## Jan 17, 2012

### .999 repeating does not equal 1

Scott Wilhelm nerd-sniped me with this video that claims infinity equals -1

The method used in the video is being basically the same method used in the false proof that .999 repeating = 1.

I believe that both of those proofs are wrong. The problem is that they are engaging in mathematical legerdemain and using an infinite series as a real number. But I don't believe it is mathematically possible to use infinite series as a real number. You can't multiply it by anything. It is infinite. You end up with terms that cannot be mixed, like a blue mile or a red 65 miles per hour. It just doesn't make sense or describe anything real.

This is here is some fucking crazy calculus, but I think it is basically saying you cannot treat an infinite series as a sum, which is what the .999.. = 1 and the infinity video are trying to do. See the part beginning with:
Before we move on to a different topic let’s discuss multiplication of series briefly.
It says that you cannot treat multiplying a series as just multiplying the constant terms, you have to distribute each term into each other and then combine them, which is literally impossible when dealing with an infinite series. The only way to do it is to represent it with notation (which is another infinite series), but you cannot derive a real number out of it. And that is the key. An infinite series is not a sequence of numbers. See the concluding two paragraphs:
We’ll leave this section with an important warning about terminology. Don’t get sequences and series confused! A sequence is a list of numbers written in a specific order while an infinite series is a limit of a sequence of finite series and hence, if it exists will be a single value.

So, once again, a sequence is a list of numbers while a series is a single number, provided it makes sense to even compute the series. Students will often confuse the two and try to use facts pertaining to one on the other. However, since they are different beasts this just won’t work. There will be problems where we are using both sequences and series so we’ll always have to remember that they are different.
In that the infinity video, the speaker is giving us an infinite series and then claiming that is identical to a sequence of digits that can be cancelled out using another sequence. But you can't do that. You can't compare an infinite series to a sequence in that way. It's an abuse of terms.

An infinite series is a single number, provided it makes sense to even compute the series. And in that case the computation of the series is a number, but that is not the same thing as saying the series itself is the number. That's the same as how a fully grown apple seed is a tree, but an apple seed itself is not a tree.

If you want to perform mathematics using series you must use only series. In that case you are using notation so you would use .999.. and never resolve it to a real number. If you have an infinite series and you want to use it with real numbers you must take the limit of it, which is more crazy calculus but as I understand it it is a way to find the real number to which the series gets the closest (if any) with infinitely diminishing margin of error. In the case of .999.. that is in fact 1, because there is no real number closer to .999.. than 1.

BUT, in this case we aren't saying that .999.. equals 1, we are saying that the limit of .999.. is equivalent to one. Those are not the same statements.

LONG STORY SHORT:

You can say that 1 = 1 or that .999 to infinity = .999 to infinity, but you cannot say that .999 to infinity equals 1. It is exactly like this:
An apple equals a fruit.
An orange equals a fruit.
If you cancel out the fruit term, you get an apple equals an orange.
But you can't compare apples to oranges!

The infinity video I believe both has the method wrong and the conclusion wrong. Infinity is not equal to -1, because the limit of infinity is not -1.

The .999 repeating equals 1 has the method wrong, because you can't perform sequential number calculations on an infinite series, and the conclusion is syntactically wrong. It's the limit of .999.. that is equal to 1.

I know some posters on Dark Taco are much better trained math wizards than I. What say you?

1. I've said it, in the context of another argument:

http://www.darktaco.com/2009/09/god-is-not-infinite.html

Your claim amounts to an outright rejection of calculus. The whole point is that you can use an infinite series to represent a theoretically derived constant. Rejecting that will put a serious dent in... well, almost everything since the mid 1600's.

From there, you dive into what accounts to eloquent numerology, culminating in a syllogism that requires two false premises to make your point. Apple is a subset of Fruit, not an equivalent.

In the end, I'll refer to a quote in that other blog from a favorite logic professor of mine:
" if you had a decimal and an infinite string of nines, then if this number really was less than one, what number would you add to it to get one? The answer, of course, is a decimal followed by an infinite string of zeros with a one at the end....which, of course, is incoherent. No infinite string of zeros can have a one at the end!"

