All language subtitles for Zero to Infinity ¦ Full Documentary ¦ NOVA ¦ PBS

foreign

but they didn't all arrive at once why

did it take so long it's utility in

mathematics is Undisputed for one number

in particular it's more like a concept

than a number to gain full numberhood I

call it a very significant number what's

so scary you divide a number by it you

blow up

about zero in Science and Mathematics

the simplest ideas end up the most

influential the most profound

from zero where do numbers lead can we

follow them all the way to infinity

infinity and zero are two sides to the

same coin can one Infinity be bigger

than another how much that is to

understand that's where all the

amazingness of infinity is what happens

when mathematicians mix the clout of

zero with the power of infinity

it's all one big principle nothing less

than our modern world

come join me to Lithia Williams

as we dance with two of the strangest

beasts all of mathematics

it's nothing and everything

zero to Infinity right now

on Nova

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[Music]

imagine if you had to explain how we

keep track of time

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to an alien

so they are an alien

it's for earth to travel around the Sun

one year

so far so good

then you explain we break a year down

into 12 months

though they don't fit exactly

and we break months into four weeks

no that's not an exact fit either

at this point the alien might think one

twelve four is there a pattern for me

but then you go on and explain a week is

made up of seven days

and that a day is made of 24 hours

and an hour is made up of 60 minutes

so that's groups of 1 12 4 7 24 and 60.

that's the system

even the aliens buddies can't figure it

out

maybe they can wrap their heads around

another number

a dance number

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it's easy to imagine that the real

universal language should be mathematics

and maybe it is

though on Earth over the course of our

history

how we represent numbers has been

anything but Universal

over thousands of years we humans have

tried out a lot of systems but there's

one that many of us use today

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with just 10 numerals zero through nine

we can in principle write out any number

we want however large or small

so writing out some may require an

eternity I'm looking at you pie

so where did all these numbers come from

and do they really go on forever

my name is talithia Williams and when

I'm not on an alien planet you can find

me

[Music]

at Harvey Mudd College in Claremont

California

I'm a professor of mathematics and a

statistician

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statistics is a mathematical science

that looks for patterns and data

here information that researchers can

gather from anywhere

but all of which is ultimately

translated into numbers

using the very digits we learn by

counting well our digits

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one two three four five and so on they

can be arranged as whole steps on a

number line that extends off into the

distance

heading towards something we learned to

call Infinity

which we shall see can be a very strange

Place indeed

though there is one number that tends to

be overlooked at least at first most of

us learn to count starting with one

but is that really the beginning

or is the start a number that isn't

there at all

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about zero

we're talking about nothing

so we start you know teaching children

here's one apple two apples three apples

and we don't think about well what about

everywhere else where there are no

apples

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0 is a special number

which makes every other number

Meaningful

these days most of us take zero for

granted

but as it turns out unlike the counting

numbers one two three and so on

zero was late to the party

maybe that's understandable numbers help

us keep track of things like the number

of sheep you have or chickens or cows

so why keep track of zero goats then

there would be an infinite number of

things that we're not counting

the number zero may seem like it's been

with us forever but ancient

civilizations had numbers and

Mathematics for thousands of years

without it for example those of

Mesopotamia

that's the historical name for an area

that includes parts of modern Iraq Iran

Syria and turkey

it was home to some of the earliest

cities and the earliest civilizations in

the world as well as an influential

numeral system based on the number 60.

first invented by the Sumerians and

later developed by the Babylonians

it survived for thousands of years

and its Legacy is with us today

in the 60 minutes in an hour

nearby and at about the same time or the

ancient Egyptians

they developed sophisticated mathematics

geometry and astronomy

they also had their own hieroglyphic

numeral system that evolved over time

and just like the Mesopotamians

the ancient Egyptians didn't use the

number zero

neither did the Greeks

nor the Romans now remember we're

talking about zero as a number

for us xero also acts as a placeholder a

way to distinguish 44.

