Zeller's congruence

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Zeller's congruence is an algorithm devised by Christian Zeller in the 19th century to calculate the day of the week for any Julian or Gregorian calendar date. It can be considered to be based on the conversion between Julian day and the calendar date.

Contents

Formula

For the Gregorian calendar, Zeller's congruence is

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 2 February 2010 (02/02/2010 in DD/MM/YYYY), the algorithm counts the date as the second day of the fourteenth month of 2009 (02/14/2009 in DD/MM/YYYY format) So the adjusted year above is:

= the actual year, for months from March to December.
= the previous year, for January and February.

For an ISO week date Day-of-Week d (1 = Monday to 7 = Sunday), use

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:

The reason that the formula differs between calendars is that the Julian 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. This is only required through 1929, as this was the last year that the Julian calendar was still in use by any country on earth, and thus is not required for 1930 or later.

The formulae can be used proleptically, but "Year 0" is in fact year 1 BC (see astronomical year numbering). The Julian calendar is in fact proleptic right up to 1 March AD 4 owing to mismanagement in Rome (but not Egypt) in the period since the calendar was put into effect on 1 January 45 BC (which was not a leap year). In addition, the modulo operator might truncate integers to the wrong direction (ceiling instead of floor). To accommodate this, one can add a sufficient multiple of 400 Gregorian or 700 Julian years.

Examples

For 1 January 2000, the date would be treated as the 13th month of 1999, so the values would be:

So the formula evaluates as .

(The 36 comes from , truncated to an integer.)

However, for 1 March 2000, the date is treated as the 3rd month of 2000, so the values become

so the formula evaluates as .

Implementations in software

Basic modification

The formulas rely on the mathematician's definition of modulo division, which means that −2 mod 7 is equal to positive 5. Unfortunately, in the truncating 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 with + 5J and J with + 6J.

For the Gregorian calendar, Zeller's congruence becomes

For the Julian calendar, Zeller's congruence becomes


One can readily see that, in a given year, the last day of February and March 1 are a good test dates.

As an aside note, if we have a three-digit number abc, where a, b, and c are the digits, each nonpositive if abc is nonpositive; we have (abc) mod 7 = 9*a + 3*b + c. Repeat the formula down to a single digit. If the result is 7, 8, or 9, then subtract 7. If, instead, the result is negative, then add 7. If the result is still negative, then add 7 one more time. Utilizing this approach, we can avoid the worries of language specific differences in mod 7 evaluations. This also may enhance a mental math technique.

Common simplification

Zeller used decimal arithmetic, and found it convenient to use J and K values as two-digit numbers representing the year and century. But when using a computer, it is simpler to handle the year as a single 4-digit number. For the Gregorian calendar, Zeller's congruence becomes

where is , defined in the section above.

In this case there is no possibility of underflow due to the single negative term because .

For the Julian calendar, Zeller's congruence becomes

The algorithm above is mentioned for the Gregorian case in RFC   3339, Appendix B, albeit in an abridged form that returns 0 for Sunday.

Other variations

At least three other algorithms share the overall structure of Zeller's congruence in its "common simplification" type, also using an m [3, 14] Z and the "modified year" construct.

Both expressions can be shown to progress in a way that is off by one compared to the original month-length component over the required range of m, resulting in a starting value of 0 for Sunday.

See also

References

  1. The every five months rule only applies to the twelve months of a year commencing on 1 March and ending on the last day of the following February.
  2. 1 2 Stockton, J R. "Material Related to Zeller's Congruence". "Merlyn", archived at NCTU Taiwan. Archived from the original on 2023-09-18. Retrieved 2021-03-10.
  3. Tøndering, Claus. "Week-related questions". www.tondering.dk.

Bibliography

Each of these four similar imaged papers deals firstly with the day of the week and secondly with the date of Easter Sunday, for the Julian and Gregorian calendars. The pages link to translations into English.