9.999… really is equal to 10

by birtanpublished on September 26, 2020

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 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

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