Ordinal date

Mission control center's board with time data, displaying coordinated universal time with ordinal date (without year) prepended, on OctoberTemplate:Nbsp22, 2013 Template:Nobr

An ordinal date is a calendar date typically consisting of a year and an ordinal number, ranging between 1 and 366 (starting on January 1), representing the multiples of a day, 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 Template:Mono, or Template:Mono, 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 Template:Mono, 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

Template:Original research 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 Template:Mvar are presented. The inputs taken are integers Template:Mvar, Template:Mvar and Template:Mvar, 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:

1. Let O be 0.
2. From Template:Math, add the length of month Template:Mvar to O, taking care of leap year according to the calendar used.
3. 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 Template:Mvar is changed to Template:Math if Template:Math. It can be shown (see below) that for a month-number Template:Mvar, the total days of the preceding months is equal to Template:Math. As a result, the March 1-based ordinal day number is Template:Math.

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 Template:Sfrac 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 Template:Math (March through December), Template:Nowrap where Template:Math is a function returning 0 or 1 depending whether the input is a leap year.
• For January and February, two methods can be used:
  1. The trivial method is to skip the calculation of Template:Math and go straight for Template:Math for January and Template:Math for February.
  2. The less redundant method is to use Template:Math, 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

$O=\lfloor 63.2n-91.4\rfloor$

giving consecutive differences of 63 (9 weeks) for Template:Nowrap 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

$O=\lfloor 63.2n-56+0.2\rfloor$

and with m and d interchanged

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

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

Table

To the day of	13 Jan	14 Feb	3 Mar	4 Apr	5 May	6 Jun	7 Jul	8 Aug	9 Sep	10 Oct	11 Nov	12 Dec	i
Add	0	31	59	90	120	151	181	212	243	273	304	334	3
Leap years	0	31	60	91	121	152	182	213	244	274	305	335	2
Algorithm	$30(m-1)+\lfloor 0.6(m+1)\rfloor -i$

For example, the ordinal date of April 15 is Template:Nowrap in a common year, and Template:Nowrap in a leap year.

Month–day

Template:Original research The number of the month and date is given by

$m=\lfloor od/30\rfloor +1$

$d=mod(od,30)+i-\lfloor 0.6(m+1)\rfloor$

the term $mod(od,30)$ can also be replaced by $od-30(m-1)$ with $od$ the ordinal date.

• Day 100 of a common year:

$m=\lfloor 100/30\rfloor +1=4$

$d=mod(100,30)+3-\lfloor 0.6(4+1)\rfloor =10+3-3=10$

April 10.

• Day 200 of a common year:

$m=\lfloor 200/30\rfloor +1=7$

$d=mod(200,30)+3-\lfloor 0.6(7+1)\rfloor =20+3-4=19$

July 19.

• Day 300 of a leap year:

$m=\lfloor 300/30\rfloor +1=11$

$d=mod(300,30)+2-\lfloor 0.6(11+1)\rfloor =0+2-7=-5$

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

Helper conversion table

ord. date	common year	leap year
001	Template:01 Jan
010	10 Jan
020	20 Jan
030	30 Jan
032	Template:01 Feb
040	Template:09 Feb
050	19 Feb
060	Template:01 Mar	29 Feb
061	Template:02 Mar	Template:01 Mar
070	11 Mar	10 Mar
080	21 Mar	20 Mar
090	31 Mar	30 Mar
091	Template:01 Apr	31 Mar
092	Template:02 Apr	Template:01 Apr
100	10 Apr	Template:09 Apr

ord. date	comm. year	leap year
110	20 Apr	19 Apr
120	30 Apr	29 Apr
121	Template:01 May	30 Apr
122	Template:02 May	Template:01 May
130	10 May	Template:09 May
140	20 May	19 May
150	30 May	29 May
152	Template:01 Jun	31 May
153	Template:02 Jun	Template:01 Jun
160	Template:09 Jun	Template:08 Jun
170	19 Jun	18 Jun
180	29 Jun	28 Jun
182	Template:01 Jul	30 Jun
183	Template:02 Jul	Template:01 Jul
190	Template:09 Jul	Template:08 Jul

ord. date	comm. year	leap year
200	19 Jul	18 Jul
210	29 Jul	28 Jul
213	Template:01 Aug	31 Jul
214	Template:02 Aug	Template:01 Aug
220	Template:08 Aug	Template:07 Aug
230	18 Aug	17 Aug
240	28 Aug	27 Aug
244	Template:01 Sep	31 Aug
245	Template:02 Sep	Template:01 Sep
250	Template:07 Sep	Template:06 Sep
260	17 Sep	16 Sep
270	27 Sep	26 Sep
274	Template:01 Oct	30 Sep
275	Template:02 Oct	Template:01 Oct
280	Template:07 Oct	Template:06 Oct

ord. date	comm. year	leap year
290	17 Oct	16 Oct
300	27 Oct	26 Oct
305	Template:01 Nov	31 Oct
306	Template:02 Nov	Template:01 Nov
310	Template:06 Nov	Template:05 Nov
320	16 Nov	15 Nov
330	26 Nov	25 Nov
335	Template:01 Dec	30 Nov
336	Template:02 Dec	Template:01 Dec
340	Template:06 Dec	Template:05 Dec
350	16 Dec	15 Dec
360	26 Dec	25 Dec
365	31 Dec	30 Dec
366	Template:N/A	31 Dec

See also

• Julian day
• Zeller's congruence
• ISO week date