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The ISO week date system is a leap week calendar system that is part of the ISO 8601 date and time standard. The system is used (mainly) in government and business for fiscal years, as well as in timekeeping. The system specifies a week year atop the Gregorian calendar by defining a notation for ordinal weeks of the year.
The Gregorian leap cycle, which has 97 leap days spread across 400 years, contains a whole number of weeks (20871). In every cycle there are 71 years with an additional 53rd week. An average year is exactly 52.1775 weeks long; months average at 4.348125 weeks.
An ISO weeknumbering year (also called ISO year informally) has 52 or 53 full weeks. That is 364 or 371 days instead of the usual 365 or 366 days. The extra week is referred to here as a leap week, although ISO 8601 does not use this term. Weeks start with Monday. The first week of a year is the week that contains the first Thursday of the year (and, hence, 4 January). ISO week year numbering therefore slightly deviates from the Gregorian for some days close to 1 January.
A date is specified by the ISO weeknumbering year in the format YYYY, a week number in the format ww prefixed by the letter 'W', and the weekday number, a digit d from 1 through 7, beginning with Monday and ending with Sunday. For example, the Gregorian date 31 December 2006 corresponds to the Sunday of the 52nd week of 2006, and is written 2006W527 (extended form) or 2006W527 (compact form).
December 2014  
Wk  Mo  Tu  We  Th  Fr  Sa  Su 
(49)  1  2  3  4  5  6  7 
(50)  8  9  10  11  12  13  14 
(51)  15  16  17  18  19  20  21 
(52)  22  23  24  25  26  27  28 
(1)  29  30  31  1  2  3  4 
The ISO weeknumbering year number deviates from the number of the Gregorian year on, if applicable, a Friday, Saturday, and Sunday, or a Saturday and Sunday, or just a Sunday, at the start of the Gregorian year (which are at the end of the previous ISO year) and a Monday, Tuesday and Wednesday, or a Monday and Tuesday, or just a Monday, at the end of the Gregorian year (which are in week 01 of the next ISO year). In the period 4 January to 28 December and on all Thursdays the ISO weeknumbering year number is always equal to the Gregorian year number.
Date  Notes  

Vulgar  ISO  
Sat 1 Jan 2005  20050101  2004W536  
Sun 2 Jan 2005  20050102  2004W537  
Sat 31 Dec 2005  20051231  2005W526  
Mon 1 Jan 2007  20070101  2007W011  Both years 2007 start with the same day. 
Sun 30 Dec 2007  20071230  2007W527  
Mon 31 Dec 2007  20071231  2008W011  
Tue 1 Jan 2008  20080101  2008W012  Gregorian year 2008 is a leap year. ISO year 2008 is 2 days shorter: 1 day longer at the start, 3 days shorter at the end. 
Sun 28 Dec 2008  20081228  2008W527  ISO year 2009 begins three days before the end of Gregorian 2008. 
Mon 29 Dec 2008  20081229  2009W011  
Tue 30 Dec 2008  20081230  2009W012  
Wed 31 Dec 2008  20081231  2009W013  
Thu 1 Jan 2009  20090101  2009W014  
Thu 31 Dec 2009  20091231  2009W534  ISO year 2009 has 53 weeks and ends three days into Gregorian year 2010. 
Fri 1 Jan 2010  20100101  2009W535  
Sat 2 Jan 2010  20100102  2009W536  
Sun 3 Jan 2010  20100103  2009W537 
The ISO 8601 definition for week 01 is the week with the year's first Thursday in it. Mutually equivalent definitions would be possible based on the following properties of this week:
If 1 January is on a Monday, Tuesday, Wednesday or Thursday, it is in week 01. If 1 January is on a Friday, it is part of week 53 of the previous year; if on a Saturday, it is part of week 52 (or 53 if the previous year was a leap year); if on a Sunday, it is part of week 52 of the previous year.
The last week of the ISO weeknumbering year, i.e. the 52nd or 53rd one, is the week before week 01. This week’s properties are:
If 31 December is on a Monday, Tuesday or Wednesday, it is in week 01 of the next year. If it is on a Thursday, it is in week 53 of the year just ending; if on a Friday it is in week 52 (or 53 if the year just ending is a leap year); if on a Saturday or Sunday, it is in week 52 of the year just ending.
The long years, with 53 weeks in them, can be described by any of the following equivalent definitions:
All other weeknumbering years are short years and have 52 weeks.
The number of weeks in a given year is equal to the corresponding week number of 28 December.
On average, a year has 53 weeks every 5.6338… years (= 7 / [365.2425 − 52×7] = 400 / 71).
The following 71 years in a 400year cycle (add 2000 for current years) have 53 weeks (leap years, with February 29, are emphasized), years not listed have 52 weeks:
These long ISO years are 43 times 6 years apart, 27 times 5 years apart, and once 7 years apart (between years 296 and 303).
The Gregorian years corresponding to these 71 long years can be subdivided as follows:
The Gregorian years corresponding to the other 329 short ISO years (neither starting nor ending with Thursday) can also be subdivided as follows:
Thus, within a 400year cycle:
The ISO standard does not define any association of weeks to months. A date is either expressed with a month and dayofthemonth, or with a week and dayoftheweek, never a mix.
Weeks are a prominent entity in accounting where annual statistics benefit from regularity throughout the years. Therefore in practice usually a fixed length of 13 weeks per quarter is chosen which is then subdivided into 5 + 4 + 4 weeks, 4 + 5 + 4 weeks or 4 + 4 + 5 weeks. The final quarter has 14 weeks in it when there are 53 weeks in the year.
When it is necessary to allocate a week to a single month, the rule for first week of the year might be applied, although ISO 8601 does not consider this case. The resulting pattern would be irregular. The only 4 months (or 5 in a long year) of 5 weeks would be those with at least 29 days starting on Thursday, those with at least 30 days starting on Wednesday, and those with 31 days starting on Tuesday.
For all years, 8 days have a fixed ISO week number (between 01 and 08) in January and February. And with the exception of leap years starting on Thursday, dates with fixed week numbers occurs on all months of the year (for 1 day of each ISO week 01 to 52) :
Month  Dates  Week numbers  

