29 and Leap Years - Numberphile

Uploaded by numberphile on 28.02.2012


Our calendar is just a mess.
It's a very complicated mismatch of different cycles
and different lengths of time.
And every once in a while, we have to make an adjustment to
those lengths of time to make things match.
And one way we do that is every once in a while adding
an extra day to our calendar.
And that extra day is February 29.
When we have an extra day, we call that year a leap year.

So there are two fundamental units of time that we use that
are tied to actual, physical events.
One is the day, and that's the time it takes for the Earth to
make one rotation around its axis.
The other is the year, and that's the time it takes the
Earth to make one revolution in its orbit around the Sun.
Every other unit we use-- the week, the second, the hour--
is fairly arbitrary, but those two are tied to actual,
physical events.
In a more well-behaved universe, our calendar would
be a lot easier to deal with if 365 of these exactly
matched one of those.
But of course, that's not the way it works.
Actually, our year is made up of about 365 and 1/4 days.
That means that every time the Earth had gone around the Sun
once, it's actually rotated 365 and 1/4 times.
If we rotate 365 and 1/4 times every year, but we're only
counting 365 of them in a year, that means every four
years, we've got an extra day that hasn't appeared in our
calendar, so we have to put it back in.
And that's where February 29 comes in.
But that's not quite enough.
The number of days in a year is not
exactly 365 and 6 hours.
It's 365, 5 hours, 49 minutes, and 16 seconds.
That means if we do what I just said, by the end of 100
years, we've accumulated too much time in our calendar, and
now we need to take a full day away.
So every 100 years, on a year that would've been a leap year
going by the four-year rule, we skip one, and we take away
that extra day.
So every time the year is divisible by 100-- that means
1900, 2000, 2100--
we skip the leap year.
And so, on those years, February has 28
days and not 29 days.
Except that's still not quite right.
There's an extra bit that we need to correct again, because
now, every 400 years, we've missed an extra day.
So now, every 400 years, we reverse that rule, and we add
the leap year back in.
So this becomes very complicated, but the
fundamental rule is for a given year, if the year's
divisible by 4, then it's a leap year.
Unless the year is divisible by 100, in which case it's not
a leap year.
Unless the year is divisible by 400, in which case it's a
leap year, and we have February 29.
The mismatch between the rotation of the Earth and the
revolution around the Sun is such that that extra amount of
time is taken care of to a fairly high precision by all
these actual corrections.
In actual fact, it still means that we will need an extra
correction at the end of 3,200 years, but that's not actually
built into our calendar.
We stop at the end of 400 years in these corrections.
Astronomers are used to dealing with all sorts of not
very convenient measurement systems.
That's, unfortunately, the legacy of our
very historical science.
So we are used to dealing with this.
And if we need to calculate the number of days that have
passed between two dates, it does become very complicated,
and you have to calculate how many months, and how many days
in that month, and whether there are leap years involved.
We simplify that by using our own calendar for the passage
of time known as Julian dates, and all that is is a single
number that counts the number of days that have gone by
since a very distant day in the past.
The zero point of that calendar is Monday, January 1,
4713 BC, 12:00 noon, Greenwich Mean Time.
And of course, being here in Nottingham,
that's our time zone.
We're in the same time zone as the Greenwich meridian.
By that reckoning, February 29, 2012
is Julian date 2455987.
Got it right.
You can add a decimal place after that date which records
purely the decimal fraction of the day that's gone by, but
one Julian date is equal to one day.
So if you hear a Julian date that ends with a 0.5, that
means half a day has passed in addition to the full day that
it's counting.
Yes, this is going to sound familiar to a lot of people,
because of course in the Star Trek universe, you often hear
"Captain's log, stardate 2455.73," which is a similar
kind of concept.
But the Star Trek writers and producers deliberately didn't
make it an official, fast and fixed rule as to what a
stardate meant, so that they could factor in all the
different concepts like speed of travel, and different parts
the galaxy, and that sort of thing.
So stardate is not the same as Julian date.
So normally, I would probably record the date of an
observation as April 3, 2010, something like that, because
the work I do is not particularly time-critical.
However, if you're recording something like a gamma ray
burst or a periodic event like the transit of a planet, then
a Julian date would be given to a very high precision.
So having just said that I don't use Julian dates very
often, here is a paper that I wrote where I did use it.
So here, I made a plot with Julian date, but I've
subtracted all of the large part of the number just to
leave the cumulative number of days here.
And so you can easily see that we've gone from 545 to 560,
which is 15 days, and that's a lot easier than saying we've
gone from January 25 to February
whatever in that reckoning.
See, I can't even do it in my head.
So a leap second is the same kind of concept in the idea
that it's correcting our calendars to make things match
up, but it's not related to the leap year at all.
It's purely adjusting for the variable rotation of the Earth
to make a day last a day.