## The Mathematical Confusion created by Leap years: This rare Day

29.02.2024 | 13:02 |Every four years we have February 29th – except for those that fall at the turn of the century, unless the year is divided by 400. One of these confusing stories about how leap years work was told by the BBC.

As a rule, leap days occur about once every four years. Nevertheless, there are exceptions to this rule. For example, at the turn of each century, we skip a leap year. Despite the fact that the year is divided into four, we do not add a leap day to the years ending at 00. However, there is an exception to this rule too. If the year is a multiple of 400, we add an additional leap day again. At the turn of the millennium, even though the year 2000 was divisible by 100, it actually had February 29th because it was also divisible by 400.

So far, everything is complicated. Nevertheless, why do we have leap years at all? In addition, why are the rules governing them so confusing? As you probably know, the answer has something to do with synchronizing things.

There are only two fundamentally defined units of time on our planet. One of them is the day: the time it takes for the Earth to make one revolution around its axis: from the direction to the Sun, then back, and then back. The second is a year: the time it takes for the Earth to make one revolution around the Sun.

If leap years occurred exactly every four years (which has been happening for the last 120 years), then the probability of being born on a leap day would be one in 1,461. This means that there are probably about 5-5.5 million such "lucky ones" in the world right now.

Coincidentally, you get about the same amount – 1,440 – extra minutes in a leap year – so that's about one extra minute per day of a four-year cycle (this is because the solar year is about 349 minutes longer than 365 days, which is almost an extra minute per day of a four-year cycle).

Unfortunately, the Earth takes 365.24219... (About 365 and a quarter) days to make a revolution around the Sun and return to its original position. Therefore, a real solar year does not actually last 365 days. It is very inconvenient. We cannot celebrate New Year's Eve at midnight one year, and then at 6 a.m. the next year and at noon the next year – the desynchronization is getting further and further away.

Back in 46 BC, Julius Caesar realized this problem and, together with his advisers, decided to find a clever solution to improve the work of his Julian calendar, which included adding additional quarter days accumulated every four years to create an entire extra day.

However, if you add a day every four years, then the average length of the year will be 365.25 days – a little too long.

When the Gregorian calendar was introduced, it was decided to improve the approximation by crossing out one of the leap days in years divisible by 100. According to this system, over the course of a century, we added 24 additional days rather than 25, resulting in an average year lasting 365.24 days.

To improve the approximation, it was decided to add an additional leap day every 400 years. Over a single 400-year period, this entails adding a total of 97 additional days, bringing the average length of the year to 365.2425 days – close enough.

This is how leap years are calculated according to the Gregorian calendar:

If a year is divided into four, then it is a leap year.

But if a year can be divided not only by 4, but also by 100, then this is not a leap year.

However, if the year is divided by 400, then it is a leap year.

This system ensures that the calendar is linked to the solar year with an accuracy of several decimal places.

It took several attempts and false starts to achieve modern calendar accuracy. To move to the next level of accuracy, we will need to remove leap days from years that are multiples of 3200. This will give us an additional 775 days over 3,200 years, resulting in an average year length of 365.2421875 days – an even higher level of accuracy.

It seems to be very troublesome, just to make sure that the days coincide with the years. Why don't we just change our definition of the year instead to make it exactly 365 days? This seems like a reasonable solution, and it really would be if it weren't for the tilt of the earth's axis.

The "Big Bang" theory suggests that about 4.5 billion years ago, a huge collision of the proto-Earth with another planet the size of Mars caused enough debris to fly out to form the Moon, but also caused the tilt of the Earth's axis. Although it is believed that this tilt has changed over the years, the fact that we have a tilt at all leads to the appearance of seasons familiar in higher latitudes: summer, when your part of the Earth is tilted towards the Sun between spring and autumn, and winter, when it is tilted the other way.

If we had not made adjustments to leap days, our calendars would not have synchronized with the seasons. After 100 years, the calendar will deviate by about 25 days. Indeed, the lack of consistency between the civil and solar calendars prompted Caesar to add a leap day, as well as introduce a 445-day year in 46 BC to correct the accumulated months-long backlog.

You may have heard about the extra seconds. You may ask, why can't we just add a few extra seconds every day so that by the end of each year we have the right amount of extra hours? This is a good idea, but of course it will mean that due to the extension of the day, our clocks will be out of sync with daylight, which will be an even more serious problem. In the middle of the year, we can have breakfast at sunset or go to bed at dawn. In fact, the extra seconds are used to avoid exactly this problem – small changes in the period of the Earth's rotation around its axis, which would otherwise confuse our time.

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