Leap years and decimals

365 days is not quite enough

A calendar year has 365 days, but one trip of the Earth around the Sun takes 365.2422 days. So when a calendar year ends, the Earth still has 0.2422 of a day to travel, nearly 6 hours (0.2422 × 24 = 5.8128 hours). That looks tiny, but it grows: after 100 calendar years the shortfall is 0.2422 × 100 = 24.22 days. The seasons depend on where the Earth is on its path, so a calendar that keeps slipping would slowly carry summer into the months we call winter.

Adjustment 1: a leap day every 4th year

Give every year divisible by 4 one extra day, 29 February. Four calendar years then have 4 × 365 + 1 = 1461 days, while four trips take 4 × 365.2422 = 1460.9688 days, which is very close. Over 100 years, 25 years are divisible by 4, so the calendar has 100 × 365 + 25 = 36,525 days. But the Earth needs only 100 × 365.2422 = 36,524.22 days. The calendar is now 0.78 day ahead every century: we have overcorrected.

Adjustment 2: skip the century years

So years divisible by 100 get no extra day. In 100 years only 25 − 1 = 24 years are now leap years, giving 24 × 366 + 76 × 365 = 36,524 days, just 0.22 day short of 36,524.22. Small, but it grows again: in 1000 years the calendar has 36,524 × 10 = 3,65,240 days, while the Earth needs 1000 × 365.2422 = 3,65,242.2 days. Now the calendar is 2.2 days behind.

Adjustment 3: keep every 400th year

To win back about two days in a thousand years, every year divisible by 400 is a leap year after all. Count the years from 1 to 1000:

Kind of year	How many
Divisible by 400 (leap)	2
Divisible by 100, not by 400 (ordinary)	10 − 2 = 8
Divisible by 4, not by 100 (leap)	250 − 10 = 240
All the rest (ordinary)	1000 − 250 = 750

So 242 years have 366 days and 758 have 365: 242 × 366 + 758 × 365 = 3,65,242 days, only 0.2 day short of 3,65,242.2. The calendar makers stopped there, and these three rules make up the Gregorian calendar used in most of the world today.

What about 10,000 years?

In 10,000 years, 2500 years are divisible by 4, 100 by 100 and 25 by 400, so there are 2500 − 100 + 25 = 2425 leap years. The calendar has 10,000 × 365 + 2425 = 36,52,425 days, while the Earth needs 10,000 × 365.2422 = 36,52,422 days, so the calendar ends up about 3 days ahead. (In the 1000 years above it was slightly behind, because 1 to 1000 holds only 2 multiples of 400, fewer than the average of 2.5.) One possible fix is to make years divisible by 4000 ordinary years. That removes 2 leap days in 10,000 years and leaves the calendar about 1 day ahead.

Why it matters

A gap of 0.2422 of a day sounds too small to matter, yet decimal multiplication shows how it builds up over the centuries, and why the leap year rule needs three parts. (Extra) Many traditional Indian calendars follow both the Moon and the Sun, and they keep festivals in their seasons by adding an extra month, called adhik maas, about once every three years. Revise the arithmetic in multiplying decimals and dividing decimals. Answers to the textbook questions are in the NCERT solutions for Part 2 Chapter 4; see all of Part 2 Chapter 4 notes.