# Ordinal date

Date 2021-10-23 2021-296

An ordinal date is a calendar date typically consisting of a year and a day of the 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.

## 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 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.

In the following text, several algorithms for calculating the ordinal date 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 date 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.[1]

### 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 date 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 date to a January 1 based ordinal date:

• 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:

For ${\displaystyle m=2n}$ and ${\displaystyle d=m}$ we get

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

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.

For ${\displaystyle m=2n+1}$ and ${\displaystyle d=m+4}$ we get

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

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

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

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

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