- Hello everyone. I'm Sarah Hart, and I'm the
Gresham Professor of Geometry and welcome to my lecture on "Where Mathematical Symbols Come From?" Now, in 1977, the Voyager
space probe was launched and on that probe, there
was a golden record. It contained information
about our human civilization, the planet earth, who we are, where we are and cultural artifacts like music from Barker and
other composers, the human voice and all sorts of wonderful
things on this record, there were over 100 pictures as well. The idea was this probe was going to pass through our solar system and an outing to wider
space into the galaxy. And one day maybe an alien
civilization would find it and they'd want to know who are the people who have created this space probe. Now, one of the important
things we wanted to communicate with that was where we
are, what we look like but where we are and
what our planet is like. But how do you actually
communicate with aliens? They don't know what symbols we use. They don't know how we write. They don't know what it would mean to say, we're 93 million miles from our sun, what's a mile? We don't have this common language. So the first job really
is to tell the aliens about how we describe numbers. We're going to give them some numbers. And this picture was one of the pictures
on the golden record. It was the third picture and it gives a lot of information
about our number systems, lots of symbols and you can see on that picture how it was described,
what our numbers are. So to begin with you for
the first few numbers you've got some blobs
here, one blob that's one. So there's a symbol for one,
two is two, three is three and so on. What are these vertical
and horizontal lines? They are the binary
representations of those numbers. So in binary, of course, we write, normally we would use ones and zeros, but this is our base
to that computers use. Using ones and zeros when
you're at the same time trying to define ones and zeros though, it's not a sensible idea, hence the vertical and horizontal lines. So our extraterrestrials looking at this, they've got lots of clues to help them understand what
the first few numbers are. So this symbol, what's this well it's the thing that's five blobs. So that's what's these
earthlings must be using for the number five. Then the numbers carry on, we drop the little blobs after a while and then you get to the first examples of our place value notation that we use. So this one and the zero,
that means one loss of 10 and no units, of course, as we know and there's a couple more examples over in the right-hand side of the picture of that
happening for 12 and 24 then you get next relation
of this exponential notation. So 100 is 10 squared, It's 10 times 10 and so we've got some examples of that. And then what else has defined addition? Some examples of addition so that our extraterrestrial friends can understand what we mean by that. We've got some fractions
shown and some multiplication. And with those basic
building blocks of symbols, then the numbers that are
needed to be communicated can be expressed and units are defined in terms of things that
the aliens will know. So the wavelength of light,
the size of hydrogen atoms something like that, we
don't write in miles. One thing to mention here,
fractions are being used, no decimal points appear
on the golden record. I suspect that's because
a little dot somewhere could easily be assumed to be a damage to the record or something like
that, a flaw in the record. And so that's why fractions
are sort of safer to use in this context. Now, why have I shown you this picture? Well, it's to make the point that our putative alien civilization it's going to have numbers, it's going to have its
own symbols for numbers and for mathematical terms like
addition and multiplication they're different from ours but the underlying mathematics
is of course the same, right? You still count one, two,
three, four, five, six in whatever language and
whatever symbols we use. So this lecture is about
mathematical symbols and notation but you might very well ask. Why does it matter? Surely the underlying mathematics is all going to be the same. And the answer to that is yes, but, the but is that it's people it's human, conscious
beings doing mathematics and the choice of notation,
symbols that we make can actually really influence
what mathematics we do. Some notational choices almost prevent us from making progress inadvertently while others allow us say open pathways that we may not have seen before and they allow us to make more progress and we'll see examples
of that as we go along. But before we do plunge in
and start talking about, where our number symbols came from, there's another symbol on this slide that we haven't discussed yet. It's one that seems so fundamental
and basic to mathematics that we perhaps haven't
even noticed it being used, It's just implicit almost. But this symbol was invented
by a man in Tudor, Britain. He was born in the little town of Tenby in Wales in about 1510. His name was Robert Recorde and he invented the equal
sign and that equal sign which you might just think
it's just part of mathematics. We could never know who
could have thought of that. It's just such a basic
thing that was invented by Robert Recorde in 1557. And now I know when I think
about the Voyager space probe which has left the solar
system by now on that record in that little piece of
machinery out in space there is a symbol that was invented in Tudor, Britain all those years ago and is now, has left our solar system. I love that thought. Lots of these other symbols
we'll discuss along the way and their origins. One other thing that we'll
talk about as well as we go is if you like language or
interested in language at all, the language of these
mathematical notations and symbol can tell us the path that they have traced through different cultures on our planet. Some things have started, over in India and then
moved via the Arab world and coming to the Western world and there's all sorts of
wonderful stories that we can find sort of do a bit of
mathematical archeology to see ideas traveling across the world and that's very appealing concept. Okay, so let's get started
then and talk about numbers. Now, ever since writing has existed, mathematical writing has existed. This is a Babylonian tablet sort of middle Babylonians,
about 3000 years old and on this tablet, clay tablet you would write with
a stylist making marks and you can see there are some marks here and these are numbers. So there's two columns of numbers. And this is a bit hard
to see on this slide, what these symbols actually are. So I've redrawn them and
made them a bit bigger. So this is the top half of the tablet. This is the bottom half. There is a reverse as
well that takes this table a bit further, but this is
enough information already. We can see this first column if you just sort of, with
a bit of detective work we can see this looks like one, two, three you've got three marks, four
marks, five marks and so on and so on. This seems to be the numbers
one, two, three, four, five and so on. It gets a bit hard by the time you get to something like seven,
you have to be perhaps starting to count the
number of little marks. So maybe that's not quite efficient. And then we get eight, nine, oh, and then there's something different. And that's something different, If we think about fingers,
one, two, three, four by the time we have 10
fingers, that's two hands and so probably this symbol
here is kind of looking like two hands together,
cause it's 10 fingers. So that's 10 and then we
carry on 11, 12 and so on. That's all very nice. What about the second column? What is being recorded there? Well, we can count up here, there are nine little wedges. So one corresponds to nine and two, what's this, it's a 10 and then
there's eight little wedges. So 10 and eight 18. So two goes with 18. If you look at the next one, three that looks like 27, four, 36. This is the nine times table written on a 3000 year old
tablet, the nine times table. It's all great. And now we can see at the
bottom of this column here and we see something that is telling us about the Babylonian