from

404

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some ancient numeral systems had

placeholders as well filling in blank

spots

but they weren't seen as a number they

were just a way to keep things organized

in fact as far as historians can tell

using xero as a number has only turned

up twice

the Mayans had the idea they represented

the number zero with a shell

but the zero that we commonly use today

Came From Another Part of the world

the Indian subcontinent has been home to

many societies cultures and traditions

some dating back hundreds if not

thousands of years for example the

colorful Festival of holy

which celebrates the Divine love of

rather and Krishna

and it was here in India about 1700

years ago that one of the most powerful

ideas in all of mathematics is thought

by some to have taken hold

zero

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to learn more about India's critical

role in Zero's history

I've traveled to Princeton University

speak with one of the most highly

regarded mathematicians in the world

manjul bhargava

also an accomplished player of the

primary percussion instrument in Indian

classical music the Tabla

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until we've had number systems for

thousands of years from the Egyptians to

the Babylonians but they didn't seem to

have a need for zero why do you think it

started in India at this time

the concept of zero started off in

philosophical works

state of zerleness

the state that we all try to achieve

when we meditate

in the Hindu and Buddhist Traditions

both with deep roots on the Indian

subcontinent the concept of emptiness

plays a key role

emptying the mind of all sensations of

all Temptations of ego of thoughts of

emotions

and so that really puts zero in the air

as as an important concept but the first

symbolic representation of a zero

actually happened in the field of

linguistics

in about the 5th Century BCE an Indian

scholar Panini laid out the linguistic

rules of what came to be called

classical Sanskrit

sometimes when you're pronouncing things

you like to leave out a sound when

you're when you're pronouncing quickly

so manani is one of the great

grammarians of India had a special

symbol when a sound gets deleted that

was called a lopa and that's like a

linguistic zero very parallel to the

modern apostrophe in the English

language

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traditional Indian music of the type

module plays is greatly influenced by

the poetic traditions of Sanskrit it too

will sometimes omit sounds

so when the Lupa came to music that void

is considered just as important as an

actual sound and can be just as powerful

so occasionally to emphasize the

downbeat you won't play it

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and so that's how the musical zero came

about and the musical zero can be very

powerful zeros like any other note that

you can use it in very important moments

and just

foreign

Ty of emptiness in Indian philosophical

traditions and the symbolic linguistic

zero may have set the stage for the

number zero

many scholars date its development to

sometime in the first half of the first

Millennium between the third and fifth

centuries

but that opinion was originally based on

indirect evidence because no hard

physical proof had ever been found

some believe that changed in 2017 when

Oxford University's bodilian libraries

made a surprising announcement about one

of their Treasures now scientists from

the University of Oxford have found a

manuscript that originated in India and

pushes back the discovery of the concept

of zero by at least 500 years

the bakshali manuscript about 70 birch

bark pages of mathematical writings in

Sanskrit had been dated to around 800 CE

but new carbon dating of one of its

Pages pushed that back about 500 years

the page shows a DOT which has been

interpreted to represent zero

there we see the zero used in the Indian

number system it's just the way that we

write them today with one difference is

that the zero is written as a DOT

if the dating is correct the manuscript

is now the earliest evidence of Zero's

use as a number

not all Scholars agree however and the

assertion that the writing is that old

is hotly contested

foreign

however there is a little question that

zero was in use in mathematics in India

by the 7th Century in the time of the

great astronomer and mathematician

brahmagupta

I'm a group that came around and he said

well zero is a number just like any

other

so he actually goes and writes down

rules for multiplication and addition

and subtraction of zero so he's the

first person to have like thought of

zero we work with zero today right right

yeah

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along with zero brahmagupta also

investigated negative numbers

today when we place zero at the center

of the number line between positive and

negative numbers that is a legacy of his

work

so when we talk about the history of the

zero for a mathematician's point of view

this was the Grand moment where zero

became a full-fledged number as part of

our mathematics and that really that

really changed mathematics do you think

it's the it's the best idea ever in

mathematics in Science and Mathematics

it's often the simplest and the most

basic ideas that end up

yeah the most influential the most

profound

like the wheel and it really did change

mathematics and science

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before the Indian system became widely

