Is Zero Even? - Numberphile

Uploaded by numberphile on 02.12.2012


That's our friend.
To a mathematician this is a number.
But it wasn't always obvious.
ROGER BOWLEY: Math existed to do financial transactions, to
sell sheep, to buy copper things, whatever.
And you would use coins.
And there would be values to the coins.
So you would say this costs 5 pounds 20
shillings, or whatever.
You don't do any trading with the number 0.
I've got no sheep here.
Oh, I'm gonna buy no sheep, I'll give you no pounds.
It doesn't make any sense.
So there was no need for shopkeepers to
invent the number 0.

JAMES GRIME: They had the idea of 5 cows.
But the idea of 5, not so much.
The idea of 5-ness they didn't quite understand.
So a number like 0, which is the absence of--
well, whatever, cows, whatever it was--
made no sense to them.
I'm going ask Brady a question.
See if I can catch him out.
Brady, is 0 an even number?

BRADY HARAN: I think the answer is no.
But it feels like it is because it is round.
JAMES GRIME: So this is an interesting--
OK, let me ask another question.
Is 16 an even number?
So people have studied this.
People have a delayed reaction.
People are not sure whether 0 is an even number.
Now I can tell you that 0,
categorically, is an even number.
It will follow every definition of an even number.
ROGER BOWLEY: So there was an Indian called Brahmagupta who
invented the idea of nothing not being nothing but existing
as an abstract quantity in the mathematics.
Well, then it goes to North Africa.
And there's a guy called Al Qasimi.
And he writes a book about the art of Hindu mathematics and
reckoning by Hindu mathematics.
And that spreads through North Africa to Spain and so forth.
And it becomes, they think, Arabic numbers.
So you have all the numbers plus 0.
That was fine.
Until 1200 when Leonardo of Pisa, also known as Fibonacci,
translates this book.
It's a wonderful idea.
But this is in the period of the Crusades.
So people think these are Arabic numbers, not Hindu
numbers because it's come from North Africa.
So the Catholic church objects to this because there are
Crusades going on and this is the work of the enemy.
So in Florence, for example, they banned the use of this
Arabic number 0.
And it was thought to be the work of the devil.
BRADY HARAN: Because my definition of an even number
is something that can be divided by 2.
JAMES GRIME: So Brady's just told me an even number is a
number that can be divided by 2.
Well, 0 can be divided by 2.
0 divided by 2 is 0.
In fact, in that sense, it is the most even number.
In ancient times they had this idea of things being singly
even or doubly even.
So 12 would be doubly even because you can divide by 2
and then 2 again.
Well, by that sense, 0 can be divided by 2 and 2 and 2 and
2-- it is the most even number.
The correct definition of an even number is it's
a multiple of 2.
Something times 2.
Something times 2.
Well, it is.
There's no problem with that.
It's 0 times 2.
Brady said it had to be divisible by 2.
Well, that works.
0 divide by 2 is 0.
OK, so it fits between two odd numbers.
That might be a definition for even.
Let's do that.
There's 0.
And over here that's 1.
Over here that's minus 1.
And then you get 2 over here, minus 2.
Perfectly fits.
There are some rules for numbers.
Some arithmetic rules.
Two even numbers, if you add two even numbers together, you
get an even number.
Well, that works.
0 plus 4 is 4, an even number again.
That's what we want.
We're saying 0 is even.
It follows every definition.
In fact, if it wasn't even, then it would break our rules
of arithmetic, which would be a disaster.
It is true to say that 0 is neither positive or negative.
It sits here between the positive numbers and the
negative numbers.
So it is neither positive or negative.
ROGER BOWLEY: The discovery of 0 was the most important
advance in mathematics of all, because it made mathematics
capable of being understood by everybody.
Everybody could do this mathematics.
JAMES GRIME: The Babylonians, the Greeks, they had the idea
of a space.
So there was a difference between 26, 2
and 6, and 206, 2-0-6.
They had a space there.
They didn't use the symbol 0, but they had a space.
It was more like a punctuation mark.
So the Babylonians would understand the difference
between 26 and 206.
But instead of using the symbol for 0, they would just
have a space.
However, if they wanted to write 260, 260 was
written like this.
And to us, that looks like 26.
To them and in context, that would make sense.
But they didn't have this idea of 0 by itself.
In the 9th century, the first instance--
or the first recorded instance of 0--
was found.
It was actually found by a gardener keeping track of the
number of flowers that his garden would produce.
And he used the number 50 as we would recognize it.
5, 0.
After that point, they started to experiment with 0.
What could it do?
If you add 0, it makes no difference.
If you times by 0, you get 0.
Dividing by 0--
that caused them some problems as it still does today.
ROGER BOWLEY: In this sense, the mathematical sense, the
number 0 does not mean nothing.
It means a quantity which you can manipulate in the
And so it's better to call it 0 rather than nothing.
Nothing is when you're counting.
There's nothing there.
0 is the abstract mathematical quantity.
CGP GREY: I'm CGP Grey and my favorite number is 0.
I like 0 because it's not an obvious number.
You can have counting systems where there's one thing, two
things, three things, four things.
But mathematics existed for a long time without having a 0
as part of it.
So it's a number, but it also isn't
anything in and of itself.