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The ISO week date system is effectively a leap week calendar system that is part of the ISO 8601 date and time standard issued by the International Organization for Standardization (ISO) since 1988 (last revised in 2004) and, before that, it was defined in ISO (R) 2015 since 1971. It is used (mainly) in government and business for fiscal years, as well as in timekeeping. This was previously known as "Industrial date coding". 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 (corresponding to the Gregorian years that contain 53 Thursdays). An average year is exactly 52.1775 weeks long; months (^{1}⁄_{12} year) average at exactly 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 sometimes referred to as a leap week, although ISO 8601 does not use this term.
Weeks start with Monday. Each week's year is the Gregorian year in which the Thursday falls. The first week of the year, hence, always contains 4 January. ISO week year numbering therefore slightly deviates from the Gregorian for some days close to 1 January.
A precise 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 Monday 23 December 2019 corresponds to Monday in the 52nd week of 2019, and is written 2019W521 (in extended form) or 2019W521 (in compact form). The ISO year is slightly offset to the Gregorian year; for example, Monday 30 December 2019 in the Gregorian calendar is the first day of week 1 of 2020 in the ISO calendar, and is written as 2020W011 or 2020W011.
Week  Mon  Tue  Wed  Thu  Fri  Sat  Sun 

W27  29  30  01  02  03  04  05 
W28  06  07  08  09  10  11  12 
W29  13  14  15  16  17  18  19 
W30  20  21  22  23  24  25  26 
W31  27  28  29  30  31  01  02 
Relation with the Gregorian calendar
English short  ISO  

Sat 1 Jan 2005  20050101  2004W536 
Sun 2 Jan 2005  20050102  2004W537 
Sat 31 Dec 2005  20051231  2005W526 
Sun 1 Jan 2006  20060101  2005W527 
Mon 2 Jan 2006  20060102  2006W011 
Sun 31 Dec 2006  20061231  2006W527 
Mon 1 Jan 2007  20070101  2007W011 
Sun 30 Dec 2007  20071230  2007W527 
Mon 31 Dec 2007  20071231  2008W011 
Tue 1 Jan 2008  20080101  2008W012 
Sun 28 Dec 2008  20081228  2008W527 
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 
Fri 1 Jan 2010  20100101  2009W535 
Sat 2 Jan 2010  20100102  2009W536 
Sun 3 Jan 2010  20100103  2009W537 
Notes:

The ISO week year number deviates from the Gregorian year number in one of three ways. The days differing are a Friday through 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 through 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 the ISO week year number is always equal to the Gregorian year number. The same is true for every Thursday.
First week
The ISO 8601 definition for week 01 is the week with the first Thursday of the Gregorian year (i.e. of January) in it. The following definitions based on properties of this week are mutually equivalent, since the ISO week starts with Monday:
 It is the first week with a majority (4 or more) of its days in January.
 Its first day is the Monday nearest to 1 January.
 It has 4 January in it. Hence the earliest possible first week extends from Monday 29 December (previous Gregorian year) to Sunday 4 January, the latest possible first week extends from Monday 4 January to Sunday 10 January.
 It has the year's first working day in it, if Saturdays, Sundays and 1 January are not working days.
If 1 January is on a Monday, Tuesday, Wednesday or Thursday, it is in W01. If it is on a Friday, it is part of W53 of the previous year. If it is on a Saturday, it is part of the last week of the previous year which is numbered W52 in a common year and W53 in a leap year. If it is on a Sunday, it is part of W52 of the previous year.
Dominical letter(s) 
Days at the beginning of January  Effect  Days at the end of December  

