Data Source and Methodology

This calculator determines the day of the week using the JavaScript internal date object, which correctly handles the Gregorian calendar reform and leap years. The underlying mathematical principle for this calculation is an algorithm known as Zeller's Congruence.

Authoritative Source: Zeller, C. (1882). "Die Grundaufgaben der Kalenderrechnung auf neue und vereinfachte Weise gelöst". Württembergische Vierteljahrshefte für Landesgeschichte (in German). 5: 313–314.

All calculations are based strictly on the formulas and data provided by this established algorithm.

The Formula Explained

Zeller's congruence is an algorithm that provides the day of the week. The formula for the Gregorian calendar (adopted in 1582) is:

$$ h = \left(q + \left\lfloor \frac{13(m+1)}{5} \right\rfloor + K + \left\lfloor \frac{K}{4} \right\rfloor + \left\lfloor \frac{J}{4} \right\rfloor - 2J \right) \mod 7 $$

The result $h$ is an integer (0-6) that corresponds to a specific day of the week.

Glossary of Variables

$h$ (Result)
The day of the week, where $0$ = Saturday, $1$ = Sunday, $2$ = Monday, $3$ = Tuesday, $4$ = Wednesday, $5$ = Thursday, $6$ = Friday.
$q$ (Day)
The day of the month (e.g., 25).
$m$ (Month)
The month. This is the trickiest part: January and February are counted as months 13 and 14 of the previous year.
  • January = 13 (of previous year)
  • February = 14 (of previous year)
  • March = 3, April = 4, ..., December = 12
$year$
The year. If the month is January or February, you must use $year - 1$.
$J$ (Century)
The century, calculated as $J = \lfloor \frac{year}{100} \rfloor$.
$K$ (Year of Century)
The year of the century, calculated as $K = year \mod 100$.
$\lfloor ... \rfloor$ (Floor)
The floor function, which means rounding down to the nearest integer (e.g., $\lfloor 20.8 \rfloor = 20$).

How It Works: A Step-by-Step Example

Let's find the day of the week for the U.S. Declaration of Independence: July 4, 1776.

  1. Identify Inputs:
    • Day ($q$) = 4
    • Month (July) = 7
    • Year = 1776
  2. Adjust Variables:
    • $q = 4$
    • $m = 7$ (It's not Jan or Feb, so no change)
    • $year = 1776$ (No change)
  3. Calculate $J$ and $K$:
    • $J = \lfloor \frac{1776}{100} \rfloor = \lfloor 17.76 \rfloor = 17$
    • $K = 1776 \mod 100 = 76$
  4. Apply the Formula:

    $h = (q + \lfloor \frac{13(m+1)}{5} \rfloor + K + \lfloor \frac{K}{4} \rfloor + \lfloor \frac{J}{4} \rfloor - 2J) \mod 7$

    $h = (4 + \lfloor \frac{13(7+1)}{5} \rfloor + 76 + \lfloor \frac{76}{4} \rfloor + \lfloor \frac{17}{4} \rfloor - 2 \times 17) \mod 7$

    $h = (4 + \lfloor \frac{13 \times 8}{5} \rfloor + 76 + 19 + \lfloor 4.25 \rfloor - 34) \mod 7$

    $h = (4 + \lfloor \frac{104}{5} \rfloor + 76 + 19 + 4 - 34) \mod 7$

    $h = (4 + \lfloor 20.8 \rfloor + 76 + 19 + 4 - 34) \mod 7$

    $h = (4 + 20 + 76 + 19 + 4 - 34) \mod 7$

    $h = (123 - 34) \mod 7$

    $h = 89 \mod 7$

  5. Find the Remainder:

    $89 \div 7 = 12$ with a remainder of $5$.

    Therefore, $h = 5$.

  6. Interpret the Result:

    Using the formula's key (where $0$ = Saturday, $1$ = Sunday, $2$ = Monday, $3$ = Tuesday, $4$ = Wednesday, $5$ = Thursday), the result $h=5$ corresponds to Thursday.

Frequently Asked Questions

Why are January and February treated as months 13 and 14 of the previous year?

This is a quirk of Zeller's algorithm. It simplifies the leap year calculation by placing the leap day (Feb 29) at the very end of the "year," ensuring the math for the $K$ term (year of the century) works consistently.

Does this calculator work for the Julian calendar?

This tool (and the formula shown) is optimized for the Gregorian calendar, which is the standard civil calendar used today. The Gregorian calendar was adopted in 1582, but different countries adopted it at different times. The built-in JavaScript logic provides the most consistent result for all dates based on modern projections.

How accurate is this for future dates?

It is perfectly accurate for future dates, assuming the current Gregorian calendar rules for leap years (divisible by 4, except for years divisible by 100 unless also divisible by 400) are maintained.

What is a "modulo" (mod) operation?

The modulo operation (abbreviated as 'mod') finds the remainder after division. For example, $10 \mod 3 = 1$ because 10 divided by 3 is 3 with a remainder of 1. In this calculator, $h \mod 7$ ensures the final answer is always a number between 0 and 6, which can then be mapped to a day of the week.

Can I find the day of the week for the year 1 AD?

Yes. This tool can calculate dates back to 1 AD by projecting the Gregorian calendar rules backward (this is known as the proleptic Gregorian calendar). For example, entering January 1, 1, will correctly return Saturday.

Tool developed by Ugo Candido.
Algorithm and content verified by the CalcDomain Editorial Board.

Last accuracy review: