All language subtitles for Magic Numbers Mysterious World of Maths 1of3 720p HDTV x264 AAC MVGroup org

There is a mystery
at the heart of our universe.

A puzzle that, so far,
no-one has been able to solve.

I can't, it's too weird!
Welcome to my world!

If we can solve this mystery,
it will have profound consequences

for all of us.

That mystery is why mathematical
rules and patterns

seem to infiltrate pretty much
everything in the world around us.

Many people have, in fact,
described maths

as the underlying language
of the universe.

But how did it get there?

Even after thousands of years,
this question causes controversy.

We still can't agree on what maths
actually is, or where it comes from.

Is it something that's
invented like a language?

Or is it something
that we've merely discovered?

I think discovered.

Invented.
It's both.

I have no idea.

Oh, my God!

Why does any of this matter?

Well, maths underpins
just about everything

in our modern world,
from computers and mobile phones

to our understanding of human biology

and our place in the universe.

My name is Hannah Fry,
and I'm a mathematician.

In this series, I will explore
how the greatest thinkers in history

have tried to explain the origins
of maths' extraordinary power.

You ruined his equation!

I'm going to look at how,
in ancient times,

our ancestors thought maths
was a gift from the gods.

How, in the 17th and 18th centuries,

we invented new mathematical systems

and used them
to create the scientific
and industrial revolutions.

And I'll reveal how,
in the 20th and 21st centuries,

radical new theories are forcing us
to question, once again,

everything we thought we knew
about maths and the universe.

The unexpected should be expected,
because why would reality down there

bear any resemblance
to reality up here?

In this episode, I go back
to the time of the Ancient Greeks

to find out where our fascination
with numbers started...

# Laaaaaa! #

You know I think I can hear
the neighbourhood cats screeching.

..and reveal why we're now looking
for maths deep inside our brains.

Our world is full of maths,
often in unusual places

like this roller-coaster.

The thrill and excitement
of this ride wouldn't be possible

without physics and engineering.

And at the heart of all of
that science...

..is mathematics. Oh, my God!

SHE SCREAMS

It's sobering to think
how much we entrust
our personal safety to maths...

..without even realising it.

My rush of adrenaline
relies on someone's calculations

of kinetic energy, momentum,
tensile strengths,

coefficients of friction,
and much, much more.

Oh, my God!

What do you make me do?!

To put it bluntly, the modern world
wouldn't exist without mathematics.

It is hiding behind almost everything
that's around us

and subtly influencing
almost everything that we now do.

And, yet, it's invisible.

It's intangible.

So, where does mathematics comes
from? Where do numbers live?

It's a question that goes
to the very heart of our world.

We often think about numbers
as something tied to objects,

like the number of fingers
on one hand or the number of petals
on a flower.

This flower has got eight petals.

If I take three away,
then it will be left with just five.

And it will look a lot less pretty.

The petals are gone,
but the number three still exists.

The idea of three,
or any other number for that matter,

is still out there, even if
we destroy the physical object.

But you can't say
that about everything.

If pencils had never been invented,
then the idea of a pencil
wouldn't exist.

But the idea of numbers
would still exist.

In every culture around the world,
we all agree on what the concept
of fourness is like.

And it doesn't matter
whether it's called four,

quatre, vier,
or even what the symbol looks like.

With numbers, I can destroy
the physical object,

burn it to a crisp, but I can't
destroy the idea of numbers.

So, here's the question
I want to answer,

is it invented or discovered?

Is there some magical
parallel world somewhere

where all mathematics lives? A place
where you have fundamental truths

that help us to understand
rules of science,

helping us put man on the moon
and to study the tiniest particles

of the universe?
Or, is maths all in our minds?

Is it just a figment
of our imagination and intellect?

Whether maths is
invented or discovered

is something we can't agree on.

It's just too extraordinary
to think that

the mathematical truths
in everything

are a sort of product entirely

of our conventions
in the human mind.

That's... I don't think
we're that inventive.

It sometimes feels like
mathematics is discovered,

especially when the work
is going really well and it feels

like the equations are
driving you forwards.

But then you take a step back,
and you realise it's the human brain
that's imposing theseideas,

these patterns, on the world
and from that perspective,

it feels like mathematics
is something that comes from us.

The number five is called fem
in Swedish, my mother tongue.

That part we invent,
the baggage, the description,

the language of mathematics.

But structure itself,
like the number five and the fact

that it's two plus three,
that's the part that we discover.

There's virtually no
part of our existence

that isn't touched by maths.

So, if it is discovered,
part of the fabric of the universe,

how can we unlock its secrets?