2. Anonymous4.3.12

.999... is infinitely unequal to 1.

Why is that so hard to accept? Your calculus problems/solutions will still work the way you want them to. Just a matter of faith really. Yeah, I know, hard to accept that last assertion, while it is the truth.

How is .000...1 incoherent? Weird that one can (pretend to) grasp .999... and then not grasp what is right in front of them and clearly written as .000...1. An infinite string of zeros can have any number on the conceived end. When utilizing the dot dot dot to represent infinity, you are clearly in the abstract / conceptual as no one reading this has done long form of 1/3 to get .333...

Believing (and let's be clear that it is belief) that .333... is equal to 1/3 is incoherent, but many stand by it as if it is 'crystal clear sense.' You can use that same faith to see 'crystal clear sense' in .000...1

3. You may be shocked by this, but faith is very rarely successful in discovering new mathematical or scientific techniques, even if you use them to try to grasp (incorrect) current ones.

I'd be delighted if you could indicate some instances where I'm wrong, of course.

4. Anonymous5.3.12

And I would be equally delighted if you could provide examples whereby faith is not invoked in the substantiation of any mathematical technique / proposition.

But to take you up on your 'suggestion' which conveniently sidesteps the truth provided for you in earlier comment, let us observe:

RS: "Your claim amounts to an outright rejection of calculus"
is invocation and substantiation of faith. That the claim represents to RS a rejection, is faith. That calculus could plausibly be rejected, is faith. That the assertion when taken as whole 'claim' that is coherent, and furthermore (superficially) true or accurate is substantiating the faith. It is not devoid of reason, while at same time, the basis for the rationale has yet to be invoked within the assertion.

RS: "The whole point is that you can use an infinite series to represent a theoretically derived constant."
holds to faith in 'whole point' as well as faith in "representation," thus substantiating faith in this assertion as well. Again, it is not devoid of reason, but it is neither invoking basis of reason, other than to in the next assertion lead to a (false) appeal to (psuedo) authority, as in...

RS: Rejecting that will put a serious dent in... well, almost everything since the mid 1600's.
... which is a most interesting claim as the claim assumes to understand what 'serious dent in... almost everything' actually entails. This assumption is based in faith, and not rigorous analyses or well defined reasoning.

Infinitely unequal is observable within the long form of 1/3, and can be demonstrated whenever a reasonable person attempts to exploit the infinite nature of the math 'problem.'

Furthermore, for "x" to equal 0.999 in the algebraic (false) proof of the central argument, the x is rationally observed as finite, while invoking correspondence to an infinite numeration. This relies on faith to make the correspondence, for it would be far more accurate / rational to represent xxx (infinitely) or xxx... as equal to 0.999..., then to use false rational that finite x can legitimately correspond to infinitely repeating decimal. It cannot (rationally) but as a matter of faith, it often does find (sense of) corresponding relation.

More accurate / rational = xxx... = 0.999...
Less accurate / faith based = x = 0.999....

5. Anonymous7.3.12

Does not 10 times anything add a zero onto the end of the result, automatically. In which way does 10 times 0.999.... add a zero onto that? Please provide that proof, such that we can accurately say (and observe) that 10x (in algebraic proof) would conceivably be equal to 0.999...

In essence we are adding a zero onto 0.999... such that it would have to at some point be seen as 0.999...0 - which is as incoherent, I argue, as the number 0.000...1

It seems all too convenient, and highly inaccurate, to truncate 0.999 (infinite) and then neglect the inevitable zero that must appear at end of 0.999...

Utilization and acknowledgement of that zero changes everything in understanding the algebraic proof. Without that utilization appropriated, the equation relies on faith, and nothing less.

