Mathematician: Usually, in math, there are lots of ways of writing the same thing. For instance:
As it so happens, 0.9999… repeating is just another way of writing one. A slick way to see this is to use:
One.
Another approach, that makes it a bit clearer what is going on, is to consider limits. Let’s define:
…
and so on.
Now, our number 
Taking 1 and subtracting all parts of the equation from it gives:
Then, we observe that:
and hence
But we can make the left hand side into as small a positive number as we like by making n sufficiently large. That implies that 1-0.9999… must be smaller than every positive number. At the same time though, it must also be at least as big as zero, since 0.9999… is clearly not bigger than 1. Hence, the only possibility is that
and therefore that
What we see here is that 0.9999… is closer to 1 than any real number (since we showed that 1-0.9999… must be smaller than every positive number). This is intuitive given the infinitely repeating 9’s. But since there aren’t any numbers “between” 1 and all the real numbers less than 1, that means that 0.9999… can’t help but being exactly 1.
Update: As one commenter pointed out, I am assuming in this article certain things about 0.9999…. In particular, I am assuming that you already believe that it is a real number (or, if you like, that it has a few properties that we generally assume that numbers have). If you don’t believe this about 0.9999… or would like to see a discussion of these issues, you can check this out.
1/10^n as n->inf
What does that equal?
.1, .01, .001, ….
There. I just defined it.
“Oh boy it sure does :)”
No, it does not. Saying it exists does not mean it does. Unfortunately, mathematics is more complicated than that. It does not exist, it will never exist, and nothing you ever say can change this.
@Anonymous:
“1/10^n as n->inf
What does that equal?”
It equals 0. This is very easy to show using the rule of L’Hôpital. You also can look at the asymptotes of 10^(-n) and see it is indeed 0.
“.1, .01, .001, ….
There. I just defined it.”
No, you did not. What you defined is 0. Not an infinitesimal, and certainly not the nonsense of 0.000…1. Infinitesimals have no decimal expansion. They cannot have one, by definition and by construction. This is why there is no such a thing as 0.000…1. And there is no world in which you can define it. The limit you just defined is equal to 0, which is exactly what we said it is, so you in fact did prove that 0.9…= 1. Thank you for proving our point.
9.999… – 0.999… = (9-0.9) + (0.9-0.09) …. = 8.999….
The “slick” property you used is number shifting which I argue the indeces must remain the same, which will result in the above.
@Mathman:
Yes, and 8.999… = 9.0.
Also, for the record, shifting is not a real property. Shifting is an artifact of how we choose to write some numbers sometimes, not a real property of operations.
I hope you understand that your argument is circular: 0.999… = 1 because 8.999… = 9.
No thanks. “0.999… = 0.999…” and “1 = 1”.
The sum is not equal to the limit. The very definition says there is an epsilon, however small it is, is irrelevant.
Take 1 mile. Then go 9/10. Now, 9/10 of the remainder, recurring. You never get the to exactly one mile. That’s how limits work. You converge on the limit. You don’t actually get there.
@ Ángel “What you defined is zero”
The limit of that sequence is zero. The actual sequence is not equal to zero. It’s what it is: a sequence. Just like “0.999…” is a sequence or a sum. It is not the limit of itself. That’s why I reject the concept. 1 is the LIMIT of the sequence 0.9, 0.99, … ; it is not the sequence.
“The limit of that sequence is zero.”
Correct. And real numbers are limits of Cauchy sequences. Hence, you have proven my statement.
“The actual sequence is not equal to zero.”
Of course not. A number cannot equal a sequence. One is an element of a vector space which is not a field. The other one is an element of both a vector space and a scalar field. No one claimed the sequence is 0.
“It’s what it is: a sequence. Just like “0.999…” is a sequence or a sum.”
0.(9) is a sum, not a sequence. It is an element of a scalar field. It is not a sequence.
“It is not the limit of itself.”
Actually, every number is the limit of itself. By definition, lim x —>c [f(x) =
a] = a.
“That’s why I reject the concept.”
Then you obviously fail to understand either what limits are or what numbers are.
“1 is the LIMIT of the sequence 0.9, 0.99, … ; it is not the sequence.”
Correct, and no one claimed it is the sequence. 0.(9) is not the sequence either. It is merely, by definition, the limit of the sequence. It must be, since infinite sums are by definition limits. A basic idea you have failed to notice is that.
“I hope you understand that your argument is circular: 0.999… = 1 because 8.999… = 9.”
I never claimed that 0.(9) = 1 because 8.(9) = 9. You are deliberately putting words in my mouth now, which is exactly what a person without validatguments or a decent understanding of the topic would do as a scapegoat. No, I merely made two claims, I never made an argument concerning the two claims themselves, because I never made inferences from those two premises.
“No thanks. “0.999… = 0.999…” and “1 = 1”.”
Yes, both claims are true. 0.(9) = 1 is also a true claim, and this has no bearing on 0.(9) = 0.(9) or 1 = 1.
“The sum is not equal to the limit.”
Yes, it is, actually. By definition, it is.
“The very definition says there is an epsilon, however small it is, is irrelevant.”
Yes, the definition of the LIMIT contains an epsilon indeed, but the epsilon is not actually number that appears in the calculation.
