# Leap Years: why do we need them?

#### Colin Sparrow, Head of Maths at Warwick, tells us a bit more...

Right! This is a bit nerdy. But it's a bit of a nerdy question. So here goes...

It takes the earth approximately 365.2421... days to go once around the sun, so almost 365 and one quarter days. If every calendar year was 365 days long, the winter weather would gradually creep earlier and earlier (by about 6 hours a year) until after about 700 years we would would have winter in July and summer in December.

To avoid this, every fourth year is a leap year and has 366 days, so the usual average length of a year over a four year cycle is 365 and a quarter days long. This (365.25) is slightly too big a correction, so every 100 years we omit a leap year. But that is slightly too big a correction in the other direction, so every 400 years we keep the leap year at the turn of the century (the year 2000 was a leap year, but 2100, 2200 and 2300 will not be).

All of this gives an average year length of 365.2425 days, which is near enough the time it takes for the earth to go round the sun, and ensures that the seasons do not move noticeably relative to the calendar over thousands of years. See? Told you it was nerdy!"

### Leap years: when is my birthday on a Friday?

In 2013 my birthday was on Friday 13th. I had waited 11 years, since 2002 to have my birthday on a Friday (which is, I think, the right day to have a birthday if you are born on the 13th). But the next one will come much quicker, and the one after that even sooner (in 2019 and 2024 respectively). In fact the gap between my Friday 13th birthdays goes in a cycle, with gaps of 11 years, 6 years, 5 years, 6 years and then back to

11 years again, repeating every 28 years.

This behaviour is caused by the leap years. An ordinary year is 52 weeks and 1 day, so usually if my birthday is on a Friday in one year it is on a Saturday the next year. Indeed, in 2014 by birthday was on Saturday and in 2015 it was on Sunday. But if there is a leap year, this is 52 weeks and 2 days, so my birthday jumps forward 2 days. So this year it goes from Sunday in 2015 to Tuesday in 2016, since the extra leap day on Feb 29th 2016 happens between the two birthdays. If there were no leap years, the Friday birthdays would recur every 7 years, as you might expect. But sometimes there is one leap year occuring somewhere in the cycle (which means I get back to Friday in 6 years), and sometimes there are two (so it takes only 5 years to get through the week). But occasionally a leap year occurs so that my birthday jumps from Thursday to Saturday and I then have to wait for it to creep all through the week a second time before I get back to Friday (and this takes a total of 11 years).

There is nothing special about my birthdate. If you like to have your birthday on a Saturday, you will have the same pattern (though you may be at a different point in the cycle). And if your birthday was on a Tuesday in the year following March 1st 2002, you will have waited 11 years to have it on a Tuesday again in the year following March 1st 2013.

The only people with a totally regular pattern are those with birthdays on Feb 29th. Anyone with a birthday on Monday 29th Feb 2016 will find their successive birthdays move backwards through the week two days at a time, in a completely regular fashion (Saturday in 2020, Thursday in 2024, returning eventually to Monday in 2044). You can check your own birthdays online here.

Caveat: All this pays no account of the special rules for the missing leap years at the end of most centuries..... but these will not trouble us till 2100, and have not troubled us since 1900.