Kind: captions
Language: en
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
[Music]
imagine if you had to explain how we
keep track of time
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
[Music]
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
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
foreign
[Music]
[Applause]
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
[Music]
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
[Applause]
[Music]
foreign
[Music]
to order this program on DVD visit shop
PBS or call 1-800 play PBS episodes of
Nova are available with passport Nova is
also available on Amazon Prime video
[Music]
thank you
foreign
[Music]