adopted the main purpose of written

numerals was for recording numbers not

calculating with them instead

calculations were done with the variety

of techniques and devices

such as abacuses or counting boards that

used Pebbles

numerals were only for storing the

results

but the Indian system uses the same

numerals for calculation and Storage

like the number zero that's a

fundamental breakthrough we all just

take for granted

the Innovative Indian system would

eventually become the most popular in

the world

but not immediately

a crucial step in that Journey came out

of the remarkable rise of the Islamic

empire

originating in the Arabian peninsula in

the 7th Century after only about a

hundred years it had reached India in

the East and Spain in the West

to learn more about the key role of

Islam in the spread of Indian numerals

in zero I'm visiting the Hispanic

Society of America in New York City

which houses perhaps the most

influential work in that Journey

I'm joined by Waleed El Ansari

he's an expert in Islamic Studies and

the intersection of religion science and

economics

and like me eager to see the rare

manuscript

its roots go back to what was then a

recently constructed City and a new

political and Cultural Center of Islam

Baghdad

so Baghdad was designed in a circular

shape after Euclid's writings

and the circle is viewed as a perfect

shape and therefore it's a symbol in a

sense of God

strategically located at the crossroads

of several trade routes the city quickly

grew

and it became the largest city in the

world

it's really quite amazing this Center

for trade on one hand as well as

intellectual trade

the transfer and transmission of ideas

Scholars translated texts that had been

gathered from across the Islamic world

and Beyond

including those about Indian mathematics

they viewed all knowledge coming from

these other civilizations that was

consistent with the unity of God as

being Islamic in the deepest sense of

the word and so it was very easy for the

Muslims to integrate that into their

world view sounds like they were also

the curators of this knowledge and and

and once they sort of brought it

together they had then built on it as

well that's right it wasn't just

Aristotle in Arabic that's right it was

more than that

foreign

in the early part of the 9th century

Muhammad IBN Musa al-qarasmi a Persian

scholar in a variety of subjects

wrote several hugely influential books

two had a powerful impact on mathematics

in one he laid out the foundations of

algebra

in fact part of the title of the book

would give the subject its name

another of his key works in mathematics

which only survives today in a 13th

century Latin translation is what's

brought us to the Hispanic Society of

America home to one of the oldest and

the most complete version

this is a gem

and so you can see here the Indian

Arabic numeral system

with zero one two three four five six

seven eight nine

and some of them are shaped very similar

to what we have today some of them are

not mathematics today the foundation is

right here in front of us that's right

which is unbelievable

the purpose of the book was to promote

the Indian numeral system

and explain its key Innovations zero

and the use of the numerals for

arithmetic

the book also included procedures for

computation that would come to be known

as algorithms a Corruption of

al-qarismi's name

so it's a little manual to show people

how to operate with these and we learned

these as kids so in some ways we take it

for granted but you're right it's

someone had to say this is the process

that we're going to use in order to

build this mathematical knowledge and

here it is that's right wow that's right

so this is very foundational

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work along with that of other Islamic

mathematicians helped spread the Indian

numeral system throughout the Islamic

world and eventually Beyond

the Islamic promotion of the Indian

numeral system was so successful the

numbers would even come to be known as

Arabic numerals somewhat obscuring their

Indian Origins

so what we're looking at here is

something that is now not only used in

the Islamic world in the west but really

is the most important numeral system for

the entire world

and so I can hardly over emphasize the

significance of this text

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in Europe the Indian Arabic numeral

system with its revolutionary zero

would eventually have a powerful role in

the advancement of science

but the earliest users were Italian

Merchants who saw its immediate

advantages for calculations and business

records

in fact in 1202 the son of a merchant

Leonardo of Pisa better known today as

Fibonacci wrote Lieber abachi

an influential book about the new

numerals advocating for their use

ultimately it would take hundreds of

years for the new numerals to displace

both the existing systems for recording

numbers such as Roman numerals

and the various devices and techniques

used for calculating

but by the late 16th century in part

aided by the Advent of the printing

press and growing literacy the new

system had been widely adopted in Europe

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because of the European Renaissance it