And if it's invented
and all in our heads,

how far can our inventive brains
take us?

I want to start with
the discovered camp, those who say,

"Maths is all around us.
You just need to know where to look."

Of all of the structures
that you get in nature,

I think one of the most
beautiful is the nautilus shell.

There's a little creature
that lives inside here
and creates all these shapes,

and it hops from one chamber
to the next as it grows.

And this shell is just
incredibly intricate,

and you might wonder
how something so small can create

something quite so remarkable.

But, actually,
there is a hidden pattern in here

that you can start to see
when you measure these chambers.

So, that one is coming out at...

..14.5 mm.

And this one...

..on the same axis, is...

..46.7.

I'm measuring how wide the shell
would have been as the nautilus grew.

I pick an angle and measure
the inner chamber,

and then a second measurement
to the outer rim.

99.5.

I do this three times
for three different angles

until I have three sets of numbers.

When you look at them, I mean,
they look pretty random, right?

It looks like there's no connection
between them at all.

But looks can be deceptive,

because if you take
each of these pairs of numbers

and divide one by the other, a very
clear pattern starts to emerge.

So here, if we do this
number divided by this,

we get 3.22.

This number divided
by this one gives 3.25,

I think. Sorry, my mental
arithmetic isn't great!

Eh, and this number divided
by this number

then gives 3.24.

And suddenly, the same
number starts to appear,

around about 3.2-ish.

It doesn't matter where
on the shell you measure,

the ratio of the width
of these chambers ends up

being pretty much constant
throughout the shell.

I've got it right
to one decimal place. Not bad.

That'll be down to my measuring
skills rather than the nautilus.

What all of this means
is that the nautilus

is growing its shell
at a constant rate.

So every time it does a complete
turn, it ends up sitting

in a chamber that is around
about 3.2 times the width

of the turn before.

And by repeating this very, very
simple mathematical rule,

it can create this beautifully
intricate spiralled shell.

Clever old nautilus!

The nautilus isn't the only living
thing that has a mathematical
pattern hidden inside it.

If you've ever counted
the petals on a flower,

you might have noticed
something unusual.

Some have three petals.

Some five.

Some eight.

Some 13.

But rarely any of
the numbers in between.

These numbers crop up
time and time again.

They seem random,
but they're all part of what's called

the Fibonacci sequence.

You start with the numbers one
and one, and from that point,

you keep adding up
the last two numbers.

So one and one is two.

One and two is three.

Two and three is five, and so on.

When looking at the number
of petals in a flower,

these numbers from the Fibonacci
sequence keep appearing,

but that's just the start.

If you look at the head
of a sunflower, you'll see

the seeds are arranged
in a spiralling pattern.

Count the number of
spirals in one direction,

and you will often find
a Fibonacci number.

Then, if you count the spirals
going in the opposite direction,

you'll hit upon an adjacent
Fibonacci number.

Why do plants do this?

Well, it turns out that this is
the best way for the flower

to space out its seeds
so they don't get damaged.

We find these spirals
so intriguing,

we've worked hard
to unlock their secrets.

We've gotten very good at copying
the patterns that we find in nature

and using them to create
things of great beauty,

like this majestic staircase.

Simple, glorious mathematical rules
found hidden in nature,

doesn't seem to me
like a coincidence.

These mathematical patterns,
once you spot them,

do feel discovered.

It's as if the maths
is already out there

just waiting for you to find it.

This fascination for finding
hidden mathematical patterns

is nothing new.

Go back over 2,000 years to the time
of the Ancient Greeks,

and you will find the philosopher
Pythagoras and his followers

were just as enthralled
by the patterns they discovered.

The Pythagoreans were
obsessed with numbers.

They were a people who believed
numbers were a gift from God,

and part of their fascination
might have been thanks

to their experiments with music.

The Pythagoreans discovered patterns
that linked the sound

of beautiful music
to the length of a vibrating string.

This, they believed,
was no accident, but a window

into gods' worlds that had been
gifted to the Pythagoreans.

Mathematician and musician
Ben Sparks is fascinated

by this age-old relationship
between music and maths.

THEY APPLAUD
Well done!

Lovely. Thank you for joining us,
Hannah, with your beautiful cello,
there.

OK, Ben, you are, eh, you're
going to have to explain this to me.

Where does the maths come in
in making this instrument sound nice?

The wobbling is what's giving you
the sound,

and if you make the string wobble,
you hear a sound. So, maybe
you could play your D string.

SHE PLAYS

Oh, that sounds lovely!
It sounds lovely, doesn't it?

