right 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 a 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 straightedge like a ruler but
not a measured ruler does a straightedge
and a compass so lines and circles those
are the rules those are the rules that
the ancient Greeks had to work by using
those rules can you construct a square
with the same area as a 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 then
they 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 I
start with B and then I mark half 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 yes do you know that this
has length one and this has length a I'm
going to scale up the triangle I scale
it up so now that this has length B the
big triangle has length B then this has
been scaled up as well so this has the
length a times three you can divide same
idea but it's kind of the reverse of
that I took a bigger triangle if that
had a length of B and the long edge here
has a length of a and if I scale it down
so now this has a length of one 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
scale
triangles as well and there's one more
thing you can do I draw a line here this
is a I'm going to add one to it that's
one and then I'm 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 us with a ruler and a
compass you can add multiply divide
subtract work out square roots and that
is it that is all you can do the numbers
you make using those rules are called
constructible numbers so all those
numbers are just repeated use of those
few rules if we had a circle if you had
a circle let's 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 well
that's 1 so the area of the circle is
just pi so I want to square with the
same area I want to square now that has
an area of pi and you may be ahead of me
here because if a square has an area of
pi the lengths of the sides are the
square root of pi now it's the square
root of pi now 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 work 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
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 sort
of roots like cube roots or v roots or
your nth roots if I'm allowed to add
those in as well then numbers that you
can make using those rules are called
algebraic numbers that's just a bit more
general same idea and it turned 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 it is
not algebraic is called a transcendental
number which is a fantastic word now to
prove something is transcendental it's
really hard
in fact mathematicians had shown that
transcendental numbers existed seven
years before they had even found an
example of a transcendental number so
you have to imagine in those seven years
people were saying to the mathematicians
alright so show me a transcendental
number come on show me a transcendental
number but it was a really hard thing to
do eventually they were found they found
a few examples of this but we still had
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 pies not
constructible and that is a proof that
you cannot square the circle can you
only not square the circle given the
parameters of these rules in this game
could a computer square a circle yeah so
you can you can make a square with sides
that have length root pi so absolutely
so in terms it so to the modern day
people with with our tools of algebra
right algebra is fantastic so 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
and they'll 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 we it's 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 you say it's easy to
make a
with the area PI but because pi has this
nature where the digits sort of continue
and we can't even get to the exact
figure sure that you can't make a square
that's exactly pi because you don't even
know what pi is or you can do a pinion
you can do it and you're you're treading
on something called Zeno's paradox there
which we're about to do next ok so
that's so for a second video in a row
we're going to tell people they have to
wait to find out more if that's the way
you want to play it Brady it is your
choice
this is Brady's decision right 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