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Template:Dablink 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 since an ISO week-numbering year (also called ISO year or week 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.

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, the Gregorian date 31 December 2006 corresponds to the Sunday of the 52nd week of 2006, and is written 2006-W52-7 (extended form) or 2006W527 (compact form).

The system has a 400-year cycle of Template:Gaps weeks (Template:Gaps days), with an average year length of exactly 52.1775 weeks (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.

Examples of contemporary dates around New Year

Date			Notes
Vulgar	ISO
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	The ISO year 2009 is 3 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, thus it is 3 days into the Gregorian year 2010.
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

First week

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:

• It is the first week with a majority (four or more) of its days in January (ISO weeks start on Monday)
• Its first day is the Monday nearest to 1 January.
• It has 4 January in it. Hence the earliest possible dates are 29 December through 4 January, the latest 4 through 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 week 01. If 1 January is on a Friday, Saturday or Sunday, it is part of week 52 or 53 of the previous year.

Last week

The last week of the ISO week-numbering year, i.e. the 52nd or 53rd one, is the week before week 01. This week’s properties are:

• It has the year's last Thursday in it.
• It is the last week with a majority (four 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 latest possible dates are 28 December through 3 January, the earliest 21 through 28 December.

If 31 December is on a Monday, Tuesday or Wednesday, it is in week 01 of the next year. If it is on a Saturday or Sunday, it is in week 52 of the same year. If it is on a Thursday, it is in week 53 and if it is on a Friday it is either in week 52 or 53.

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 ED) and any leap year starting on Wednesday (DC)
• any year ending on Thursday and any leap year ending on Friday
• years in which 1 January and 31 December (in common years) or either (in leap years) are Thursdays

All other week-numbering 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 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):
  • 14 starting on Thursday, hence ending on Friday, and
  • 13 starting on Wednesday, hence ending on Thursday;
• 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).

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.

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) :

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

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 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.

Equal weeks

Week triplets Template:ISOCALENDAR/weekTemplate:ISOCALENDAR/weekTemplate:ISOCALENDAR/week Template:ISOCALENDAR/weekTemplate:ISOCALENDAR/weekTemplate:ISOCALENDAR/week Template:ISOCALENDAR/weekTemplate:ISOCALENDAR/weekTemplate:ISOCALENDAR/week

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.

The weeks 09, 19–26 and 31–35 never share their days of the month with any other week of the same year.

Advantages

• 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.

Disadvantages

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 2004-W12-7 and 2010-W11-7.

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 400-year 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.

Calculation

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
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.

$week(date)=\lfloor \frac{ordinal(date)-weekday(date)+10}{7}\rfloor$

Example: Friday 26 September 2008

• Ordinal day: 244 + 26 = 270
• Weekday: Friday = 5
• 270 − 5 + 10 = 275
• 275 / 7 = 39.28…
• Result: Week 39

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.

$ordinal(date)=week(date)*7+weekday(date)-(weekday(year(date),1,4)+3)$

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, 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 year-week-weekday 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.

See also

• Week
• 4-4-5 Calendar

Notes

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External links

• 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