Zeller's congruence
Encyclopedia
Zeller's congruence is an algorithm
devised by Christian Zeller to calculate the day of the week for any Julian
or Gregorian calendar
date.
for the Julian calendar it is
where
NOTE: In this algorithm January and February are counted as months 13 and 14 of the previous year. E.g. if it is February 2, 2010, the algorithm counts the date as the second day of the fourteenth month of 2009 (14/02/2009 in MM/DD/YYYY format)
For an ISO week date
Day-of-Week d (1 = Monday to 7 = Sunday), use
division, which means that −2 mod 7 is equal to positive 5. Unfortunately, the way most computer languages implement the remainder function, −2 mod 7 returns a result of -2. So, to implement Zeller's congruence on a computer, the formulas should be altered slightly to ensure a positive numerator. The simplest way to do this is to replace − 2J by + 5J and − J by + 6J. So the formulas become:
for the Gregorian calendar, and
for the Julian calendar.
One can readily see that, in a given year, March 1 (if that is a Saturday, then March 2) is a good test date; and that, in a given century, the best test year is that which is a multiple of 100.
Zeller used decimal arithmetic, and found it convenient to use J and K in representing the year. But when using a computer, it is simpler to handle the modified year Y, which is Y - 1 during January and February:
for the Gregorian calendar, and
for the Julian calendar.
For the Gregorian calendar, the various parts of this formula can therefore be understood as follows:
The reason that the formula differs for the Julian calendar is that this calendar does not have a separate rule for leap centuries and is offset from the Gregorian calendar by a fixed number of days each century.
Since the Gregorian calendar was adopted at different times in different regions of the world, the location of an event is significant in determining the correct day of the week for a date that occurred during this transition period.
The formulae can be used proleptically, but with care for years before Year 0. To accommodate this, one can add a sufficient multiple of 400 Gregorian or 28 Julian years.
So the formula evaluates as (1 + 36 + 99 + 24 + 4 − 38) mod 7 = 126 mod 7 = 0 = Saturday
(The 36 comes from (13+1)*26/10 = 364/10, truncated to an integer.)
However, for March 1, 2000, the date is treated as the 3rd month of 2000, so the values become
so the formula evaluates as (1 + 10 + 0 + 0 + 5 − 40) mod 7 = −24 mod 7 = 4 = Wednesday
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
devised by Christian Zeller to calculate the day of the week for any Julian
Julian calendar
The Julian calendar began in 45 BC as a reform of the Roman calendar by Julius Caesar. It was chosen after consultation with the astronomer Sosigenes of Alexandria and was probably designed to approximate the tropical year .The Julian calendar has a regular year of 365 days divided into 12 months...
or Gregorian calendar
Gregorian calendar
The Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...
date.
Formula
For the Gregorian calendar, Zeller's congruence isfor the Julian calendar it is
where
- h is the day of the week (0 = Saturday, 1 = Sunday, 2 = Monday, ...)
- q is the day of the month
- m is the month (3 = March, 4 = April, 5 = May, ..., 14 = February)
- K the year of the century (
). - J is the century (actually
) (For example, in 1995 the century would be 19, even though it was the 20th century.)
NOTE: In this algorithm January and February are counted as months 13 and 14 of the previous year. E.g. if it is February 2, 2010, the algorithm counts the date as the second day of the fourteenth month of 2009 (14/02/2009 in MM/DD/YYYY format)
For an ISO week date
ISO week date
The ISO week date system is a leap week calendar system that is part of the ISO 8601 date and time standard. The system is used in government and business for fiscal years, as well as in timekeeping....
Day-of-Week d (1 = Monday to 7 = Sunday), use
Implementation in software
The formulas rely on the mathematician's definition of moduloModulo operation
In computing, the modulo operation finds the remainder of division of one number by another.Given two positive numbers, and , a modulo n can be thought of as the remainder, on division of a by n...
division, which means that −2 mod 7 is equal to positive 5. Unfortunately, the way most computer languages implement the remainder function, −2 mod 7 returns a result of -2. So, to implement Zeller's congruence on a computer, the formulas should be altered slightly to ensure a positive numerator. The simplest way to do this is to replace − 2J by + 5J and − J by + 6J. So the formulas become:
for the Gregorian calendar, and
for the Julian calendar.
One can readily see that, in a given year, March 1 (if that is a Saturday, then March 2) is a good test date; and that, in a given century, the best test year is that which is a multiple of 100.
Zeller used decimal arithmetic, and found it convenient to use J and K in representing the year. But when using a computer, it is simpler to handle the modified year Y, which is Y - 1 during January and February:
for the Gregorian calendar, and
for the Julian calendar.
Analysis
These formulas are based on the observation that the day of the week progresses in a predictable manner based upon each subpart of that date. Each term within the formula is used to calculate the offset needed to obtain the correct day of the week.For the Gregorian calendar, the various parts of this formula can therefore be understood as follows:
represents the progression of the day of the week based on the day of the month, since each successive day results in an additional offset of 1 in the day of the week.
represents the progression of the day of the week based on the year. Assuming that each year is 365 days long, the same date on each succeeding year will be offset by a value of 
.
- Since there are 366 days in each leap year, this needs to be accounted for by adding an additional day to the day of the week offset value. This is accomplished by adding
to the offset. This term is calculated as an integer result. Any remainder is discarded.
- Using similar logic, the progression of the day of the week for each century may be calculated by observing that there are 36524 days in a normal century and 36525 days in each century divisible by 400. Since
and 
, the term :
accounts for this (again using integer division and discarding any fractional remainder). To avoid negative numbers, this term can be replaced with :
with equivalent results.
- The term
can be explained as follows. Zeller observed that, by starting each year on March 1, the day of the week for each succeeding month progressed by multiplying the month by a constant value and discarding the fractional remainder.
- The overall function,
, normalizes the result to reside in the range of 0 to 6, which yields the index of the correct day of the week for the date being analyzed.
The reason that the formula differs for the Julian calendar is that this calendar does not have a separate rule for leap centuries and is offset from the Gregorian calendar by a fixed number of days each century.
Since the Gregorian calendar was adopted at different times in different regions of the world, the location of an event is significant in determining the correct day of the week for a date that occurred during this transition period.
The formulae can be used proleptically, but with care for years before Year 0. To accommodate this, one can add a sufficient multiple of 400 Gregorian or 28 Julian years.
Examples
For January 1, 2000, the date would be treated as the 13th month of 1999, so the values would be:- q = 1
- m = 13
- K = 99
- J = 19
So the formula evaluates as (1 + 36 + 99 + 24 + 4 − 38) mod 7 = 126 mod 7 = 0 = Saturday
(The 36 comes from (13+1)*26/10 = 364/10, truncated to an integer.)
However, for March 1, 2000, the date is treated as the 3rd month of 2000, so the values become
- q = 1
- m = 3
- K = 0
- J = 20
so the formula evaluates as (1 + 10 + 0 + 0 + 5 − 40) mod 7 = −24 mod 7 = 4 = Wednesday
External links
- The Calendrical Works of Rektor Chr. Zeller: The Day-of-Week and Easter Formulae by J R Stockton, near London, UK. The site includes images and translations of the above four papers, and of Zeller's reference card "Das Ganze der Kalender-Rechnung".