started becoming impossible to really

make those huge scientific leaps without

switching over

to zero and get in the Indian system of

numeration the system that allowed you

to really do computations easily

and so instead of becoming impossible

not to use them and so by the 17th

century they started becoming in regular

use in Europe and then around the world

in the rest is history

[Music]

treating zero as a number transforms

mathematics but it did take some getting

used to because in some ways xero isn't

like any other number

first of all it has unique properties

zero has some properties of number but

also some properties that make it more

like a concept than a number

in addition subtraction and

multiplication

zero behaves differently than every

other number

but where zero really creates Havoc is

in division

you get to division and all of a sudden

it's the first time that you're sort of

told like well that's impossible

can divide any number by every other

number except zero

then you divide a number by zero for

example you blow up three two one zero I

have no apples and I share that among

six students would everybody gets zero

apples they had no apples to share

but if I have six apples and they are

shared among zero students

the concept becomes personal

how do we make sense of that

the problem is you can't

think of it this way

dividing six by zero is the same thing

as asking what number multiplied by zero

will give you six

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since everything multiplied by zero

always equals zero there's no solution

so mathematicians officially consider

the answer as undefined

now you might wonder is that sort of

hole in the bucket of division a problem

does it get you into trouble

turns out it certainly does under the

right circumstances

in fact a Greek philosopher who lived

thousands of years ago before zero even

came to be invented a paradox that

captures the problem

his name was Zeno of Elia

and the Paradox was about an arrow

[Music]

demonstrate xeno's Paradox I've turned

to Eric Bennett from Surprise Arizona

DF is what we're looking for he's a

physics and Engineering teacher at a

local high school

and he's a paralympian in archery four

times over

so Eric what does it feel like to

participate in the Paralympics four

times

um it makes me feel old a little bit but

um it's amazing I've been competing at a

really high level for uh 15 years wow

wow so how far away is the target here

the target is the standard Olympic

competition distance of 70 meters

which is about three quarters of a

football field no way Yes actually it's

pretty far okay all right I want to see

you shoot this

at 15 years old Eric lost an arm in an

automobile accident

so he draws the bow string back with his

teeth

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the arrow Finds Its Mark

wow that's awesome

all right so you're gonna show me how to

use one of these absolutely okay yep

from from 70 meters

I just want to make sure that you're

super successful in your first try I

appreciate that yeah I appreciate it

Eric offers me a try with a beginner's

bow and a Target about 20 yards away so

I go and it'll go right in the bullseye

[Laughter]

so I Channel my inner Katniss Everdeen

from The Hunger Games

[Music]

May the odds be ever in my favor

[Music]

whoa what I don't know where to go

that is like 100 yards down there we'll

go find it

a lot of work to do here yeah well I

think it's going to be a while before

I'm ready to compete

I had a lot of power yeah you know and

so but back to Zeno and that paradox

all of xeno's original writings have

been lost but according to a later Greek

philosopher Xeno suggested that we

consider an arrow in flight at any

instant in time

and at that instant that thou moment

the arrow is Frozen in space

Motionless

it's neither arriving

nor leaving

and if you consider the entire flight

there's an Infinity of those motionless

Frozen moments in time and space

so Zeno asked is the Flight of the arrow

and all motion really just an illusion

his radical conclusion is that motion is

impossible at a given instant

that arrow is someplace and then click

time forward

it's at some other place but at no

moment was it moving

you're ready to go

[Music]

