail Vladimir Click to send private message to Vladimir Click to view user profileClick to add this user to your buddy list  
4. "RE: Leap Year"
In response to message #3
 
   LAST EDITED ON Aug-20-03 AT 09:50 PM (EST)
 
This is not correct, sir. We know the solar year more accurately than anybody else in history. The calendar year in a particular time zone must start and end at midnight, i.e., the calendar year must have an integer number of days. We can always skip a leap year if the calendar falls out of the step with the solar year by more than, say, 0.5 of a day. This flexibility allows the calendar to be in step with the Sun 0.5 day and you cannot do better than that.

Babylonian calendar was a kind of mix between solar and lunar calendars. It was based on the accidental fact that 235 lunar months (23529.5306 days = 6939.691 days) is approximately equal to 19 solar years (19365.2422 = 6939.602 days). This calendar lost about 0.089100/19 = 0.47 days/century to the solar century, which is less than the Julian calendar (0.78 days/century) but more than the Gregorian calendar (0.03 days/century). Moreover, the length of year was jumping around in a fixed pattern during this 19-year cycle, because 1 lunar month had to be added to 7 out of the 19 years. The calendar was periodicaly gaining on and falling behind the Sun and it was never more than about 15 days (half a month) out of step with the Sun.

Moslem calendar, on the other hand, is strictly a lunar calendar. It is based on the accidental fact that 30 lunar years (12 lunar month each) almost exactly equal to an integer number of days: 301229.5306 = 10631.016 days. Compare this with 30 lunar calendar years, where the length of a calendar month alternates between 29 and 30 days, for 29.5 days on the average. 30 lunar calendar years equal to 301229.5 = 10620 days, which is just 11 days short of the 30 true lunar years. Again, these 11 days are scattered through the 30 lunar calendar years in a fixed pattern. The calendar gains 0.016100/30 = 0.053 days to a lunar century (approximately 97 solar years). The Gregorian calendar with a loss of 0.03 days/century is still better than that.


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
Graham C
Member since Feb-5-03
Aug-25-03, 11:03 PM (EST)
Click to EMail Graham%20C Click to send private message to Graham%20C Click to view user profileClick to add this user to your buddy list  
6. "RE: Leap Year"
In response to message #4
 
   None of what you say is connected to the Persian calendar, which has incidentally currently been restored in Afghanistan, replacing the Muslim calendar which the Taleban had enforced.

You might like to google for it, perhaps including Omar Khayyam in the search terms, since he is popularly supposed to have invented it.

I'm currently holidaying in the US, so don't have my reference books with me, so can't add much more now.


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
Vladimir
Member since Jun-22-03
Aug-26-03, 08:10 AM (EST)
Click to EMail Vladimir Click to send private message to Vladimir Click to view user profileClick to add this user to your buddy list  
7. "RE: Leap Year"
In response to message #6
 
   The word Persian reminded me of history, while the present day countries in the area are called Iran, Afganistan, etc. I did not know about any modern Persian calendar.

Persian calendar
The modern Persian calendar uses quite complicated leap year rules, defining a 2820-year cycle with 683 leap years, which results a in mean length of a year of 365 683/2820 = 365,2422 days. Considering the length of the tropical year as being constant, the remaining error would amount to a day in more than 2 million years.

The 2820-year cycle is divided into 21 subcycles of 128 years each, and a 132-year subcycle at the end of each 2820-year cycle. A 128-year subcycle consists of a 29-year sub-subcycle, followed by 3 sub-subcycles of 33 years each. Finally, the 132-year subcycle consists of one sub-subcycle of 29 years, followed by two 33-year sub-subcycles and a final sub-subcycle of 37 years.
--------------------------------------------------------------------

The 1st paragraph stands, is relevant to any calendar, including Persian, and I repeat it:

We know the solar year more accurately than anybody else in history. The calendar year in a particular time zone must start and end at midnight, i.e., the calendar year must have an integer number of days. We can always skip a (century) leap year if the calendar falls out of the step with the solar year by more than, say, 0.5 of a day. This flexibility allows the calendar to be in step with the Sun 0.5 day and you cannot do better than that.

Compare this to the complicated and rigid pattern of leap years in the Persian calendar. Moreover, making calendar for more than a few millenia is quite unjustified, certainly with the present experimental data (which is better than any other experimental data in history). We even know how the Earth rotation is slowing down - about 5.33 sec/millenium at present:

Scientific article:
One of the latest (1990) experimetal formula for the length of tropical year is

t = 365.242189669781 - 6.16187010-6T - 6.4410-10T2.

where T is the time reckoned from the year 2000 and measured in Julian centuries of 365.25 ephemeris days. The formula valid over about 8000 years centered at the present. A comparison of the Gregorian calendar with a perfect solar calendar suggests that the former will be adequate at least during the nearest one to two thousand years. Due to high uncertainty in the Earth rotation it is premature at present to suggest any reform that would reach further than a few thousand years into the future.


