ISO week date
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 uses the same cycle of 7 weekdays as the Gregorian calendar. Weeks start with Monday. ISO week-numbering years have a year numbering which is approximately the same as the Gregorian years, but not exactly (see below). An ISO week-numbering year has 52 or 53 full weeks (364 or 371 days). The extra week is referred to here as a leap week (note that ISO 8601 does not use this term).
A date is specified by the ISO week-numbering 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, 2006-W52-7 (or in compact form 2006W527) is the Sunday of the 52nd week of 2006. In the Gregorian system this day is called 31 December 2006.
The system has a 400-year cycle of 146097 days (20871 weeks), with an average year length of exactly 365.2425 days, just like the Gregorian calendar. In every 400 years there are 71 years with 53 weeks.
The first week of a year is the week that contains the first Thursday of the year. It is also (equivalently) the week containing the 4th day of January.
Relation with the Gregorian calendar 
The ISO week-numbering 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 week-numbering year number is always equal to the Gregorian year number.
First week 
Mutually equivalent definitions for week 01 are:
- the week with the year's first Thursday in it (the ISO 8601 definition)
- the week with the Thursday in the period 1 – 7 January
- the week starting with the Monday in the period 29 December – 4 January
- the week starting with the Monday which is nearest in time to 1 January
- the week ending with the Sunday in the period 4 – 10 January
- the week with 4 January in it
- the first week with the majority (four or more) of its days in the starting year
- If 1 January is on a Monday, Tuesday, Wednesday or Thursday, it is in week 01. If 1 January is on a Friday, Saturday or Sunday, it is part of week 52 or 53 of the previous year.
- the week with the year's first working day in it (if Saturdays, Sundays, and 1 January are not working days).
Last week 
The last week of the ISO week-numbering year is the week before week 01; in accordance with the symmetry of the definition, equivalent definitions are:
- the week with the year's last Thursday in it
- the week ending with the Sunday which is nearest in time to 31 December
- the week with 28 December in it (therefore the number of weeks in a given year is equal to the corresponding week number of 28 December)
- the last week with the majority (four or more) of its days in the ending year
- the week starting with the Monday in the period 22 – 28 December
- the week with the Thursday in the period 25 – 31 December
- the week ending with the Sunday in the period 28 December – 3 January
- If 31 December is on a Monday, Tuesday, or Wednesday, it is in week 01 of the next year, otherwise in week 52 or 53.
Number of ISO weeks in an ISO year 
The 53-week ISO week-numbering years can be described by any of the following equivalent definitions:
- years with the dominical letter ED, D, or DC;
- all years starting on Thursday, and leap years starting on Wednesday;
- all years ending on Thursday, and leap years ending on Friday;
- years in which either 1 January or 31 December is a Thursday (in leap years), or in which both are Thursdays (in common years).
All other week-numbering years have 52 weeks.
|day month year||iso date||iso week date|
|Sat 1 Jan 2005||2005-01-01||2004-W53-6|
|Sun 2 Jan 2005||2005-01-02||2004-W53-7|
|Sat 31 Dec 2005||2005-12-31||2005-W52-6|
|Mon 1 Jan 2007||2007-01-01||2007-W01-1||Both years 2007 start with the same day.|
|Sun 30 Dec 2007||2007-12-30||2007-W52-7|
|Mon 31 Dec 2007||2007-12-31||2008-W01-1|
|Tue 1 Jan 2008||2008-01-01||2008-W01-2||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||2008-12-28||2008-W52-7||For 2008/2009 where the ISO week-numbering year is three days into the previous Gregorian year.|
|Mon 29 Dec 2008||2008-12-29||2009-W01-1|
|Tue 30 Dec 2008||2008-12-30||2009-W01-2|
|Wed 31 Dec 2008||2008-12-31||2009-W01-3|
|Thu 1 Jan 2009||2009-01-01||2009-W01-4|
|Thu 31 Dec 2009||2009-12-31||2009-W53-4||ISO year 2009 has 53 weeks, extending the Gregorian year 2009, which starts and ends with Thursday, at both ends with three days. For 2009/2010 the ISO week-numbering year is three days into the next Gregorian year.|
|Fri 1 Jan 2010||2010-01-01||2009-W53-5|
|Sat 2 Jan 2010||2010-01-02||2009-W53-6|
|Sun 3 Jan 2010||2010-01-03||2009-W53-7|
Weeks per month 
The ISO standard does not define any association of weeks to months. A date is either expressed with a month and day-of-the-month, or with a week and day-of-the-week, 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.
