ISO week date

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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 20871 weeks (146097 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.

February 2014
WkMoTuWeThFrSaSu
(5)272829303112
(6)3456789
(7)10111213141516
(8)17181920212223
(9)242526272812

Relation with the Gregorian calendar[edit]

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
DateNotes
VulgarISO
Sat 1 Jan 20052005-01-012004-W53-6
Sun 2 Jan 20052005-01-022004-W53-7
Sat 31 Dec 20052005-12-312005-W52-6
Mon 1 Jan 20072007-01-012007-W01-1Both years 2007 start with the same day.
Sun 30 Dec 20072007-12-302007-W52-7
Mon 31 Dec 20072007-12-312008-W01-1
Tue 1 Jan 20082008-01-012008-W01-2Gregorian 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 20082008-12-282008-W52-7The ISO year 2009 is 3 days into the previous Gregorian year.
Mon 29 Dec 20082008-12-292009-W01-1
Tue 30 Dec 20082008-12-302009-W01-2
Wed 31 Dec 20082008-12-312009-W01-3
Thu 1 Jan 20092009-01-012009-W01-4
Thu 31 Dec 20092009-12-312009-W53-4ISO year 2009 has 53 weeks, thus it is 3 days into the Gregorian year 2010.
Fri 1 Jan 20102010-01-012009-W53-5
Sat 2 Jan 20102010-01-022009-W53-6
Sun 3 Jan 20102010-01-032009-W53-7

First week[edit]

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:

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[edit]

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:

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[edit]

The long years, with 53 weeks in them, can be described by any of the following equivalent definitions:

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:

The Gregorian years corresponding to the other 329 short ISO years (neither starting nor ending with Thursday) can also be subdivided as follows:

Thus, within a 400-year cycle:

Weeks per month[edit]

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[edit]

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
MonthDatesWeek numbers
January0411182501–04
February0108152205–08
March010815222909–13
April0512192614–17
May031017243118–22
June0714212823–26
July0512192627–30
August020916233031–35
September0613202736–39
October0411182540–43
November010815222944–48
December0613202749–52

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[edit]

Week triplets
(6)567891011
(10)567891011
(45)567891011


(7)12131415161718
(11)12131415161718
(46)12131415161718


(8)19202122232425
(12)19202122232425
(47)19202122232425

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[edit]

Disadvantages[edit]

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[edit]

Calculating the week number of a given date[edit]

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:JanFebMarAprMayJunJulAugSepOctNovDec
Add:0315990120151181212243273304334
For leap years:0316091121152182213244274305335

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

Example: Friday 26 September 2008

Calculating a date given the year, week number and weekday[edit]

This method requires that one know the weekday of 4 January of the year in question.[1] 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)

Other week numbering systems[edit]

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[edit]

Notes[edit]

  1. ^ 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.

External links[edit]