Day of the Week Calculator

What is a Day of the Week Calculator?

A Day of the Week Calculator is a mathematical tool that determines the day of the week for any given date. It uses a specific formula to convert a calendar date into a numerical representation of the day of the week. This calculator is useful for historical research, planning events, or simply satisfying curiosity about past or future dates.

The Formula

The Day of the Week Calculator uses Zeller's Congruence, a mathematical algorithm developed by Christian Zeller in the 19th century. The formula is:

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

Where:

• \(h\) is the day of the week (0 = Saturday, 1 = Sunday, ..., 6 = Friday)
• \(q\) is the day of the month
• \(m\) is the month (3 = March, 4 = April, ..., 14 = February)
• \(K\) is the year of the century (\(year \mod 100\))
• \(J\) is the zero-based century (\(\lfloor year/100 \rfloor\))
• \(\lfloor \cdot \rfloor\) denotes the floor function

Calculation Steps

1. Adjust the month and year:
   • If the month is January or February, subtract 1 from the year and add 12 to the month
   • For other months, use them as is
2. Calculate \(q\), \(m\), \(K\), and \(J\) based on the given date
3. Apply the formula: \(h = (q + \lfloor\frac{13(m+1)}{5}\rfloor + K + \lfloor\frac{K}{4}\rfloor + \lfloor\frac{J}{4}\rfloor - 2J) \mod 7\)
4. The result \(h\) corresponds to the day of the week

Example

Let's calculate the day of the week for July 4, 1776 (U.S. Independence Day):

1. July is month 7, so no adjustment needed
2. \(q = 4\), \(m = 7\), \(K = 76\), \(J = 17\)
3. \(h = (4 + \lfloor\frac{13(7+1)}{5}\rfloor + 76 + \lfloor\frac{76}{4}\rfloor + \lfloor\frac{17}{4}\rfloor - 2(17)) \mod 7\)
4. \(h = (4 + 21 + 76 + 19 + 4 - 34) \mod 7 = 90 \mod 7 = 6\)
5. Result: 6 corresponds to Friday

Visual Representation

This visual representation illustrates the date July 4, 1776, with the calculated day of the week (Friday) highlighted in the center of a circular calendar representation. The significance of this date as U.S. Independence Day is noted, demonstrating how the Day of the Week Calculator can be used to find the day of the week for important historical dates.