method of writing numbers because this is for wedge shapes, but it's not the number,
can't be the number four, from context, it ought to be
seven times nine, which is 63 and that's what it is. The Babylonians used a base 60
mathematical writing system. So we have a base 10 system, we write tens and hundreds
of pounds of 10 like that. They had sixties, so this means 60. Then they leave a little
space and then three. So it means 63, which is what it ought
to mean from the context. So this is a quite
efficient way of writing but there were some problems with it. One problem is how do you
just write the number 60? Because you can't leave a space 'cause there's nothing
else that's goes after it. So if you just see a wedge on its own you have to kind of work out from context whether that's the number
one or the number 60. So this is sort of
problematic with this system. However, it was a useful
system and lasted a long time. Let's look at another one
or two early number systems. So what were the ancient Egyptians doing? Well, they didn't have to worry about can I see a space? Or is there a space, what's going on? Because they had a symbol
for every power of 10 and then you just have as
many of those as you need it. So here are some of the
symbols, and here's an example, there on the slide at the top, you can see one
Lotus flower, that's a 1000. Two coiled ropes, 200. Three bent sticks, that's 30 and four straight sticks, that's four. So that's number, there is 1,234. Because there was a different
symbol for each pair of 10, you didn't have to worry about
leaving spaces or things. There wasn't an ambiguity between, so the next two ancient
addiction numbers here you've got four pointing
fingers, that's 40,000 and then you've got you, haven't got any one thousands,
but you've got some hundreds there are nine coiled ropes
there, if I've done it right. So that's 40,090, no, it's 40,905, right? You don't have to worry
about the missing thousands because you just don't
see a Lotus flower symbol, so there aren't any. Though number on the bottom completely, obviously, a different number that has got 495. So they can't be confused, you have different symbols. This is, again, it's an okay solution but it's quite unwieldy for big number and you do have to offer
them stop and count how many of each little symbol there are. So that's the ancient Egyptian system. There were lots of different systems. I'm just mentioning a few. The ancient Chinese
system had powers of 10 but it solved the problem of
counting lots of little symbols by also incorporating a
different symbol for each number one to nine. So if you wanted 400, you'd
write your symbol for four and then your symbol for 100. Ancient Greece, now, this wasn't quite the
first system they developed but this alphabetic system
was quite an ingenious way of writing at least for small
numbers, it was efficient because there was a number
from one up to nine. They were just the first nine digits of the Greek alphabet or
actually an early version of it, 'cause this dye gamma
is in between Epsilon between zeta. And so you've got Alpha
is one, Beta is two, Gamma is three and so on. But then once you get up to
10, you then have numbers, 10, 20, 30, 40, 50, 60 up to 90 are the next letters
of the Greek alphabet. And then you have symbols, again, carrying on for 100, 200 and so on. So you can write 444 with, what's that? Upsilon and mew and Delta. So mew is 40 and delta is four. Now that worked, as I say,
it's reasonably efficient for small numbers, but
what are you going do when you want to write 1,000,073 it's tricky and you've run out of letters. I mean, even with these couple of extra letters that were dropped from the Greek alphabet of various points, you still, you start
to run out of letters. So that's sort of problematic as well. Why didn't they realize, oh, instead of writing mew for 40 and delta for four, why can't
we just have a place system like that Babylonians had started to do and just have just the delta meaning four and then you can put
things in different places. Well, one reason, the ancient Greeks were not stupid. They were doing great mathematics. It wasn't that they weren't
able to think of this. I guess it's more that they weren't using
the written numbers to do calculations in the way we do now. So they, somehow they didn't need to develop that technology yet because they were using counting boards just like the Romans were as well, they had counting boards
or they'd have an Abacus. So we do have some kind of archeological evidence
for what they were doing. There's a tablet called the
Salamis tablet, which survives and there are columns for different denominations,
I guess, of numbers. So the biggest one you
would have would be talents and then minae, drachmas, obols. So six obols and a drachma and then you'd have fractions of obols. So that survives. But we also have references in writing to how these things were used. So you have these little
stones that you'd move around on the counting board and they would have different values depending
which column they were in. So Polybius second
century BC or thereabouts he's got this thing, he says, "The courtiers who surround kings are exactly like the counters on the lines of a counting board, for, depending on the
will of the reckoner, they may be valued either at
no more than a mere chalkos," which is, I think an eighth of an obols which is a very small amount, "Or else at a whole talent." So actually the difference in those, it's something like 288,000 times as much value if you
are a talent compared. And so he's giving this description everyone knows what he means because everyone knows
about counting boards. It's the same in Rome. Nobody's actually doing arithmetic. Nobody's trying to do
multiplication with Roman numerals. They are doing the calculations
on the counting board with their little stones and
the Latin for that is calculi, hence the word calculate
and also calculus. They're moving them around. They are not trying to work out what, how to multiply MDXCVI
by XVL or something. They're not trying to do that. So they aren't needing
to grapple with this. They also had expressions that tell us that they're
using these things. So they had a expression
"Vocare aliquem ad calculos" which has meaning to
settle up with someone to kind of pay your debts. But what it literally
means is to call someone to the calculating table
or the counting board. So this is what they were doing and so they were not
held back in that sense by the cumbersome numbers,
although they were a bit as we'll see, it took longer
to develop various technologies because of it. Okay, so those are kind of
some of the old number systems. What about our numbers that we now use? Where did they come from? So they were developed in India. We don't know exact dates, we don't know for sure there
are various manuscripts that have found the hard to date. The manuscripts themselves may be older than the writing that's on them. But what we do think happened is that there was an
early Brahmi number system that originally would have had
the numbers for one to nine and then probably 10, 20, 30 and so on. But at some point, those
later ones were dropped and we just kept the
symbol for one to nine, and then crucially, getting round the problem
faced by the early Babylonians. Although they did invent a
placeholder symbol at some point at some point in the
first few centuries AD and I'm not competent as to when 'cause no one is, this happened. So a little dots started to
be used for a placeholder. So, if you want to write 105 and you want to distinguish
that from watching 15 you have to say, there's
no tens happening. So on your counting board, that's fine, you can just leave a gap but if you're writing it
down, you need to signify that in some way, otherwise
things get confusing and this is what they did, they put a little dot
and that became our zero. So to begin with it was
just a place holder, it wasn't a number in the same
way that we've got numbers that people were considering them. However, it later did
start to become treated as a number in its own, right? So you have people like Brahmagupter the Indian mathematician around 730AD was giving rules for dealing with zero. So he, for example said, "Any number subtracted from