2. Anonymous17.8.12

That isn't entirely true though... it turns out to be a non-trivial problem:
http://www.blog.republicofmath.com/archives/1412

6. this is a disgrace.

i'll say what i've said to everyone who disputes something as silly as this. first look up the definition of infinity, next the definition of a real number.

i'll be nice:

Infinity: never ending
Real number: a quantity that can be represented along a continuous line.

in other words if two numbers have zero difference between them than as far as real numbers are concerned (that's what we're discussing right? don't dispute this at the very least) THEY ARE THE SAME NUMBER

now, what's 1-.999...?
.0000...1 i'd think you'd say, but i'll ask, how the hell can that one exist? there are infinite nines in .999...so in return there are infinite 0's, what's the definition of infinite again? NEVER ENDING. so how can you put a 1 after a line of zeros that never end? YOU CANNOT. 1-.999 is in fact 0.000... = 0. there is no difference between them and they are the same number, end of the story.

1. Anonymous5.12.12

You said, "how how can you put a 1 after a line of zeros that never end?"

Done rather easy when one is confronted with faith based arguments whereby .999... must equal to 1. You say "you cannot." I say "I did."

There is a difference between the 2 numbers being representative of conceived values. That difference may (or may not be) irrational, but the current proofs available currently show that .999... is infinitely unequal to 1.

End of story.
(That has no end)

7. also LOL at faith

8. Just think, blogs that disable anonymous comments are missing out on all of this.

Threads like this are why I keep Darktaco going. Even though I squarely hold one of the two competing viewpoints, an open discourse on it can never be a bad thing.

Cheers around!

9. Anonymous2.5.12

This post is entirely wrong. There is no argument. 0.999... is equal to one.

1 divided by 3 = 1/3
1/3 = 0.333...
1/3 multiplied by 3 = 1 or 0.999...
0.999... = 1

1. Anonymous5.12.12

1/3 is infinitely unequal to 0.333....

10. Anonymous2.5.12

"Does not 10 times anything add a zero onto the end of the result, automatically. In which way does 10 times 0.999.... add a zero onto that? Please provide that proof, such that we can accurately say (and observe) that 10x (in algebraic proof) would conceivably be equal to 0.999..."

No. It moves the decimal place over one. 2.4 * 10 = 24. No zeroes added.

1. Anonymous5.12.12

Zero added then dropped, for convenience of representation. The added zero doesn't change the implied value in this case.

11. Whoop3.5.12

1/3 isnt 0.333..., its just not.
1 - 0.999... doesnt equal 0.000...1 or 0.000..., it actually equals 0.000[as much zero's as (you chose) 9's minus 1]1.

Hopefully everyone will get this but its about the limit you choose, as soon as you choose a limit, you lost infinity, and to perform any truly valid operation you have to put a limit, because you cannot operate with true infinity. This limit will also be the same limit in the result of the operation, thus giving you a really small number. 0.999... is not 1, we have to use it as 1 at some point in our math because we define a limit to the amount of decimals we use. This may still be a conceptual limit, but a limit nonetheless.

12. Whoop3.5.12

In other words, the same goes for all these nifty calculations with 'infinity' 0.999... like saying 0.999... + 0.999... = 1.999...; its really not. No its not. No really. Stop. its not. We must calculate like that because otherwise none of calculus will work, but the truth is that at the moment you performed that operation, you technically lost infinity. You took a moment in time and tried to pin down at least one of the numbers that you previously wrote down as infinity. You can never add something that you dont know the value of, and you surely dont know the value of infinity.

So to even ever perform this operation, you were forced to limit yourself, and thus somewhere, at the end is now an 8. You just didnt define how many decimals you limited yourself to use, so the true answer is; 1.999[some amount of 9's]8. But how can we calculate without even knowing how many decimals we are using? We can't, so we're forced to stop and forced to use 1 instead of 0.999... This whole discussion is pointless because both sides are right. You can use calculus with sort of infinity, and you have to say 0.999... is 1 to be able to use math, but.. thats not a true representation of infinity. a theoretical zero with truly infinite recurring 9's never equals 1, but its completely useless in math or life for that matter.

13. Whoop3.5.12

Maybe another way to further explain: Infinity is an inherently unknown number. By definition its always outside of what you can put a number on.

