Zeller's congruence

Zeller's congruence is a mathematical way of finding the day of the week on a given date . The mathematician and theologian Christian Zeller published a formula for this in 1882.

Formulas

Let h be the day of the week to be determined, q the day, m the month (whereby March to December have the numbers 3–12 as usual, January and February correspond to the months 13 and 14 of the previous year), J the century (these are the first two) Digits of the four-digit year) and K the last two digits of the four-digit year (for January and February the number of the previous year), the following applies:

1. for a date in the Gregorian calendar :

${\displaystyle {\mathit {h}}_{greg}=\left({q}+\left\lfloor {\frac {(m+1)\cdot 13}{5}}\right\rfloor +K+\left\lfloor {\frac {K}{4}}\right\rfloor +\left\lfloor {\frac {\mathit {J}}{4}}\right\rfloor -2\cdot {\mathit {J}}\right){\bmod {\,}}7}$

2. for a date in the Julian calendar :

${\displaystyle {\mathit {h}}_{jul}=\left({q}+\left\lfloor {\frac {(m+1)\cdot 13}{5}}\right\rfloor +K+\left\lfloor {\frac {K}{4}}\right\rfloor +5-{\mathit {J}}\right){\bmod {\,}}7}$

The expression ( Gaussian bracket ) returns the largest integer . The mod 7 (pronounced modulo 7) at the end means that the determined value is divided by 7 and the remainder that remains after this whole-number division by 7 is determined. This results in a number between 0 and 6 for h , which indicates the weekday of the date: ${\displaystyle \lfloor x\rfloor }$ ${\displaystyle \leq x}$

Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
1	2	3	4th	5	6th	0

If the result is negative (depending on the modulo function used), add 7 to produce a positive number. This number then corresponds to the day of the week. To get a positive number in every case, simply replace with or with in the formula . ${\displaystyle -2{\mathit {J}}}$ ${\displaystyle +5{\mathit {J}}}$ ${\displaystyle -{\mathit {J}}}$ ${\displaystyle +6{\mathit {J}}}$

Explanation

The variable q flows with its actual value into the variable h for the day of the week. The integration of the month becomes more complicated because the length of the individual months does not follow a uniform pattern. With the term , i.e. the increase of the value m for the month by 1, the multiplication by and the subsequent rounding off, the inconsistent sequence of the length of the individual months is generally incorporated into the formula. The term takes into account the year and the leap days to be inserted in the century up to the relevant year. Both formulas only differ in the last term, which takes into account the different leap year regulations of both calendar systems. ${\displaystyle \left\lfloor {\frac {(m+1)\cdot 13}{5}}\right\rfloor }$ ${\displaystyle {\frac {13}{5}}}$ ${\displaystyle K+\left\lfloor {\frac {K}{4}}\right\rfloor }$

Examples

Two examples to illustrate this:

1. On what day of the week was Friedrich II of Prussia born ( January 24, 1712 )?

The values ​​are: q = 24, m = 13 (January is the 13th month of the previous year), J = 17, K = 11 (January is treated as belonging to the previous year.) The following applies:

${\displaystyle {\mathit {h}}=\left(24+\left\lfloor {\frac {(13+1)\cdot 13}{5}}\right\rfloor +11+\left\lfloor {\frac {11}{4}}\right\rfloor +\left\lfloor {\frac {17}{4}}\right\rfloor -2\cdot 17\right){\bmod {\,}}7}$

${\displaystyle {\mathit {h}}=\left(24+36+11+2+4-34\right){\bmod {\ }}7}$

${\displaystyle {\mathit {h}}=43{\bmod {\ }}7=1}$

Friedrich II of Prussia was born on a Sunday.

2. What day of the week did Christopher Columbus discover the new world ( October 12, 1492 )? (Since the date is before the introduction of the Gregorian calendar, the formula for the Julian calendar is used here.)

The values ​​are: q = 12, m = 10, J = 14, K = 92. The following applies:

${\displaystyle {\mathit {h}}=\left(12+\lfloor ((10+1)\cdot 13)/5\rfloor +92+\lfloor 92/4\rfloor +5-14\right){\bmod {\,}}7}$

${\displaystyle {\mathit {h}}=\left(12+28+92+23+5-14\right){\bmod {\,}}7}$

${\displaystyle {\mathit {h}}=\left(146\right){\bmod {\,}}7=6}$

Christopher Columbus landed in America on a Friday.

Use in mental arithmetic

Zeller's congruence can also be used to determine the day of the week in the head. To make the formula easier to work with in your head, it can be simplified a bit by calculating the values ​​for the months and memorizing them:

January	February	March	April	May	June	July	August	September	October	November	December
1	4th	3	6th	1	4th	6th	2	5	0	3	5

Instead of recalculating the second term for each date, simply insert the corresponding number from the table above. The same applies here: January and February are treated as belonging to the previous year.

Perpetual calendars are a simple and reliable method of checking the results .

See also

• Weekday calculation
• Gaussian weekday formula

literature

• Christian Zeller: The basic tasks of calendar calculation have been solved in a new and simplified way. In: Württembergische Vierteljahrshefte für Landesgeschichte 5 (1882), 313f. ( Available on the Internet. )
• Christian Zeller: Calendar formulas . In: Mathematical-scientific communications of the mathematical-scientific association in Württemberg 1 (1885), 54–58. ( Available on the Internet. )
• Christian Zeller: Calendar formulas . In: Acta Mathematica 9 (1887), pp. 131-136. ( available online from Springer Verlag or here on the Internet. )

Individual evidence

1. ↑ Cf. Christian Zeller: The basic tasks of the calendar calculation solved in a new and simplified way. In: Württembergische Vierteljahrshefte für Landesgeschichte 5 (1882), 314f.
2. ↑ Cf. Christian Zeller: Calendar formulas. In: Acta Mathematica 9 (1887), 131-136. These formulas are a revised version of the formulas published in 1882. The ancient Roman calendar is to be used for the months, i.e. March = 1, April = 2 ... December = 10, January = 11 and February = 12.
3. ↑ Zeller also used these two dates to illustrate his formulas. See ibid. 132.