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 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 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.

Relation with the Gregorian calendar[link]

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[link]

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 4 January 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
• the week ending with the Sunday in the period 4 – 10 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.^{[clarification needed]}

Last week[link]

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.

The 53-week ISO week-numbering years can be described by any of the following equivalent definitions:

• all years starting with Thursday, and leap years starting with Wednesday
• all years ending with Thursday, and leap years ending with Friday
• years in which 1 January or 31 December is a Thursday or in which both are Thursdays (usual case)
• years with the dominical letter D, DC or ED

All other week-numbering years have 52 weeks.

Examples[link]

Weeks per month[link]

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[link]

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. Years not listed have 52 weeks, years with February 29 are emphasized.

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.

Long years are 5 or 6 years apart, except that 296 and 303 are 7 years apart.

There are 13 28-year subcycles with 5 long years (53-week years) each, and 6 remaining long years in the remaining 36 years (the absence of leap days in 100, 200, and 300 interrupts the subcycles). The long 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 long years can be subdivided as follows:

• 27 Gregorian leap years:
  • 13 leap years starting on Thursday - Doomsday is Sunday, dominical letter is DC.
  • 14 leap years starting on Wednesday, hence ending with Thursday) - Doomsday is Saturday, dominical letter is ED.
• 44 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.

Dates with fixed week number[link]

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.

Advantages[link]

• 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[link]

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.

Calculation[link]

Calculating the week number of a given date[link]

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.

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 belongs 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[link]

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.

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[link]

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, while correspondence of lexicographical order and chronological order is preserved.

See also[link]

• Week
• 4-4-5 Calendar

Notes[link]

External links[link]

• 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
• ISO Date and time format FAQ

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