You can never calculate A+B if you are never told what either is, they will always remain A+B, just like
somethingwithinfinity + somethingelsewithinfinity
Always remains
somethingwithinfinity + somethingelsewithinfinity
There is no way to put something behind the = sign.

In these examples with .999... its hard to grasp because its a number close to a number we are familiar with and it tricks us into making technical mistakes without realizing it.

Imagine a number that consists of a zero and then a random infinite pattern of digits, so
0.[infinite random digits recurring]

And I ask you to multiply it by 2, you will have no clue, except for that its below 2 and above 0. Your answer would always be wrong. However a situation may occur where its enough to know that its between 0 and 2.

The problem arises because you dont KNOW what the number is. If we do the same with 0.666... the only difference is that you have a much better estimate where it is, you have an upper limit of 0.667 (you know its somewhere below that) and a lower limit of say 0.666. So i can ask you to define 0.666... within 3 decimals and you obviously will say 0.667, because you know its 6 recurring and that will always make it lean more towards that 7. However you dont KNOW its 0.667, actually you know its NOT 0.667. What you know is that the more decimals you add, the closer you will be to the real number, but you will never be able to define it (which is what makes it infinite). So if you want to use the number in any calculation, you will have to use as many decimals as possible to minimize impact of your 'unknown' information. So 0.66666666666667 or with trillions of 6's and the 7 hidden at the end. It may make a huge difference depending on where you apply it.

Now the confusion about 1 and 0.999... is that where 0.666[maximum number of 6's]7 is the ideal representation for 2/3, the round 1 is the ideal representation of 0.999..., there will never be any amount of decimals you can use to get a better representation of what you do know, than by using the simple 1. So where a gazillion of decimals would be ideal for representing 2/3, you can create the most amount of decimals ever known to man to represent 0.999... but youll still be 9 times further away from the correct representation than using 1. Thus you are forced to use 1.

But ONLY because you have to work with an estimate of where the number could be, not because it IS 1.

14. It's not a matter of simply a repeating digit, though. Of course 0.666... doesn't equal 0.667, because I can tell you that the difference between those two is at least .0003. I challenge you to give me a lower bound on the value that lies between .999... to 1.000.

Imagine if you divided 1 by the difference between 1 and .999...

It's a loaded question, but it does offer one more way to try to imagine the significance of zero in relation to infinity. What is 1/0? Well, there's at least one zero in 1, as by taking zero away from one, I still have more than zero. If I take another one away, this is still the case, so the answer must be more than two. If I take yet another zero out of one, the persistence of the remainder to be greater than zero suggests that the solution must be greater than three.

Simply because you can keep going doesn't mean that there's something to find at the end. Calculus is not just galactic rounding.

15. whoop7.5.12

Don't get me wrong though, I understand the comparison you're making. But zero is just nothing so infinitely adding nothing to nothing would always nothing. However the difference between 1 and 0.999... could be called the infinitely smallest value that's not 0, in which case if you could infinitely add those to eachother you would have something (arguably an infinitely large number, or 1). And just thinking about that and in relation to 1, shows how little we can actually say about infinity using math.

16. Whoop7.5.12

Hmm weird, something went wrong with my initial reply. (before the "dont get me wrong")

What I responded on your challenge was that the lower bound is 0.999 with as much 9's as you can use (much like 0.667) and the upper bound is always 1. However because you always know that the next digit would be 9, which is closer to 10 than 0, you would always round to 1 instead of 0.99999. That doesn't mean there is no difference though, it just means that we have no way of representing (or using) that difference in math.
The zero question is semantics and nothing like the problem above. You're just showing impossible use of the language math.

17. some of you idiots are claiming that 0.000...1 (an infinite string of 0s followed by a 1 at the end) is perfectly coherent. it's not. you completely misunderstand decimal representation.

take any decimal expansion you want, like 51.29385958493... There is the tenths place (where the 2 is located), the hundredths place (where the first 9 is located), the thousandths place, and finally the '10^n'ths place for every natural number n.

In other words if you give me a decimal expansion and want me to describe it, then for every whole number n > 0, ANY place 10^n, I should be able to tell you which number among 1, 2, ... 9, 0, goes into that place.