Weeks per year 
On average, a year has 53 weeks every 5.6338… years (= 7 / [365.2425 − 52×7] = 400 / 71).
The following 71 years in a 400-year cycle (add 2000 for current years) have 53 weeks (leap years, with February 29, are emphasized), years not listed have 52 weeks:
- 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.
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:
- 27 Gregorian leap years (366 days, and whose corresponding Julian years are also Julian leap years):
- 44 Gregorian common years (365 days, and whose corresponding Julian years are also Julian common years) starting, hence also ending on Thursday.
The Gregorian years corresponding to the other 329 short ISO years (neither starting nor ending with Thursday) can also be subdivided as follows:
- 70 are leap Gregorian years (all their corresponding Julian years are also Julian leap years), and
- 259 are non-leap Gregorian years (but the corresponding Julian years corresponding to 3 of them are Julian leap years : 100, 200 and 300).
Thus, within a 400-year cycle:
- 27 long ISO years (53 weeks or 371 days) are 5 days longer than the corresponding leap Gregorian years (366 days),
- 44 long ISO years (53 weeks or 371 days) are 6 days longer than the corresponding common Gregorian years (365 days),
- 70 short ISO years (52 weeks or 364 days) are 2 days shorter than the corresponding leap Gregorian years (366 days), and
- 259 short ISO years (52 weeks or 364 days) are 1 day shorter than the corresponding common Gregorian years (365 days).
Dates with fixed week number 
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) :
|January||4, 11, 18, 25||01-04|
|February||1, 8, 15, 22||05-08|
|March||1, 8, 15, 22, 29||09-13|
|April||5, 12, 19, 26||14-17|
|May||3, 10, 17, 24, 31||18-22|
|June||7, 14, 21, 28||23-26|
|July||5, 12, 19, 26||27-30|
|August||2, 9, 16, 23, 30||31-35|
|September||6, 13, 20, 27||36-39|
|October||4, 11, 18, 25||40-43|
|November||1, 8, 15, 22, 29||44-48|
|December||6, 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 400-year 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.
- All weeks have an integral number of days (i.e. there are no fractional weeks).
- All years have an integral number of weeks.
- The date directly tells the weekday.
- All week-numbering years start with a Monday and end with a Sunday.
- When used by itself without using the concept of month, all week-numbering 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.
Each equinox and solstice varies 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 2004-W12-7 and 2010-W11-7.
It does not replace the Gregorian calendar, which it uses to define the new year day (Week 1 Day 1). However, it could be defined without reference to Gregorian. One needs at most a defined start and a table of year-lengths over the 400-year 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.
Calculating the week number of a given date 
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|
|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 (week-based) 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
- Ordinal day: 244 + 26 = 270
- Weekday: Friday = 5
- 270 - 5 + 10 = 275
- 275 / 7 = 39 plus an irrelevant fraction
- Result: Week 39
More algorithms which are able to handle special cases (the given date may belong to the preceding or following year) are shown in the discussion page of this article.
Calculating a date given the year, week number and weekday 
This method requires that one know the weekday of 4 January of the year in question. 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)
- Correction for 2008: 5 + 3 = 8
- (39 * 7) + 6 = 279
- 279 - 8 = 271
- Ordinal day 271 of a leap year is day 271 - 244 = 27 September
- Result: 27 September 2008
Other week numbering systems 
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. 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 year-week-weekday numbering), but partial weeks make some computations of weekly statistics or payments inaccurate at end of December of 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.
See also 
- Either see calculating the day of the week, or use this quick-and-dirty method: Subtract 1965 from the year. To this difference add one-quarter of itself, dropping any fractions. Divide this result by 7, discarding the quotient and keeping the remainder. Add 1 to this remainder, giving the weekday number of 4 January. Do not use for years past 2100.
- The Mathematics of the ISO 8601 Calendar
- A generic Excel Calendar with ISO Week numbers
- ISO week day calendar
- A simple website giving the current ISO week date
- A website giving you the current ISO 8601 week number
- (Broken)ISO Date and time format FAQ