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# Ending the 0.999~ = 1 debates

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In case anyone here hasn't noticed, there have been a lot of debates on this online between people on other forums, some with a good grasp of mathematics and some without. The debate is over whether 0.999 does or does not equal 1.

There are many different proofs to prove that it does, but because some people understand some proofs better than others, and some not at all the debates still go on. I've personally been involved in these sort of debates and I can think up at least five different proofs, all based on different sorts of logic, that show that they are equal. The more forms we can find, the better the chances are that someone will understand it.

I thought that perhaps we could do a service to the internet and try and sort this out once and for all, by compiling the most comprehensive list of proofs on the internet. There are already some sites like this, but they only cover one or two forms of the proof and many are based on questionable things (like 0.333 = 1/3 even though it's just an approximation). If this thread eventually appears high enough up on Google, people might find it for themselves and we can stop arguments caused by a poor knowledge of maths across the globe. Anyone think that this is a worthy goal?

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I'll post a few as soon as I get dave's mimetex module fixed :-D

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We did a proof for this in my analysis module. Went something along the lines of this:

Let x = d0.d1d2d3... with di in the set {0,1,...,9}.

From our definition of the decimal, we have that x = sumn=0 -> infinity(dn*10-n). So for 0.999 we can say that for i > 0, di = 9. Hence if we let x = 0.999, then we have:

x = sumn=1 -> infinity(9/(10^n)).

This is a very easy geometric progression, and if you work it out, you get x = 1.

The other proof (which I'm somewhat dubious about) I learnt at GCSE (when I was 16):

x = 0.999, so 10x = 9.999. Now subtract one equation from the other, so you get 9x = 9 => x = 1. As I said, I'm dubious to say the least

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Also, moving this to the Number Theory forum because I think it's probably a better place for it.

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This proof works with infinite geometric sequences.

.999... = 9/10 + 9/100 + 9/1000 ...

so you have a geometric sequence.

You put it into the formula. a1/(1-r) where a1 is the first term of the sequence and r is the ratio.

(9/10)/(1-1/10). .9/.9= 1

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That's what I just proved

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Oh, hehe, I haven't ever seen that notation before.

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No problem, it's exactly the same argument, just in a more formal styling.

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there are no irrational numbers between 0.99999... and 1.

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If i remember correctly the decimal representation is defined to have this property.

Thus making 0.4999999999999999etc = 0.5 and 0.9999999999etc = 1.

The geometric mean argument is i think the most easily understood.

A pseudo-intituive argument could be something like :

Let x = 0.99999999999999999etc. It is easily seen that x <= 1 and also that for any n > 1, 9*sum_m=1^n (1/10)^m < x, So for any eps > 0, 1 - eps < x <= 1. It follows that x could only be equal to 1.

Which basicaly comes down to the geometric mean argument.

Hey Radical edward your argument is not finished yet...you have to show why it is impossible for a rational to be in between x and 1, not using the fact they are equal;

Mandrake,

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I was personally under the impression that most of these proofs go along the same kind of lines; anyone got a different one?

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I remember that infinite series form of the proof... I accidentally came up with it one lunch and probably annoyed my friend because we were meant to be starting a game of chess. But I wondered whether it counted as I've heard that the formula for the sum of an infinite series (if it's finite) was only an approximation, though I don't see how.

There's another form of the proof that's a bit less mathematical but hopefully understandable by most people. Say that x = 0.999. In that case, 1 - x = 0.0001. There's an infinite number of zeroes before the 1. But infinity can be defined as a number with no endpoint, so you never get to the 1 and therefore 0.0001 = 0.000 = 0, and 0.999 = 1.

Ah, yes they are mostly similar dave. It's just that most people online don't understand the majority of mathematical proofs (at least on the forums where this is argued over) and so a lot of different versions are needed before one might see one that they understand. Mostly it's due to people having only done maths at a level where every number follows the same mathematical rules but who are interested in terms like infinity, hence they assume things like 1/0 = infinity, and don't know what undefined answers are. So they think that you can do things like multiply by infinity and divide by zero (unless you really can and I'm just the wrong one... I've heard that 0/0 = 0 but that's another thing altogether)

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forgive my nievetity here, but after reading all this, it looks an awefull lot like "Rounding Up"?

like to the nearest whole number etc...

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It's something along those lines, yes, but just defined in a more rigourous way.

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so its kinda like youre demonstrating HOW and WHY we "round up" then?

as opposed to just doing it because the teacher or exam paper tells you to, sorta thing?

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Basically what we're saying is that as you add more and more 9's to the end of 0.9, eventually (i.e. in the limit) you'll get 0.999 = 1. There's been lots of debate about this in the mathematical community (as there usually is with rather small and little things such as notation).

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so effectively 10/3 may be said to be just 3.3 or 3.3333 depending on how accurate you need it to be, but when taken to infinity, 0.999... may as well be just 1 as near as damnit, and for the sake of sanity we just say 1 to avoid it getting ridiculous!

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I remember arguing with fafalone over this years ago.

looks like I win, dear fafalone

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Don't we just say that 0.999..=1 because it doesn't really matters in calculations since you can ignore the small mistake you make? Same thing in chemistry, if something is small enough compared to another value we can neglect it, which doesn't mean it doesn't exist.. I think that 0.999.. is definitely NOT the same thing as 1 (it simply isn't, same thing like 1+1=1), but since the difference is so small you can say 0.999.. is EQUAL to 1 in calculations. It all depends how you look at it or what you have to compare it to..

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well personaly 0.999... is just that, its never 1. but for the sake of sanity, its often prudent to consider it to be 1 unless a huge degree of accuracy is required, a bit like 3.14 will do just fine when using Pi for many applications. 3.1415926535897932384626 blah blah would be a triffe excessive

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think that 0.999.. is definitely NOT the same thing as 1

Thats the thing though, according to all our calculations, it is.

Here's one:

X = .999~

10X = 9.999~

Subtracting the original equation:

10X = 9.999~

- X = .999~

____________

9X = 9

X = 1

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thats just because a calc rounds up

try this:

10/3 =3.333...

ans X 3 = 10

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Nooo, ^ thats just because a calc rounds up

3.333~ * 3 = 9.999~

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in all honesty it sounds a bit like that old bell bow and the 27 notes and wheres the missing note type thing, its a quirk.

Im not sure WHAT the quirk is exactly, but I feel sure it`s something quirky/odd like that

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Hmm, maybe we'd better sort it out amongst ourselves first and just give a "presentation" about our conslusions after, or people might get worried about solidarity.

I think the main problem in sorting this out all lies in the fundamental properties of infinity, and what happens when you divide by it. At first glance one would say that 0.999 is certainly less than 1. But if it's less than 1, what is the difference? I'd say it's 0.0001, as that's all that makes sense (although a number after an unending number of zeroes doesn't). Since any number divided by any other number, no matter how large, always leaves a vanishingly small part, it can't be zero. But that only applies to finite numbers. Infinity intrinsically defies normal rules of maths, as in you can't actually multiply by it (or at least in my level of maths that's the dogma we've been fed). But you can divide by it, and when you divide by it you get zero. Since 0.0001 is an infinitely small number, it would be zero, right? Otherwise the infinite series formula would not work, and something like Zeno's paradox would be true, in which case movement would be impossible.

DISCLAIMER: I may be wrong.

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