29 and Leap Years  Numberphile  Summary and Q&A
TL;DR
Our calendar and time measurement systems are a complicated mismatch of different cycles and lengths, requiring adjustments like leap years and leap seconds to make them align with physical events.
Questions & Answers
Q: Why do we have leap years?
We have leap years to account for the extra 1/4 day it takes for the Earth to complete a revolution around the Sun every year. By adding an extra day (February 29) every four years, we align our calendar with this additional time.
Q: Do all years divisible by 4 have a leap year?
No, not all years divisible by 4 have a leap year. If a year is also divisible by 100 but not by 400, it is not considered a leap year. This rule accounts for the slight inaccuracy in the length of a year.
Q: Why do we need leap seconds?
Leap seconds are necessary to adjust our clocks for the variable rotation of the Earth. These additional seconds help keep our time measurement systems aligned with the Earth's rotation, ensuring a day lasts approximately 24 hours.
Q: How do astronomers calculate the passage of time between two dates?
Astronomers use Julian dates, which count the number of days since a specific date in history. By using a single number, they can simplify calculations involving months, days, and leap years, making it easier to calculate the passage of time accurately.
Summary & Key Takeaways

Our calendar consists of different cycles and lengths, resulting in the need for adjustments like leap years to keep it in sync with actual events.

The Earth takes approximately 365 and 1/4 days to complete a revolution around the Sun, leading to the need for an extra day every four years.

However, the actual length of a year is approximately 365 days, 5 hours, 49 minutes, and 16 seconds, requiring additional corrections every 100 and 400 years.