itself gives you zero." So this is the first beginnings of it being treated as a
number like other numbers. It took a long time for that to happen. So I've the title this slide has got Arabic crossed out and Indian written in for Indian numerals. These are Indian numbers. So why do we call them Arabic numbers? Well, it's because they
didn't come straight to us from India. What happened was that
Arabic mathematicians like Al Khwarizmi in the ninth century had translations from the Sanskrit texts into Arabic of various mathematical works including works by Brahmagupter and Al Khwarizmi found
out about these symbols and about how they were being
used in calculation as well. It's not just the symbols it's a new method of calculation that can be done that doesn't require
counting boards or an Abacus. So these symbols were
brought into Arab mathematics and about this time, and Al
Khwarizmi wrote about them he wrote a book about these new symbols and the Hindu mathematics. He is known to us as well because he will also wrote
about how to solve problems that involved what we now call algebra and this word here that features in the title of one of his
books, Aljabr is what we now, this is where we get our word algebra from and Al Khwarizmi his name himself that has become our word algorithm. So he's very influential for mathematics in general Western
mathematics in particular. So he's working on these things, he's seeing the new Indian numerals, he's got them into Arabic and from there not directly,
I think from his work but from the Arab world
and from our interactions. So traders and merchants that are working in Italy and places
like that are traveling and meeting along trade routes. They're talking to Arab
traders and they are learning about these new numbers
that are really efficient and you can do calculations with them and they are a great
new piece of technology that are going to supplant the Abacus and the counting board. It's really through trade that these pieces of technology
are transmitted, algorithms work was translated into Latin, it was seen in the West
actually a lot earlier than when the numbers really caught on, but it really was through trade routes that these numbers spread. And one of the people who were
spreading them was this chap, Leonardo of Pisa, who is
better known as Fibonacci and he, his father was a merchant, he learned about these new numbers and about how you couldn't express any number in terms of them and how you can calculate with them directly without using any
kind of board or Abacus. And he wrote about them in a
book called Liber Abaci in 1202 I've of quoted from it here, you don't need to be able to speak Latin to get the gist of this, but he's saying, here are the nine Indian figures, the nine figures of the Indians, and he writes them down, nine, eight, seven, six,
five, four, three, two, one. But I like, these are written probably as they would have been
seen in an Arabic book because, of course in Arabic,
you read from right to left and so that they would be
in ascending order then, he's preserved that way of writing, even though it's a little
bit counter-intuitive to write them in descending order, but that must be why he did it. Just sort of without thinking about it. So he's written down these nine numbers but just notice he says, with these nine figures
and with this sign zero. So, it's not a number like the others, it's a different thing got the nine figures and this sign zero which the Arabs called Zephyr. So Zephyr that gives us our word cipher. That's an old word for zero and from cipher, we get ciphering which is an old word for kind
of doing or calculations. So with these nine
figures and the sign zero which is the Arabs called Zephyr we can write whatever number we like and then you give some examples
of what, how to do that. So this is here, it is happening. Now, these amazing numbers which allow you to do
calculations by writing them down and without using little stones, we not doing using our
calculator to calculate we're using our cipher to do ciphering or algorithmus is another word for this new way of calculating they spread but it took a while for
them, to it wasn't just like, everyone said, oh, yes, this is good we're going to immediately adopt this. There was resistance. It took a long time. If you are happy and your friend is happy and you trust each other that you've got your
financial calculations are done in a way that you're familiar with the symbols you're
familiar with, it takes a while for that trust to be
gained for a new system. So it did take time and we have instances like in 1299 the Florence City Council
banned writing things in these Indian Arabic
numbers in financial records you had to write it out in words. The worry was not that they didn't think they expressed numbers but that it was too
easy to tamper with them potentially you could turn this
weird zero thing into a six or a nine just by adding
a little tail or a hook. And so potentially you could
tamper with financial records. So you had to act it out in full in words, just as, on the rare occasions nowadays where we write a cheque, you do that exact thing for that same reason that
you want a clear record of what these things are. So, that those numbers took
a while to be accepted. There are various woodcuts books published about using this new system how to do ciphering here
is arithmetic personified. She's sort of, disdainfully looking away from the counting board and she's pointing to the wonderful new
Indian Arabic numbers here. This is the way forward. There they are one, two,
three, four, five, six, seven, eight, nine, and then 10 and 100 and 200 and so on. This is the new system and it did catch on but it took a while. And particularly we could say that or notice that this zero, you saw when Fibonacci introduced
it, it was nine figures and this weird zero thing. Zero really was treated
with suspicion, with caution for an awfully long time, centuries really, even by Shakespeare's time,
you've got in King Lear there's a scene where King
Lear has been brought low and his fool is taunting
him and he's saying, thou at an O without a figure, so your nothing a worthless your an O, a zero with no figure attached, you haven't got a number attached to you, so you don't mean it. If there's a number attached to a zero then it might turn one into 10 but on its own, it's nothing, I mean, that's okay, that's true. But it's the point that he's
saying here without a figure. So the zero somehow isn't a figure it's not a number like the others. And so this view
persisted for a long time, and we see that even when
people are talking about trying to solve algebra
problems still for a long time even when they're doing
quite complicated algebra that requires the use of things
like even the square root of negative numbers, which
is a difficult concept. Even when they're doing that they're still not happy about zero and they're still not happy
even about negative numbers. So it really takes a long time for these things to get acceptance. So we've seen where our numbers come from and how this new technology helped that our number symbols to be adopted. But there are, of course, lots of other mathematical
symbols, algebraical symbols the plus sign and things like this we've mentioned the equal sign. We'll talk about Robert Recorde
a bit more in due course but let's have a little tour of algebra. Now, the beginnings of algebra, were what is called rhetorical algebra everything was expressed
in words, and this is true the Babylonians they had things that we might now call algebraic problems and it would be, I have
a length and a width and the length and the width add together to make this number and
the length and the width the difference between
the length and width is it's other number, find
the links in the width. So it was all done in words, this is a particularly beautiful example. So in India, the tradition was for Sanskrit writing to be done in poetry. So you have lots of people who are mathematicians
and poets and astronomers. It's all part of the same
intellectual thing really, this is a beautiful problem. This is why Bhaskara in
