[Music]
all right so the other day I said with
my kids watch TV you probably familiar
with these reality shows where they put
a bunch of clueless people put them in
an island and make them do funny things
and somebody wins a prize at the end
well there's a kid's version of this and
animated kids versions called total
drama island and the episode they were
watching was called no pain no gain
there was one segment in there which
really you know left an impression with
me so don't show you that that's section
and if your victim can last 10 seconds
without saying uncle you get eliminated
instead
[Music]
nine and nine one hundred thousandth of
a million whatever it's not ten okay
close shave close shave well that's what
a really really close look here so
there's two main characters really
there's Jeff the guy with the head who
tries to stay in the barrel for ten
seconds and as the game host okay what
does the game uh say well first thing he
says is nine and nine one hundred
thousands of a millionth well if you
actually write down his number it looks
like this which has some resemblance but
it's not really the same and also you
know it was the other thing he says it's
not ten well I'm not so sure maybe if
they had invested in a more expensive
clock there on the wall you know one
we're kind of digits go on forever what
we would have seen is you know the nines
going on forever and if that's the case
well then nine point nine nine nine was
the nines going on forever
that's a she ten in maths so then Jeff
would have won right so happy Jeff well
problem was this is even if it had been
like this mmm
the game horse is probably not convinced
I mean he seems to be pretty clueless
when it comes to Matt so he probably
never heard of you know ten being equal
to nine point nine nine nine in fact
most people haven't heard of this and
even if they have heard of it they don't
believe it right so nobody believes that
ten is equal to nine point nine nine
nine nine nine and so on so what I'd
like to do today is help Jeff win the
prize by convincing somebody like this
that 10 is reka really equal to nine
point nine a nine so how we're going to
do this well I mean this guy's clueless
right
so probably anytime something Mattie
comes his way he reaches for the
calculator let's give him a calculator
there's a calculator and ask him to
punch in ten so this will punch in ten
and then he's going to divide by three
and we get the familiar 3.3333
and of course I mean even that guy knows
that if he's invested in a more
expensive calculator one with digits
going on forever
this would actually look like this so
what we've just done here is basically
say ten divided by three is equal to
three point three three three going on
forever and he'll probably be pretty
okay with this so what do we do next
well we're almost there
we just have to multiply by three now
right so what happens to the left side
if you multiply ten divided by three by
three get ten and on the right side well
everybody one of those threes gets
multiplied by three that it gives nines
all over and so we get our result so
Jeff's happy and can probably claim the
prize except if the game host is a
little bit Chloe
he'll actually have noticed that we just
kind of tricked him so a lot of people
are actually okay with this all right so
if if you actually what just want to
shut up somebody you know somebody
doesn't believe that ten is equal to
nine point nine nine you tell them this
they're probably okay ways that you move
on but actually you've cheated a bit
because I mean who says that this is
actually equal ten divided by three why
is this equal to three point three three
three just because the calculator kind
of spits out more and more threes right
I mean you would think well you know
okay there's more and more threes but we
really we get closer and closer to ten
divided by three but you really really
never get there somehow all right the
same sort of thing with this ten and
nine point nine and nine yeah goes on
forever it's not a real argument so
basically these two things are really
saying the same right so if you believe
this you should believe this but if you
don't believe this
you also shouldn't believe that so it's
really saying the same thing anyway so
well maybe maybe that's not quite it
right that's not quite it that's
probably good enough for this guy but
for a bit more Chloe game host he needs
a bit more convincing and you know some
other people also need a bit more
convincing so there is another argument
which is actually a lot nicer is that
one you actually start with just saying
well nine point nine and nine that's
some sort of number okay some sort of
number now we want to figure out what it
is all right so
we'll just call it for the moment
because we don't know what it is we call
it the mystery number so the mystery
number is nine point nine nine nine and
what comes next is I've actually tried
it so my kids the one who said next to
me and you know they buy it but anyway
let's just go for it right so we've got
this here and I would just do some kind
of familiar manipulations so we will
just multiply this thing by ten okay so
then we get of course 10m ten times the
mystery number what do we get on the
other side where whenever you see a
number like this with a decimal point
and you multiply that by ten the nest of
a point kind of moves right to the right
one position to the right so all of a
sudden see a 99 and then a tail of nines
okay
so we see this and now it's really
really important to a note that these
nines go on forever there's no last nine
right so when this shift occurs whether
the decimal point moves or whether all
the lines move depending on how you look
at it you know there's no end to it
there's nothing kind of shifting forward
it's kind of all staying an infinite
tail of nines same up here right so that
really on the on the right here we see
exactly the same this infinite tail is
is repeated down here okay now we we we
subtract the bottom from the top okay
and then what we get well on that side
we get 9m and on this side we're ninety
nine minus nine is ninety and then the
two tails are identical so that is kind
of cancel out so what we get here is
ninety point lots of zeroes and we don't
worry about the zero so we can just take
them away and now we just divide by nine
and we get M is equal to ten all right
and well just happy because we've just
shown that nine point nine and nine is