motion of an arrow looks real enough for

me that's right Katniss got nothing on

me

but you can see why Zeno's Timeless

Frozen moments are so problematic

our whole notion of speed depends on

time

here's the formula

distance traveled divided by length of

time equals speed

but xeno's Frozen moment has a length of

time of zero

that means trying to divide by zero

which is against the rules of division

but at the same time we often want to

know the speed of Something In Motion at

a particular instant

one solution to the problem of

instantaneous speed is a concept called

a limit

let's consider a stick figure who walks

half the distance to a wall

and does that again and again and again

if the stick figure keeps going half the

distance to the wall they'll get closer

and closer but the steps will get

smaller and smaller and they'll never

reach the wall

the wall is an example of a limit as the

number of steps heads to Infinity the

distance to the wall decreases towards

zero but the figure will never reach the

wall

you're getting infinitely close to a

limit as far as you're going to get but

you never actually get there which yeah

it's one of those Concepts that bothers

a lot of people even mathematicians it

bothers I think I can never start with a

whole number and divide it by something

to get zero

there's nothing there's no way for me to

ever get to zero even if you have an

issue and you divide it in half you

still don't have zero

harnessing the power of infinity through

limits gives mathematicians a workaround

to the problem of dividing by zero and

in turn opens the door to a world of

solutions to some extremely difficult

problems

it helped create a new field of

mathematics calculus and that's really

the big idea at the heart of calculus as

understood in modern terms this idea of

the limit that you're supposed to think

how far did I go over a microsecond

that gives me an approximation to my

instantaneous velocity you know the

distance traveled divided by that

duration but that's not yet an instant

so rather than a microsecond I think now

a nanosecond a thousand times shorter

how far did I travel then that gives me

a better approximation and then this

limit as the duration of time goes to

zero

you often find you'll get a well-defined

limiting answer for the for the speed

and that limit is What's called the

instantaneous velocity

it sounds like a clever trick but does

it get the job done to find out I

traveled to New York City

to the national museum of mathematics

MoMA may I please thank you take your

picks here Cornell University

mathematicians Steve strogatz is

enjoying a year as a distinguished

visiting Professor 13 points thank you

very much

shows me around

but I'm here for a specific reason

Steve is going to demonstrate the

problem-solving power of limits and

infinity

though as it turns out whoa we're

missing the key component

if you want to understand what Infinity

can do we're going to need pizza

there's a science there

we don't typically associate pizza with

infinity

so how can New York City's most famous

food help solve one of the most elusive

mysteries of early mathematics

[Music]

so Steve how is this pizza gonna help us

understand Infinity huh I would say it

the other way infinity and the pizza are

going to help us understand one of the

oldest problems in math what's the area

of a circle which is not intuitive no

you know what's hard about it you might

think a circle is a beautiful simple

shape but actually it's got this nasty

property that it doesn't have any

straight lines in it ancient

civilizations didn't know how to find

the area of a shape like that foreign

how to find the exact area of a circle

isn't obvious

for a square or rectangle you just

multiply the sides

but what do you do with a circle so what

do they do well they came up with an

argument that you can convert a round

shape into a rectangle if you use

Infinity so we're basically going to

kind of deconstruct this pizza make it

into a rectangle beautiful and then

we're going to know the area that's it

so I'm going to start with four pieces

okay

to do that I'm going to go one point up

and one point down and then one point up

and one point down

and yeah like that

how'd you do in Geometry

you don't think that looks like a

rectangle it's not closer no it's not

it's not but come on I'm only using four

pieces if I use more I can get closer

okay so we got to cut these babies in

half let's cut them

let's rearrange them same trick

alternating point up and point down

one up and one down

one up and one down

that is looking a lot better

what do you think is that a rectangle

um it's it's not quite a rectangle but

it's getting closer it is right yeah

in both the four piece and eight piece

versions

half the crust sits at the top and half

at the bottom but with eight pieces The

Edge becomes less scalloped closer to a

straight line

so we need to go at least a step further

let's go more we got to do 16.

so we have to just change every other

one am I going to mess this up I mean

that's that's a parallelogram that's

aspiring to be a rectangle that's got

aspirations

[Music]

four slices

to eight slices

[Music]

to 16 slices

and even 32 slices there's a clear

progression towards a rectangle with one

piece out of 32 cut in half to create

vertical sides the rectangle is almost

complete except for the wavy top and

bottom

but as the number of slices increases

the straighter and straighter those

edges would become

and the argument here is that if we

could keep doing this all the way out to

Infinity so that this would be

infinitely many slices infinitesimally

thin this really would become a

rectangle yeah and we can read off the

area it's this radius

that's the distance from the center out

to the crust times

half the circumference which is half the

crust half the curvy stuff and that's a

famous formula a half the crust times

the radius one half CR

usually see for circumference but you

can see it's crust so at the limit once

we got all the way out there it's going

to look like a it would be a rectangle

and that is actually the first calculus

argument in history like 250 BC to find

the area of a circle who knew you could

learn so much from Pizza

Infinity is your friend in math and

that's the great Insight of calculus

that you can you can rebuild the world

out of much simpler objects as long as

you're willing to use infinitely many of

them

[Music]