a book that he dedicated to his daughter, "Lilivati"
in the 12th century and wouldn't it be lovely if our algebra problems were like this. So it's about a swarm of bees and a fifth part of them
comes to rest at one flower and then three times the
difference between that and the third of it flew to another flower and then there's one bee
alone remained in the air attracted by the perfume
of a Jasmine and the bloom. Tell me beautiful girl, how
many bees were in the swarm? So this is a very poetic description of a problem that can
then it's all in words and the solution is given also in words, so this, yes, it's algebra, but it's, there are no
symbols yet involved. So it's not symbolic algebra. It's rhetorical algebra. Now, there's an intermediate step which is where you start to
maybe use a few abbreviations in what you're doing and that's got the very jazzy
name of syncopated algebra and that's the kind of
an intermediate step which is not, it's not purely abstract but there were some,
somewhere along the way and an example of this is
in the work of Diophantus. So he was a Greek mathematician working ish the third century AD and he wrote very influential
book called "Arithmetica" which you may happen to know that it was inferma's copy of
arithmetic that he wrote that famous marginal note about
having a wonderful theorem and the margin is too small to contain it. So, Arithmetica as medical
was used for a long time it's really a great achievement
of early mathematics. He's did use some abbreviations,
maybe one or two symbols. He had a symbol for minus I believe but it was basically words
with some abbreviations. And there's a problem about
that if you're writing in Greek in ancient Greek, because
we've seen that the symbols for numbers were just
letters of the alphabet. So alpha is one, beta is two and so on. So if you then want to start using symbols for variables letters for variables like you would now that's
going to be problematic. Example, he was looking at
problems about the squares and the cubes of things, all,
very geometrically based. So a square, x squared which he wouldn't have
written is you think of it as a square of side x and something cubed is a
cube of side, that thing. So it's all kind of couched in geometry but there is some algebra happening. It's still all in words, if
you want to write a symbol, an abbreviation for cube,
the cube of something, well, the word is kubos, and the first letter of kubos is kappa, but Kappa is the number 20. So that's going to be very
confusing 'cause if you look at it, you won't know whether
it's 20 or it's the cube. So then what do you do? Well, maybe use the first
two letters, kappa upsilon and then that's problematic because that actually
represents the number 20,400. So you can see this is really difficult and somehow all of these things
are cut from the same cloth. They are all using
letters, alphabet letters which also mean numbers and now they also have to
mean something else as well. Very difficult to sort of
pause and expression like that. This helps to explain why it wasn't, it didn't really catch
on kind of using symbols because the symbols are
having to do too much work. The same symbols. What eventually seemed to have happened is that you have an expression a little thing like this, which is kappa, and then sort of a superscript
little upsilon on the top and that's your abbreviation
for the kubos or cube. So, because of all
these issues around this and abbreviations, perhaps not
really helping you much more than just writing in words, writing your mathematics and
your algebra inwards persisted and carried on for at
least the next 1000 years. And the next, the sort of
beginnings of new progress on this are in Renaissance, Italy where you've got people like Cardan. So he's very famous. He's written a book called "Ars Magna" and in that book, he explains how to solve
any cubic equation. So, we at school, we learned
the quadratic formula for solving quadratic equations, things with x squared in them, basically. The cubic equation is
a bit harder to solve but it was worked out at around this time. There were partial
solutions given by others and lots of arguing about
who'd come up with what first but Cardan did put how to
solve any cubic in this book, "Ars Magna" of 1545. Now, it's a great achievement and it's even more of great
achievement when you realize he still, he doesn't like zero. He doesn't like negative numbers. And because of that, it wasn't
just one cubic equation. So if we were nowadays, when we try and solve
a quadratic equation, what we do is our first
step before anything else, we bring everything over
to the left-hand side and equals note on the right. Well, the equal sign
hadn't been invented yet. He didn't like zero and he didn't like any negative numbers. So you can't do that. Normally not even writing
down equations at all. You're saying in words that things are equal to other things. So just an example here I've written that something
like an expression like, again, note the notation
that would be used, but something like x squared plus ax, the number plus some multiple of its root is a constant that's viewed
as a different kind of thing. It's a different kind from to solve from something that looks like the number is equal to some multiple of
its root plus another number. They were thought and
classified as different things. And with cubic equation you can imagine there's many more different sub types of cubic. So this meant from made for
much longer descriptions of how to solve all of these things but it was done, in spite of the suspicion around even zero and negative numbers, nevermind imaginary numbers
like square root of minus one. Geometrical arguments were
largely what we're being used and Cardan, then that made it
very difficult to do anything with equations where you
might have higher powers like something to the power four 'cause that doesn't make no
there's no force dementia of course, now, we
suspected there might be, but more importantly we don't insist that anything involving a fourth power is somehow has to be
corresponding to geometric thing. We think differently nowadays. So this terminology and the kind of the
mindset, amazing though the achievement was there ways in which, we hadn't seen further at that stage. So the next steps really we start to get this one or two symbols but really they really
coming from abbreviations. The next steps really, I would say are coming in Germany in the
late part of the 16th century where you have writers like Michael Stifel and Christoff Rudolff and they, so this word Coss, I've got there and the Cossic Art. So the Italian word for unknown is Cossa and that became Coss. So Rudolph's book, "D-coss"
is about finding the unknown like doing algebra and in England this was
called the Cossic Art often. So Michael Stifel here he is, he was the first to use in
print the symbols plus and minus or addition and
subtraction, as you can see that before then you'd get
expressions like this one. So what's this thing? This looks very difficult for us to read. Again, the problem is everything
looks the same somehow except we now at least
have different symbols for the numbers. So they're easy to pick out. So this is one lot of the zenzus. That's the number we trying, the unknown that we're trying to find, p plus, p for plus, five lots of R. What's R, it's the, Radix the root. So this is the root of the
zensus that we're trying to find. Already implicit in this
is that there is a root on this sort of assumed
to be a positive number and so we all with the
notation where accidentally not be allowing ourselves we need to consider other possibilities like quadratic equations
have two roots usually. M, minus six. So that's what things were
looking like to begin with. Then a few years later,
you get different editions of books and the denotation
gradually changes with the different editions
of D-Coss, for example. Then within a few years, that expression would
become something like this. So one lot of Q, the square
plus five lots of N, the number minus six. So that's to our modernized
much easier to read. And I think to everyone's eyes because you're starting to
