Definition:Calendar
Definition
A calendar is a technique to keep track of the passing of time throughout the course of a year.
Its objective is to provide a more-or-less standard way of identifying the time of year by associating a name and number to each day in the year.
Julian Calendar
The Julian calendar is the calendar which was introduced by Julius Caesar in $45 \, \text{BCE}$.
It divides the year into:
and:
The years themselves are given an index number, and are known by that number.
A standard numbering system was introduced by Dionysus Exiguus.
He identified a particular year as being $525 \, \text{A.D.}$, where $\text{A.D.}$ is an abbreviation for Anno Domini, Latin for in the year of the Lord.
The year $1$ was conventionally supposed to identify the year of the birth of Jesus of Nazareth, although the accuracy of this has since been questioned.
Years before $1 \, \text{A.D.}$ are counted backwards and assigned the label $\text{B.C.}$.
However, note that the year immediately prior to $1 \, \text{A.D.}$ is $1 \, \text{B.C.}$, not the intuitive year $0$, a discrepancy that can cause confusion.
Using this system of numbering, a leap year is identified by this number being divisible by $4$.
The $365$ or $366$ days in the year are divided into $12$ approximately equal sections called months, which are assigned both names and index numbers:
| \(\ds 1:\) | \(\) | \(\ds \) | January | \(\quad\) $31$ days | ||||||||||
| \(\ds 2:\) | \(\) | \(\ds \) | February | \(\quad\) $28$ days, or $29$ days in a leap year | ||||||||||
| \(\ds 3:\) | \(\) | \(\ds \) | March | \(\quad\) $31$ days | ||||||||||
| \(\ds 4:\) | \(\) | \(\ds \) | April | \(\quad\) $30$ days | ||||||||||
| \(\ds 5:\) | \(\) | \(\ds \) | May | \(\quad\) $31$ days | ||||||||||
| \(\ds 6:\) | \(\) | \(\ds \) | June | \(\quad\) $30$ days | ||||||||||
| \(\ds 7:\) | \(\) | \(\ds \) | July | \(\quad\) $31$ days | ||||||||||
| \(\ds 8:\) | \(\) | \(\ds \) | August | \(\quad\) $31$ days | ||||||||||
| \(\ds 9:\) | \(\) | \(\ds \) | September | \(\quad\) $30$ days | ||||||||||
| \(\ds 10:\) | \(\) | \(\ds \) | October | \(\quad\) $31$ days | ||||||||||
| \(\ds 11:\) | \(\) | \(\ds \) | November | \(\quad\) $30$ days | ||||||||||
| \(\ds 12:\) | \(\) | \(\ds \) | December | \(\quad\) $31$ days |
Thus, for example, the day following the $31$st of January is the $1$st of February, and the $30$th of June is followed by the $1$st of July.
Gregorian Calendar
The Gregorian calendar is the calendar which was introduced by Pope Gregory XIII in $1582 \, \text{CE}$.
Over the course of the next few centuries it was gradually adopted by the various subcultures of Western civilization.
It was derived as a refinement of the Julian calendar to correct for the discrepancy between the Julian calendar and the tropical year.
The years themselves are assigned the same numbers as their Julian counterparts.
An evolving modern convention is to refer to $\text{A.D.}$ and $\text{B.C.}$ as $\text{CE}$ and $\text{BCE}$, for common era and before common era respectively.
Like the Julian calendar, it divides the year into either $365$ days or $366$ days, according to the year number.
The months, similarly, are kept the same as for the Julian calendar.
The only difference is in the determination of which years are classified as leap years.
Let $y$ be the year number.
Then $y$ is a leap year if and only if
- $y$ is divisible by $400$
or:
Thus, for example:
Muslim Calendar
The Muslim calendar is a calendar whose period is based on $12$ full cycles of phases of the moon.
Each Muslim year lasts either $354$ or $355$ days, depending on precisely when the relevant new moons happen
The years are numbered from the year when Muhammad and his followers migrated from Mecca to Medina in $622 \, \text{A.D}$ by the Julian calendar.
Years are denoted with the suffix $\text{A.H.}$, meaning Anno Hegirae, literally in the year of the Hejira.
The start and end of the Muslim year are $10$ to $12$ days earlier every civil year.
The $354$ or $355$ days in the Muslim year are divided into $12$ approximately equal sections called months, each either $29$ or $30$ days, depending on the phases of the moon.
They are assigned both names and index numbers:
| \(\ds 1:\) | \(\) | \(\ds \) | Muḥarram | |||||||||||
| \(\ds 2:\) | \(\) | \(\ds \) | Ṣafar | |||||||||||
| \(\ds 3:\) | \(\) | \(\ds \) | Rabī‘ al-awwal | |||||||||||
| \(\ds 4:\) | \(\) | \(\ds \) | Rabī‘ ath-thānī | |||||||||||
| \(\ds 5:\) | \(\) | \(\ds \) | Jumādá al-ūlá | |||||||||||
| \(\ds 6:\) | \(\) | \(\ds \) | Jumādá al-ākhirah | |||||||||||
| \(\ds 7:\) | \(\) | \(\ds \) | Rajab | |||||||||||
| \(\ds 8:\) | \(\) | \(\ds \) | Sha‘bān | |||||||||||
| \(\ds 9:\) | \(\) | \(\ds \) | Ramaḍān | |||||||||||
| \(\ds 10:\) | \(\) | \(\ds \) | Shawwāl | |||||||||||
| \(\ds 11:\) | \(\) | \(\ds \) | Dhū al-Qa‘dah | |||||||||||
| \(\ds 12:\) | \(\) | \(\ds \) | Dhū al-Ḥijjah |
Jewish Calendar
The Jewish calendar is a calendar based on both the phases of the moon and on the orbital period of earth around the sun.
The whose period is based on $12$ full cycles of phases of the moon.
The basic Jewish year is $354$ or $355$ days of $12$ months, each alternately $29$ and $30$ days long.
When the error between the Jewish year and tropical year becomes approximately $30$ days, a $13$th month is inerted into the year.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $365 \cdotp 2422$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $365 \cdotp 24219 \, 878$