DR. .JAMES GRIME: So today we're
going to talk about one of the great unsolved problems
in mathematics that went back to the Ancient Greeks, thousands
of years ago, which was eventually solved in 1882. And the problem is called
squaring the circle. And you may have even heard of
the problem as a metaphor for something that's impossible
to do. The question is, can you make a
square with the same area as the circle? Now you have to understand
the rules. In Ancient Greece they
didn't have algebra. So they could only
construct numbers using lines and circles. So you could only make things
using a straight edge-- like a ruler, but not a
measured ruler, just a straight edge-- and a compass. So lines and circles,
those are the rules. Those are the rules the Ancient
Greeks had to work by. Using those rules, can you
construct a square with the same area as the circle? Let's have a look at
what you can do with rules and compasses. You can add numbers. Here's a line, and
it has length a. And I add another line
of length b. And the whole thing
is a plus b. So you can add numbers together quite easily with lines. You can subtract numbers,
as well. If I start with b, and then I
mark off a length, which I'm going to call a, then this
bit here, that's going to be b minus a. We can multiply. If I draw a little triangle
here, this has length 1. And this has length a. I'm going to scale
up the triangle. I scale it up, so now this has
length b, the big triangle has length b, then this has been
scaled up as well, so that this has a length a times b. You divide. Same idea, but it's the
reverse of that. If I took a bigger triangle, if
that has a length of b, and the long edge here has a length
of a, if I scale it down so now this has a length
of 1, then this has been scaled down as well. And you scale it down so
it has a length of a divided by b. So you can divide by scaling
triangles as well. And there's one more
thing you can do. If I draw a line here,
this is a. I'm going to add 1 to it. That's 1. I'm then going to draw
a semicircle. And this length, here, is
the square root of a. And that's all you can do. That is it. With a ruler and a compass, you
can add, multiply, divide, subtract, workout square
roots, and that is it. That is all you can do. The numbers you make using
those rules are called constructible all numbers. So all those numbers are
just repeated use of those few rules. If we had a circle-- let's this make this easy. Let's say this circle
has a radius of 1. The area of the circle is pi
times the radius squared. And the radius squared,
that's 1. So the area of the circle
is just pi. So I want a square with
the same area. I want a square now that
has an area of pi. And you may be ahead of me here,
because a square has an area of pi, the lengths of the
sides are the square root of pi Now it's at the square
root of pi. We can do square roots. That's allowed. So there's nothing yet to say
that we can't do this. So what's the problem? Well we know that pi
is irrational. Actually, that wasn't
easy to work out. We only worked that out
quite recently. By quite recently I mean 1761. But being irrational
is not enough. There are constructible numbers
that are irrational. You can make the square root
of 2 quite easily. And you can make the square
root of the square root of 2, and so on. So being irrational is not
enough to say that this can't be done. If I'm allowed to make this
a bit more general, if I'm allowed to use other roots,
like cube roots, or fifth roots, or nth roots, if I'm
allowed to add those in as well, the numbers that you can
make using those rules are called algebraic numbers. That's just a bit more
general, same idea. And it turns out that they were
able to prove that pi was not an algebraic number. And if it's not an algebraic
number, then it's definitely not a constructible number. Now this is what I like. The word for a number that is
not algebraic is called a transcendental number, which
is a fantastic word. Now to prove something is
transcendental is really hard. In fact, mathematicians had
shown that transcendental numbers existed seven years
before they even found an example of a transcendental
number. So you have to imagine in those
seven years, people were saying to the mathematicians,
show me a transcendental number, come on, show me a
transcendental number. But it was a really
hard thing to do. Eventually they found a
few examples of this. But we still have to wait until
we were able to prove that pi itself was
transcendental. And then that was
done in 1882. Once we've shown that, we've
shown that pi is not constructible. And that is a proof that you
cannot square the circle. BRADY: Can you only not square
this circle given the parameters of these rules
in this game? Could a computer square
a circle? DR. .JAMES GRIME: You can make
a square with sides that have length root pi. Absolutely. So to the modern day people,
with our tools of algebra-- algebra is fantastic. People complain about algebra,
and having to learn algebra at school. But if we didn't have algebra,
we would be doing maths like the Ancient Greeks did maths. I promise you, that would
be much, much worse. Your textbooks would
be ridiculous. They'd be very wordy. Algebra is an amazing, powerful
tool in mathematics. It's not there to cause
you problems. It's there for a reason. It's brilliant. So this is easy to make a square
with the area of pi. But using the Greek rules,
straight lines and circles, it's impossible to do. BRADY: You say it's easy
to make a square with the area of pi. But because pi has this nature
where the digits continue, and we can't even get to the exact
figure, surely you can't make a square that's exactly pi. Because you don't even
know what pi is. Or you can do it, can you? DR. .JAMES GRIME: You can do
it and you're treading on something called Zeno's Paradox
there, which we're about to do next. BRADY: OK. So for a second video in a row
we're going to tell people they have to wait to
find out more? DR. .JAMES GRIME: If that's the
way you want to play it, Brady, that's your choice. This is Brady's decision. He's stringing you along here. The representatives
actually passed it unanimously, 67 to 0. They passed it. Now this is the weird bit. On the same day that they
were passing the bill--
I would say that since ancient mathematics did not allow for infinite sums, they could not accurately construct pi. Even though sqrt(2) can be constructed, it can also be given an infinite sum. Pi is no different. Thus with some added rules, pi can easily be constructed.
I am not the biggest fan of Numberphile. When he says that the ancients didn't have algebra, he is only technically correct. They didn't have symbolic algebra, but they for certain manipulated unknown values using geometry. Moreover, the introduction of abstraction in algebra does not guarantee that we can find pi. As I said above, we need to use more complicated tools, like sums, that need more complicated topics, like infinity.
At minute 5:33
Can you clarify what you mean?