What they also noticed is another
note really related to that,

which, if you make it wobble
twice as fast, and to do that,

you can make the string
half the length. OK.

So, you're putting your finger here -

well, I guess it pretty much is
actually halfway along. OK.

SHE PLAYS NOTES

What's weird about these two notes
is they sound kind of the same,

but they're definitely different,
and this is what the Greeks noticed.

We call it an octave.

But if you play them together,
does it sound nice?

SHE PLAYS CHORD

Hm! It was delightfully pleasant.

In the octave, the length
of the vibrating string

creates a relationship,
or ratio, of two to one.

So, that's if you chop it in half.

Are there other fractions
that make it sound nice?

Exactly what the Greeks thought.

"Well, can we find other notes

"that sound even nicer together,
more interesting together?"

Can you play us..? This is what
they called a perfect fifth.

SHE PLAYS NOTE

What happens when you play
those two together, then?

SHE PLAYS TWO NOTES

Very pleasant. Felt like you were
about to launch into a jig there!

In a perfect fifth, the ratio of the
vibrating string is three to two.

The high note is two thirds of
the length of the low note.

What happens when you play
a note that isn't one

of these neat fractions?

When notes aren't in these
nice simple ratios,

we tend to notice it even if we're
not aware of mathematics.

Could you play us a really
horrible harmony together -

maybe like a semitone apart?

SHE PLAYS DISCORDANT NOTES

When the strings are not
in a simple ratio,

the harmony sounds
distinctly unpleasant.

The Greeks were obsessed
with having simple ratios

describing the notes,
so they get nice harmonious noises.

How does this work for other
instruments? This is very clear,

you've got this sort of string here.

But what about, I don't know,
the human voice?

Every noise you ever hear
is things wobbling

somehow, whether it's your vocal
cords in there or a string or...

Not my vocal cords? Well...really?

Have you never used your
vocal cords for a bit of music?

Can we try? Oh, no!

I'm such a bad singer.
Please, don't make me do this.

OK...
Have you got your earplugs in?

Let's try. Can you pitch us a note?
I mean, something nice and low.

If you just do it to "la"
then I've got a chance to copy you.
OK. All right. OK, OK.

SHE CLEARS THROAT

# Laaaaaa! #

Just like the cello,
it's the length of mine and Ben's

vocal cords that's changing
the pitch of these notes.

So, that was me singing a perfect
fifth, a Chariots Of Fire note.

# Laaaa, laaaaa! #

You know, I think I can hear
the neighbourhood cats screeching.

So, I think that's enough of that.

These patterns convinced the Ancient
Greeks that they'd been gifted

a glimpse into this godly realm.

Why else would these patterns exist?

Pythagoras and his followers
were in little doubt that maths

was just as real as the music was,
and it was even neater

and more elegant that anything
the human mind could conceive of.

The Pythagoreans were by no means
the first people to use

some form of maths.

There's some evidence
that marks found cut into bones

from the Upper Palaeolithic era
37,000 years ago,

were tally marks
used for counting.

But it was the Pythagoreans who were
the first to look for patterns.

It does feel to me as if maths
is all around us and something

we discover, a fundamental part
of the world we live in.

And yet, somehow, very strangely
separate from it.

Trying to make sense
of this apparent paradox

is at the heart of this battle
about where maths really lives.

The philosopher Plato is one
of the most important figures

of the Ancient Greek world.

But what he said about the origins
of maths is still the basis

for what many mathematicians
believe today.

He was fascinated by the geometric
shapes that could be produced

by following
the rules of mathematics,

rules that he believed came from God.

I'm going to try and draw a circle
really, really carefully.

It takes me back to my school days,
this.

Now, I'm not doing bad.

That's pretty good.
But if you look really closely,

it's just not quite perfect,
the circle.

But I'm not going to beat myself
up about it because even if

I had access to the most accurate
computer in the world,

the circle that it would draw
still wouldn't be perfect.

Zoom in close enough,
and any physical circle

will have bumps and imperfections.

That's because, according to Plato,

flawless circles don't
exist in the real world.

He believed the perfect circle lives

in a divine world
of perfect shapes,

a kind of mathematical heaven
where all of maths can be found,

but only if you're a true believer.

He was convinced that everything
in the cosmos could be represented

by five solid objects
known as the Platonic solids.

So, the earth was
the rock solid cube.

Fire was the
very pointy tetrahedron.

And then, with eight triangular
sides, air was the octahedron,

while the icosahedron, with its 20
triangular sides, represented water.

