A calendar has to turn the leftover part of each year into extra days, and each calendar spreads that leftover with a different fraction.
Worked example: how far apart the Julian and Gregorian calendars drift
The Julian calendar adds 1 day every 4 years, which is 1/4 of a day per year. The Gregorian calendar adds 97 days every 400 years, which is 97/400. What is the difference per year, and how much does it add up to in 400 years?
- Find a common denominator. 400 is a multiple of 4, so 1/4 = 100/400.
- Subtract. 100/400 − 97/400 = 3/400.
- Multiply by 400 years. 400 × 3/400 = 3 days.
- Convert to a decimal. 3/400 = 0.0075 days per year.
The difference is 3/400 of a day per year, which is 3 days every 400 years.
Check: in 400 years the Julian calendar counts 100 leap years and the Gregorian counts 97. The difference is 3.
Those 3 days are an average over a full cycle, not three fixed dates on the calendar. Further down you can see when each day of difference built up.
Why almost a quarter of a day is left over
The Earth takes about 365.2422 days to complete the cycle of the seasons, called the tropical year. A 365-day calendar leaves a remainder of 0.2422, a little less than 1/4. The remainder looks small, but it piles up: in 4 years it nearly reaches a whole day, and in a century it passes 24.
Julius Caesar introduced one extra day every four years in 45 BC. In 4 years there are 4 × 365 + 1 = 1461 days, so the mean Julian year is 1461/4 = 365.25 days. Since the tropical year is 365.2422, the Julian year runs over by 365.25 − 365.2422 = 0.0078 days per year. That is small, but at 0.0078 per year, over the 1257 years from the Council of Nicaea in 325 to 1582 it adds up to 1257 × 0.0078 = 9.8046 ≈ 9.8 days (error under 0.02 days); the source speaks of about 10 days of drift in 1582.
The value 365.2422 is an approximation. With more digits the tropical year is 365.242189; the difference from 365.2422 is about 0.000011 days, less than 0.0001, which is why it is written here with ≈. To convert any of the fractions on this page to a decimal, the fraction to decimal calculator shows whether the decimal terminates or repeats; the three fractions that appear here all terminate.
The Gregorian rule and its 97 leap years in 400
The Gregorian calendar keeps the extra day every 4 years and adds one exception. Years ending in 00 are leap years only if they are divisible by 400. Of the 100 multiples of 4 in 400 years, three end in 00 and are not divisible by 400, which leaves 97. So 400 years contain 400 × 365 + 97 = 146,097 days, and the mean Gregorian year is 365 + 97/400 = 365.2425 days.
To decide whether a year is a leap year, ask these questions in order and stop at the first answer that decides.
- If it is not divisible by 4, it is not a leap year.
- If it is divisible by 4 and not by 100, it is a leap year.
- If it is divisible by 100, it is a leap year only if it is also divisible by 400.
If you need a refresher on telling when one number divides another without calculating, the guide to divisibility rules collects the rules for 4, 100 and 400.
| Year | Divided by 100 | Divided by 400 | Leap year? |
|---|---|---|---|
| 2024 | does not divide | not needed | Yes |
| 1900 | 19 | 4.75 | No |
| 2000 | 20 | 5 | Yes |
| 2100 | 21 | 5.25 | No |
Only years ending in 00 need the third question, and a decimal quotient such as 4.75 or 5.25 shows the division is not exact.
When each day of difference built up
Each year that the Julian calendar counted as a leap year and the Gregorian did not, the gap between the dates grows by one day. That happened in 1700, 1800 and 1900, and it will happen again in 2100. That is why the gap was 10 days in 1582, 11 in 1752 and 13 in 1918, rather than growing steadily at 3/400 per year.
The Gregorian calendar took effect in October 1582, when Thursday, October 4 was followed by Friday, October 15. Each country adopted it on its own date, and at each jump as many days were skipped as the gap had accumulated.
| Place | Year | Date jump | Days skipped |
|---|---|---|---|
| Iberian Peninsula and Canary Islands | 1582 | October 4 to October 15 | 10 |
| Great Britain | 1752 | September 2 to September 14 | 11 |
| Russia | 1918 | January 31 to February 14 | 13 |
| Greece | 1923 | February 15 to March 1 | 13 |
The skipped days come from counting the numbers that are jumped over: between September 2 and September 14, the dates 3 through 13 are left out, which is 11 days. The Spanish colonies in the Americas applied the change a year later, in October 1583.
How far off the Gregorian calendar still is
The mean Gregorian year, 365.2425, exceeds the tropical year, 365.2422, by 0.0003 days per year. That difference, also computed with the rounded 365.2422, adds up to one day in 1/0.0003 ≈ 3333 years (error under 1 year). Sources give figures between 3030 and 3333 years, depending on how the tropical year is rounded (3030, 3323 and 3333 all appear), so it is best to keep the order of magnitude: an error of about one day every 3300 years.
The Julian calendar was off 26 times faster: 0.0078 versus 0.0003. To see what fraction of a unit each remainder is and how they compare, the compare fractions tool helps.
| Year | How it is obtained | Days |
|---|---|---|
| Julian | 365 + 1/4 | 365.25 |
| Gregorian | 365 + 97/400 | 365.2425 |
| Tropical | measured, not a simple fraction | ≈ 365.2422 (error under 0.0001 days) |
Between 1/4 and 97/400, only the fraction of extra days per year changes. The tropical year cannot be written as a fraction with a small denominator, so calendars choose the closest one that gives a rule that is easy to apply. The subtraction done above can be repeated with the subtract fractions calculator, which finds the common denominator for you. To review how a common denominator is found, see the least common denominator. The article on why an hour has 60 minutes covers another case where splitting a unit into whole parts decides how we measure time, and more reading of this kind is on the blog.
Practice
Three quick checks
- Was 1800 a leap year? What about 2400?
- How many leap years are there from 2001 to 2100, both included?
- If a calendar added 1 day every 5 years, how many days would it add in 400 years?
Show the answers
- 1800 was not, because it is divisible by 100 and 1800/400 = 4.5. 2400 is, because 2400/400 = 6.
- There are 24, the multiples of 4 from 2004 to 2096, and 2100 is left out because it is not divisible by 400.
- 80 days, because 400/5 = 80, quite a bit less than the 97 of the Gregorian calendar (and the 100 of the Julian).
Sources
- Wikipedia, Gregorian calendar: mean year of 365.2425 days with 97 leap years in 400, Thursday, October 4, 1582 followed by Friday, October 15, the drift of 10 days since the Council of Nicaea, and the removal of 11 days from September 1752 (accessed October 17, 2026).
- Wikipedia (Spanish), Calendario gregoriano: an error of about one day every 3333 years and the 10-day shift in the equinox (accessed October 17, 2026).
- Wikipedia (Spanish), Año bisiesto: tropical year of 365.242189 days, the Julian calendar of 45 BC, the cases of 1800, 1900, 2000 and 2100, and an error of one day every 3323 years (accessed October 17, 2026).