# Ending the 0.999~ = 1 debates

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Regarding the quirk in the proof

x = .99999

10x = 9.9999

The quirk is this: If you define a set number of 9's, it doesn't work. When you say it goes on forever, you side-step this.

Use 4 nines, for instance:

.9999

9.999

9.999 - .9999 = 8.9991, not 9

to make that work, you have to pretend that x has two different numbers of 9's

when you multiply by 10 you add a nine

It's all moot, though, because the difference between .999 repeat and 1 is infinitely small, by definition.

however, in real terms, there is a difference. It is just so small that it is impossible to measure OR prove mathematically, because we are using a finite type of math to describe an infinite (non-real?) quantity

So technically that is not a proof of anything but the limitations in our math system, in my opinion.

At the same time, I'd say that .999 repeat is equal to one. It's a moot point to say otherwise.

Of course, math is not my specialty. I agree with YT though.

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I have to disagree with the fact that, repeting decimals are not a finite number. I think they are well defined by the integer ratios they turn into through the method I described. It works for simple ratios such as 1/3 = .333, and even more and more complex ones, what is wrong with it working or .999

You are saying if you have X nines after the decimal point, then there will be X - 1 nines if you multiply by ten. That leaves 1 nine behind when you are done with the subtraction.

The problem here is that you are saying infinity - 1. For all practical purposes infinity * 50 = infinity.

For example there is the problem of the infinately large hotel. There are rooms labeled from 0 ... infinity. One person wants a room but all the rooms are full. So in solution to this, the hotel calls up everyone in the hotel and tells them each to move to their room + 1. This makes room 0 vacant and the new arival sleeps there. If 2 people arrive at the hotel, you can have everyone move to their room + 2. Now what about the case where an infinate number of people want rooms in the hotel. In that case the hotel can once again call up everyone in the hotel and tell them each to move to their room * 2. This frees every odd numbered room in the hotel, and there is room for an infinate number of people. This shows how finite numbers and infinite numbers can have relationships, and how infinate variables can do strange things in word problems.

Another problem that was described to me is if you have two buckets. You do the following steps an infinate number of times.

Have 2 variables, X and Y both at zero.

Place X and X + 1 in the first bucket, and increment X by 1.

Take Y from the first bucket and place it in the second bucket and increment Y by 1.

Repeat.

When you are done, what numbers are in each bucket? Since you do it an infinate number of times, there are no numbers that can be in the first bucket, and all numbers will be in the second bucket, dispite the fact that logic will state that there should be the same number of numbers in the first bucket as the second. This shows how logical assupmtions and resonable truths can be bent by the rules of infinity.

In the case of the repeating decimal, we are not talking about a finite number. Instead think of it as saying that after each repetition there is another repetition.

For example if you take 9999 and you take off a nine from the front, if there is always another repetition it is still 9999 you take off 100 million, or more and you still get 9999

All fractions are exactly equal to some number with repeating decimals. Many of them have repeating zeroes however, and it that case the zeroes are simply not written.

Infinity is a fickle thing and hard to understand. When I was first posed with the bucket example, I tried to find some way to get an equal size of each of the buckets, when I did so I got it wrong. When the answer was revealed however it made perfect sense. I think that is where I really began to understand infinity.

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"debates"? What debate? That makes the issue sound as if it were debatable.

Sadly, idiots v. sensible, intelligent people is a debate that will never end.

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I do not believe there are any idiots in the world. Just different forms of Inteligence.

Since the real question about the difference between 1 and .999 is what infinity means, I would like to show you this page that I found a few minutes ago. http://diveintomark.org/archives/2003/12/04/infinite-hotel It shows the significant difference between thinking of infinity and thinking of real numbers is that infinity is the lack of a bound. Infinity means there is no number larger than it, even itself, and even itself * itself, or itself to the power of itself.

