Calendars

QUIZ

Calendar

The solar year consists of
365 days, 5 hours, 48 minutes, 48 seconds.
This means that one complete revolution of the Earth around the Sun does not take an exact whole number of days. The extra 5 hours 48 minutes 48 seconds accumulate every year and must be adjusted in the calendar to keep dates aligned with seasons.

In 47 BC, Julius Caesar introduced the Julian calendar to correct this problem. In this calendar:

  • An ordinary year was taken as 365 days.
  • Every year has an excess of nearly $\tfrac{1}{4}$ day.
  • After four years, this excess becomes nearly 1 full day.
  • To account for this, one extra day was added once every fourth year.
  • This extra day was added to February.
  • Such a year is called a leap year.

Hence:

  • $1\ \text{ordinary year} = 365\ \text{days} = 52\ \text{weeks} + 1\ \text{day}$
  • $1\ \text{leap year} = 366\ \text{days} = 52\ \text{weeks} + 2\ \text{days}$

Gregorian correction

The length of the solar year is approximately $365.2422$ days, which is not exactly $365$ days.
If every year were taken as $365$ days, the calendar would lose about $0.2422$ day each year, causing the seasons to slowly shift.

To correct this, an extra day was added every fourth year, giving an average year length of

\[365 + \frac{1}{4} = 365.25 \text{ days}.\]

This system was used in the Julian calendar. However, $365.25$ days is still slightly longer than the actual solar year. The small error of about $0.0078$ day per year accumulates and becomes significant over centuries.

To remove this accumulated error, the Gregorian calendar introduced a further correction:

  • Every year divisible by $4$ is a leap year.
  • Every year divisible by $100$ is not a leap year.
  • Every year divisible by $400$ is a leap year.

Thus, century years such as $100$, $200$, and $300$ are not leap years, while $400$ is a leap year.

Because of this rule, in a cycle of $400$ years:

  • Leap years = $97$
  • Ordinary years = $303$

The average length of a Gregorian year becomes

\[365 + \frac{97}{400} = 365.2425 \text{ days},\]

which is very close to the actual solar year.
This correction keeps the calendar aligned with the seasons and ensures that the pattern of days of the week repeats every $400$ years.

India adopted the Gregorian calendar for civil use in 1753. Therefore, for all Indian dates after 1753, standard Gregorian rules and the odd-days method give correct weekdays. 1st January 1953 was Monday.

A.D and B.C

The years of the calendar are divided into two parts based on the birth of Jesus Christ.
The years after the birth of Christ are called A.D. (Anno Domini), meaning “in the year of the Lord”.
The years before the birth of Christ are called B.C. (Before Christ).

There is no year zero in the calendar.
The year 1 B.C. is immediately followed by 1 A.D.

While counting years:

  • In A.D., years increase forward in time (1 A.D., 2 A.D., 3 A.D., …).
  • In B.C., years increase backward in time (1 B.C., 2 B.C., 3 B.C., …).

This difference is important while calculating odd days.

In A.D., odd days are counted forward from the reference date
(1 January 1 A.D., which was a Monday).

In B.C., odd days are counted backward, because time moves in the reverse direction.

Example (A.D.)
From 1 January 1 A.D. to 1 January 2 A.D.:

  • Number of days = 365
  • Odd days = 1
    So the day moves forward by 1 day.

Example (B.C.)
From 1 January 1 A.D. to 1 January 1 B.C.:

  • Number of days = 1
  • Odd days = 1
    So the day moves backward by 1 day.

Thus, while solving calendar problems, special care must be taken to:

  • Remember that there is no year zero
  • Count odd days forward for A.D. dates
  • Count odd days backward for B.C. dates

Note:
1 January 1 A.D. was a Monday. This fixed reference date allows us to determine the day of the week for any future or past date.

Odd Days

A week has 7 days. Any number of days greater than 7 can be expressed as complete weeks plus some extra days.
These extra days are called odd days.

For example:

  • 7 days = 1 week → 0 odd days
  • 8 days = 1 week + 1 day → 1 odd day
  • 9 days = 1 week + 2 days → 2 odd days

Odd days determine how many days the calendar moves forward from a known reference day.

To find the day corresponding to a given date:

  • Count the total number of days from the reference date up to that date.
  • Divide this total by 7.
  • The remainder obtained is the number of odd days.
  • Move forward that many days from the reference day.

Worked Example (Real-world)
1 January 1 A.D. was a Monday.
Find the day of the week for 1 January of the next year.

  • An ordinary year has 365 days
  • $365 = 52 \times 7 + 1$
  • Number of odd days = 1

So the day moves forward by 1 day:

  • Monday → Tuesday

Therefore, 1 January of the next year is Tuesday.

In a leap year:

  • Total days = 366
  • $366 = 52 \times 7 + 2$
  • Number of odd days = 2

So the day moves forward by 2 days:

  • Monday → Wednesday

The day of the week corresponding to the number of odd days is given below.

Odd Days Day
0 Sunday
1 Monday
2 Tuesday
3 Wednesday
4 Thursday
5 Friday
6 Saturday