1 Mon 
2 Tue 
3 Wed 
4 Thu 
5 Fri 
6 Sat 
7 Sun 
W011^{[a]}  01 Jan week  …  31 Dec week  1 Mon^{[b]} 
2 Tue 
3 Wed 
4 Thu 
5 Fri 
6 Sat 
7 Sun  
G(F)  01  02  03  04  05  06  07  01 Jan  W01  …  W01  31 (30)^{[c]}  (31)  
F(E)  01  02  03  04  05  06  31 Dec  W01  …  W01  30 (29)  31 (30)  (31)  
E(D)  01  02  03  04  05  30 Dec  W01  …  W01 (W53)  29 (28)  30 (29)  31 (30)  (31)  
D(C)  01  02  03  04  29 Dec  W01  …  W53  28 (27)  29 (28)  30 (29)  31 (30)  (31)  
C(B)  01  02  03  04 Jan  W53  …  W52  27 (26)  28 (27)  29 (28)  30 (29)  31 (30)  (31)  
B(A)  01  02  03 Jan  W52 (W53)  …  W52  26 (25)  27 (26)  28 (27)  29 (28)  30 (29)  31 (30)  (31)  
A(G)  01  02 Jan  W52  …  W52 (W01)  25 (31)  26  27  28  29  30  31 
Last week
The last week of the ISO weeknumbering year, i.e. W52 or W53, is the week before W01 of the next year. This week’s properties are:
 It has the year's last Thursday in it.
 It is the last week with a majority (4 or more) of its days in December.
 Its middle day, Thursday, falls in the ending year.
 Its last day is the Sunday nearest to 31 December.
 It has 28 December in it.
Hence the earliest possible last week extends from Monday 22 December to Sunday 28 December, the latest possible last week extends from Monday 28 December to Sunday 3 January.
If 31 December is on a Monday or Tuesday it is in W01 of the next year. If it is on a Wednesday, it is in W01 of the next year in common years and W53 in leap years. If it is on a Thursday, it is in W53 of the year just ending. If on a Friday or Saturday it is in W52 of the year just ending. If on a Sunday, it is in W52 of the year just ending in common years and W01 of the next year in leap years.
01 Jan  W011  Common year (365 − 1 or + 6)  Leap year (366 − 2 or + 5)  

Mon  01 Jan  G  +0  −1  GF  +0  −2 
Tue  31 Dec  F  +1  −2  FE  +1  −3 
Wed  30 Dec  E  +2  −3  ED  +2  +3 
Thu  29 Dec  D  +3  +3  DC  +3  +2 
Fri  04 Jan  C  −3  +2  CB  −3  +1 
Sat  03 Jan  B  −2  +1  BA  −2  +0 
Sun  02 Jan  A  −1  +0  AG  −1  −1 
Weeks per year
The long years, with 53 weeks in them, can be described by any of the following equivalent definitions:
 any year starting on Thursday (dominical letter D or DC) and any leap year starting on Wednesday (ED)
 any year ending on Thursday (D, ED) and any leap year ending on Friday (DC)
 years in which 1 January or 31 December are Thursdays
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, because it is the only date that is always in the last week of the year since it is a week before 4 January which is always in the first week of the following year.
Using only the ordinal year number y, the number of weeks in that year can be determined:^{[1]}
004  009  015  020  026 
032  037  043  048  054 
060  065  071  076  082 
088  093  099  
105  111  116  122  
128  133  139  144  150 
156  161  167  172  178 
184  189  195  
201  207  212  218  
224  229  235  240  246 
252  257  263  268  274 
280  285  291  296  
303  308  314  
320  325  331  336  342 
348  353  359  364  370 
376  381  387  392  398 
On average, a year has 53 weeks every ^{400}⁄_{71} = 5.6338… years, and these long years are 43 × 6 years apart, 27 × 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:
 27 Gregorian leap years, emphasized in the list above:
 44 Gregorian common years starting, hence also ending on Thursday.
The Gregorian years corresponding to the other 329 short years (neither starting nor ending with Thursday) can also be subdivided as follows:
 70 are Gregorian leap years.
 259 are Gregorian common years.
Thus, within a 400year cycle:
 27 week years are 5 days longer than the month years (371 − 366).
 44 week years are 6 days longer than the month years (371 − 365).
 70 week years are 2 days shorter than the month years (364 − 366).
 259 week years are 1 day shorter than the month years (364 − 365).
Weeks per month
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 explicitly. 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.
Dates with fixed week number
Month  Dates  Week numbers  