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I agree that, in conceptual terms, .999 repeat is 1. And who can say that .999 repeat exists in anything outside of conceptual terms, anyway? Of course, if such a thing can exist, so too can an infinitely small number that is the difference between 1 and .9999. If infinity can exist, then so too can a difference, that is infinitly small, between .999 repeat and 1. And if there exists such a difference, then we can say that the two are not equal. In one sense, at least.

I also think that people who go around calling others idiots, simply because they attempt to understand things in a different (and in this case, unpopular) way, are most often the idiots. But then, I am being a bit abrasive here, too. I suppose I am just trying to say that I took offense to being called an idiot, indirectly, for adding my thoughts to this thread.

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Do you consider infinity a number. Because for any number X, X != X + 1 infinity does.

Anything that is small but has any size at all has a finite size. Infinity is the lack of a bound, so 1 / infinity would equal zero.

If you have 1 / X, then X (1 / X) = 1 but in that case 2X ( 1 / X ) = 2. That means that since infinity * 2 = infinity then infinity ( 1 / infinity ) = 1 and infinity ( 1 / infinity ) = 2 a contradiction.

Math cannot be done with infinity as a number, however when you use limits you can, in that case 1 / infinity = 0 and 1 / 0 = infinity, meaning that something infinately small is equal to nothing.

Infinity is not a number that math can be done with.

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Not quite. If you're using limits (as I hoped you were in the last question), then [imath]\lim_{x\to 0}\frac{1}{x}[/imath] is undefined. You need to take either the left or right limit to get plus or minus infinity respectively.

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I agree that' date=' in conceptual terms, .999 repeat is 1. And who can say that .999 repeat exists in anything outside of conceptual terms, anyway? Of course, [i']if such a thing can exist, so too can an infinitely small number that is the difference between 1 and .9999[/i]. If infinity can exist, then so too can a difference, that is infinitly small, between .999 repeat and 1. And if there exists such a difference, then we can say that the two are not equal. In one sense, at least.

An infinitely small number can exist in another system, like the hyperreals (*R), but not in R. Once you agree that we are talking about the real numbers, then you must play by R's rules. There is no such thing as a real non-zero number with an infinitesimal value in R -- it follows directly from the Archimedean property. Further, that 1 and .999_ are equal follows directly from that, since there is no infinitesimal value you can get by subtracting them.
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In my opinion:

if it isn't a 1, it is not a 1.

If it doesn't look like a 1 it isn't a 1.

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There's no way that you or anyone else could possibly reach any sensible mathematical conclusion without falling back on mathematical proof, and proofs that establish the truth of the statement 0.999...=1 abound both in textbooks and on the internet.

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In my opinion:

if it isn't a 1' date=' it is not a 1.

If it doesn't look like a 1 it isn't a 1.[/quote']

That's a bit silly really. What's the difference between saying "2/2 doesn't look like a 1, therefore it isn't a 1"? The two are mathematically equal, and I'm willing to bet that you won't be arguing that fact.

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you have two parts of something that was divided into two, making it into a whole one.

it may be one.. however.. it isn't exactly one nor is it originally whole. sure it may be together.. but not originally whole since it is divided and then combined. Of course then standard for being whole is an abstract idea of what whole means..

I'm done here. Talking about the earth number system is pointless. It's all an abstract way of putting things into symbols anyways. who cares?

anyways.. .001 + 0.999 = 1

.0 (infinite 0s) 1 + 0.0 (infinite 9s) 9 = 1

i think since humans are analog (i will not believe of infinite as a concept.. some of you already know what I think of infinite anyways).. i believe it is a reality.. and since a feature of analog in the universe is that analog beings can reach any high number achievable infinite is a reality. However digitally infinite is constrained..

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anyways.. .001 + 0.999 = 1

.0 (infinite 0s) 1 + 0.0 (infinite 9s) 9 = 1

What you're saying isn't possible. You can let the last digit be recurring' date=' but not another one. So we're talking about 0.99[u']9[/u] and not 0.999, not that it really matters in this case. It does matter with 0.001, which isn't possible. You cannot have an infinite number of recurring 0's, and then still 'add a 1'.