just get different things, the operations are symbols. You can spot what they are. You can see what the numbers are. We don't get though. It's not easy to just
see by glancing at this that Q and N are related in any way. So that it doesn't help our intuition. So the final kind of step
in this progression is that we start to see
explicitly look, here's an A and here is A times A and they are, that's that number? It's one single number and it's there but it's also over here squared. So this, it makes it much easier to it doesn't sort of get
rid of the starting gates the possibility that A might
be negative or something like this, or even imaginary,
this is a better notation in the sense that it it's more amenable to generalization and obstruction. So there was some good progress here. I will just say before, moving on another bit of development that happened in Germany at this time
was with square roots. So the square root symbol was coming in around this time and it appeared I believe first in Germany,
our modern square root system but the notation for
cube roots was terrible. Let's see if we can see it on the slide. So the cube root symbol,
you take your square root but because it's cube
root and cube is three, you have three little wiggles very hard to see in a handwritten text. And then, okay the fourth root does that have four little wiggles? No, it does not. It has two little wiggles, why? Because it's a square
root of the square root. So this is very confusing and if you get someone like the 12th root, well, that's the cube
root of the square root of the square root. So is that going to have five wiggles? And then what's the
fifth root going to have? It's just impossible. And you know, it totally prevents you from being able to talk about,
the nth root to anything. You just can't do it. And you have to, how many
wiggles am I going to write? So something like this, they
stopped progress being made. Whereas the good notation
allows progress to be made. And we're going to see that
now with the start of something which later becomes an exponent notation. So things x to the five
or something like that. So kind of precursor of
this was Bombelli in 1579. So what's he writing here? Well, we saw, we start to be able to have, A and A times A and A times A times A, of course, being humans, we would very soon start to
say, how many A's have I got and can I just write
that down as a number? And this is what Bombelli is doing. Unfortunately, the
notation sort of obscures what the unknown is, It's hidden a little bit. So he's got lots of expressions like this and he's broke a whole page
full of examples, really of how this thing works. And what is this thing? And he's not saying four plus
seven is 11, it's not that, that little kind of curved line underneath that's representing the unknown. So what he's actually
saying, we now write it as x to the four times x to
the seven equals x to the 11. He's introducing that. The fact that we know about powers now that when you multiply powers you add those coefficients together. So he gives lots of examples of this but the issue here is that,
again, it doesn't allow for you to do things that
you might later what to do. It's not his fault. He deeds the trick for what he wanted to
do, but it's difficult. Then you've got this unknown that you haven't got a letter for. And so what if you wanted to
have two unknowns or something? What if you wanted to talk about something like a graph that we talk
about now, y equals x squared, the parabola graph that we perhaps drew at one point in our lives, you can't do that with this notation. But happily, Descartes comes along in his book, "Geometry" in 1637 and he introduces and gives us the exponent notation that we use today. And actually this book,
I mean, it's in French it's not in Latin, it's in French. The mathematical expressions
really do look quite modern, they look like how we might write things now, mostly, mostly. So he writes, he's got things like where he defines
a squared as a times a, a cubed, a times a times a. And so on to infinity, you can carry this on as long as you like. Another really crucial
thing about this book is he's stepping away from the idea that, something cubed has to
involve an actual cube, three dimensions. Instead he talks about curves that are, they're two dimensional things
often, and you express them in terms of relationships
between two variables. So that like was x squared,
not how he would wrote it not he would write it, but that kind of relationship
between two variables you can plot on the plane
allows you to move away and so then there's nothing
to stop you writing, y equals x to the five,
and looking at that you don't have to be thinking
it's all five dimensional or something, and then dismissing it as being beyond nature as
Cardan would have done. So, as you know, there's a
lot to talk about in this book but he gives us the exponent notation. Another thing he gives
us is the convention of having variables being
at the end of the alphabet. So x, y, z, there was a variables and constants terms being at
the start to the alphabet, a, b and c. So, when we did the
quadratic formula in school we learn how to solve ax squared plus bx plus c equals zero. That convention is from Descartes. Now the idea of viewing curves as sort of dynamic things that
were expressing relationships between, your y and your
actual other variables, and somehow they're moving and then you could have
a rate of change of them. That is what laid the groundwork really for calculus to come into existence. Now I'm not going to talk
about calculus really. I'll give you one minute just to show you some notation and Newton
and leibniz of course, battle it out over who had priority, who had come up with the idea first. The notation that comes from
sort of Newtonian outlook is not in my view, as good as
the notation that had comes from a leibniz outlook. So if you're trying to do what Newton might have written. So this dot means you differentiate once. Don't worry if you don't know what that is we moving on in a moment. Two dots mean you do it twice and so if you want to do it eight times. You've just have to have
horrible panoply of dots above your x, and you have
to count how many there are and you can miss some with
an ink spot or something. So this is becomes unwieldy. Whereas this location
that we sort of developed from what Leibniz was doing, where it looks like fractions
that actually it fits in well with how differentiation
and it's reverse integration seem to work. So that's good. And you can then see immediately how many times you're doing this process. So there's an eight, do it eight times. That's much more easy to see
the, not that the eight dots and you can also give yourself
a latitude to do it n times if you want to. So many people would take the view and I'm one of them that
this kind of Leibniz view notation is superior. Although there are places for the other. So, we've seen how the
development of certain notations can allow new ideas to emerge and they're helpful sort of
a seed bed for new concepts. Once you have the concept of x to the n, well, n has been a positive
whole number, positive integer, x, x squared, and so
on, but could you have x to a non whole number of, how could you have x to the a
half maybe, could you raise it to some other pal that
isn't a whole number? Well, if you remember that, take an x, the square root of x times
the square root of x equals x. So square root of x squared equals x and since we know about the
addition rule of exponents maybe we start to use the
notation x to the power a half for the square root. So you can make these kinds
of, you can generalize a bit and start to use this
notation more broadly. And of course, understanding that multiplying, what
you do add the exponents. That's a way of turning
multiplication into addition and this is really what underlies, although Napier wouldn't
have thought of it like this, it kind of what underlies
eventual development of things like logarithms. So, this innocent bit of initial progress that
makes it a bit easier but then leads to whole
new avenues of thought. And actually, there's a quote. I'm going to move you from
