Ordinal date

An ordinal date is a calendar date typically consisting of a year and a day of year ranging between 1 and 366 (starting on January 1), though year may sometimes be omitted. The two numbers can be formatted as YYYY-DDD to comply with the ISO 8601 ordinal date format.

Calculation

Computation of the ordinal date within a year is part of calculating the ordinal date throughout the years from a reference date, such as the Julian date. It is also part of calculating the day of the week, though for this purpose modulo-7 simplifications can be made.

For these purposes it is convenient to count January and February as month 13 and 14 of the previous year, for two reasons: the shortness of February and its variable length. In that case the date counted from 1 March is given by

floor ( 30.6 ( m + 1 ) ) + d − 122

which can also be written

floor (30.6 m − 91.4 ) + d

with m the month number and d the date.

The formula reflects the fact that any five consecutive months in the range March–January have a total length of 153 days, due to a fixed pattern 31–30–31–30–31 repeating itself some more than twice.

"Doomsday" properties:

For m = 2n and d=m we get

floor (63.2 n − 91.4 )

giving consecutive differences of 63 (9 weeks) for n = 2, 3, 4, 5, and 6, i.e., between 4/4, 6/6, 8/8, 10/10, and 12/12.

For m = 2n + 1 and d=m + 4 we get

floor (63.2 n − 56.8 )

and with m and d interchanged

floor (63.2 n − 56.8 + 118.4 )

giving a difference of 119 (17 weeks) for n = 2 (difference between 5/9 and 9/5), and also for n = 3 (difference between 7/11 and 11/7).

The ordinal date from 1 January is:

• for January: d
• for February: d + 31
• for the other months: the ordinal date from 1 March plus 59, or 60 in a leap year

or equivalently, the ordinal date from 1 March of the previous year (for which the formula above can be used) minus 306.

Modulo 7

Again counting January and February as month 13 and 14 of the previous year, the date counted from 1 March is modulo 7 equal to

floor (2.6 m − 0.4 ) + d

with m the month number and d the date.

This is the weekday relative to "Doomsday."

Table

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
Leap years	0	31	60	91	121	152	182	213	244	274	305	335

For example, the ordinal date of April 15 is 90 + 15 = 105 in a common year, and 91 + 15 = 106 in a leap year.

References and external links

• Abstract for article on standard "REPRESENTATION FOR CALENDAR DATE AND ORDINAL DATE FOR INFORMATION INTERCHANGE", Federal Information Processing Standards Publication 4-1, 1988 JANUARY 27, National Institute of Standards and Technology
• Perpetual Julian/ordinal date chart (browser needs to be able to read PDF files).

See also

• Julian day - Calculation
• Zeller's congruence