Day of the Week Calculator

How to Calculate the Day of the Week

Determining the day of the week for any given date is a fascinating mathematical process that combines calendar knowledge with algorithmic thinking. This calculation is useful in various fields, including history, event planning, and even in understanding certain cultural or religious practices tied to specific days of the week.

Formula and Components

The most commonly used formula for calculating the day of the week is Zeller's Congruence:

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

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 (\(year \bmod 100\))
• \(J\) is the zero-based century (\(year \div 100\))
• \(\lfloor \rfloor\) denotes the floor function
• \(\bmod\) is the modulo operation

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 the century and year of the century.
3. Apply Zeller's Congruence formula.
4. Take the result modulo 7 to get the day of the week.
5. Map the result to the corresponding day (0 = Saturday, 1 = Sunday, etc.).

Example Calculation

Let's calculate the day of the week for July 20, 1969 (the date of the first moon landing):

1. Date: July 20, 1969
   q = 20, m = 7, year = 1969
2. K = 69 (year of century), J = 19 (zero-based century)
3. Apply the formula:
   \(h = (20 + \lfloor\frac{13(7+1)}{5}\rfloor + 69 + \lfloor\frac{69}{4}\rfloor + \lfloor\frac{19}{4}\rfloor - 2(19)) \bmod 7\)
   \(h = (20 + 21 + 69 + 17 + 4 - 38) \bmod 7\)
   \(h = 93 \bmod 7 = 2\)
4. Result: 2 corresponds to Sunday

Therefore, July 20, 1969, was a Sunday.

Visual Representation

This diagram visually represents the day of the week (Sunday) for the given date (July 20, 1969). The day is prominently displayed in the center of the circle, emphasizing its significance in relation to this historic date.