The last Platonic solid,
the dodecahedron,

this one was supposed to encapsulate
the entire universe.

It's the whole universe sitting
in your hands there.

It's kind of a neat idea.

There's something special
about the Platonic solids.

They're the only objects
where every side is the same shape,

and there are only five.

Try as you might,
you will never find another object

with these unique
mathematical qualities.

All of these shapes,
Plato believed, existed

in a world of perfect shapes

beyond the reach of us mere mortals,

a place we call the Platonic world.

I know that these ideas might seem
they're a bit bonkers,

but there are actually quite a few
people who believe them,

and those people come
across as though they're sane.

Ooh!

My third most favourite
mathematical structure,

the octahedron.

A-ha!
The Platonic solids, I presume?

It's a dodecahedron!
I love dodecahedra.

I have a misspent youth
making models of polyhedra.

Oh, my goodness!

These are the Platonic solids!

Oh, guys!

OK. Very beautiful.

You know, at 67, this is Christmas.

Can I keep these two, please?

These Platonic solids, to me,
are a great example

of how mathematics is discovered
rather than invented.

When Ancient Greeks discovered
that this one existed,

they were free to invent
the name of it.

They called it the dodecahedron.

But the pure dodecahedron itself,

it was always out there
to be discovered.

I have this kind of Platonic view
that there are triangles out there.

There are numbers, there are these
circles I'm seeking to understand.

So, for me, they feel
like quite tangible things.

They're all part of this
mathematical landscape
that I'm exploring.

But not everyone believes
in this Platonic world

of mathematical truths.

I think that the Platonic world

is in the human head.

It's a figment of our imaginations.

I get that there are people
who really buy into this other

realm of reality,

and, especially,
if your days and nights are spent

thinking about it, investigating,
researching this realm.

That doesn't mean that it's real.

Plato would have strongly disagreed.

He encouraged us to believe
in this other world

where all of maths could be found,
and not to be fooled into thinking

the world around us is all there is.

What we perceive as reality,
he cautioned,

is no more than shadows cast on the
walls of a cave.

Plato had a very lively
and quite dark imagination.

of a group of humans
locked in a cave.

These people would have been
imprisoned since childhood,

and they were shackled by their necks
and their legs, and trapped,

staring at a blank wall
directly in front of them.

In his mind's eye,
Plato pictured a fire

burning high above
the prisoners' heads.

But they have no idea it's there.

On top of the wall is a path
along which all manner of people

and objects are travelling,
but the only thing

the prisoners can see of them

is the shadows
they cast down the wall.

Those shadows are
the prisoners' reality.

According to Plato,
we are no different to the prisoners

in the cave who mistake
the shadows for reality.

If Plato is right,
what does this mean for you and me?

Is what we think of as reality
and maths just an illusion?

Are we living in Plato's cave, erm,

and just see the shadows?

It is not impossible
that that is the case.

You know, we are maybe just all...
We are just some simulation

in some world of some
more intelligent being.

This is all possible.

I mean, if you think that there's
some world of mathematical objects.

It's different from ours. It's not
the physical world we live in,

But that doesn't make the physical
world any less real.

So, I don't think there's anything
to me in the philosophy of maths

that would force you to think
that our world is an illusion
of any kind.

Our senses evolved,
really, for one purpose, survival.

But survival and the true nature of
reality are two different subjects.

So, the fact that we have been able
to survive by thinking

about the world one way
does not, in any way, say

that that way of thinking
about the world is truly
what's happening out there.

Over 2,000 years ago,
Plato took the geometry of shapes

as evidence of God's influence,
ideas that were limited

to the senses and imagination.

Today, geometry is at
the cutting edge of science.

New technologies have allowed us
to look at the world

beyond our senses, and once again,
it seems the natural world

really is written
in the language of maths.

This is a model of a virus.

Straightaway, you notice
its geometric shape.

Plato would have recognised this
shape as one of the platonic solids.

If there's one person
who understands geometry,

it's a mathematician.

Reidun Twarock is a professor
of mathematics

at the University of York.

She's trying to work out
how viruses use maths

to form their geometric shapes.

If you know that,
you can find a way to stop them.

That's why Reidun and
her colleagues have designed

a computer simulation
that puts the mathematician

at the heart of the virus.

What we try to understand
is how this virus forms,

and in order to do that, we will
create the illusion of being

inside of the virus, in the position

where the genetic material
normally is.

Reidun has discovered that the virus
harnesses the power of maths

to build its shell in the quickest
and most efficient way possible.