January  04  11  18  25  01–04  
February  01  08  15  22  05–08  
March  01  08  15  22  29  09–13 
April  05  12  19  26  14–17  
May  03  10  17  24  31  18–22 
June  07  14  21  28  23–26  
July  05  12  19  26  27–30  
August  02  09  16  23  30  31–35 
September  06  13  20  27  36–39  
October  04  11  18  25  40–43  
November  01  08  15  22  29  44–48 
December  06  13  20  27  49–52 
During leap years starting on Thursday (i.e. the 13 years number 004, 032, 060, 088, 128, 156, 184, 224, 252, 280, 320, 348, 376 in a 400year cycle), the ISO week numbers are incremented by 1 from March to the rest of the year (this last occurred in 1976 and 2004 and will not occur before 2032; these exceptions are happening between years that are most often 28 years apart, or 40 years apart for 3 pairs of successive years: from year 088 to 128, from year 184 to 224, and from year 280 to 320).
The day of the week for these days are related to Doomsday because for any year, the Doomsday is the day of the week that the last day of February falls on. These dates are one day after the Doomsdays, except that in January and February of leap years the dates themselves are Doomsdays. In leap years the week number is the rank number of its Doomsday.
(6)  5  6  7  8  9  10  11 
(10)  5  6  7  8  9  10  11 
(45)  5  6  7  8  9  10  11 
(7)  12  13  14  15  16  17  18 
(11)  12  13  14  15  16  17  18 
(46)  12  13  14  15  16  17  18 
(8)  19  20  21  22  23  24  25 
(12)  19  20  21  22  23  24  25 
(47)  19  20  21  22  23  24  25 
The pairs 02/41, 03/42, 04/43, 05/44, 15/28, 16/29, 37/50, 38/51 and triplets 06/10/45, 07/11/46, 08/12/47 have the same days of the month in common years. Of these, the pairs 10/45, 11/46, 12/47, 15/28, 16/29, 37/50 and 38/51 share their days also in leap years. Leap years also have triplets 03/15/28, 04/16/29 and pairs 06/32, 07/33, 08/34.
The weeks 09, 19–26, 31 and 35 never share their days of the month with any other week of the same year.
Solar astronomic phenomena, such as equinox and solstice, vary over a range of at least seven days. This is because each equinox and solstice may occur any day of the week and hence on at least seven different ISO week dates. For example, there are spring equinoxes on 2004W127 and 2010W117.
The ISO week calendar relies on the Gregorian calendar, which it augments, to define the new year day (Monday of week 01). As a result, leap weeks are spread across the 400year cycle in a complex, seemingly random pattern. There is no simple algorithm to determine whether a year has 53 weeks without tabular lookup. Most calendar reform proposals using leap week calendars are simpler in this regard, although they may choose a different leap cycle.
Not all parts of the world have a work week that begins with Monday. For example, in some Muslim countries, the work week may begin on Saturday, while in Israel it may begin on Sunday. In the US the work week is often defined to start on Monday, although the week itself is usually considered to start on Sunday.
The week number of any date can be calculated, given its ordinal date (i.e. position within the year) and its day of the week. If the ordinal date is not known, it can be computed by any of several methods; perhaps the most direct is a table such as the following.
To the day of:  Jan  Feb  Mar  Apr  May  Jun  Jul  Aug  Sep  Oct  Nov  Dec 

Add:  0  31  59  90  120  151  181  212  243  273  304  334 
For leap years:  0  31  60  91  121  152  182  213  244  274  305  335 
Method: Using ISO weekday numbers (running from 1 for Monday to 7 for Sunday), subtract the weekday from the ordinal date, then add 10. Divide the result by 7. Ignore the remainder; the quotient equals the week number. If the week number thus obtained equals 0, it means that the given date belongs to the preceding (weekbased) year. If a week number of 53 is obtained, one must check that the date is not actually in week 1 of the following year.
Example: Friday 26 September 2008
This method requires that one know the weekday of 4 January of the year in question.^{[1]} Add 3 to the number of this weekday, giving a correction to be used for dates within this year.
Method: Multiply the week number by 7, then add the weekday. From this sum subtract the correction for the year. The result is the ordinal date, which can be converted into a calendar date using the table in the preceding section. If the ordinal date thus obtained is zero or negative, the date belongs to the previous calendar year; if greater than the number of days in the year, to the following year.
Example: year 2008, week 39, Saturday (day 6)
For an overview of week numbering systems see week number.
The US system has weeks from Sunday through Saturday, and partial weeks at the beginning and the end of the year, i.e. always 53 weeks. An advantage is that no separate year numbering like the ISO year is needed. Correspondence of lexicographical order and chronological order is preserved (just like with the ISO yearweekweekday numbering), but partial weeks make some computations of weekly statistics or payments inaccurate at end of December or beginning of January.
A variant of this US scheme groups the possible 1 to 6 days of December remaining in the last week of the Gregorian year within week 1 in January of the next Gregorian year, to make it a full week, bringing a system with accounting years having also 52 or 53 weeks and only the last 6 days of December may be counted as part of another year than the Gregorian year.
The US broadcast calendar counts the week containing 1 January as the first of the year, but otherwise works like ISO week numbering without partial weeks.