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it may be one.. however.. it isn't exactly one nor is it originally whole.

No' date=' 2/2 is exactly 1.

anyways.. .001 + 0.999 = 1

.0 (infinite 0s) 1 + 0.0 (infinite 9s) 9 = 1

No, 0.001 is not a real number.

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• 2 weeks later...

Every proof used has a problem. You are treating 0.999... as a discrete number and comparing it to 1. 0.999... has no value in terms of discrete numbers such as 1. How much is 0.999...? Infinitely close to 1. It is not a discrete number and is not therefore comparable.

Intuition tells us that 0.999... is less than 1. Arithmetic tells us that 0.999... is equal to 1. However, logic tells us that 0.999... is not arithmetically related to 1 because it is infitecimal. The proofs apply discrete arithmetic to a non-discrete number.

Now you're goin to prove that 1.000.... = 0.999... with the same silly arithmetic. Stop dividing infinitely, stop adding infinitely. You can't make apple pie from florida oranges.

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Every proof used has a problem. You are treating 0.999... as a discrete number and comparing it to 1. 0.999... has no value in terms of discrete numbers such as 1. How much is 0.999...? Infinitely close to 1. It is not a discrete number and is not therefore comparable.

Intuition tells us that 0.999... is less than 1. Arithmetic tells us that 0.999... is equal to 1. However' date=' logic tells us that 0.999... is not arithmetically related to 1 because it is infitecimal. The proofs apply discrete arithmetic to a non-discrete number.

Now you're goin to prove that 1.000.... = 0.999... with the same silly arithmetic. Stop dividing infinitely, stop adding infinitely. You can't make apple pie from florida oranges.[/quote']

Logic? Logic tells me that $.\=(9)$ is equal to one. There is a very simple, mathematically rigorous proof for this.

All real numbers are BY DEFINITION defined by the infinate LIMITS of their decimal representations, not the representations by themselves. Many times the limit and the representation are one in the same, however other times it is not. One blatently obvious case is the set of all irrational numbers: $\pi$, $e$, $\phi$, $sqrt(2)$, etc, which, if they weren't defined as limits, couldn't be irrational in the first place, because no matter how many decimals you added to the sequence you would still end up with a rational number.

So with that in mind, let us examine the number $.\=(9)$

By definition, this decimal is defined as:

$lim_{x\to\infty}\sum_{n=1}^{x}\frac{9}{10^n}$

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I noticed that if I type .9999999999 (ten nines) into my TI-83 calculator and press enter, it gives me the answer to be .9999999999 (ten nines), But if I type .99999999999 (eleven nines) into my calculator and press enter it gives the answer to be one. I suppose my TI-83 rounds to the 10th decimal place.

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Every proof used has a problem. You are treating 0.999... as a discrete number and comparing it to 1. 0.999... has no value in terms of discrete numbers such as 1. How much is 0.999...? Infinitely close to 1. It is not a discrete number and is not therefore comparable.

If by "discrete number" you meant "integer' date='" I fail to see how that makes your argument meaningful. All I see is question begging. "They're different, so they're not equal."

Intuition tells us that 0.999... is less than 1. Arithmetic tells us that 0.999... is equal to 1. However, logic tells us that 0.999... is not arithmetically related to 1 because it is infitecimal. The proofs apply discrete arithmetic to a non-discrete number.

Now you're goin to prove that 1.000.... = 0.999... with the same silly arithmetic. Stop dividing infinitely, stop adding infinitely. You can't make apple pie from florida oranges.

