# Ordinal date

Date 2024-02-19 2024-050

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 JD, or JDATE, 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 JDN, which was in prior use and which remains ubiquitous in astronomical and some historical calculations.

## 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 O is presented. The inputs taken are integers y, m and d, 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 i = 1 .. m - 1, add the length of month i to O, taking care of leap year according to the calendar used.

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 m is changed to m + 12 if m ≤ 2. It can be shown (see below) that for a month-number m, the total days of the preceding months is equal to ⌊(153 * (m − 3) + 2) / 5⌋. As a result, the March 1-based ordinal day number is OMar = ⌊(153 × (m − 3) + 2) / 5⌋ + d.

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 153/5 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 m ≤ 12 (March through December), O = OMar + 59 + isLeap(y) , where isLeap 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 OMar and go straight for O = d for January and O = d + 31 for February.
2. The less redundant method is to use O = OMar − 306, 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 ${\displaystyle m=2n}$ and ${\displaystyle d=m}$ gives

${\displaystyle O=\left\lfloor 63.2n-91.4\right\rfloor }$

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.

${\displaystyle m=2n+1}$ and ${\displaystyle d=m+4}$ gives

${\displaystyle O=\left\lfloor 63.2n-56+0.2\right\rfloor }$

and with m and d interchanged

${\displaystyle O=\left\lfloor 63.2n-56+119-0.4\right\rfloor }$

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).

## Table

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

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

## Month–day

The number of the month and date is given by

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

the term ${\displaystyle {\bmod {\!}}\!(od,30)}$ can also be replaced by ${\displaystyle od-30(m-1)}$ with ${\displaystyle od}$ the ordinal date.

• Day 100 of a common year:
${\displaystyle m=\left\lfloor 100/30\right\rfloor +1=4}$
${\displaystyle d={\bmod {\!}}\!(100,30)+3-\left\lfloor 0.6(4+1)\right\rfloor =10+3-3=10}$
April 10.
• Day 200 of a common year:
${\displaystyle m=\left\lfloor 200/30\right\rfloor +1=7}$
${\displaystyle d={\bmod {\!}}\!(200,30)+3-\left\lfloor 0.6(7+1)\right\rfloor =20+3-4=19}$
July 19.
• Day 300 of a leap year:
${\displaystyle m=\left\lfloor 300/30\right\rfloor +1=11}$
${\displaystyle d={\bmod {\!}}\!(300,30)+2-\left\lfloor 0.6(11+1)\right\rfloor =0+2-7=-5}$
November - 5 = October 26 (31 - 5).

## Helper conversion table

ord.
date
common
year
leap
year
001 01 Jan
010 10 Jan
020 20 Jan
030 30 Jan
032 01 Feb
040 09 Feb
050 19 Feb
060 01 Mar 29 Feb
061 02 Mar 01 Mar
070 11 Mar 10 Mar
080 21 Mar 20 Mar
090 31 Mar 30 Mar
091 01 Apr 31 Mar
092 02 Apr 01 Apr
100 10 Apr 09 Apr
ord.
date
comm.
year
leap
year
110 20 Apr 19 Apr
120 30 Apr 29 Apr
121 01 May 30 Apr
122 02 May 01 May
130 10 May 09 May
140 20 May 19 May
150 30 May 29 May
152 01 Jun 31 May
153 02 Jun 01 Jun
160 09 Jun 08 Jun
170 19 Jun 18 Jun
180 29 Jun 28 Jun
182 01 Jul 30 Jun
183 02 Jul 01 Jul
190 09 Jul 08 Jul
ord.
date
comm.
year
leap
year
200 19 Jul 18 Jul
210 29 Jul 28 Jul
213 01 Aug 31 Jul
214 02 Aug 01 Aug
220 08 Aug 07 Aug
230 18 Aug 17 Aug
240 28 Aug 27 Aug
244 01 Sep 31 Aug
245 02 Sep 01 Sep
250 07 Sep 06 Sep
260 17 Sep 16 Sep
270 27 Sep 26 Sep
274 01 Oct 30 Sep
275 02 Oct 01 Oct
280 07 Oct 06 Oct
ord.
date
comm.
year
leap
year
290 17 Oct 16 Oct
300 27 Oct 26 Oct
305 01 Nov 31 Oct
306 02 Nov 01 Nov
310 06 Nov 05 Nov
320 16 Nov 15 Nov
330 26 Nov 25 Nov
335 01 Dec 30 Nov
336 02 Dec 01 Dec
340 06 Dec 05 Dec
350 16 Dec 15 Dec
360 26 Dec 25 Dec
365 31 Dec 30 Dec
366 31 Dec