Armed with this knowledge, she's
trying to find a way to stop viruses

such as hepatitis B,
and even the common cold,

from developing in the first place.

Once you understand
how this mechanism works,

you can turn tables on viruses
and actually prevent that process.

That is what makes this
research so exciting.

If you know the mathematics
of how the virus forms its shell,

you can work out
a way to disrupt it.

No shell, no virus, no infection.

Today, mathematicians like Reidun
are joining the front line

in the fight against disease.

Far beyond the realm of human
senses, it really does seem

like the universe
somehow knows maths.

It really is amazing how often
these patterns seem to crop up.

They're in plants,
they're in marine life,

they're even in viruses.

There really is an awful lot
that we can explore and exploit

using the mathematics that we have.

It does lend weight to the idea
that there is some natural order

underpinning the world around us.

So far, it does feel like the idea
that maths is discovered

is leading the charge.

But perhaps we've been looking for
patterns in the wrong places.

If it's all in our heads,
then the brain feels

like a good place to look.

Is there evidence in there
of maths being an invention

of the human mind?

I've got a real treat
in store for me today.

I am heading over to UCL,
the university that I work at,

where some colleagues are going to
scan my brain

and see which bits of it are working
whenever I do mathematics.

Neuroscientist Professor Fred Dick
is going to place me

inside an FMRI scanner.

If you'd just put your feet up,
please.

He'll measure my brain activity
by tracking where the blood flows

when I'm answering questions ranging
from language to maths.

OK, how was that? All right.
Excellent.

If my brain treats the mathematical
problems in the same way

as any other problem,

then it suggests there's nothing
special about maths.

It's the same as any other language.

A clue, perhaps,
that it's an invention.

I'll have ten seconds to think
about each question.

I don't need to answer out loud -

I just have to work out the answer
in my head.

OK, Hannah, how was that? Good.

Some of those questions
were really hard!

Well, we didn't want you to relax
in there, really.

I've answered all the questions
to the best of my ability.

After a few hours of processing,

Prof Sophie Scott has my results.

Is that my brain? That's your brain.

Let me make sure I understand
what I'm seeing, then. OK.

So this is like you've cut my brain
in half... Yes.

..and I've got the left-hand
side there. Is that right? Yes.

And the right-hand side is that.

So, it's like you've chopped my
head down the middle

and then split it out.

Exactly.

So what you can see here, Hannah,
is the pattern of activity

in your brain when you're hearing
straightforward language,

and here you can see,
in the left hemisphere,

very classic language areas
activated.

The bright yellow areas are where
there's increased blood flow,

an indication that the neurons
in the left-hand side of my brain

are working harder.

This is a side of the brain
that we know is linked to language.

Compare that to the right-hand side
of my brain where there's hardly

any yellow areas, which means there's
far less activity taking place.

So, can we see maths, please?

Oh, hold on.

And look at that.
This whole bit here. Yeah,

and also down at the bottom there.

This time, when I'm thinking
about maths, there ARE yellow areas

in the right-hand side of my brain.

This is very different
to the lack of activity seen

when I was thinking about language.

These scans reveal there seems
to be a place in our brains

where maths lives.

What we're definitely able to say
is this is not just the meanings
of the words that you were reading.

We're not just looking at you
thinking about the meaning of words.

You're seeing something that does
seem to be qualitatively different
for the maths.

Maths is real. Maths is real.
At least in my head. Yes.

THEY LAUGH

I tell you what really struck me
about that conversation with Sophie

just then, is that...

..it doesn't matter whether you're
doing two plus two equals four

or whether you're answering these
much higher-level maths questions.

It's the same bit of your brain
that's doing the grunt work.

It's not the same thing
that does words or language.

You're seeing these problems,
and you're manipulating them

in your mind.

Research with similar experiments
shows it's broadly the same

for all of us.

In your brain and mine,
there is a specific place

where we do maths.

But this doesn't prove that maths
is something we discover.

It could still be an invention -
just one that we learn at school.

To get to the bottom of this
question, I need some

volunteers who've never had a maths
lesson in their lives.

We need to just put one of those on
there. Very nice!

That worked very nicely, didn't it?

Yeah, you liked that, did you?

Dr Sam Wass is an experimental
psychologist

at the University of East London.

Helping him with some experiments
are six-month-old Ira

and Leo, who's just under a year.

To begin with, each child is placed
in a room where they're shown

a series of images.

Sam uses a battery of tests
to analyse how they react

to different situations.

The first experiment uses
eye-tracking technology

to see how the baby follows the
movement of a piggy puppet.

Was that good?

