16 - Numberphile

Uploaded by numberphile on 23.11.2011


RIA SYMONDS: We're talking about the number 16.
16's a very special number to me.
It's my actual favorite number.
MATT PARKER: One of my favorite numbers is 16.
And a lot of people seem to like 16.
Even though it is an even number, it wins out for being
a square number.
RIA SYMONDS: It's partially favorite because I was born on
the 16th day of December, to be precise.
And I always liked it because when I was sweet 16 on the
16th, I could a lot of things by then.
I could go off and get married if I wanted to.
I could leave school legally.
I didn't, of course.
But 16 is also mathematically very special.
It's the square of a perfect square.
Did you know that?
Go on then.
RIA SYMONDS: Would you like me to explain?
BRADY HARAN: Yeah, of course.
BRADY HARAN: Square of a perfect square.
What does that mean?
RIA SYMONDS: So a perfect square, a good
example is 2 squared.
2 squared, you get 4.
It's perfect because I've got 2 to the power of 2.
It's the same digit.
If I square this, so all of that squared, so this is 4
squared, lo and behold, we get 16, my My favorite number.
MATT PARKER: And I was at a pub once with some friends of
mine, and we were drinking and having fun.
And we thought we'd order some food.
We were on table 16.
I went up to the bar.
And I said, to get food, could we get delivered to 16?
And I went, oh, wait, a minute, which is 4 squared.
Can I order food to the table 4 squared?
And it ended up the poor girl behind the
bar looked very confused.
And I went, no, actually, we'll make it easier.
How about I order it 2 to the 4?
Because not only is 16 4 times 4, it's also a 2 times 2
times 2 times 2.
And then at this point, she looked very bewildered, and
later on, we managed to intercept our food on the way
to table 24.
RIA SYMONDS: Let's say I have the relationship x raised to
the power of y.
So x and y are two different integer numbers.
So x cannot equal y.
So x could be 2, y could be 3.
We'd have 2 cubed.
Now, they only relation that is satisfied by this number 16
is that x to the y is equal to y to the x.
So the number on this side of your equal sign is going to be
exactly the same as the number on this side
of the equal sign.
To do that, I'm going to take the number x is going to be,
let's say 4.
Let's take y to be 2.
If we interpret those onto the other side of the equation, y
was 2, and x was 4.
4 to the power 2, we've already seen it, 16.
2, raise that to the power 4, you put it into your
What do you get?
And 2 and 4 are the only numbers that can be
interpreted as x and y in order to get this
relationship, this special number, 16.
MATT PARKER: And we thought there must be other numbers.
And we had a bit of a chat, and we
realized none of us knew.
And we all do math things for a living.
And we thought, well, let's work it out, we decided.
Because you could look it up.
We could go online and see if anyone else had tried this.
And we thought, no.
It's as much fun to work this out as if no
one else ever has.
And some people went away and started trying to write
computer software to see if there were bigger ones.
And the rest of us were like--
some people were drawing graphs, some people were doing
number theory stuff.
A lot of prime factors suddenly appeared.
And other people--
because this went across many evenings.
Other people were like, what are you guys doing?
I'm like, we're trying to work out if 16 is the only number
where you could write it x to the y, or y to the x.
RIA SYMONDS: You cannot get any other number using any
other integers to get a number that equals itself.
MATT PARKER: 16 is the only number, and we've since
realized other people do know about this, and it's a nice
property of 16.
But we had the joy of working it out for ourselves.
So if you get bored, you can check why.
And it's fascinating why 16 is the only
number that does that.
BRADY HARAN: That's pretty cool.
I think so.
MATT PARKER: Initially, it was just a lot of
arguing over food.
So people walking by just saw us eating.
And someone would go, well, what about 64?
And then there would be a bit of I don't know.
Then someone else wold go, no, not 64.
Oh, OK, and we're back to the food.
BRADY HARAN: Is that why it's your favorite number?
Or is it just because of your birthday?
RIA SYMONDS: Because it's my birthday.
This is just a cool relationship.
MATT PARKER: Often during lessons, we'd be like, what if
it's, and then someone would go, I've got a log.
Check out this log graph.
This is it.
And we'd go, oh, I don't know.
And actually, this was the beginning--
if I can get distracted briefly, is the beginning of
something called Math Jam, which is an organization I run
where people get together in a pub and do math.
Because we realized we started off in this pub drinking, and
we actually did math.
And then later on, we were doing more and more, and other
puzzles started to show up.
We realized that 36 is both a square number and a triangle
number, but that's a whole different video.
And we suddenly thought, why are we having these debates
when we're meant to be teaching school kids?
Why don't we get rid of the school kids and just replace
them with tasty beverages?
And so ever since--
and now it's all around the world.
Last count, 14 or 15 different cities in the UK and other
countries have Math Jams, where people get together on
the second to last Tuesday of the month and talk about math,
and do problems, and play with toys.
And yeah, all because I sat down in a pub at table 16.