First week

The ISO 8601 definition for week 01 is the week with the first Thursday of the Gregorian year (i.e. of January) in it. The following definitions based on properties of this week are mutually equivalent, since the ISO week starts with Monday:

• It is the first week with a majority (4 or more) of its days in January.
• Its first day is the Monday nearest to 1 January.
• It has 4 January in it. Hence the earliest possible first week extends from Monday 29 December (previous Gregorian year) to Sunday 4 January, the latest possible first week extends from Monday 4 January to Sunday 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 W01. If it is on a Friday, it is part of W53 of the previous year. If it is on a Saturday, it is part of the last week of the previous year which is numbered W52 in a common year and W53 in a leap year. If it is on a Sunday, it is part of W52 of the previous year.

Notes

Last week

The last week of the ISO week-numbering year, i.e. W52 or W53, is the week before W01 of the next year. This week's properties are:

• It has the year's last Thursday in it.
• It is the last week with a majority (4 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 earliest possible last week extends from Monday 22 December to Sunday 28 December, the latest possible last week extends from Monday 28 December to Sunday 3 January.

If 31 December is on a Monday, Tuesday, or Wednesday it is in W01 of the next year. If it is on a Thursday, it is in W53 of the year just ending. If on a Friday it is in W52 of the year just ending in common years and W53 in leap years. If on a Saturday or Sunday, it is in W52 of the year just ending.

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 DC) and any leap year starting on Wednesday (ED)
• any year ending on Thursday (D, ED) and any leap year ending on Friday (DC)
• years in which 1 January or 31 December 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, because it is the only date that is always in the last week of the year since it is a week before 4 January which is always in the first week of the following year.

Using only the ordinal year number y, the number of weeks in that year can be determined from a function, ${\displaystyle p(y)}$, that returns the day of the week of 31 December:^[1]

${\displaystyle {\begin{aligned}p(y)&=\left(y+\left\lfloor {\frac {y}{4}}\right\rfloor -\left\lfloor {\frac {y}{100}}\right\rfloor +\left\lfloor {\frac {y}{400}}\right\rfloor \right){\bmod {7}}\\{\text{weeks}}(y)&=52+{\begin{cases}1{\text{ (long)}}&{\text{if }}p(y)=4{\text{, i.e. current year ends Thursday}}\\&{\text{or }}p(y-1)=3{\text{, i.e. previous year ends Wednesday}}\\0{\text{ (short)}}&{\text{otherwise}}\end{cases}}\end{aligned}}}$

On average, a year has 53 weeks every 400⁄71 = 5.6338... years; there are 43 times when these long years are 6 years apart, 27 times when they are 5 years apart, and once they are 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, emphasized in the list above:
  • 14 starting on Thursday, hence ending on Friday, and
  • 13 starting on Wednesday, hence ending on Thursday;
• 44 Gregorian common years starting, hence also ending on Thursday.

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

• 70 are Gregorian leap years.
• 259 are Gregorian common years.

Thus, within a 400-year cycle:

• 27 week years are 5 days longer than the month years (371 − 366).
• 44 week years are 6 days longer than the month years (371 − 365).
• 70 week years are 2 days shorter than the month years (364 − 366).
• 259 week years are 1 day shorter than the month years (364 − 365).

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, a fixed length of 13 weeks per quarter is usually chosen in practice. These quarters may then be 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 8601-1 does not consider this case explicitly. The resulting pattern would be irregular. There would be 4 months of 5 weeks per normal, 52-week year, or 5 such months in a long, 53-week year. They meet one of the following three criteria:

• The first day of the month is a ...
  • Thursday and the month has 29 through 31 days.
  • Wednesday and the month has 30 or 31 days.
  • Tuesday and the month has 31 days, ending on a Thursday.
• Equivalently, the last day of the month is a ...
  • Thursday and it is not the 28th.
  • Friday and it is not in February.
  • Saturday and it is the 31st.

Dates with fixed week number

For all years, 8 days have a fixed ISO week number (between W01 and W08) in January and February. With the exception of leap years starting on Thursday, dates with fixed week numbers occur in all months of the year (for 1 day of each ISO week W01 to W52).

