An ordinal date is a calendar date typically consisting of a ''

year A year is a unit of time based on how long it takes the Earth to orbit the Sun. In scientific use, the tropical year (approximately 365 Synodic day, solar days, 5 hours, 48 minutes, 45 seconds) and the sidereal year (about 20 minutes longer) ...

'' and an ''

ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...

'', ranging between 1 and 366 (starting on January 1), representing the multiples of a ''

day A day is the time rotation period, period of a full Earth's rotation, rotation of the Earth with respect to the Sun. On average, this is 24 hours (86,400 seconds). As a day passes at a given location it experiences morning, afternoon, evening, ...

'', called day of the year or ordinal day number (also known as ordinal day or day number). The two parts of the date can be formatted as "YYYY-DDD" to comply with the ISO 8601 ordinal date format. The year may sometimes be omitted, if it is implied by the context; the day may be generalized from integers to include a decimal part representing a fraction of a day.

Nomenclature

''Ordinal date'' is the preferred name for what was formerly called the ''" Julian date"'' or , or , which still seen in old programming languages and spreadsheet software. The older names are deprecated because they are easily confused with the earlier dating system called ' Julian day number' or , which was in prior use and which remains ubiquitous in astronomical and some historical calculations. The U.S. military sometimes uses a system they call the "Julian date format", which indicates the year and the day number (out of the 365 or 366 days of the year). For example, "11 December 1999" can be written as "1999345" or "99345", for the 345th day of 1999.

Calculation

Computation of the ordinal day within a year is part of calculating the ordinal day 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. In the following text, several algorithms for calculating the ordinal day are presented. The inputs taken are integers , and , for the year, month, and day numbers of the Gregorian or Julian calendar date.

Trivial methods

The most trivial method of calculating the ordinal day involves counting up all days that have elapsed per the definition: # Let ''O'' be 0. # From , add the length of month to ''O'', taking care of leap year according to the calendar used. # Add ''d'' to ''O''. Similarly trivial is the use of a lookup table, such as the one referenced.

Zeller-like

The table of month lengths can be replaced following the method of encoding the month-length variation in Zeller's congruence. As in Zeller, the is changed to if . It can be shown (see below) that for a month-number , the total days of the preceding months is equal to . As a result, the March 1-based ordinal day number is . 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 twice. This is similar to encoding of the month offset (which would be the same sequence modulo 7) in Zeller's congruence. As is 30.6, the sequence oscillates in the desired pattern with the desired period 5. To go from the March 1 based ordinal day to a January 1 based ordinal day: * For (March through December), where is a function returning 0 or 1 depending whether the input is a leap year. * For January and February, two methods can be used: *# The trivial method is to skip the calculation of and go straight for for January and for February. *# The less redundant method is to use , where 306 is the number of dates in March through December. This makes use of the fact that the formula correctly gives a month-length of 31 for January. " Doomsday" properties: With

m = 2n

and

d=m

gives :

O = \left \lfloor 63.2 n - 91.4 \right \rfloor

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

m = 2n + 1

and

d = m + 4

gives :

O = \left \lfloor 63.2 n - 56+0.2 \right \rfloor

and with ''m'' and ''d'' interchanged :

O = \left\lfloor 63.2 n - 56 + 119 - 0.4 \right\rfloor

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

Table

For example, the ordinal date of April 15 is in a common year, and in a

leap year A leap year (also known as an intercalary year or bissextile year) is a calendar year that contains an additional day (or, in the case of a lunisolar calendar, a month) compared to a common year. The 366th day (or 13th month) is added to keep t ...

Month–day

The number of the month and date is given by :

m = \left \lfloor od/30 \right \rfloor + 1

d = \bmod\!\! (od, 30) + i - \left \lfloor 0.6 (m + 1) \right \rfloor

the term

\bmod\!\! (od, 30)

can also be replaced by

od - 30 (m - 1)

with

od

the ordinal date. *Day 100 of a common year: ::

m = \left \lfloor 100/30 \right \rfloor + 1 = 4

d = \bmod\!\! (100, 30) + 3 - \left \lfloor 0.6 (4 + 1) \right \rfloor = 10 + 3 - 3 = 10

:April 10. *Day 200 of a common year: ::

m = \left \lfloor 200/30 \right \rfloor + 1 = 7

d = \bmod\!\! (200, 30) + 3 - \left \lfloor 0.6 (7 + 1) \right \rfloor = 20 + 3 - 4 = 19

:July 19. *Day 300 of a leap year: ::

m = \left \lfloor 300/30 \right \rfloor + 1 = 11

d = \bmod\!\! (300, 30) + 2 - \left \lfloor 0.6 (11 + 1)\right \rfloor = 0 + 2 - 7 = - 5

:November - 5 = October 26 (31 - 5).

Helper conversion table

References

{{DEFAULTSORT:Ordinal Date Calendars Ordinal numbers