[Applause]
my name is Katie and I am a
mathematician one of the things I find
most fascinating about maths is the fact
that it is made up of ideas it's just
this bunch of ideas and concepts and
structures and things that are kind of
the way I see it just sort of floating
out there in the universe waiting for
someone to come and discover them and
play with them and do something
interesting with it and what this means
is that the same mathematical idea the
same piece of maths can be picked up and
played with by many different
mathematicians all over the world and
throughout time as well an example of
this is this piece of maths here which
is from China in 1303 and you can see
there are numbers in there if I show you
what those numbers are you can see this
is a triangle of numbers that are
arranged with a pattern so each number
is the sum of the two numbers above it
you've got four is three plus one and
six is three plus three and you might
see this you know you might have seen
this before potentially at school but
you may not know this as the yanqui
triangle you might think of this as
being called Pascal's triangle and
that's because that's what it's called
but Pascal wasn't born until the 17th
century so this triangle actually
existed several hundred years before
Pascal and in fact he even existed
before that this is the exact same
structure from the Year 900 and in fact
this kind of thing happens all the time
mathematical ideas get kind of picked up
by different people in different places
and what I'd like to do is tell you the
story of one particular piece of maths
that goes from very long time in the
past all the way through to the modern
day and it's based on a simple idea so
here I have some cans and I could use
these cans if I wanted to kind of create
a display for my supermarket where I
sell cans to create a nice little
display like this little kind of
triangular shape like that and a
reasonable question you might ask
yourself if you're building this kind of
shape is how many cans do I need to
build different sizes of stack you know
if I've got four rows here I've got
plus two plus three plus four cans in
general what do these numbers do what's
the pattern in these numbers and you
might reasonably see that this is you
know one plus two plus three plus four
and you could calculate them this way so
if I've got you know different numbers
of cans in my pyramid this is how I
could calculate them but you can also
see that if I take the one can from the
top row and add it to the four cans from
the bottom row I get five and if I take
the two and add it to the three I also
get five and this is not a coincidence
this happens with any size of stack that
you use if you've got numbers that
increase like this you take the top one
and add it to the bottom one you get the
same thing as if you add the second two
top one to the second the bottom one
because one of them's gone up and the
other ones gone down so you get the same
total so you can do this all the way
down to the middle and it means that you
can pair up all the numbers so you have
pairs of numbers each of which adds up
to one more than the biggest number and
you have the number of pairs is half the
number of layers in your stack so in
this case I've got four layers so I've
got four divided by two pairs that's two
and I've got four plus one is five in
each pair two times five is ten cans and
I knew that because these are my cans
but if I had a stack that was say 20
cans high I would have 20 plus 1 21 cans
in each pair and 20 divided by 2 is 10
pairs of cans so I'd have 210 cans
altogether and that means I've got a
pattern for this I know that the number
of cans divided by 2 times the number of
cans plus one will always give me the
total number of cans and this is the way
that you see this formula often written
and if I wanted to do this for something
more interesting so not just a flat
triangle of cans if I wanted to make
kind of a 3d pyramid
I could do this as well so I have one
can on the top underneath that I have
four cans arranged in a square
underneath that I have nine cans
arranged in a square and so on and so
forth and there is a general pattern for
this as well if I want to know how many
cans I use all together it's this and
it's a bit more complicated than the
first one but it still gives you a
general rule and if I wanted to go on
further than this because of course
mathematicians always do I could have
one can on top I don't know why I would
do this but I could have a 2 by 2 by 2
cube of cans and denieth that and
cube that's three by three by three cans
underneath that this is now just the sum
of the cube numbers and there is a
general formula for this as well and it
looks like this and these kinds of ideas
have been thought about for a very long
time people have played with this
particular bit of maths since a very
very long time ago
so for instance Pythagoras you may have
you may be familiar with Pythagoras his
greatest hits but the followers of
Pythagoras were known to arrange pebbles
in triangles on the floor to try and
work out how many you could get in a
triangle shaped like this so they were
looking at these kinds of questions this
is a mathematician from around about a
thousand years later from India and
aryabhata's widely considered one of the
first physicists and he knew that pie
was 3.1416 which is pretty good for the
time and Aryabhatta also considered
these formula the sums and the sum of
squares in the sum of cubes and this is
a mathematician called
al qiraji who was based in Baghdad again
and in the 500 years later also worked
on this particular type of mathematics
and also discovered Pascal's triangle
several hundred years before Pascal and
has quite a popular thing to do and but
at the time maths was done in a quite a
different way to the way that it's done
now so they wouldn't necessarily have
had these formulae in the same form that
we're familiar with so this is from
aryabhata's writing on the topic and
it's it's quite dense there's a lot
going on here but I'll put up the
equation as well so you can see what's
going on here
the sixth part that's the the six on the
bottom of the product of three
quantities consisting of the number of
terms that's n the number of terms plus
one that's n plus one and twice the
number of terms plus one is the sum of
the square so this is kind of a word
version of that same mathematical
formula and at the time this was the way
that people did mathematics we didn't
have this kind of algebraic notation
that we use today so I guess the next
person to come into this story is going
to be the first person to write down
these formulae properly I guess in a
numerical form and that was a