Ernst Mach in 1895, he said, "The student of mathematics
often finds it hard to throw off the uncomfortable
feeling that is science in the person of his pencil,
surpasses human intelligence." In other words, you come up
with some bit of notation and then, or some new bit of
exploration, and then you find that it's leading you off
into all sorts of new places. It goes much further than you thought linked with other bits of mathematics. So this is kind of an often things like Descartes and wonderful
thing about mathematics. Okay, now, I want to give you a little
further anecdote about how the language we use for
notation and symbols can tell us about where that
mathematics has come from. So we've seen with our
numbers, they started in India. Another thing that started in
India was, well, trigonometry. So where does this word sign come from? It's a lovely story and this archer is going
to tell us the story. So the Indian mathematicians
who were writing in Sanskrit they wanted to explain
what they meant by sine. So we know sine in a triangle
it's opposite over hypotenuse right sine of that angle. So if you imagine here, this guy imagine this being an arc of a circle. It's not quite, but it's pretty much and there's his bow string there. And then when he pulls
back to fire an arrow, we get a drawing a bit like this. And this angle here, what's
the sine of that angle? It's the opposite over the hypotenuse. So the opposite bit is
this bow string, right? All, you know half the bow string. And so if you have that diagram
in mind, it's really natural that the word for sign was
the word for string jiva in Sanskrit my pronunciation it's not guaranteed to be correct. So this is how it first started. And then just as with the numbers, these work found their
way to the Arab world and were translated but there was no word for sine in Arabic. And so what happened was
they used the word jiva which didn't have a meaning in Arabic, in its own right, it now It
now just came to me in sine and it became jiva in Arabic. So that isn't a word already Arabic and so then what happened was
it was a bit sort of corrupted over time into something that was a word and that word was jayb which means cavity. Go so then fast forward a
little while the Arab texts come to the Western world and
get translated into Latin. And now the word cavity
is translated into Latin. What does that? In Latin it's sinus? Like we have sinuses, you know now as sculpts, we have sinus and so then in English, that sine, right? So this is this word and you can see originally
the lovely bowstring analogy and it's traveled across all
these languages to get to us which is just a nice story there of some of these tricky metric terms. Okay, we going to, we've
got about 10 minutes left. I want to briefly mention
Robert Recorde again 'cause I said I would, he's a great guy. He's a Tudor working for
the Tudors born about 1510. He was a big proponent of explaining things clearly
so people can understand them. And he wrote in English he wrote in the vernacular,
he didn't write in Latin. He wrote several books. The one we want to talk about is called " The Whetstone of Witte" it's out of shelf on your
wits by doing algebra, which is always a good plan but he had views about the importance of explaining things
so that people who were in charge of making the rules
actually have a basic grasp and understanding of
science and mathematics which we probably all can agree with still being a challenge. So he said," Oh, in how miserable cases that realme where the
ministers and interpreters of the laws are destitute
of all good sciences which are the keys of the laws. How can they either make
good laws or maintain them? That lack the true knowledge
whereby to judge them?" Yes, indeed. Sorry he writing his book,
"The Whetstone of Witte," he was explaining in
English for the first time, really a lot of these rules about the Cossic art algebra,
how to take square roots. How do we understand about powers and things like this? He bought a lot of words into English. So he was reading books like by the books by Stifel and Rudolph that
we mentioned, he read French, Latin, German, Italian,
Greek, bit of Arabic. He was very widely read and he brought words that he
thought chose the best words from all these things, he
brought them into English. And if there wasn't a
good word, he made a word. So he gave us the words
binomial and commensurable for example, I can't
quite myself mentioning some words that didn't stick. So what do you think these words mean? Well, absurde numbers. That's what he called negative numbers. Still the old prejudice
they're absurde, gemowe lines. So that has the same
root as a gemo or gemini. Twin lines that means parallel lines and his decision about
calling the equal sign as a pair of parallel lines, he said, "These are because of no
two things can be more equal than two parallel lines. So that was why he, when he wrote in "The Whetstone of Witte" I'm going to use this sign. This is why he said he'd chosen it. Nouelike triangles, we call them scaling. Now none of the sides
are like, and cinkangles I like this one. Cinkangles that's pentagons
for obvious reasons, I guess. He also had a brilliant bit of notation that I want to tell you because it helps the record in the Oxford English
dictionary for the longest, no, for the word in English
with the most dense in it. So here we go. So we had zenzus earlier,
meaning the number. So then zenzike, it means the square and zenzizenzike, obviously,
then means the fourth power, the square of the square. So then of course we
have zenzizenzizenzike, which is the eighth power and take a big run up,
zenzizenzizenzizenzike which means the 16th power. So those are lovely and I think we should bring them back into the classroom immediately and find these
zenzizenzizenzike of things. So those are some words that
perhaps didn't catch up. But of course his legacy
is the equal sign. That's what we remember him mostly for, this young man from 10B. And I say, this is the world's
oldest equation, right? Why do I say that? I mean, of course people doing algebra I'm not saying that, people weren't solving things,
but what's an equation. It's something equals something. Can't do that if you don't have the equal sign. So this equation, this was the first use of it in his book, "The
Whetstone of Witte," 1557. Here it is. So if you can solve this, you can tell all your friends and family that you have solved the
world's oldest equation. This symbol is the unknown with now, y, x. This just means the number, so a constant. So translating, we'd now write this as 14x plus 15 equals 71. So I'm sure you have solved
it in your head already. If, for just bring the 15 over to the right hand side, take it away. We get 14x equals 56. And so x is four which I liked because four
is my favorite number. So this week I'm saying this
is the world oldest equation. So that's real quick record. So I wanted to finish this part before I spend like five
minutes, right at the end with just a little bit of
mathematical fun with language I wanted to mention just
a couple more symbols you might be interested in and
to make the point it changes and often takes time
for a choice to be made, a convention to be established
as to what our symbol is. So these symbols here, these
five symbols were all used before 1930 to represent
a particular number set. And that number set is what we now call Z is the integers. So the whole numbers that Z for Zahlen which is the German for number
that was chosen by a group of French mathematicians in
about 1940, they got together and decided they wanted
to have a series of books that were going to describe
the elements of mathematics and that's what the series was called. And they wrote individually anonymously, but under the pseudonym, Bourbaki 'cause he's a very influential
books and they said, "It's high time symbol for
this, that we all agree." And this is the one they
chose and this is what stuck. So again, only in 1940 it feels like it's been around
forever, but not actually within the last century,
it was finally chosen. Another thing is I want to
mention this wonderful equation. So this is often cited as the most beautiful