This is why 0.999... is a perfectly reasonable and coherent number. For any place, tenths, hundredths, thousandths, ten-thousandths, or (10^n)ths for any n > 0, I can tell you what goes in that place: it's a 9.

What about 0.000...1? In which place does the 1 lie? The tenths place? The millionth place? The billionth? Nope, you said yourself it doesn't lie in any of those places. It seems to me that since for each '10^n' place there is a zero registered there, that when you talk about the number 0.000...1, you are really just talking about zero.

And don't bring up that bullshit about "well durr its like infinitely close to 0 but not quite lol." you don't know what you're talking about.

1. Anonymous3.9.12

it;s not infitly close but the limit is one. it never quite reaches one no matter how much i expand it. for all mathematical purposes, i understand where you are coming from. (0.999...)does in fact equal 1 the same way (0.666...) equals (0.667) however realize that this only applies because we are rounding. after all, the only reason (0.0...111) doesn't exist is because we can't round zero to one. it's the same reason (0.333...) doesn't have an upper limit. so while (0.999...)=1 for all mathematical computations, .999... does not ever actually equal 1.

2. Anonymous4.12.12

I'm glad there are people who actually know what they're talking about. Never did Shankman talk about limits. If you use limits, then yes technically .999repeating does not equal 1, BUT WE AREN'T USING LIMITS.

We're talking theoretically if the 9s repeat infinitely then it does, but ONLY if they repeat infinitely. We aren't rounding, we aren't using limits, we are talking about a theoretical approach to viewing differences between two numbers that are in fact equal to each other.

3. Anonymous5.12.12

Theoretical approach = faith based
When able to show realistic approach, I will remove the claim of faith from this equation.

18. gavin22.9.12

Mr. Houchin, I agree with your argument here--although I'm not a mathetician so I don't know if my agreement is comforting to you.

I also believe this to be a type of sleight of hand that works well because modern mathematics has successfully added many layers of obfuscation to the reality of what we are actually describing. We get confused by our own convenient representations of the truth, and sometimes our shorthand causes us to lose some of the original "information".

Let's first assume (as you point out) that infinity is not a real number. It is shorthand for expressing an idea--something we have invented in our minds. The idea describes something like "an amount so large than it cannot be quantified". But of course, something which cannot be quantified is not a number, therefore cannot be "mixed" with numbers.

Think of another idea: love. Now, let's assume (incorrectly) that we can mix love with a real number to achieve a single result. For instance, 100 x love = love00 , or love / 100 = lo.ve. You can obviously see absurdity of this assumption.

Secondly, let's express .999 repeating as an expression of something closer to reality. But before we do that, let's take away the "repeating", and add it on afterward.

.999 is really just shorthand for 999/1000 (999 divided 1000 ways). Multiplying that expression by 10, and the result becomes 999/100 or 9+99/100. But how did we get there? We achived that expression by reducing the denominator by a factor of 10. That is easy to achieve because the denominator is a number. But what happens when we express the original shorthand of .999 repeating in a way that expresses is essential "truth"?:

9 repeated-an-infinite-times / 10^infinity

and after multiplying by 10?

10 x (9 repeated-an-infinite-times / 10^infinity)

Can that denominator (10^infinity) be reduced by a factor of 10? No it cannot, because we cannot quantify the denomiator, and cannot express the reduced result as a single number. But it is clumsy to have to write: 10 x (9 repeated-an-infinite-times / 10^infinity), so understandably we have learned to express this expression as .999 repeating.

I hope this crude argument doesn't distract readers from the argument I'm trying to convey...

1. He is wrong here though

19. Yes. It is true that 0.999... is not equal to 1 for the very reasons you stated, the most important being that 0.999... is an ill-defined concept and one runs into problems with arithmetic when treating 0.999... as a number. Arithmetic is guaranteed only to function correctly when the objects are rational numbers. Real numbers do not exist.