by embracing Infinity through calculus

mathematicians created one of their most

powerful tools

for this professor of Applied

Mathematics it is part of how he sees

the world

thank you

do you remember that movie The Sixth

Sense where the kid says

I see dead people

sort of what I feel like except I see

math

well I go out and see the New York

skyline

see all the rectangles and pyramids and

the skyscrapers

[Music]

patterns of geometry I see hidden

algebraic relationships

there is traffic flow and the cars look

like corpuscles which makes me think

about blood flow and arteries laws of

fluid dynamics and aerodynamics

patterns of cylinders and the rings on

the cylinders are spaced unevenly

because of the way hydrostatic pressure

works

there's so much math in the real world

and it's all one big principle

whole world runs on calculus and math is

everywhere just can't help but notice it

I see math

actually I see dead people too

[Music]

[Laughter]

calculus is applied everywhere

comes into play in the modern world you

need search no further

but even with the Advent of calculus

Infinity itself in mathematics remains

poorly understood

it was only in the late 19th century

that new mind-bending ideas helped tame

that strange Beast Infinity

[Music]

when I asked my friend author and

mathematician Eugenia Chang to discuss

her thoughts on Infinity

she suggested that we visit the

imaginary Hilbert's Hotel

a thought experiment first proposed by

mathematician David Hilbert in the 1920s

to demonstrate some of the odd

properties of infinity

this hotel is definitely an odd property

well the helmet hotel is a pretty

amazing Hotel it has an infinite number

of rooms wouldn't that be great you

might think that you could always fit

more people in but what if an infinite

number of people showed up and then the

hotel would be full

oh dear then if another person came

along what Would You Do Well if you

weren't very astute then you might just

say sorry we're full

that's one solution

or you might think

given there are an infinite number of

rooms you can just assign the late guest

the room that comes after the one given

to the last guest that checked in you

know just farther down the hall

to put this person at the end of the

line why can't we do that where is the

end of the line sounds like a

philosophical question but the thing is

you can't just tell them to go to the

end you have to give them a room number

and all the rooms are full

hmm seems unsolvable

but luckily any manager of a hotel with

an infinite number of rooms and an

infinite number of guests has to have an

infinite number of tricks up their

sleeve

okay how about the person in room one

moves into room two and the person in

room 2 moves into room 3 and the person

in room 3 moves into room 4 and so on

[Music]

