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 2019) 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 and end on Sunday. 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 usually deviates by 1 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.
Relation with the Gregorian calendar
English short  ISO  

Sat 1 Jan 1977  19770101  1976W536 
Sun 2 Jan 1977  19770102  1976W537 
Sat 31 Dec 1977  19771231  1977W526 
Sun 1 Jan 1978  19780101  1977W527 
Mon 2 Jan 1978  19780102  1978W011 
Sun 31 Dec 1978  19781231  1978W527 
Mon 1 Jan 1979  19790101  1979W011 
Sun 30 Dec 1979  19791230  1979W527 
Mon 31 Dec 1979  19791231  1980W011 
Tue 1 Jan 1980  19800101  1980W012 
Sun 28 Dec 1980  19801228  1980W527 
Mon 29 Dec 1980  19801229  1981W011 
Tue 30 Dec 1980  19801230  1981W012 
Wed 31 Dec 1980  19801231  1981W013 
Thu 1 Jan 1981  19810101  1981W014 
Thu 31 Dec 1981  19811231  1981W534 
Fri 1 Jan 1982  19820101  1981W535 
Sat 2 Jan 1982  19820102  1981W536 
Sun 3 Jan 1982  19820103  1981W537 
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)^{[a]} 
Days at the start of January  Effect^{[a]}^{[b]}  Days at the end of December^{[a]}  

1 Mon 
2 Tue 
3 Wed 
4 Thu 
5 Fri 
6 Sat 
7 Sun 
W011^{[c]}  01 Jan week  …  31 Dec week  1 Mon^{[d]} 
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)  (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 
Notes
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, Tuesday, or Wednesday it is in W01 of the next year. If it is on a Thursday, it is in W53 of the year just ending. If on a Friday it is in W52 of the year just ending in common years and W53 in leap years. If on a Saturday or Sunday, it is in W52 of the year just ending.
01 Jan  W011  Common year (365 − 1 or + 6)  Leap year (366 − 2 or + 5)  

Mon  01 Jan  G  +0  −1  G F  +0  −2 
Tue  31 Dec  F  +1  −2  F E  +1  −3 
Wed  30 Dec  E  +2  −3  E D  +2  +3 
Thu  29 Dec  D  +3  +3  D C  +3  +2 
Fri  04 Jan  C  −3  +2  C B  −3  +1 
Sat  03 Jan  B  −2  +1  B A  −2  +0 
Sun  02 Jan  A  −1  +0  A G  −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 from a function, , that returns the day of the week of 31 December:^{[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, a fixed length of 13 weeks per quarter is usually chosen in practice. These quarters may then be 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 86011 does not consider this case explicitly. The resulting pattern would be irregular. There would be 4 months of 5 weeks per normal, 52week year, or 5 such months in a long, 53week year. They meet one of the following three criteria:
 The first day of the month is a …
 Thursday and the month has 29 through 31 days.
 Wednesday and the month has 30 or 31 days.
 Tuesday and the month has 31 days, ending on a Thursday.
 Equivalently, the last day of the month is a …
 Thursday and it is not the 28th.
 Friday and it is not in February.
 Saturday and it is the 31st.
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 1986 is ISO 1987W011, i.e., it is in year 1987 instead of 1986. 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.
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. (However, a relatively simple algorithm to determine whether a year has 53 weeks from its ordinal number alone is shown under "Weeks per year" above.) 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).
woy = (10 + doy − dow) div 7
where
doy = 1 → 365/366, dow = 1 → 7 og div means integer division (i.e. the remainder after a division is discarded).
When using serial numbers for dates (e.g. in spreadsheets) doy = serial number for a date − serial number for 31st December of the previous year (or the serial number for 1st January the same year + 1).
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  Add  

Common year  0  31  59  90  120  151  181  212  243  273  304  334  dom  
Leap year  0  31  60  91  121  152  182  213  244  274  305  335 
 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:
Find the week number of Saturday 5th November 2016 (leap year):
woy = (10 + (305 + 5) − 6) div 7 woy = (10 + 310 − 6) div 7 woy = (320 − 6) div 7 woy = 314 div 7 = 44. woy = (10 + (42679 − 42369) − 6) div 7 woy = (10 + 310 − 6) div 7 woy = (320 − 6) div 7 woy = 314 div 7 = 44.
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. 52 full and 1 partial week of 1 or 2 days if the year starts on Sunday or ends on Saturday, 52 full and 2 singleday weeks if a leap year starts on Saturday and ends on Sunday, otherwise 51 full and 2 partial 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.
References
 ^ Gent, Robert H. "The Mathematics of the ISO 8601 Calendar".