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The '''[[International_Organization_for_Standardization|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 year]]s, 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 here called a leap week (ISO 8601 does not use the 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 146 097 days (20 871 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 a year.'''

==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&ndash;28 December and on all Thursdays the ISO week-numbering year number is always equal to the Gregorian year number.

Mutually equivalent definitions for week 01 are:
*the week with the year's first Thursday in it (the ISO 8601 definition)
*the week starting with the Monday which is nearest in time to 1 January
*the week with the year's first working day in it (if Saturdays, Sundays, and 1 January are not working days)
*the week with January 4 in it
*the first week with the majority (four or more) of its days in the starting year
*the week starting with the Monday in the period 29 December - 4 January 
*the week with the Thursday in the period 1 - 7 January
*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 in week 52 or 53 of the previous year.

Note that while most definitions are symmetric with respect to time reversal, one definition in terms of working days happens to be equivalent.

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:{{CURRENTCALENDAR}}
*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 December 28 in it
*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.

The 53-week ISO week-numbering year years can be described by any of the following equivalent definitions:
* all years starting with Thursday, and leap years starting with Wednesday
* years with the [[dominical letter]] D, DC or ED
* years in which 1 January and/or 31 December is a Thursday
All other week-numbering years have 52 weeks.

===Examples===
*2005-01-01 is {{ISOWEEKDATE|2005|01|01}}
*2005-01-02 is {{ISOWEEKDATE|2005|01|02}}
*2005-12-31 is {{ISOWEEKDATE|2005|12|31}}
*2007-01-01 is {{ISOWEEKDATE|2007|01|01}} (both years 2007 start with the same day)
*2007-12-30 is {{ISOWEEKDATE|2007|12|30}}
*2007-12-31 is {{ISOWEEKDATE|2007|12|31}}
*2008-01-01 is {{ISOWEEKDATE|2008|01|01}} (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)
*2008-12-29 is {{ISOWEEKDATE|2008|12|29}}
*2008-12-31 is {{ISOWEEKDATE|2008|12|31}}
*2009-01-01 is {{ISOWEEKDATE|2009|01|01}}
*2009-12-31 is {{ISOWEEKDATE|2009|12|31}} (ISO year 2009 has 53 weeks, extending the Gregorian year 2009, which starts and ends with Thursday, at both ends with three days)
*2010-01-03 is {{ISOWEEKDATE|2010|01|03}}

====Example where the ISO week-numbering year is three days into the next Gregorian year ====
*2009-12-31 is {{ISOWEEKDATE|2009|12|31}}
*2010-01-01 is {{ISOWEEKDATE|2010|1|1}}
*2010-01-02 is {{ISOWEEKDATE|2010|1|2}}
*2010-01-03 is {{ISOWEEKDATE|2010|1|3}}
*'''2010-01-04 is {{ISOWEEKDATE|2010|1|4}}'''

====Example where the ISO week-numbering year is three days into the previous Gregorian year ====
*2008-12-28 is {{ISOWEEKDATE|2008|12|28}}
*'''2008-12-29 is {{ISOWEEKDATE|2008|12|29}}'''
*2008-12-30 is {{ISOWEEKDATE|2008|12|30}}
*2008-12-31 is {{ISOWEEKDATE|2008|12|31}}
*2009-01-01 is {{ISOWEEKDATE|2009|1|1}}

The system does not need the concept of month and is not well connected with the Gregorian system of months: some months January and December are divided over two ISO years.

