Why months don't have a fixed length (and time math is messier than it looks)

Every other unit on this site divides evenly. A kilometer is always exactly 1,000 meters, no exceptions. Time doesn't get that luxury. A month can be 28, 29, 30, or 31 days. A year needs an extra day bolted on every few years just to stay lined up with the seasons. For decades, we even added extra seconds by hand. Time math looks simple and mostly isn't.

Why months don't have one length

Calendar months are a compromise between two things that don't divide into each other cleanly: a roughly 29.5-day lunar cycle and a roughly 365.24-day solar year. Twelve lunar months add up to about 354 days, eleven short of a full solar year, so at some point history just stopped trying to make months track the moon exactly and instead stretched and squeezed them (28 to 31 days each) to force 12 of them to add up to a year. That's why, when this converter needs a single number for "one month," it has to use an average, about 30.4 days, rather than a real fixed value. It's a genuinely reasonable approximation for rough duration math, but it will never match a specific named month exactly, because no single number could.

Leap years exist to fix a decimal, basically

A real solar year, the time it takes Earth to orbit the sun, is about 365.2422 days, not a clean 365. Ignore that extra roughly quarter-day and the calendar slowly drifts out of sync with the seasons, a few decades of it and July calendars start looking like winter. The fix is the leap year: add a day back every 4 years to absorb that quarter-day. That overcorrects very slightly, though, which is why the rule has an exception for century years (no leap day) and an exception to the exception for years divisible by 400 (leap day after all). The year 2000 got a leap day; 1900 and 2100 don't. It's a small patch on top of a small patch, but it keeps the calendar accurate to within about 26 seconds a year.

Leap seconds: the one time we tried to patch the Earth itself

Since 1972, official timekeeping has occasionally added a single leap second to Coordinated Universal Time, not to fix the calendar, but to fix the clock. Modern time is measured by atomic clocks counting vibrations of a cesium atom, incredibly precise and completely indifferent to the Earth's actual rotation, which is gradually and irregularly slowing down. Every so often the gap between atomic time and the Earth's real rotation got big enough that a second had to be inserted by hand to keep clocks matching sunrise and sunset where they're expected to be. There have been 27 of these since the practice started. It's also being phased out: international timekeeping bodies voted in 2022 to stop adding leap seconds by 2035, mostly because they wreak havoc on computer systems that don't expect a minute to ever have 61 seconds in it.

Even the week is a little arbitrary

Unlike the day (Earth's rotation) or the year (Earth's orbit), the 7-day week isn't tied to anything astronomical at all, it's a cultural convention with roots in ancient Babylonian and biblical tradition. It's persisted mostly out of sheer momentum. France actually tried to replace it during the Revolution with a 10-day "décade" as part of a broader push to decimalize everything, including time itself. It lasted about 12 years before being abandoned in 1805, largely because nobody wanted to give up a day of rest every 10 days instead of every 7.