So, here we can see a feed out of
what the child is looking at,

and those two red dots are where the
baby is looking.

What we're presenting is a puppet
that jumps up and then disappears,

and it jumps up and disappears two
times in a row and then it stops.

We present this same sequence
again and again.

And as the baby watches
it again and again,

their looking times, the amount of
attention that they're paying

to the screen diminishes. And that
tells us that the child has learned
this sequence.

Now, instead of popping up twice,
as expected,

the puppet appears three times.

Does the child notice a difference
between the two-ness of it

popping up twice in a row and the
three-ness of it popping up
three times?

And if it does, then that tells us
that the child understands

the difference
between two and three.

These tests reveal that the child
is surprised

when the puppet appears more often.

When larger scale experiments were
carried out by researchers

in the US, the results suggested that
infants DO have a sense of quantity.

So, this research is really
important because it's suggested

that even infants as young as
five months old can do the basics

of addition and subtraction.

They know the difference between one
plus one equals two

and one plus one equals one, which
is an incorrect conclusion.

That, though, is a really, really
strong, provocative finding.

This idea that the concept of
mathematics and the basics

of mathematics rules might be
hard-wired, might be innate

our genetic code.

This research isn't conclusive,
but it does suggest we all come

preprogrammed to do maths.

Some argue that we evolved
this maths part of our brains

to discover the world
of mathematical truths.

The evidence for maths being
discovered is compelling.

We've found patterns in nature, the
latest technology has uncovered

startling patterns in viruses,

and scans reveal there's a part
of our brains where maths lives.

But this question is too important
to leave the evidence here

and move on.

If it IS discovered, if it lives
in this other world,

can we trust what it's telling us?

How do we know that our idea
of numbers is right?

How do we know someone isn't just
going to come along at some point
and say, "Well, actually,

"you've got that completely wrong,
and one plus one doesn't
equal two after all"?

How do we know that we can rely on
the maths that we take for granted?

What you need to be sure
of is your foundations.

If they're shaky, then all of your
carefully constructed ideas

come crashing down.

And there was one mathematician
who understood this only too well.

His name was Euclid.

Around 300BC in Alexandria,

he wrote one of the most famous and
important books of all time -

The Elements.

He was trying to go right back
to the beginning to find

the smallest elements on which you
can build the vast,

gigantic structure of mathematics.

If you have a little flick through,
you can see the kind of things

that Euclid was considering.

So, here it says that you can draw
a straight line between

any two points, which seems
blindingly obvious.

And here, it says that all right
angles are the same.

Now, these are quite simple concepts,
but I think they really illustrate

just how exhaustive Euclid had to be
to build the foundations

for what was to come.

He took statements like these, which
mathematicians assumed were true,

and put them to the test.

He then set out to prove a whole
host of other theories

based on these fundamental
building blocks.

This was really the first time
that someone had written down

formal proofs for mathematical
assumptions.

Now, mathematical proof isn't like
scientific proof

or proof in a court of law.

There's no room
for reasonable doubt here.

Instead, if something is true
mathematically once,

then it is true forever.

And that is why this book
is so important.

It's the reason why Euclid's Elements
is still relevant today.

Every page within it is as true
now as it ever was.

And from that point of view, it
really does feel like we're tapping
into a world that already exists.

Unless, of course, you throw
a spanner in the works,

change the language of maths,
and invent a better way

of doing things.

Suddenly, this rock solid
world of God-given truths

might fill decidedly shaky.

Uno...
ALL: Dos, tres, cuatro, cinco.

Well done. Good counting.

Muy bien!

One thing we know about languages
is that they never stand still.

They're constantly evolving
to meet the challenges

of a changing world.

47? Cuarenta y siete.

49? Cuarenta y nueve.

Muy bien!

Let's go for another tricky one.

For centuries, the language
of maths was thought

to be fixed and unchangeable.

That is, until something
was found to be missing.

It is the number zero.

What exactly is zero?

A zero means nothing.

If you've got zero flowers,
you've got no flowers.

And if you've got zero of something,
you have got nothing.

So, you can't really do anything
with the zero.

I don't really use it
when I'm counting in numbers.

Before the 7th century,
neither did anyone else.

Though people have always understood
the concept of having nothing,

the concept of zero
is relatively new.

We had numbers and could count,
but zero didn't exist.

If you think about it
for long enough,

zero is actually quite a strange
concept.

It's almost as though the absence
of anything becomes something.

Is it just a number or an idea?

And how can something with no value
have quite so much power?

It's not exactly clear who first
thought of zero.

It might have originated
in China or India.