equal to ten I wish we start out with
saying we don't know what it is you know
and then we've got these kind of logical
steps and at the end of this chain of
logical steps we come to the conclusions
actually M is equal to ten something
we've seen before okay pretty
bulletproof convinced as many people
convinced mathematicians there's
actually no cheat in here right there is
you know
is you know infinity in there but if we
want to make sense of infinity in some
sort of way I mean this is a this is a
proper way of doing it but still I mean
there's still like this guy he he
watches me do it is and he's probably
still not absolutely comfortable with
what I've been saying here and he is
he's sort of right and so what I would
like to do is well I mean I've actually
been looking at lots of people kind of
do explanations of this and I'm not
saying anything different here so at
this point in time when when people are
finished with this they kind of think
well after now I made it absolutely
crystal clear that you know we're
talking about the same thing ten and
nine point nine or nine but there's a
lot of you know a lot of people who
would somehow get lost here in the
process so what I'd like to do is kind
of go to mathematical slow motion okay I
want to do the whole thing again in
mathematical slow motion and really take
any step like really really slowly and
make absolutely sure that everybody you
know is on the same boat here so let's
let's just do it one more time okay so
the first thing that kind of confuses
people is well they look at this
expression here and well there there's
there's all sorts of things going on I
mean they know about nine point nine
nine nine is but what is actually this
this infinite thing there all right what
is this infinite thing so this is kind
of a name for something infinite and
let's just let's just really really
write down what it is all right so we're
kind of one step removed from what the
actual object you the mathematical
object that we're talking about so let's
just go again mystery number is equal to
that and then let's see what are the
individual nines actually standing for
okay so the first nine world a stent for
nine so there's nothing special here the
second nine what's the second nine the
second nine is basically zero point nine
or nine tenths so the second nine really
stands for nine ten so we have gets nine
and then we add to it 9/10 okay so we
know what branch ants are so just add to
it what about the third one right the
third nine the third nine is nine
hundredths so we have to add this and
then we've got nine thousandths next one
and we keep on going like this right so
keeps on going like this so that's what
it really is that's what we're talking
about we're talking about a nine and to
this nine we add nine tenths and then
nine hundredths and so on and we keep on
going right keep on going we don't stop
anywhere where there's no last last term
eeeh okay that's what it is
so let's just deal with this huh so
we've got a pretty good idea what this
is you can actually visualize it all
right okay now we said I wanna multiply
this by 10 okay so we multiply this by
10 so again left side is 10 times okay
right side and I would take every single
one of those guys and multiply it by 10
and see actually what happens okay okay
first one 9 times 10 is 90 okay
need a bit more space here I'll put the
90 up there all right now next one 9/10
times 10 let's just write it down okay
so 9/10 times 10 that's what it is all
right you can see there's a there's a 10
at the top 10 at the bottom they just
cancel out what's left over is a 9 all
right so there's a 9 now we take the
next term here 9 divided by 100 times 10
let's just write it down okay 100 is 10
times 10 okay then 10 at the top 10 in
bottom cancel out what's left over 9 10
just do this one more time we're in
mathematical slow motion okay 9 divided
by 1000 okay times 10,000 is 10 times
100 so there we go
10 goes away 10 10 goes away and we've
got nine hundredths and I mean at this
point in time should be pretty clear and
this is keeps on going right forever
right forever that's really important to
see that whatever you see down there
you'll also see at the top all this
stuff here whatever you see here to the
right will also come up on the top line
so for example so 9 divided by 1 million
is somewhere there it's also up there
now let us ask well a 9 divided by
googolplex is up here yes it's up there
because down here we've got
/ 10 times Googleplex which when you
multiply by 10 gives 9/1 Google place
okay so that these tails are exactly
same way in mathematical emotion okay
now we're doing the - business right so
bottom from the top 9m on this side on
that side well everything here is
doubled up right everything is doubled
up so it just goes away when we do the
subtraction and the only thing that
stays over here is is the 19 okay
and then we divide by nine and we get
again 10 since we started obvious point
one nine point nine nine nine it means
that 10 is equal to this and this is a
pretty good argument so I mean there
still is this business of you know
having an infinite sum and you have to
be careful with infinite silence all
kinds of terrible things can happen but
as far as infinite sums go you know what
we're actually dealing with here is
about as tame as it can be and
mathematicians have given this a very
very you know close look and I can
promise you I'm a mathematician you know
we've been very careful and what we're
doing here is it's okay okay anyway so
he believes it now Jeff gets his prize
but also he's also convinced now for
example that one is equal to 0.99 999
and for example point three is equal to
point two nine nine nine ninety nine and
also any any kind of terminating number
like this can be actually written in the
second way with lots of nine so you just
take this last digit here and you know
it decreases by one so that turns into
six and then add lots of nines so all
the same stuff all the same stuff and
well I've succeeded in convincing this
guy and he's actually so convinced that
he's now agreed to changing the whole
episode to give it an alternative ending
so from now on whenever you see this
episode on TV the alternative ending to
the scene is actually going to look like
like this
close shave Jeff nine and nine nine nine
whatever
it's ten
just happy I'm happy hopefully everybody
is happy if not let me know
you