Why was 2000 a leap year but 1900 wasn't?

Century years follow a special exception: divisible by 100 disqualifies a year from being a leap year, unless it's also divisible by 400, in which case it's a leap year after all. The year 2000 is divisible by 400, so it kept its leap day; 1900 is divisible by 100 but not 400, so it did not.

How many days are between two dates, counting both the start and end day?

That depends on which counting convention you need. Subtracting the dates gives the exclusive gap between them; adding 1 to that result gives the inclusive count, which includes both the start date and the end date as full days.

What happens when I add one month to January 31?

There's no single correct answer, since February doesn't have 31 days. Some systems clamp the result to the last day of February, either the 28th or the 29th; others count the extra days forward into March. Both are common, defensible conventions, which is why the specific system or calculator you're using matters.

Why isn't a year exactly 365.25 days?

The true figure is closer to 365.2422 days, slightly less than 365.25. Using 365.25 alone, the simple divisible-by-4 leap year rule, would overcorrect over centuries, which is why the calendar adds the divisible-by-100 and divisible-by-400 exceptions to stay aligned with the actual solar year over long spans of time.

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Time and date calculations cover leap years, elapsed time between two dates, and adding a duration to a calendar date, all of which look simple until an edge case exposes an off-by-one error or an ambiguous rule. This category exists because calendar arithmetic has more exceptions than most people expect.

The leap year rule has three parts, not one

A year divisible by 4 is a leap year, except a year divisible by 100 is not a leap year, except a year divisible by 400 is a leap year after all. The year 2000 was a leap year because it's divisible by 400, but 1900 was not, because it's divisible by 100 but not 400. This three-tier rule exists because a year is not exactly 365.25 days, it's closer to 365.2422 days, and the extra century-level exception keeps the calendar from drifting out of sync with the actual solar year over many centuries.

Skipping the 100 and 400 year exceptions and applying only the divisible-by-4 rule gets every ordinary year right but silently miscounts century years three times out of four, an error that only surfaces when a calculation spans a century boundary.

Inclusive versus exclusive day counting

The most common source of an off-by-one error in a date calculation is not knowing whether the start date, the end date, or both should be counted. The number of days between January 1 and January 10 is 9 if you're counting the gap between them, but 10 if you're counting every calendar day from the 1st through the 10th inclusive, both endpoints included. Contracts, billing cycles, and hotel stays often specify which convention applies, and a calculator that assumes the wrong one will be off by exactly one day, which matters most when the count determines a fee or a deadline.

Why 'one month later' is ambiguous

Adding a calendar month to a date sounds straightforward until the starting date is near the end of a shorter month. January 31 plus one month has no clean answer, because February doesn't have 31 days, and different systems resolve it differently: some clamp the result to February 28, or 29 in a leap year, while others roll the excess days into March, landing on March 2 or 3 instead. Neither approach is objectively wrong, but they produce different dates from the identical starting point and operation, which is why 'add one month' needs a defined rule behind it rather than an assumed one.

Frequently asked questions

Why was 2000 a leap year but 1900 wasn't?: Century years follow a special exception: divisible by 100 disqualifies a year from being a leap year, unless it's also divisible by 400, in which case it's a leap year after all. The year 2000 is divisible by 400, so it kept its leap day; 1900 is divisible by 100 but not 400, so it did not.
How many days are between two dates, counting both the start and end day?: That depends on which counting convention you need. Subtracting the dates gives the exclusive gap between them; adding 1 to that result gives the inclusive count, which includes both the start date and the end date as full days.
What happens when I add one month to January 31?: There's no single correct answer, since February doesn't have 31 days. Some systems clamp the result to the last day of February, either the 28th or the 29th; others count the extra days forward into March. Both are common, defensible conventions, which is why the specific system or calculator you're using matters.
Why isn't a year exactly 365.25 days?: The true figure is closer to 365.2422 days, slightly less than 365.25. Using 365.25 alone, the simple divisible-by-4 leap year rule, would overcorrect over centuries, which is why the calendar adds the divisible-by-100 and divisible-by-400 exceptions to stay aligned with the actual solar year over long spans of time.