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Claus Tøndering –
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Week-related questions

This page deals with the week in the context of the Christian calendar. More information about the week can be found in the section entitled “the Week”.

The Solar Number of 2014:

7

What is the Solar Cycle?

In the Julian calendar the relationship between the days of the week and the dates of the year is repeated in cycles of 28 years. In the Gregorian calendar this is still true for periods that do not cross years that are divisible by 100 but not by 400.

A period of 28 years is called a Solar Cycle. The Solar Number of a year identifies the position of the year within the Solar Cycle. The Solar Number is calculated thus:

Solar Number = (year + 8) mod 28 + 1

In the Julian calendar there is a one-to-one relationship between the Solar Number and the day on which a particular date falls.

The leap year cycle of the Gregorian calendar is 400 years, which is 146,097 days, which curiously enough is a multiple of 7. So in the Gregorian calendar the equivalent of the “Solar Cycle” would be 400 years, not 7×400=2800 years as one might be tempted to believe. However, the terms Solar Cycle and Solar Number are normally only used to refer to the 28-year cycle mentioned above.

The Dominical Letter of 2014:

E

What is the Dominical Letter?

Each ordinary (non-leap) year is assigned a letter in the range A to G which describes what days of the year are Sundays. This letter is called the “Dominical Letter” (“Sunday Letter”) of the year.

It works in this manner: Assign the letter A to 1 January, B to 2 Jan, C to 3 Jan, ... G to 7 Jan, A to 8 Jan, B to 9 Jan, and so on, using the letters A to G and omitting the leap day.

In a year with Dominical Letter A, all days marked A are Sundays. In a year with Dominical Letter B, all days marked B are Sundays. And so on.

Leap years have two Dominical Letters, one which is used from the start of January until the leap day, and another one which is used for the rest of the year.

Thus, if you know the Dominical Letter(s) of a year, you know how the days of the year relate to days of the week.

What day of the week was 2 August 1953?

To calculate the day on which a particular date falls, the following algorithm may be used. (Click here for a description of the symbols   and   and the operator ‘mod’.)

a =
 14 – month 
12
y = year – a
m = month + 12a – 2
For the Julian calendar:
d = ( 5 + day + y + 
 y 
4
 + 
 31m 
12
) mod 7
For the Gregorian calendar:
d = ( day + y + 
 y 
4
 – 
 y 
100
 + 
 y 
400
 + 
 31m 
12
) mod 7
The value of d is 0 for a Sunday, 1 for a Monday, 2 for a Tuesday etc.

Example: On what day of the week was the author born?

My birthday is 2 August 1953 (Gregorian, of course).

a=
 14 – 8 
12
= 0
y=1953 – 0 = 1953
m=8 + 12 × 0 – 2 = 6
d=( 2 + 1953 + 
 1953 
4
 – 
 1953 
100
 + 
 1953 
400
 + 
 31 × 6 
12
) mod 7
=(2 + 1953 + 488 – 19 + 4 + 15) mod 7
=2443 mod 7
=0

I was born on a Sunday.

When can I reuse my 1992 calendar?

Let us first assume that you are only interested in which dates fall on which days of the week; you are not interested in the dates for Easter and other irregular holidays.

Let us further confine ourselves to the years 1901-2099.

With these restrictions, the answer is as follows:

  • If year X is a leap year, you can reuse its calendar in year X+28.
  • If year X is the first year after a leap year, you can reuse its calendar in years X+6, X+17, and X+28.
  • If year X is the second year after a leap year, you can reuse its calendar in years X+11, X+17, and X+28.
  • If year X is the third year after a leap year, you can reuse its calendar in years X+11, X+22, and X+28.

Note that the expression X+28 occurs in all four items above. So you can always reuse your calendar every 28 years.

But if you also want your calendar’s indication of Easter and other Christian holidays to be correct, the rules are far too complex to be put to a simple formula. Sometimes calendars can be reused after just six years. For example, under the Easter rules of the Roman Catholic and Protestant churches, the calendars for the years 1981 and 1987 are identical, even when it comes to the date for Easter. But sometimes a very long time can pass before a calendar can be reused; if you happen to have a calendar from 1940, you won’t be able to reuse it until the year 5280!

In calendrical calculations, we frequently use an operation call integer division.

Ordinarily we would say that, for example, 14 divided by 5 is 2.8. But when we use integer division, we discard the decimal fraction and simply state that 14 divided by 5 is 2.

We indicate that we use integer division by enclosing the division between the symbols   and   , for example thus:

  14/5   = 2

or thus:

 14 
5
= 2

When we perform integer division, the division leaves a remainder. In the case of 14 divided by 5, the remainder is 4: We can subtract 5 twice from 14, and this leaves us with 4.

We use the mathematical operator ‘mod’ to indicate the remainder. This is known as the modulo operator. We therefore have:

14 mod 5 = 4

If you want to use integer division in computer programs, you must realize that different programs use different notations for the operations. The following table shows how integer division and the modulo operator may be written:

Integer division Modulo
The Calendar FAQ   14/5   14 mod 5
Microsoft Excel (English) INT(14/5) MOD(14,5)
Visual Basic 14 \ 5 14 Mod 5
C, C#, C++ 14 / 5 14 % 5