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
bluediamond
guest
Aug-26-03, 02:13 PM (EST)
 
8. "RE: Leap Year"
In response to message #7
 
   I'm not sure that a "flexible" calendar is such a good thing. It'seems to be about the same as no real calendar at all; just looking at the sun and deciding that today is a good day to start the new year. The point of a calendar is its rigidity; if one understands the calendar system, then one can figure out the exact date 2000 years ago, and one can figure out the exact date 2000 years in the future. Well "years" is a bad day, but my point is that a calendar should let us say that 2,000,000 days from today, it will be the 20th day of the fifth month of the year 50000 (or whatever). Saying that the date will be 20-5-50000 plus or minus however many days we decide to skip along the way is somewhat contradictory to the utility of a calendar, in my opinion. I know this doesn't help with the changes in Earth's revolution or orbit or whatnot, but then why bother with a calendar at all? We could just name days according to the position of the Sun, determining each as we go along, and use the 7 day week for planning.

Hold on, I think I'm on to something here.


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
Vladimir
Member since Jun-22-03
Aug-27-03, 02:07 PM (EST)
Click to EMail Vladimir Click to send private message to Vladimir Click to view user profileClick to add this user to your buddy list  
9. "RE: Leap Year"
In response to message #8
 
   LAST EDITED ON Aug-28-03 AT 02:26 PM (EST)
 
All right. The whole business of proposing an accurate calendar in step with the sun is to find 2 integers L and P such that

L/P t - 365 0.2421897 days

as close as possible. Then P is the period of the calendar in years and L is the number of leap years in the calendar period P. There are some reasonable requirements on the calendar period:

1. The period should be comparable to or greater than the average human lifespan. Then the exception from the simple rule of the Julian calendar "one leap year in 4 years" occurs at most once in anyone's lifetime.

2. The simple rule of the Julian calendar "years divisible by 4 are leap years" should be valid in any particular period. Then even if the exception from the rule "one leap year in 4 years" occurs someone's lifetime, things are the same after the exception.

3. The period should preferably be a simple round number. Then the calculations of remote dates and time periods between 2 remote dates are simple.

The Gregorian calendar uses L = 97 and P = 400, L/P = 0.2425. All 3 requirements are satisfied, although one could do better on the requirement #1 - the accuracy is a loss of 0.031 days/millenium = 7 hours 27 minutes/millenium.

The modern Persian calendar was proposed in 1925. At that time, the best experimental value of the solar year length was 365.242199 days. So the 2 integers L and P of the Persian calendar are selected as L = 683, P = 2820 years, L/P = 0.2421986. The accuracy (requirement #1) is excellent - loss of 12 min 49 sec/millenium. However, neither of the remaining 2 requirements are satisfied (128 and 132 years cycles, 29, 33, and 37 years subcycles).

It turns out that the best present day experimental value for the solar year length is a little shorter - about 365.2421897 days. Let's propose a new CTK Exchange calendar. On the first attempt, L = 15, P = 62 years, L/P = 0.2421935 - after 60 years of "one leap year in 4 years", there is a gap of 1 leap year in 6 years. The accuracy is excellent (loss of 5 min 28 sec/millenium, better than the Persian calendar), the period is simple and comparable to the average human lifespan. However, the rule "years divisible by 4 are the leap years" or "double even years are leap years" has to alternate with the rule "single even years are leap years" every 62 years, which would be inconvenient. The correct this defect, let's double the calendar period to P = 124 years and the number of leap years to L = 30. You simply skip a leap year every 124 years. What could be simpler than that?


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top
Vladimir
Member since Jun-22-03
Aug-23-03, 06:01 PM (EST)
Click to EMail Vladimir Click to send private message to Vladimir Click to view user profileClick to add this user to your buddy list  
5. "RE: Leap Year"
In response to message #2
 
   There is still a simple and better way to improve the Gregorian calendar by not making leap years the century years divisible by 3200. Such improved calendar would have the average calendar year

(365.2425 * 3200 - 1)/3200 = 365.2425 - 1/3200 = 365.2421875 days

and it would gain only

(365.24220 - 365.2421875)1000 = 0.0125 days/millenium = 18 minutes/millenium

to the solar millenium and a one full day to 80 solar millenia:

(365.24220 - 365.2421875)80000 = 1 day/80 millenia


  Alert | IP Printer-friendly page | Edit | Reply | Reply With Quote | Top

Conferences | Forums | Topics | Previous Topic | Next Topic

You may be curious to have a look at the old CTK Exchange archive.
Please do not post there.

|Front page| |Contents|

Copyright © 1996-2018 Alexander Bogomolny

66162481