equation in mathematics. It combines the very basic
building blocks of numbers, zero and one, it combines pi, very important mathematical constant, then e, another very important
mathematical constant, and i, the square to minus one with which you can express a solution of any polynomial equations. There's no reason to imagine
they will be linked together and yet they are with
this beautiful equations. This is a fantastic equation. So when did we get these
symbols, e, i and pi? So well, you're on Johann
Christoph Sturm in 1689 was the first to use a letter for one on mathematical constant. Now we used e, but the one
in mathematical constant he was using a letter for,
with what we now call pi. So I love that because the first letter used
for pi, it was actually e. And what was the first
letter used for e, it was b. So if we were to follow
historical precedent, this equation would be written b to the ie plus one equals zero which just looks very
strange to our modern eye. So we don't always use the first
bit of notation for things. Just to go back to this, what we now use. I want you to sort of point out that this is a beautiful equation. It's belongs to the whole world. If you were to try and pick out which nationalities like its DNA. Well, e and i, they are due to Euler. He was the first to use those
letters for those quantities, equals that's Robert Recorde, he's Welsh. Pi was first used by another
Welshman, William Jones in 1706, I think it's the first letter
of the word periphery. And plus, we've seen
that a German origination and then zero and one, Indian. So you could, you could
say, if this is 28% Swiss and 28% Welsh, 28% Indian and 14% German. But of course we don't do that, mathematics belongs to us all and this equation and is 100% it's belongs to mathematics
and to everybody. So, we obviously can't
talk about every symbol but I hope I've shown you a few of these. I just want to spend a few minutes now at the end being a little bit frivolous, it's my last lecture of the year. I want to you some homework
to keep you going until, my next lecture starts
up again in October. So I'm going to tell you what
the incomprehensibility graph is kind of thing that gets
mathematicians, a bad name but it's a bit of fun. Which languages are the hardest? Very important question if
you're learning languages. So in English, we have an expression. If we don't understand something, it's Greek to me, it
goes back to Shakespeare. But if you're French,
things aren't Greek to you. They're Hebrew to you, that's first quote, I think, in Le Malie. So obviously, this means that
Greek is harder than English but Hebrew is harder than French and you can start making a
graph of this, a directed graph. So you have an arrow from a language to the thing that it doesn't understand. Okay, so we can start doing this thing and we can try and find out
what are the hardest languages or is there a unique chorus language? So we've got our English and French going into Greek and Hebrew. Stumbling block, potentially English. We can also say we don't
understand something it's double Dutch, right? We don't understand it. So we also ought to have an arrow to Dutch but turns out not understanding dutch is really just a stepping stone on the way to not understanding Greek because the Dutch don't understand Latin. And if you're in Latin you can say, it can't be read it's Greek. So actually we end up at Greek both ways. What about Hebrew? Well, both Greek and Hebrew
don't understand Chinese. So we get to Chinese. And what do, does Chinese not understand? There's a lovely
expression, heavenly script. It's heavenly script to me. So the writing of heaven is the ultimate
incomprehensible language. Now, I've put something in
the transcript about this. There's are people who have
drawn much bigger graphs and taken this much further. And you can explore this
idea further if you wish but it's quite a fun idea. Not really math. It's sort of looks a bit mathsy. Another mathsy thing which I've called numeral linguistics. What's the next number in this sequence? 84, 11, six, three, five. If you're watching this
lecture on recording you should pause this now
and do your due diligence and try and work out
what the next number is. For us, here, it will help you to see the words written down in English. So let's look at this 84
E-I-G-H-T-Y dash F-O-U-R, 11 characters needed, 11. How many letters in 11, six. How many lessons in six, three. How many letters in three, five. So the next number is how
many letters in five, four. How many letters in four? Oh, it's four. That's interesting. It's four again, my favorite
number is four course. It was fate. Let's try starting at
different points there. You can start with 84, you get to four and then you're in a loop. If you start anywhere else at all any other number, you end up at four and then you're stuck, you stay there. So we can draw another kind
of graph, the graph of English following this pattern and
a function of a number. It's the number of characters required. I'm counting hyphens,
I don't count spaces. That's my rule, you can make
your own rule if you prefer. You end up in the same place,
you end up at four in English. Well, there are lots of languages. This is the graph, English. Everything is four the best number. In Italian, all roads lead to tre. The word for three has three letters. In Danish, there are
three numbers like this. There are three fixed points, two, three and four all have that
respective number of letters. Now, look at French. This is brilliant. So, un has deux letters
and deux has quatre letters and quatre has six letters and six has trois letters
and trois has cinq letters and cinq, five has quatre letters. We've got a cycle. It's fantastic. So in French, there
aren't any fixed points. There's just a cycle. Now so your homework is going
to be take some languages you like, and that you know or even you don't know, what happens in your favorite language. Now, point of caution. How far up do you have to go before you've captured all the behavior? Can you, for example, go to one of the many websites that
tell you all the numbers in different languages from one to 10 is it safe to stop at 10? Glad you asked. No, it is not, (speaks
foreign language) in Russian that's the Russian for 11 has 11 letters. You can't stop at 10. So be careful, you might
need to go a bit higher. So here's your homework. Keep you going. What's a safe testing threshold? I've been going up to 100 in the languages I've looked at. That seems to be pretty safe book. What's a good place to stop? What's the highest fixed point? So four the fixed point in English, it always will get back to four. What's the highest one you
can find in any language. Is it 11? I'm really unsure whether
anyone is going to be able to tweet me
saying they can beat Zulu because in Zulu 27, as of course no is amashumi amabili nesikhombisa
that has 27 letters. Can you beat that? I don't know. Maybe you can. And then can you find a longer
cycle than the one we found in French that has four in it? So that's a little bit of
fun to keep you occupied until my next lecture in October, which is on Mathematics and Art. And then my second year
of Gresham lectures the first few lectures are
going to be on the many many links between mathematics and art. You can watch all the past lectures. You can sign up for upcoming
ones at the Gresham website or if you follow me at Gresham on Twitter of course we'll let you
know about upcoming events but I will stop there. Thank you very much for listening. (audience applause) - Professor Hart, thank you very much for a really, really interesting lecture. We've got a few questions
from the online audience. I'm not going to be able to get to all the questions, but we will try. We'll try to do as many as possible. Okay, I read somewhere
that when we were trading with other countries in the world by sale, mathematical puzzles would be exchanged with various mathematicians because even though they couldn't
speak the other's language they could all understand
the maths symbols. Is that true? - Well, so I think we've
seen that there have been a lot of different symbols used over time. It had helped of course,
