# Day of the Week Calculator

Created by Davide Borchia
Last updated: Jul 04, 2022

If you've ever wondered what day of the week you were born, or when your birthday will fall this year, our day of the week calculator will come in handy!

In this article, you will discover how to calculate the day of the week without checking the calendar. If you are willing, you can learn how to do so in as little as a few seconds!

Instead, if you wonder how is Easter calculated, we have a tool designed precisely for that purpose. Don't miss it!

## Days of the week

While the day and the year are quantities somehow rooted in the Natural world, corresponding to a rotation and a revolution of Earth, the subdivision of time into months and weeks is purely arbitrary. This led to many different calendars: from the one we know so well, the Gregorian or Julian calendar to the Chinese one or ancient calendars like the Mayan one.

What's more, you can even ask how old is my dog in human years. Is it a dog's calendar then :)

## The Gregorian calendar: the easiest way to say what day it is.

It is possible to find the day of the week of each date, thanks to the regularity of the modern Gregorian calendar.

The Gregorian calendar, adopted in 1582 and bearing the name of Pope Gregory XIII, introduced a correction to the previously used Julian calendar, taking into account the correct duration of a day, leaving us with the current pattern of the duration of a year:

• If the year is not divisible by $4$ is a "normal year" and lasts $365$ days;
• A leap year (with year number divisible by $4$) has $366$ days;
• A year divisible by $100$, but not by $400$\$ is a regular year.

This way, the calendar accumulates only a small error due to the precession of equinoxes, equal to roughly one day every $7700$ years.

## How to calculate the day of the week — without looking at the calendar

To know what day of the week is it for any date, you can either gather a lot of calendars or try to use some math to calculate what day it is in a few short passages.

🙋 This problem was covered many times in the past, maybe most noticeably by Lewis Carroll: the author of Alice in Wonderland. You can find his version of the calculations in this pretty old .

To calculate the day of the week, take a generic date:

$\text{d}_1\text{d}_2/\text{m}_1\text{m}_2/\text{y}_1\text{y}_2\text{y}_3\text{y}_4$

Follow these steps to calculate the day of the week:

1. Take the last two digits of the year ($\text{y}_3\text{y}_4$), and divide by $4$ keeping only the quotient:
$\text{day} = \lfloor\text{y}_3\text{y}_4/4\rfloor$
2. Add the day of the month:
$\text{day} = \lfloor\text{y}_3\text{y}_4/4\rfloor +\text{d}_1\text{d}_2$
3. Add the month's key value $\text{mk}$ according to the table below;
$\text{day} =\lfloor\text{y}_3\text{y}_4/4\rfloor +\text{d}_1\text{d}_2 + \text{mk}$

🙋 Here is the table for the month's key value $\text{mk}$:
$\text{J}\text{F}\text{M}\ \text{A}\text{M}\text{J}\ \ \text{J}\text{A}\text{S}\ \ \text{O}\text{N}\text{D}$
$1\ 4\ 4\ \ 0\ 2\ 5\ \ 0\ 3\ 6\ \ 1\ 4\ 6$

1. If you are dealing with a leap year, subtract $1$ if the month is either January or February.
2. Starting from the 1700's, add a factor given in the table below. For the other centuries, follow the pattern of subtracting/adding multiples of $400$ to the year and checking which one of the four centuries below the date falls.
\begin{align*} \text{for}\ 1700\text{'s}\rightarrow\ \text{add}\ 4\\ \text{for}\ 1800\text{'s}\rightarrow\ \text{add}\ 2\\ \text{for}\ 1900\text{'s}\rightarrow\ \text{add}\ 0\\ \text{for}\ 2000\text{'s}\rightarrow\ \text{add}\ 6\\ \end{align*}
1. Add the last two digits of the year, $\text{y}_3\text{y}_4$.
2. Finally, take the remainder of the division of the result by $7$.

The resulting number corresponds to the day of the week, assuming that $1$ is Sunday, $2$ is Monday, and so on.

## What day of the week was I born? Calculate the day of the week of your birthday.

Were you born on the 17th of July 1996? No? It's not important 😁 let's calculate!

Follow the steps laid down before:

1. $\text{day} = \left\lfloor96/4\right\rfloor = 24$.
2. $\text{day} = 24+17=41$.
3. The month's key value for July is $0$:
$\text{day} = 41+0=41$
4. We don't subtract anything (though the year is a leap year).
5. We add $0$ since the year belong to the $1900\text{'s}$.
$\text{day}=41+0 = 41$
6. $\text{day}= 41+96 = 137$.
7. $\text{day} = 137\ \text{mod}\ 7 = 4$

Which corresponds to Wednesday. Use our birthday paradox calculator to find out the probability of sharing the birthday with someone else at a party!

Davide Borchia
Date
Day of the week: Sunday
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