Weekday Calculator

Every quantity here derives from one underlying fact: the Gregorian calendar is a fixed arithmetic system with a 400-year period. The leap rule adds a day when the year is divisible by 4, except for century years, which must also be divisible by 400, giving 97 leap years per 400 years and a mean year of 365.2425 days against a tropical year of about 365.2422, an error near one day in 3,000 years. Those 400 years contain 146,097 days, a multiple of seven, so weekdays repeat exactly every four centuries. The ordinal day of year runs 1 to 365 or 366 and appears in aviation and military date formats as YYYY-DDD and in manufacturing lot codes. The ISO 8601 week number follows a separate rule: weeks start on Monday and week 1 contains the first Thursday of January, equivalently 4 January. A consequence is that 29 to 31 December can belong to week 1 of the following year and 1 to 3 January to week 52 or 53 of the previous one, so the ISO week-numbering year is not always the calendar year. The Julian Day Number is a continuous count from noon Universal Time on 1 January 4713 BCE in the proleptic Julian calendar; 1 January 2000 is JDN 2451545. Two caveats apply. Dates before the Gregorian reform of 1582 are proleptic, projected backwards rather than historically observed. And in JavaScript a bare YYYY-MM-DD string is parsed as UTC while formatting is local, so users west of Greenwich can see a weekday shifted by one day.

Calculating a weekday by hand was a recognized mathematical problem long before software. Christian Zeller, a German mathematician and clergyman, published the congruence that bears his name in papers appearing between 1882 and 1887, giving a compact modular formula for the weekday of any Gregorian or Julian date. Nearly a century later John Horton Conway devised the Doomsday algorithm, described in the early 1970s, which exploits the fact that certain dates such as 4 April, 6 June, 8 August, 10 October and 12 December all fall on the same weekday within a given year, making mental calculation practical. The calendar those algorithms operate on was promulgated by Pope Gregory XIII in the bull Inter gravissimas of 24 February 1582, which corrected the accumulated drift of the Julian calendar by removing ten days, so that 4 October 1582 was followed directly by 15 October, and refined the leap rule to its present form. Adoption was slow and politically fractured: Britain and its colonies converted in 1752, dropping eleven days, Russia only after the revolution in 1918, and Greece in 1923. The Julian Day Number system used here for continuous day counting was proposed by Joseph Justus Scaliger in De emendatione temporum in 1583 and named after his father, Julius Caesar Scaliger; and John Herschel helped popularize it among astronomers in the mid nineteenth century. The ordinal-date and week-date notations that the calculator reports were standardized internationally under ISO 8601, first published in 1988.