You and all the other proponents of 0.9_ not equaling 1 really need to get over this idea that if some expression is to be evaluated numerically in an infinite number of steps that it must therefore be some sort of phantom number. Consider that the square root of 2 has an infinite number of digits. If you multiplied it by itself, you'd be multiplying two infinitely long numbers, yet even a grade schooler can see that the product is 2 and wouldn't argue to "stop [multiplying] infinitely."

Likewise, the solutions to other infinite operations can be deduced without resorting to numerical/empirical methods that involve actually carrying out an infinite number of steps. As has already been shown a zillion times, 0.9_ is easily written as an infinite geometric series, with a sum that is rigorously deduced to be 1.

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I noticed that if I type .9999999999 (ten nines) into my TI-83 calculator and press enter, it gives me the answer to be .9999999999 (ten nines), But if I type .99999999999 (eleven nines) into my calculator and press enter it gives the answer to be one. I suppose my TI-83 rounds to the 10th decimal place.

Yup, I think most (If not all) calculators do at some point.

I know you guys have gone through everything in the link but maybe you cna find something there you missed to mull over

http://mathforum.org/dr.math/faq/faq.0.9999.html

Cheers,

Ryan Jones

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Every proof used has a problem. You are treating 0.999... as a discrete number and comparing it to 1. 0.999... has no value in terms of discrete numbers such as 1. How much is 0.999...? Infinitely close to 1. It is not a discrete number and is not therefore comparable.

Intuition tells us that 0.999... is less than 1. Arithmetic tells us that 0.999... is equal to 1. However' date=' logic tells us that 0.999... is not arithmetically related to 1 because it is infitecimal. The proofs apply discrete arithmetic to a non-discrete number.

Now you're goin to prove that 1.000.... = 0.999... with the same silly arithmetic. Stop dividing infinitely, stop adding infinitely. You can't make apple pie from florida oranges.[/quote']

So .999... isn't a number? Then how can it be less than or equal to 1?

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I think there is a slight difference between 0.99 and 1 or 0.999 and 1 etc…

Take an example of a system that succeeds 99.999% of the time to do whatever task and fails the remaining 1*10^-3%. This effectively means that the system will fail once every 100,000 attempts. This sounds a bit faulty since you guys are debating 0.999 and not 0.999 but a system or law of nature that succeeds 99.999% keeps that very slight possibility of failure while a 100% successful system eliminates anything other than success. This may sound stupid but there is a difference because that 0.0001% failure exits in a 99.999% successful system while the 100% leaves no margin for debate! (Failure wise)

An example of this is a quantum fluctuation whereby the possibility of you randomly chosen (by nature) to be tunneled through a concrete wall to another room exits! although very small. The very nature of that small probability existing allows me to debate it, while otherwise I wouldn’t be able to.

Another example I can think of is a trans-planetary missile that is 99.999% accurate. When launching that missile to Pluto for example, a tolerance between where it is aimed and where it will hit exists and is certain.

I have to admit that reading your posts reminded me when I got worried about birth control pills because I thought they were 99% effective (actually 99.99%) and thought the law of large numbers will get me! Hence my 1st post.

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This sounds a bit faulty since you guys are debating 0.999 and not 0.999 but a system or law of nature that succeeds 99.999% keeps that very slight possibility of failure while a 100% successful system eliminates anything other than success.

It is faulty' date=' for that very reason. It is clear that [imath']0.999 \neq 1[/imath].

Also, why bother to bring up physical systems or quantum fluctuations? They have nothing whatsoever to do with determining a mathematical truth.

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Because we happen to live on earth.

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uh, you say 0.0001% failure exists... well it doesn't. You can't have repeating zero's, then a 1. If it could equal anything it would be 0. Which would just prove the point that it'd be a 0% failure and that 99.9% = 100%.

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Because we happen to live on earth.

Mathematics is an a priori discipline. All mathematical theorems follow from mathematical definitions by mathematical rules of inference. The physical universe plays no role in deciding mathematical truths, apart from the fact that material brains are needed to carry out the thought processes.

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