that for a long time learned people would
speak or write in Latin. So that helped people of different nationalities
communicate with each other certainly within Europe. There are examples of puzzles and games that you see in different places even if we can't necessarily trace the pattern don't have
records of them traveling but there there's a
puzzle about 17 camels, for example, that crops
up all over South Asia and this has clearly been
passed on between people. So I think that has
happened, but probably, various languages that
have been the lingua franca various times I've helped with that, yeah. - Okay, we do time in base 60. Is that related to the Babylonian system? - Exactly absolutely is. Yes, so the Babylonian system with base 60 it wasn't just the counting system their. Their currency was measured in sixties and it came from the kind
of 360 days ish in a year. Now, the Babylonians are
much better astronomers than anyone who would think that they really were sit 360 days. But what they said was they
kind of had been 360 days and then in the ancient world,
there were various myths about how those extra few days came in. So the ancient Egyptians had
a myth that the sky goodness won those five extra days in a
bet with the moon god Khonsu. and so there were five extra days but they somehow were
extra intercalary days and were not part of
the months of 30 days. the 16th and the 360 days in a year, which of course also gave rise to our 360 degrees in a
circle, which we still have and divided into each degree
into minutes and seconds. 60 is a great number because you can divide
it up in so many ways. It's the smallest number that is divisible by one, two, three, four, five, and six. So that makes fractions much easier. You're much more likely to
find a fraction that doesn't that exists then in our base 10 where even a third is
very difficult to write, as a decimal or something. So 60 is a good number but it does go back to
the Babylonians, yeah. - Great, I've got another
person who's saying numbers of quantities are our first
record of numerical symbols. The purpose being to levy
city tax on goods for sale, is this true or false? - Oh, golly. The first ever record. I, the honest answer is I don't know the first ever we probably won't know
because things have been dug out of the ground all the
time, but certainly I have read that even the earliest
writing at all is numbers. So these things are
there right from the off, even if it's just making
marks very simply crudely one two, three, four, five, just marks in a piece of clay, they
are really with us, right? From the beginning of any
kind of writing, yeah. - You spoke at the beginning
of the lecture about Voyager, were the plaques on the
NASA launches informs other than audible/visible to humans? - So they were put onto a record. And so you can put
information onto a record, if you have a, I guess the most recent bit of technology might be the
CD that still does this. And there were instructions
put onto that golden record as to how it was to be used and the very first
picture was just a circle. And that would allow
you to realize you had you were doing it right. Because of course, you
couldn't, a random bit of thing is not going to be a circle. So every civilization
would recognize that. So their instructions on how to read it but it was a record. Yeah, so it had music and pictures. - Okay, this is a big question. (both laughing) Are mathematical symbols free of white colonialism influence? Hopefully yes. But in case they are not what would be the
alternative way forward.? - Yeah, that's a really great question. I think, usually over time we have, converged on eventually the best, the thing that's convenient, the things that most powerful,
that exponent notation that makes sense and it
works and it can take time. And I think what can happen
is that things are misnamed. And so you saw, the Arabic numerals were
really Indian numerals. I've seen other examples where, so I remember reading about
this Nigerian mathematician in the 18th century Ibn Muhammad, I think his name was, and he came up with this method for constructing magic squares but we don't know it by that name. It is now known as you know, and I'm I don't like this really, but it I read about it as being
called the Siamese method. So for an ancient old word Siam, for one day, it's Thailand,
isn't it Siam now, but this was, it was because this method was not really paid any attention to until this French diplomat
who happened to be in Siam found out about it and he wrote it down. and so then it's called
the Siamese method, right? But those kinds of things where things are mislabeled
according to the, like dead white guy who wrote them down because he got them from someone else, we got to be careful about those things. So I think ultimately probably
the best symbols went out but we have to be careful about how we who we credit them to. and that's why it's
really important to study the history of these things. - There's another international question. How did South American
civilizations count? Is this the topic of another lecture? - Yeah, so yeah, I did not talk about, the the Chinese methods, of Japanese methods there
are, so yeah, that there I have got quite a few
links in the transcript and there are some good
books about this topic, but yeah, there are... I don't want to make a mistake but I believe there was a
base 20 system that was used. So not based 60, not
based 10, but based 20. and that's a very interesting and quite sophisticated counting method or number system, but yeah. Share crunch of time. I thought I had to refine
it a little bit, but yeah all over the world, they
were counting systems. - Here is another interesting one. Why was our old money
system not based on 10, but on 12 pennies and the shilling and 20 shillings in the pounds? - Yeah, so I think the 12-ness, it's the same thing as for sixties, it's much easier to divide 12. 12 got a lot of factors. You couldn't have half
and a quarter and a third, we have had threatening
bits and things like this. So you could divide up
a shilling into much many more things easily than you can divide 10
pence up into a five pence. So when we were doing all
our arithmetic in our heads that make life much easier as it became less important
to be able to do everything in your head, when we had cash registers and things that motivation for dividing
things into 12 lessened. So yeah, that's probably why we switched decimal eventually, yeah. - It's really interesting. And another money question, which culture assigned money symbols, the Euro the pound sign,
the dollar sign to numbers? Where did that begin? - So having those signs, I
mean, they will have come in I guess, if you're just,
within your own culture you know what money
you're using, but as soon as you start trading with
other cultures, you need to say what am I measuring? So I know it was a long
discussion on the dollar sign in Florian Cajori's book
about mathematical symbols The pound sign comes
from, so the Latin word for pound Libra, right, is an L. So it's a sort of
deformed L pound sign Euro obviously to do with Europe so it's an E. So these things do develop
and they need to develop when you've got, imagine being in a market and someone's writing how
much things are in dollars or how much they are in pounds,
how much they own francs and lira and all of those kinds of things. You need a symbol to say
what money you're using. - Professor Hart thank you so much and thank you for your generosity in addressing the questions. Thank you to our audience for attending and I would like to encourage
you to come back tonight and listen to our professor of the environment who
will be giving a lecture at 6:00 PM called " A just and rights-based
framework for nature." So we hope to see you later on. Thanks very much.
Sorry automod. The actual math discussion in this video is elementary. What might interest this subreddit is the connection between the what and the how of communicating a concept. If a shorthand is made for describing the measure of a square or cube, it might lead to exponentiation. Or is that putting the cart before the horse?
Excellent. I love this maths history.