A calendar is a system of recording or organizing days for social, religious, commercial or administrative purposes. It is based on considerations of astronomical nature. This is done by giving names to periods of time viz. days, weeks, months and years. A date is the designation of a single, specific day within such a system.

Our existing calendar, which is known as the Gregorian calendar, is the modification of the Julian calendar, which itself was a reform of the ancient Roman calendar. The names of the months are derived from the Roman Deities, Rulers, Festivals and Numbers. The Roman year originally had ten months (Martius i.e., March to December), which was ascribed to the legendary first king, Romulus in 738 BCE (Before Common Era).

**Figure 1.1**

This calendar consisted of only 304 days and was followed by unaccounted 61 days of winter. To account for left over days, January and February were added to to the end of the year during Numa’s reign around 713 BCE, February remained the last month and being the last month of the year, contained left over days of the year. In 452 BCE, a small council of Romans, moved February to follow January and it became the second month of the year.

By the 40s BCE the Roman civic calendar was three months ahead of the solar calendar. Caesar, advised by the Alexandrian astronomer and Mathematician Sosigenes, introduced the Julius Ceaser calendar, taking the length of the solar year as 365 1/4 days. The year was divided into 12 months all of which had either 30 or 31 days except February, i.e., 28 days in common (365 day) years and 29 days in every fourth year (a leap year, of 366 days) with repeating 23 February without 29th February.

Actually solar year consists of 365 days, 5 hours, 48 minutes and 48 seconds. In Julian calendar, the span of year was taken as 365 days and 6 hours and in order to get rid of odd quarter of a day, an extra day was added once in every fourth year called Leap year.

But, as the solar year is 11 minutes 12 seconds less than a quarter of a day, the Julian calendar became inaccurate by several days and in 1582 C.E, this difference amounted to 10 days.

Pope Gregory XIII, Ruler of Papal States Rome, determined to rectify this and devised calendar known as Gregorian calendar. He dropped or cancelled 10 days − October 5th being called 15th October.

This modification brought the Gregorian system into such close exactitude with solar year that there is only a difference of 26 seconds which amounts to a day in 3323 years. The Gregorian calendar is currently used by most of the world.

**Gregorian calendar have basic rules to understand:**

- Leap years have 366 days, the extra day is added to February i.e., February having 29 days in a leap year. A year is considered as a leap year if the year number completely divided either by 4(for non-century years) or divided by 400 (for century years). e.g., 1676, 1896, 1904, 1948, 2004, 2016, 2020, 1200, 1600, 2000 etc. are leap years.
- Common years have 365 days. February having 28 days in common year. A year is considered as a common year if the year number neither divided by 4 (for non-century years) nor divided by 400 (for century years). e.g., the years 100, 200, 1900, 2001, 2002, 2003, 2005, 2006, 2007, 2009, 2018, 2019, 2100 etc. are common years.
- The day of the week repeats after every seven days.
- In a given period, the number of days more than the complete weeks are called odd days.

**Procedure of counting odd days**

a) Odd Days in Years

1 ordinary year = 365 days, 365 dividing by 7 gives (Quotient = 52; Remainder = 1)

= 52 weeks + 1 odd day

so, number of odd days in 100 years = 100 + 24 (leap years in 100 years)

= 124, when divided by 7, gives remainder 5 i.e., 5 odd days.

Similarly, Number of odd days in 200 years = 2 × 5 = 10 days, dividing by 7, gives 3 odd days.

Number of odd days in 300 years = 3 × 5 = 15 days, dividing by 7, gives 1 odd day.

Number of odd days in 400 years = (4 × 5) + 1(year 400 is leap year) = 21 days, dividing by 7, gives 0 odd day.

**Steps to extract day from any date**

- To start with, consider the completed years up to the given date by subtracting 1. e.g., if the given date is August 25, 2020, numbers of years completed are 2019, and 2020 is in progress.
- Split the completed number of years into the sum of multiple of 400, multiple of 100 and remaining years. e.g., 1984 = 1600 + 300 + 84.
- Count the total number of odd days up to the specified date staring from the beginning year of the calendar.
- Divide the total sum of odd days by 7 (because days repeat after 7 days)
- Remainder will always be any one out of 0, 1, 2, 3, 4, 5 and 6.
- For finalizing the day, refer the table of mapping as under:

Exhibit-1 How to find the day on 1st October, 1960 (Nigeria’s independence date)

For 1st October, 1960, complete years are 1960 – 1 = 1959.

Split 1959 as 1600 + 300 + 59

Calculate the number of odd days up to 1st October, 1960 as shown:

Dividing 97 by 7, gives remainder 6. By referring Table-3, 6 corresponds to Saturday. So, the Day on 1st Oct, 1960 was Saturday.

Exhibit-2 Find the year nearest to 2020 in future having same calendar as it is. For the calendars of two different years to be the same, both years must be of the same type. i.e., both years must be common years or both years must be leap years and 1st January of both the years must be on the same day. Calculate the cumulative number of odd days, starting from year 2020. When the total number of odd days, become a multiple of 7, and the next year is also a leap year (because 2020 is a leap year), then that year will have same calendar year as of 2020.

Exhibit-3 Which days of the week cannot be the last day of a century years?

Construct the following table possible number of odd days and their corresponding day mapping:

It is observed from the table-6 that week days Sunday or Monday or Wednesday or Friday are repeatedly falling on last day of all century years. Thus, the last day of a century cannot be Tuesday, Thursday or Saturday.

All the computer programmers apply Gregorian calendar algorithm to create a calendar application for all mobile phones, laptops, iPod, digital cameras, etc. We are just using Calendar Apps to unambiguously identify any day in past, present and future by a specific date in order to record or organize any event.

**References:**

- https://en.wikipedia.org/wiki/Gregorian_calendar
- https://en.wikipedia.org/wiki/Tropical_year
- https://www.wonderopolis.org
- https://www.indiatoday.in/origin-of-month-names
- https://en.wikipedia.org/wiki/Calendar_reform
- Swerdlow, N. M. , “The Length of the Year in the Original Proposal for the Gregorian Calendar”, Journal for the History of Astronomy, Vol.17(49), 1986

Dr. Anil Tanaja, is a Professor of Mathematics in Skyline University Nigeria. He has PhD Mathematics from M.D. University Rohtak, India.

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