During leap years starting on Thursday (i.e. the 13 years numbered 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 next occur in 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. They will never be 12 years apart. The only leap years that can occur 12 years apart are leap years starting on Sunday, Tuesday, Wednesday and Friday.

The day of the week for these days are related to the "Doomsday" algorithm, which calculates the weekday that the last day of February falls on. The dates listed in the table are all one day after the Doomsday, 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

Some pairs and triplets of ISO weeks have the same days of the month:

• W02 and W41 in common years
• W03 with W42 in common years and with W15 and W28 in leap years
• W04 and W43 in common years and with W16 and W29 in leap years
• W05 and W44 in common years
• W06 with W10 and W45 in common years and with W32 in leap years
• W07 with W11 and W46 in common years and with W33 in leap years
• W08 with W12 and W47 in common years and with W34 in leap years
• W10 and W45
• W11 and W46
• W12 and W47
• W15 and W28
• W16 and W29
• W37 and W50
• W38 and W51

Some other weeks, i.e. W09, W19 through W26, W31 and W35 never share their days of the month ordinals with any other week of the same year.

Calculating the week number from an ordinal date

The week number (WW or woy for week of year) of any date can be calculated, given its ordinal date (i.e. day of the year, doy or DDD, 1–365 or 366) and its day of the week (D or dow, 1–7). When using serial numbers for dates (e.g. in spreadsheets), doy is the serial number for a date minus the serial number for 31st December of the previous year, or alternatively minus the serial number for 1st January the same year plus one.

Algorithm

1. Subtract the weekday number from the ordinal day of the year.
2. Add 10.
3. Divide by 7, discard the remainder.
   • 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.

Formula

${\displaystyle {\begin{aligned}y&={\text{year}}(date)\\w&=\left\lfloor {\frac {10+{\text{doy}}(date)-{\text{dow}}(date)}{7}}\right\rfloor \\{\text{woy}}(date)&={\begin{cases}{\text{weeks}}(y-1),&{\text{if }}w<1\\1,&{\text{if }}w>{\text{weeks}}(y)\\w,&{\text{otherwise.}}\end{cases}}\\{\text{woy}}(date)&\in [1,53]\\{\text{doy}}(date)&\in [1,366]\\{\text{dow}}(date)&\in [1,7]\\{\text{weeks}}(year)&\in [52,53]\\\end{aligned}}}$

Calculating the week number from a month and day of the month

If the ordinal date is not known, it can be computed from the month (MM or moy) and day of the month (DD or dom) by any of several methods; e.g. using a table such as the following.

Example

Find the week number of Saturday 5th November 2016 (leap year):

• Find the ordinal day number first:

moy = 11
dom = 5
leap = 1
add = 305, from table lookup
doy = 305 + 5 = 310.

• Alternatively, use spreadsheet serial day numbers instead:

off = 42369, i.e. 31st December 2015
day = 42679
doy = 42679 − 42369 = 310.

• Finally, find the week number:

dow = 6, i.e. Saturday
woy = (10 + 310 − 6) div 7
woy = (320 − 6) div 7
woy = 314 div 7 = 44.

Calculating an ordinal or month date from a week date

Algorithm

1. Multiply the week number by 7.
2. Then add the weekday number.
3. From this sum subtract the correction for the year:
   • Get the weekday of 4 January.
   • Add 3.
4. The result is the ordinal date, which can be converted into a calendar date.
   • If the ordinal date thus obtained is zero or negative, the date belongs to the previous calendar year;
   • if it is greater than the number of days in the year, it belongs to the following year.

Formula

${\displaystyle {\begin{aligned}y&={\text{year}}(date)\\d&={\text{woy}}(date)\times 7+{\text{dow}}(date)-({\text{dow}}(y,1,4)+3)\\{\text{doy}}(date)&={\begin{cases}d+{\text{days}}(y-1),&{\text{if }}d<1\\d-{\text{days}}(y),&{\text{if }}d>{\text{days}}(y)\\d,&{\text{otherwise.}}\end{cases}}\\{\text{days}}(year)&\in [365,366]\\\end{aligned}}}$