==Dates with fixed week number==

Overview of dates with a fixed week number in any year other than a [[leap year starting on Thursday]]:

{| class="wikitable sortable"
!Month!!Dates!!Week numbers
|-
| January
| 4, 11, 18, 25
| &nbsp; 01-04
|-
| February
| 1, 8, 15, 22
| &nbsp; 05-08
|-
| March
| 1, 8, 15, 22, 29
| &nbsp; 09-13
|-"
| April
| 5, 12, 19, 26
| &nbsp; 14-17
|-
| May
| 3, 10, 17, 24, 31
| &nbsp; 18-22
|-
| June
| 7, 14, 21, 28
| &nbsp; 23-26
|-
| July
| 5, 12, 19, 26
| &nbsp; 27-30
|-
| August
| 2, 9, 16, 23, 30
| &nbsp; 31-35
|-
| September
| 6, 13, 20, 27
| &nbsp; 36-39
|-
| October
| 4, 11, 18, 25
| &nbsp; 40-43
|-
| November
| 1, 8, 15, 22, 29
| &nbsp; 44-48
|-
| December
| 6, 13, 20, 27
| &nbsp; 49-52
|}

The day of the week for these days are related to [[Doomsday_%28weekday%29|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 [[Doomsday (weekday)#Overview of all Doomsdays|the rank number of its Doomsday]].

==Advantages==
*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.

==Disadvantages==
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 other Muslim countries it may begin on Sunday.

==The calendar cycle==
There are 13 28-year subcycles with 5 leap years (53-week years) each, and 6 remaining leap years in the remaining 36 years (the absence of leap ''days'' in the Gregorian calendar in 2100, 2200, and 2300 interrupts the subcycles). The leap years are 27 times 5 years apart, 43 times 6 years, and once 7 years. (A slightly more even distribution would be possible: 26 times 5 years apart, and 45 times 6 years.) 

The Gregorian years corresponding to the 71 ISO leap years can be subdivided as follows:
*27 Gregorian leap years:
**13 [[Leap year starting on Thursday|leap years starting on Thursday]] - Doomsday is Sunday, dominical letter is DC.
**14 [[Leap year starting on Wednesday|leap years starting on Wednesday]], hence ending with Thursday) - Doomsday is Saturday, dominical letter is ED.
*44 [[Common year starting on Thursday|common years starting, hence also ending, with Thursday]] - Doomsday is Saturday, dominical letter is D.

Thus 27 ISO years are 5 days longer than the corresponding Gregorian year, and 44 are 6 days longer. Of the other 329 Gregorian years (neither starting nor ending with Thursday), 70 are Gregorian leap years, and 259 are non-leap years, so 70 ISO years are 2 days shorter, and 259 are 1 day shorter.

==Calculation==
===Calculating the week number of a given date===
One can calculate the week number of any date given its [[ordinal date]] (i.e. position within the year) and its [[day of the week]]. If one does not know the ordinal date, 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 (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, September 26, 2008
* Ordinal day: 244 + 26 = 270
* Weekday: Friday = 5
* 270 - 5 + 10 = 275
* 275 / 7 = 39 plus an irrelevant fraction
* Result: Week 39

===Calculating a date given the week number and weekday===
This method requires that one know the weekday of January 4 of the year in question. <ref>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 January 4. Do not use for years past 2100.</ref> 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, Friday (day 5)
* Correction for 2008: 5 + 3 = 8
* (39 * 7) + 5 = 278
* 278 - 8 = 270
* Ordinal day 270 of a leap year is day 270 - 244 = 26 of September
* Result September 26, 2008

==Other week numbering systems==
For an overview of week numbering systems see [[Seven-day week#Week_number|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, while correspondence of lexicographical order and chronological order is preserved.

==See also==
* [[Week]]
* [[ISO 8601]]

==External links==
*[http://www.phys.uu.nl/~vgent/calendar/isocalendar.htm The Mathematics of the ISO 8601 Calendar]
*[http://www.probabilityof.com/excel.shtml#21 A generic Excel Calendar with ISO Week numbers]
*[http://www.personal.ecu.edu/mccartyr/isowdcal.html ISO week day calendar]
*[http://www.weekofyear.com/ A simple website giving the current ISO week date]

{{Time measurement and standards}}
{{Chronology}}
{{calendars}}

[[Category:Weeks]]
[[Category:Calendaring standards]]
[[Category:Leap Week Calendars]]

[[fr:Numérotation ISO des semaines]]
[[nl:NEN 2772]]
[[sh:ИСО седмични датум]]

{{reflist}}