Ordinal date

Key takeaways

Ordinal date belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ordinal date to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ordinal date from memory before moving on to harder problems.

Reference excerpt

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. 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 O are 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:

Let O be 0. From i = 1 .. m - 1, add the length of month i 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 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: The trivial method is to skip the calculation of OMar and go straight for O = d for January and O = d + 31 for February. 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 m = 2 n {\displaystyle m=2n} and d = m {\displaystyle d=m} gives

O = ⌊ 63.2 n − 91.4 ⌋ {\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.

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

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

and with m and d interchanged

O = ⌊ 63.2 n − 56 + 119 − 0.4 ⌋ {\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

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

m = ⌊ o d / 30 ⌋ + 1 {\displaystyle m=\left\lfloor od/30\right\rfloor +1}

d = mod ( o d , 30 ) + i − ⌊ 0.6 ( m + 1 ) ⌋ {\displaystyle d={\bmod {\!}}\!(od,30)+i-\left\lfloor 0.6(m+1)\right\rfloor }

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

Day 100 of a common year:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Ordinal date

Start with the simplest possible case. Write down what Ordinal date claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ordinal date before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ordinal date ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ordinal date

In research: Ordinal date appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ordinal date in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ordinal date is common in secondary-school and first-year university syllabi. It links to neighbouring topics Calendars, Ordinal numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Ordinal date outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Ordinal date in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ordinal date means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ordinal date out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ordinal date in simple terms?

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 f…

Why does Ordinal date matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ordinal date?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ordinal date.