20. In fact, pages 29-31 of my article proof_that_0.999_not_equal_1.pdf prove that 1/3 is NOT equal to 0.333...

21. John, that proof requires the premise "Any process involving an infinite number of steps is ill-defined."

You use the term "ill-defined" 23 times in that paper, without defining it.

Even granting that, this premise would require calculus to be "ill-defined" and yet it seems to be pretty well accepted by the academic majority for the past few hundred years.

22. No one here has defined what 0.999... means. The decimal representation of a real number is defined as the limit of a sequence (see http://en.wikipedia.org/wiki/Decimal_representation and http://en.wikipedia.org/wiki/Limit_of_a_sequence). Unless you understand those two concept, you will not understand why 0.999... = 1 in the Real Number System. Take a while to learn and understand more deeply these concepts of everyday mathematics, and it will become clear to you.

Mathematics is axiomatic; this means that theorems just like this can be proven from definitions, axioms, lemmas, and other theorems (notice how faith is not involved here...). Now, if you choose to use different axioms and definitions, you can get different results. For example, 0.999... = 1 for real numbers, but /not/ for hyperreal numbers (see http://en.wikipedia.org/wiki/Hyperreal_number). This is because there is a hyperreal number which is greater than 0 and less than all the positive numbers. There is no real number that is greater than 0 and less than all the other positive numbers!

Summary: 0.999... is 1 by the definition of 0.999.... You can use a different definition of 0.999..., but then you cease to argue against people who claim 0.999... = 1 and begin to show that, in another system, 0.999... does not equal 1.

23. Bryan summed it up well. .999... is a rational (and therefore real) number. We know this simply because of the definition of a rational number (it ends in a repeating decimal). Darktaco's argument that it is not a number makes no sense. Once you accept this basic fact, there's no escaping the conclusion that .999... = 1. If one wants to talk about matters of faith or false proofs, those only apply to people who start their argument with an outright rejection of facts (i.e., those who say .999... is not a real number).

1. Correct, although the appeals to faith are a bit of a reaction to the original article (http://www.darktaco.com/2009/09/god-is-not-infinite.html), where I was using the assertion to make a philosophical argument. I still have a pretty serious suspicion that Richard merely pulled of the most successful, and long-lasting, troll in DarkTaco history with this article.

24. I came across a Scientopia article that seems relevant to this discussion. Numberphile asserted than an infinite series of numbers (1-1+1-1...) converges to a specific number, and Scientopia refutes that by noting that an infinite series of numbers does not converge to a specific number -- even if you can reasonably round it, it is not in fact equal to the number you rounded it to.

Much like S1, 0.999 is a non-converging series. It doesn't converge to 1, it doesn't converge at all. It can be rounded or summed to a value but as an infinite series, 0.999... itself does not converge. There fore, 0.999... is not = to 1.

1. "an infinite series of numbers does not converge to a specific number"

You should reread the article you linked to, as I don't think you understand it. Scientopia is saying that this particular series (S1) does not converge, even though it has a Cesaro sum. But there are clearly convergent series that exist. In the case of convergent series, the sum of the series is equal to the Cesaro sum.

"Much like S1, 0.999 is a non-converging series."

First, you refer to a number and call it a series without any justification. 0.999... is quite clearly a rational number. I can also specify an infinite series that has 0.999... as its sum: 9/(10^i), i=1 to infinity. Or, we can rewrite as 9*((1/10)^i). In this case, it is clear that it is a convergent geometric series, the sum of which is 1.

25. 0.(9) ≠ 1 for several reasons. For all the proofs, they are erroneous.
The algebra proof is circular. It assumes 0.(9) = 1 TWICE in the proof:

Here is the fake proof
1) x = 0.(9)
2) 10x = 9.(9)
3) 10x - x = 9.(9) - 0.(9)
4) 9x = 9.0
5) x = 1

I used line numbers to refer to all the mistakes in this so-called proof.

First off, alarms should have rang when we ended up with a different value for the constant that was already defined before the math began. For instance, if we started with x = 2, and ended up with x = 5 at the end, we know there was an error.

There are 2 problems with the above proof. Let's start with the first one.