What we do know is that zero arrived
in Europe from the Middle East

at about the same time as the
Christian Crusades

against Islam, when ideas
coming out of the Arab world

were often met with suspicion.

The West already had a numerical
system - Roman numerals.

They did the job, but were
a bit unwieldy.

For example, the number
1958 is written as

MCMLVIII.

And no matter where you place,
say, the letter C,

it will always represent
the number 100.

It was good for its time,
but times change,

and a better system was needed.

Zero was different.

Where you placed zero could change
the values of the numbers around it.

Think of the difference
between 11 and 101.

Although the concept of zero
might've been created elsewhere,

it was in India that zero started
to be accepted as a proper number.

This is a page from the Indian
Bakhshali manuscript

from around AD 225,

which shows the dots above the
characters representing zero.

This is the earliest known
use of the symbol zero

that we know today.

For almost 1,000 years, Indian
mathematicians worked happily

with zero, while their Western
counterparts ploughed on

with the Roman numerals.

That was until Italian
mathematician Fibonacci

recognised its potential.

Now, he'd been educated
in North Africa,

so he'd seen this number system
working first-hand.

Zero is a placeholder signifying
the absence of a value.

Zero is also a number
in its own right.

It allowed you to write down numbers
and manipulate them

much more quickly and easily
than Roman numerals.

Realising all this, Fibonacci
championed the new number

and brought it to the attention
of Western Europe.

Zero wasn't something
that we discovered,

so much as something
that was created as part

of a new language
to describe numbers.

That's not to say it isn't useful -

the whole of modern technology
is literally built on ones and zeros.

But, suddenly, maths feels like
something we've come up with.

Something we've invented.

We needed a more user-friendly
numerical system,

so someone came up with
the clever idea of zero.

Not a gift from the gods,
but a smart way to make

counting more convenient.

This is intriguing evidence that
maths might be invented after all,

a product of our intellect and
imagination.

Once the idea of zero
had been widely accepted,

mathematicians could relax.

All conceivable numbers
lay out on a single line

with no holes and no gaps
to speak of.

Over here you have the positive
numbers.

One sheep, two donkeys,

the kind of stuff you find
in real life.

And in the other direction,
all the negative numbers.

It's a bit trickier to imagine
what negative one sheep looks like.

SHEEP BAAS

The number line stretches out in both
directions all the way to infinity,

and zero sits proudly in the middle.

Everything was well
in Numberland... Or was it?

This is where it all starts to get
a bit strange,

because there are some numbers
that are simply weird.

There are some fundamental rules
of maths that you learned at school.

2 x 2 = 4.

3 x 3 = 9.

A positive number multiplied
by itself

equals another positive number.

Nothing controversial so far.

Curiously, a negative number
multiplied by itself

also gives a positive number.

Why is that?

Well, this is not a maths lecture,
so let's just accept it

as a fact and move on.

In fact, if you take any number and
multiply it by itself or square it,

then the answer is always going
to be positive.

If +2 squared gives me +4,

and -2 squared gives me +4, what do
I have to square to get -4?

But it's a question
without an answer.

There is no number that when
multiplied by itself

gives a negative answer.

That is unless you invent your own.

Meet "i", a number
we simply made up.

a deliberately chosen derogatory
term to scoff at its existence.

It turns out "i" is really useful,

especially when it comes to
simplifying problems with things

like electricity or wireless
technologies, things that otherwise

would seem impossible to solve.

Essentially, if you're working
with waves, you will use "i".

This imaginary number
broke all the rules.

It didn't come from this
world of ethereal numbers,

it wasn't God-given.

It was very definitely invented.

If you can have one imaginary number,

why can't you have two or three

or infinitely many of them?

Why can't you have negative
imaginary numbers, as well?

Why can't, in fact, imaginary numbers
have their very own number line?

Exactly the same as the real one,
just on a different axis.

The number line isn't
a single line at all.

Numbers are two-dimensional.

You might think this all
sounds a bit airy-fairy.

Imaginary numbers
that we just made up.

But if you've ever
flown in an aircraft,

you've already trusted your life
to these strange numbers.

AIR TRAFFIC RADIO CHATTER

At Gatwick Airport, air traffic
controllers here rely on radar

to keep everything moving
safely and quickly.

Once we get busy,
we definitely need radar.

So the busier the tower,
the busier the operation,

you need radar.

Radar works by sending out radio
waves and examining that part

of the signal that's reflected back.

The complex equations that allow us
to filter out the correct signal

from other conflicting frequencies
is heavily dependent

on imaginary numbers.

