Ordinal date

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

${\displaystyle \lfloor 30.6(m+1)\rfloor +d-122}$

which can also be written

${\displaystyle \left\lfloor 30.6m-91.4\right\rfloor +d}$

or

${\displaystyle \left\lfloor 30.6(m-3)+0.4\right\rfloor +d}$

with m the month number and d the date. ${\displaystyle \left\lfloor x\right\rfloor }$ is the floor function.

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.

"Doomsday" properties:

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

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

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

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

and with m and d interchanged

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

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).

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.

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	${\displaystyle 30(m-1)+\left\lfloor 0.6(m+1)\right\rfloor -i}$

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

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).

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

• "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