Problem 1:
in the second step (see line 2) x is assumed to be 1, and then carried through to the end, and then is concluded that it equals 1.
I will refer to the left hand side of the equation as LHS and right hand side as RHS.

In step 2, multiplying by 10, we end up with 10x on the LHS, and 9.(9) on the RHS.

This is a difference of 9x (from step 1) on the LHS, and 9 on the RHS, that is,
10x = x + 9x, and
9.(9) = 0.(9) + 9. If we added 9x on the LHS and 9 on the RHS, then x = 1 from this step forward. If x is assumed to be 1 during the proof, then we didn't prove anything, and the logic is circular.

Problem 2:
Assume that we didn't catch the error seen in problem 1. We get to step 4 and see that 9x = 9. Wait a minute...
→ if x = 0.(9), then 9 * 0.(9) = 8.(9).
In this step, we are assuming 8.(9) = 9.(0), which is what we are trying to prove. Again, we are assuming our conclusion in the proof, therefore it is, yet again, circular.

Let's use well defined numbers and use the same algorithm:
x = 5
10x = 50
(x + 9x = 5 + 45) In this step, notice that 9x = 45 assumes x STILL equals 5)
10x - x = 50-5
9x = 45
x = 5
The algorithm works fine for well defined numbers.

In the Faux-proof, we should be able to use the value of 'x' at the end, and repeat the process and get 0.(9):
x = 1
10x = 10
10x - x = 10-1
9x = 9
x = 1
Nope. If x = both 1 and 0.(9), then we should be able to get 0.(9) back, the same way we got 1, but we can't unless we assume it, just like we assumed 0.(9) was 1 in the faux-proof.

→ So what can we conclude? Basic arithmetic doesn't work on "numbers" with infinite repeating decimals. This same debunk can be used to show that 1/9 ≠ 0.(1), 2/9 ≠ 0.(2), etc etc...

I also like one of the very first comments. He said it was wrong because the algebra proof uses "x" as a constant and THEN a VARIABLE in the same equation. This is big NO NO!

1. "Basic arithmetic doesn't work on 'numbers' with infinite repeating decimals."

I love the scare quotes there. Hilarious. Numbers with infinite repeating decimals ... aka, rational numbers. Please, have an elementary understanding of math before commenting on stuff like this.

2. You don't get even basic understanding of equality. You do circular stuff because you assume they are different.

"First off, alarms should have rang when we ended up with a different value for the constant that was already defined before the math began."
Here you assume they are different values, rather than do the intellectual thing of wondering what it means and how it happened you assert they are different from the begining.

If you get 2=5 the question is where did it go wrong? We know from definition that they are distinct, but we do not know from construction of real numbers that 0,(9) and 1 are distinct, so we gotta check out why it happened and low and behold! division by 0 is the common culprit

26. Older discussions in this subject if you are interested.:
http://lasu2string.blogspot.com/2011/03/secret-behind-0999-1.html

27. " But I don't believe it is mathematically possible to use infinite series as a real number. You can't multiply it by anything. It is infinite. "
And you're wrong there, Iti s demonstrably finite because you can find a finite number greater and lesser than the one with decimal expansion
3 < pi < 4
so Pi is finite

28. This comment has been removed by the author.

29. I've been aware of this argument for years and I absolutely despise it. ...in essence I view the people that insist that .(9) = 1 are math trolls.

I maintain and always will maintain that if .(9) was meant to signify 1 then it would have been written as one. By insisting that .(9) = 1 then you lose a value representing something that is infinitesimally less than one and will never be one.

When rounding numbers you have to start at the right most value and work your way back to the decimal, with infinity there is no right most value to begin rounding from- ie: with 0.456789 you start with the 9, which makes the 8 a 9, the 7 an 8, the 6 a 7, the 5 a 6, the 4 a 5 and the the 0 a 1.

Therefore- to argue that .(9) = 1 is to argue that anything greater than .(4)=1. Slightly less than a half is not 1, and should never be considered as such.

Stupid, stupid debate. Fucking trolls.