mathematician called Thomas Harriot
who was also a map maker and a navigator
and a translator and an ethnographer and
one of the things that he did was write
down these equations so you had the sum
of numbers that sum of squares the sum
of ki
and the one in the bottom corner there
is this sum of 4th powers so the patent
carries on and the next person who comes
along is this mathematician Johannes
foul Harbor who was from Germany and he
wrote down formally for the first
seventeen powers which sort of feels
like maybe there's something kind of
general here maybe we could actually get
a formula for any power if I if I choose
how many dimensions my supermarket
display is going to be in I can then
work out how many cans I'm gonna need
and the person that is the next person
that kind of comes along in this story
is a mathematician called Jakob
Bernoulli what Bernoulli realized was
that these formula can be rewritten so
that they all look very similar so
instead of saying n over 2 times n plus
1 you can spread it out so that you've
got the sum of first powers I guess the
sum of numbers is 1/2 n plus 1/2 N
squared this is just the same equation
but rewritten and if I write the sum of
the squares I get a similar thing I've
gotten a fraction in front of an N a
fraction in front of an N squared and a
fraction in front of an N cubed and I
can do this for all of them you just end
up with more and more powers of n in
this case there's no fraction in front
of the end because it's zero but you get
a sequence of numbers and these numbers
essentially tell you what these formulae
are and Bernoulli discovered a way to
find these numbers a pattern in these
numbers that meant that you could create
these formulae and what Bernoulli found
was a set of numbers which we call the
Bernoulli numbers that's the the next
Bunch they're quite a lot of them as
something over six that's we don't know
why so these are the Bernoulli numbers
and these are a set of fractions again a
lot of them are zero there are patterns
in these as well but Bernoulli realized
that if you take these numbers and
combine them with some numbers from
Pascal's triangle it turns out you can
generate those fractions which means you
can create these formulae for any power
that you like and you can also generate
the Bernoulli numbers and in order to
create the binary numbers you need the
previous binary number so you've got the
first one which is one you can then work
out the second one you can then use
those two to work out the third one and
so on
and now we caught the Bannu who was very
smug about this discovery he he said
with the help of this table it took me
less than
half of a quarter of an hour to find
that the tenth powers of the first 1000
numbers being added together will yield
the sum this which is quite impressive
you know this is definitely a step
forward and it's a brilliant discovery
although potentially not necessarily
Bernoulli was the first person discover
this as always this is a mathematician
from Japan Sachi Tucker Kazi who
discovered these numbers and published
about a year before Bernoulli did but
for some reason they're called the
Bernoulli numbers so who knows anyway
this could be in some sense the end of
the story because we have an answer to
this question in general if I want to
build any number of stacks in any number
of dimensions how many counts do I need
I can work out the Bernoulli numbers and
then I can work out the formula using
them but it's not quite the end and the
next person who comes along with this
particular bit of maths is a person
called Ada Lovelace and she's a very
very famous example of someone who's
connected potentially if you've heard of
her you might have heard of her in the
context of computing because Ada
Lovelace worked with this machine and I
should qualify this is not the entire
machine this is one small part of the
machine and the whole thing was never
actually built so technically Ada
Lovelace worked with an imaginary
computer that never really existed this
is just kind of one small bit of it and
it was designed by Charles Babbage and
ADA was brought in to help with some of
the writing of this so she understood
how this machine worked
she was translating some papers and
wrote a big appendix on the end just to
show what this machine could do and it
was again a hypothetical thing but you
could program this machine to do
anything so you you kind of you
controlled it using a punch card system
so a bit like the jacquard looms of the
time you would punch some holes in a
thing feed it into the machine and it
would run a program and it could
calculate in theory anything you wanted
it to and Ada wanted to show off what
this machine was capable of doing and
she wrote this program and the title of
there is a diagram for the computation
by the engine of the numbers of
Bernoulli so this was the thing that she
did this was the piece of maths that she
chose to show off using what could
potentially have been the very first
ever computing machine
and you can you know this has got in it
the Bernoulli numbers there's a little
be zero and a b-1 popping out there on
the right and a b2 and you can see that
it is code for generating the numbers
and I'm incredibly impressed by this
because this was a computer program that
was written for a computer that didn't
actually exist
you can't debug this right this is just
this is so impressive and it happened
that she picked this particular bit of
maths it was something that she'd been
playing around with herself to try and
calculate the numbers and to actually
show what this could do and I don't
think this is the end of the story
either because the Bernoulli numbers
connect to various other bits of maths
as well so there's a piece of math
called the Riemann hypothesis which is
currently one of the most famous
outstanding problems in mathematics if
you can solve the Riemann hypothesis
there is a prize of a million dollars
available to the mathematician that
first solves it which makes it quite an
exciting prospect but it does connect to
this same piece of maths so if someone
were to make progress with that they
might potentially be the next person in
this story as well but I think what this
has shown me I don't normally talk about
mathematicians this much I mostly just
talk about maths but I thought it was
interesting to think about the people
who did this because these mathematical
ideas that float around in the universe
just waiting for someone to come along
without mathematicians that's all they
are they're just ideas and until someone
comes and grabs it and plays with it and
uses it to do something interesting or
useful it will only ever just be an idea
thank you very much
you