everybody has another room they can move

into because everyone just adds one to

their room number and that will leave

room one empty so new person comes

welcome you know what we're just gonna

have everybody scoot over for you just

scoot

goes in room one and then what if two

people showed up that's fine everyone

moves up two rooms

what if five people show up that's fine

but what if an infinite number showed up

say because of a fire

at a second nearby completely full

Hilbert Hotel

is there room for a second Infinity of

gifts

I've got an infinite number of people

people you can't just get everyone to

move up an infinite number of rooms

because where would they go

there is a solution the manager asks

each person checked into a room to

multiply their room number by two and

move there so one goes to two two goes

to four three goes to six and so on

which means they will all move into an

even numbered room and that will leave

all the odd numbered rooms and that's an

infinite number of rooms and so all the

new infinite number of people can move

into the odd numbered rooms

so then it feels like we've got twice

the number of rooms although we're still

at Infinity

in fact the hotel can accommodate all

the guests from an infinite number of

infinite hotels but you'll have to stop

in to learn how

I guess here at Hilbert's Hotel there's

always room

for one more

while Hilbert's hotel is named for the

person who conceived of it the ideas it

plays with came from Georg cantor a

German mathematician who in the late

19th century introduced a radically new

understanding of infinity

he built that understanding based on

another area of mathematics he created

set theory a set is a well-defined

collection of things like all the bright

red shoes you own or all the possible

outcomes from rolling a typical

six-sided die

Canter used sets as a way of comparing

quantity

if you can match up the die roll

possibilities in a one-to-one

correspondence with your shoes with none

left over in either set then you know

they have the same quantity

all of this may seem Elementary like

counting with your fingers but they are

ideas that will carry you to some

strange places

counting in pure math is very profound

and it doesn't just mean it lists

everything and label them one two three

it often means find some perfect

correspondence in the ideas so that you

don't have to list them all but you can

know that they match up perfectly

without listing them all and so there

are some really counter-intuitive things

we can do consider this which Infinity

is bigger the set of counting numbers 1

2 3 4 Etc or the set of just the even

numbers 2 4 6 and so on

and intuitively we might go well that's

half that's half right yeah but we could

still perfectly match them up with all

the numbers because all we have to do is

multiply each of the ordinary numbers by

two

and that will make a perfect

correspondence

so the set of counting numbers

and the set of even numbers are both

infinite in both the same size

Canter called these kinds of Infinities

with a one-to-one correspondence to the

counting numbers countable

and he investigated other kinds of

infinities

like that of the prime numbers whole

numbers greater than one that can only

be evenly divided by themselves or one

Cantor found the Infinity of the prime

numbers was also countable

and even the Infinity of the rational

numbers all the negative and all the

positive integers plus all the fractions

that can be made up from them even that

Infinity was countable and the same size

as the others

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but now for the ultimate challenge

if you take all the rational numbers

add in the irrational numbers like Pi or

the square root of two numbers you can't

represent as fractions using integers

you know the ones that have decimals

that go on forever

without repeating

then you have the real numbers the

complete number line

every possible number in decimal

notation

so is the Infinity of the real numbers

just like the others countable

well since the other sets of numbers are

this one has to be two right

in Canter's work for an Infinity to be

countable it has to have a one-to-one

correspondence with the counting numbers

like we saw with the Infinity of the

even numbers

so to do that you need to be able to

list the Infinities members not

literally it's infinite and would take

forever but just the way the list of all

the counting numbers marches off toward

Infinity adding one with each step

a way to list all the real numbers to

prove that they're countable Canter

demonstrated the answer is

no with an ingenious argument

imagine you presented Cantor with what

you think is a complete list of all the

real numbers

to keep it simple we will only do the

ones between zero and one

for consistency an a number that

terminates exactly

like five

will receive an endless series of zeros

after the last digit

the list of course goes down the page

infinitely and off the page to the right

because the numbers are infinitely long

Canton looks at your list and starts to

construct a new number he takes the

first digit of the number in the first

row and adds one to it if it's a 9 it

becomes a zero

now he knows his new number won't match

the one in the first row

next he takes the second digit of the

second rows number and does the same

now he knows his new number won't match

the one in the second row

and he does the same thing with the

third rows number

he continues down the list moving

diagonally building the new number

making sure that in at least one

position a digit will be different when

compared to any other number on the list

this famous diagonal proof shows that

any attempt to list all the real numbers

will always be incomplete

and if you can't create a complete list

of the real numbers

they can't be counted

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Canter called The Infinity of the real

numbers uncountable

a bigger size Infinity than all those

countable infinities

well the idea of infinity had been

around for a long time

but the idea that some infinities could

be bigger than others

that's what canters diagonalization

argument demonstrated and his argument

is so simple it's when again one of

those simple ideas

that is just so profound it's one of the

most ingenious innovative ideas ever

inserted into the study of numbers and

our understanding of infinity is forever

changed because of cantor's incredible

work

for humankind the journey from zero to

Infinity has been extraordinary zero

introduced thousands of years after the

birth of mathematics revolutionized it

enabling a new means of calculation that

helped the advancement of science

harnessing the power of zero and

infinity together through calculus made

many of the technological breakthroughs

that we take for granted possible

and cantor's work on Infinity

he unveiled a new strange vision of it

for all to see

his ideas and methods laid a foundation

for the development of mathematics in

the 20th and the 21st centuries

but for me personally

I think his imagination helps us

appreciate that we live in a universe of

infinite possibilities

No Doubt new Wonder still await us on

the road from zero to Infinity

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thank you

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