January  04  11  18  25  W01 – W04  
February  01  08  15  22  W05 – W08  
March  01  08  15  22  29  W09 – W13 
April  05  12  19  26  W14 – W17  
May  03  10  17  24  31  W18 – W22 
June  07  14  21  28  W23 – W26  
July  05  12  19  26  W27 – W30  
August  02  09  16  23  30  W31 – W35 
September  06  13  20  27  W36 – W39  
October  04  11  18  25  W40 – W43  
November  01  08  15  22  29  W44 – W48 
December  06  13  20  27  W49 – W52 
For all years, 8 days have a fixed ISO week number (between W01 and W08) in January and February. With the exception of leap years starting on Thursday, dates with fixed week numbers occur in all months of the year (for 1 day of each ISO week W01 to W52).
During leap years starting on Thursday (i.e. the 13 years numbered 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 again 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 the “Doomsday” algorithm, which calculates the weekday that the last day of February falls on. The dates listed in the table are all one day after the Doomsday, 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.
Equal weeks
Some pairs and triplets of ISO weeks have the same days of the month:
 W02 and W41 in common years
 W03 with W42 in common years and with W15 and W28 in leap years
 W04 and W43 in common years and with W16 and W29 in leap years
 W05 and W44 in common years
 W06 with W10 and W45 in common years and with W32 in leap years
 W07 with W11 and W46 in common years and with W33 in leap years
 W08 with W12 and W47 in common years and with W34 in leap years
 W10 and W45
 W11 and W46
 W12 and W47
 W15 and W28
 W16 and W29
 W37 and W50
 W38 and W51
Some other weeks, i.e. W09, W19 through W26, W31 and W35 never share their days of the month ordinals with any other week of the same year.
Advantages
 All weeks have exactly 7 days, i.e. there are no fractional weeks.
 Every week belongs to a single year, i.e. there are no ambiguous or doubleassigned weeks.
 The date directly tells the weekday.
 All weeknumbering years start with a Monday and end with a Sunday.
 When used by itself without using the concept of month, all weeknumbering years are the same except that some years have a week 53 at the end.
 The weeks are the same as used with the Gregorian calendar.
Differences to other calendars
Solar astronomic phenomena, such as equinoxes and solstices, vary in the Gregorian calendar over a range spanning three days, over the course of each 400year cycle, while the ISO Week Date calendar has a range spanning 9 days. For example, there are March equinoxes on 1920W126 and 2077W115 in UT.
The year number of the ISO week very often differs from the Gregorian year number for dates close to 1 January. For example, 29 December 2014 is ISO 2015W011, i.e., it is in year 2015 instead of 2014. A programming bug confusing these two year numbers is probably the cause of some Android users of Twitter being unable to log in around midnight of 29 December 2014 UTC.^{[2]}
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, extra 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 from its ordinal number alone. Most calendar reform proposals using leap week designs strive to simplify and harmonize this pattern, some by choosing a different leap cycle (e.g. 293 years).
Not all parts of the world consider the week to begin with Monday. For example, in some Muslim countries, the normal work week begins on Saturday, while in Israel it begins on Sunday. In much of the Americas, although the work week is usually defined to start on Monday, the calendar week is often considered to start on Sunday.
Algorithms
Calculating the week number from a month and day of the month or ordinal date
The week number (WW or woy for week of year) of any date can be calculated, given its ordinal date (i.e. day of the year, doy or DDD, 1–365 or 366) and its day of the week (D or dow, 1–7). If the ordinal date is not known, it can be computed from the month (MM or moy) and day of the month (DD or dom) by any of several methods; e.g. using a table such as the following.
Month  Jan  Feb  Mar  Apr  May  Jun  Jul  Aug  Sep  Oct  Nov  Dec 

Common year  000  031  059  090  120  151  181  212  243  273  304  334 
Leap year  000  031  060  091  121  152  182  213  244  274  305  335 
Algorithm:
 Subtract the weekday dow from the ordinal date doy.
 Then add 10.
 Divide the result by 7.
 Ignore the remainder.
 The quotient equals the preliminary 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.
Calculating an ordinal or month date from a week date
Algorithm:
 Multiply the week number woy by 7.
 Then add the weekday number dow.
 From this sum subtract the correction for the year:
 Get the weekday of 4 January.
 Add 3.
 The result is the ordinal date, which can be converted into a calendar date.
 If the ordinal date thus obtained is zero or negative, the date belongs to the previous calendar year;
 if it is greater than the number of days in the year, it belongs to the following year.
Other week numbering systems
The US system has weeks from Sunday through Saturday, and partial weeks at the beginning and the end of the year, i.e. 53 or 54 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 the end of December or the beginning of January or both.
The US broadcast calendar designates the week containing 1 January (and starting Monday) as the first of the year, but otherwise works like ISO week numbering without partial weeks. Up to six days of the previous December may be part of the first week of the year.
A mix of those, wherein weeks start Sunday and all 1 January is part of the first one, is used in US accounting, resulting in a system with years having also 52 or 53 weeks.
Notes
References
 ^ Gent, Robert H. "The Mathematics of the ISO 8601 Calendar".
 ^ Hern, Alex (29 December 2014). "Twitter kicks Android app users out for five hours due to 2015 date bug". the Guardian. Retrieved 29 May 2019.