In this case, separating out moving
objects like planes from flocks

of birds or stationary objects.

Imaginary numbers are a very
efficient tool to be able

to manipulate radio waves.

Imaginary numbers are fundamental
to the operation.

Imaginary numbers allow us to track
planes in real time.

Without them, we never would've been
able to use radar in our skies.

AIR TRAFFIC RADIO CHATTER

When I started this investigation
going back to the time

of the Ancient Greeks,
it did seem like maths

could only be discovered.

There were too many coincidences,
too many mathematical patterns

popping up all over the place.

But if we can invent the rules
and create new numbers

and they work, then
perhaps I got it wrong.

Maybe maths IS invented after all.

The concept of zero or negative
numbers or complex numbers

or imaginary numbers,
they caused great consternation

to the cultures that first invented
or encountered them.

There are some conjectures that zero
came because someone constructing

noticed that as you dig a piece
of earth out to make a hole,

there's something, an indentation
left there, that should have a name.

Zero kind of maybe came
from that observation.

The power of mathematics lies
in the way its language and symbols

have allowed us
to manipulate the world.

But this was a world that followed
the rules of God and the Church.

By the 17th century,
a new breed of intellectual

was merging, not afraid
to challenge authority.

There was one man who dared
to question all of the philosophical

and scientific assumptions
that had gone before.

This was someone who was trying
to promote a new way of thinking,

using reason, experimentation
and observation.

This was the young Frenchman
called Rene Descartes.

It was while in a restless sleep
in 1619 that Descartes experienced

a series of dreams that would change
his life - and mathematics.

The first two could be better
described as nightmares.

But the third dream...

The third dream was intriguing.

As his eyes scanned the room,
he saw books on the bedroom table

that appeared and then disappeared.

He opened one book of poems
and at random caught the opening line

of one, which read,
"What road shall I pursue in life?"

Then someone appeared out of thin
air and recited another verse,

saying simply, "What is and is not?"

As with dreams, it's all
about the interpretation

you place upon them.

In Descartes's case, the effect of
these dreams was profound.

He was convinced that the dreams
were pointing him

in a single direction,

bringing together the whole of human
knowledge by the means of reason.

He was nothing if not ambitious,
but his genius led to

perhaps one of the greatest advances
ever in the field of mathematics.

As with so many brilliant ideas,
it was deceptively simple.

Let's say that I'm meeting
a friend for a coffee.

Now, I'm standing at the end
of Endsleigh Gardens,

and they are somewhere over
on Gordon Street.

It's very easy for me to work
out how to get there.

All I need to do is go on a map
and check the route.

In this case, three streets
down and one along.

It sounds like an incredibly
simple idea,

but, actually, it revolutionised
mathematics.

He showed that a pair of numbers
can uniquely determine the position

of a point in space.

It sounds trivial,
but this was just the start.

It gets more interesting
when you apply this idea to curves.

As this point moves around a circle
its coordinates change,

and we can write down an equation
that precisely and uniquely

characterises this circle.

For the first time, shapes could be
described by a formula.

By uniting the language of numbers
and equations and symbols

with shapes, Descartes was able
to expand the horizons

of mathematics,
thus laying the foundations

for the modern scientific world.

What Descartes and the other
trailblazers like him did

was to question the accepted
wisdom of the time.

They thought differently, and the
result was that they delivered

monumental breakthroughs for our
understanding of the universe.

Descartes lived in a time when many
philosophers backed up

their arguments with appeals to God.

But Descartes preferred to place his
trust in the power

of human logic and maths.

He believed all ideas should have
their foundations

in experience and reason
rather than tradition and authority.

It still feels like maths belongs
to a discovered world,

but after Descartes, it's a world
that is increasingly devoid

of a divine influence.

We started this episode
with just one question -

is mathematics invented
or discovered?

And, based on the evidence so far,
I'm leaning quite heavily

towards discovered, because it
doesn't seem to me to be possible

that something so all-encompassing
could be the product

of the human mind alone.

Next time, I see how
new mathematical systems

allowed Newton to create
his laws of gravity...

..and even started describing the
existence of things

we didn't know were there.

All more evidence for
maths being a discovery.

Now, this couldn't be a coincidence.

But I'm forced to think again
when I confront one of

the strangest mathematical
concepts there is -

infinity.

And it makes the question of whether
maths is invented or discovered

a lot more difficult to answer.

What makes our world work
the way that it does?

Explore more about the magic
and mystery of mathematics

and how it impacts
our everyday life.

Just go to bbc.co.uk/maths
and follow the links

to the Open University.