“Take 1 mile. Then go 9/10. Now, 9/10 of the remainder, recurring. You never get the to exactly one mile. That’s how limits work.”
Not quite. If your function is a constant, then you are already at the limit. Your basic lack of understanding of the a limit is concerning. Anyhow, I know what a limit is as I literally have a degree on the subject, but what a limit is has no bearing on your claim, because your claim is that 0.(9) cannot be 1 because 1 is merely a limit, which is strictly false because numbers are defined via limits of sequences.
“You converge on the limit. You don’t actually get there.”
I could care less, it does not prove your point. Also, you are presenting an inconsistent set of premises, especially because you seem fine accepting infinitesimal numbers, which by necessity implies you accept infinite values, which literall cannot be reached by definition. Fun fact: in the surreal numbers, 0.(9) = 1 is still a true statement, and it is provable from the axioms too. You do not have a case. You are unable to demonstrate that 0.(9) – 1 = 0 is a false statement. You are also still unable to define a number equal to 0.000…1. This is because no such type of number can exist.
“Correct. And real numbers are limits of Cauchy sequences. Hence, you have proven my statement.”
→Not at all. The “number” is the limit of the sequence. 0.999… is a representation of the sequence (or infinite sum). Then by magic, it (0.999…) is also declared as the number itself. The limit is 1. That is the number. The symbol “0.999…” I will accept as a representation of the sequence or infinite sum; not the limit. LIMITS REMOVE THE INFINITY CONCEPT IN THE SOLUTION. Therefore the limit should not have a representation of infinity in the answer (limit) itself! (Logically, a number cannot be a never ending addition of other numbers. It’s senseless.)
“Actually, every number is the limit of itself. By definition, lim x —>c [f(x) =
a] = a.”
→Yes, if you accept that the symbols 0.999… are a number, but it represents a sequence/infinite sum, ie, {0.9, 0.99, … } or (0.9 + 0.09 + ….)
I haven’t failed to notice anything other than “0.999…” being called a number. I have easily completed advanced and engineering calculus, doing these kinds of problems. I “understand” what you want me to do, and I understand what you WANT me to believe. Just because I reject certain things doesn’t make it less understandable. Those are just plain ol’ empty attacks. Am I attacking you?
““You converge on the limit. You don’t actually get there.””
“I could care less, it does not prove your point.”
Sure it does, in a very logical sense.
” especially because you seem fine accepting infinitesimal numbers”
→I do not. They are not numbers since they are really a function or representation of a sequence…not a number. The limit of the infinitesimal sequence is 0.
“You are unable to demonstrate that 0.(9) – 1 = 0 is a false statement.”
It is false because 0.999… is not a number. Horse – 1 = 0 is a false statement just the same.
0.999… – 1 does not equal a “number”.
1 – 0.999… also does not equal a number. I suppose it equals: 1 – (0.9 + 0.09 + …)
which equals: 1 – 0.9 – 0.09 – … which would give a sequence: {1, 0.1, 0.01, …}
Maybe you want to represent that as 0.000…1. It’s not a number though, just the same 0.999… isn’t. They are on the same playing field. Either both not numbers, or both numbers.
Even in the original post:
“Let’s define:
p_{1} = 0.9
p_{2} = 0.99
p_{3} = 0.999
p_{4} = 0.9999
…
and so on.
Now, our number 0.9999… is bigger than … ”
Wait, what? “OUR NUMBER” ? 0.999… is a collection of (infinite) operands (numbers). There lies the problem right off the get-go.
“Not at all. The “number” is the limit of the sequence.”
Yes.
“0.999… is a representation of the sequence (or infinite sum).”
It is NOT a representation of a sequence. By definition, it cannot be. It is an infinite sum, and infinite sums are numbers, not sequences. More precisely, infinite sums are, once again, limit of sequences, not sequences themselves. They cannot be.
“Then by magic, it (0.999…) is also declared as the number itself.”
No. There is no magic in mathematics. 0.(9) is a number, and there is nothing magical about this. 0.(9) is NOT a sequence, and you declaring it a sequence has nothing mathematical about it.
“The limit is 1. That is the number.”
Yes. 0.(9) is an infinite sum by your own admission, and the sum evaluates to 1, because the sum is defined as a limit. Therefore, 0.(9) = 1. Thank you for contradicting yourself and proving what I had already proven.
“The symbol “0.999…” I will accept as a representation of the sequence or infinite sum; not the limit.”
Then you are rejecting mathematics themselves. By definition, a positional digital string represents a sum, not a sequence. Sequences are different from strings of symbols. 0.(9) is, by construction, a positional digital string, hence it is a sum, not a sequence, and if you reject this, then by logical consequence and deduction, you reject mathematics themselves. You can be dishonest and change the definition of every terminology in mathematics and words in the English language, but doing so invalidated your argument since, by using those words, you are no longer referring to the objects we were originally referring to, so any argument you present cannot disprove our claims because such arguments will be a straw man arguments by construction. Your claims are not mathematical, and this is evident because you reject mathematics. Your claims are about some unrecognizable construct only you understand and only you use, because no one else will agree with such a construct you made and there will never be applications to such a construct precisely because it is limited by your definitions, whereas mathematics are not uselessly and unnecessarily restricted by such dishonest ideas.
“LIMITS REMOVE THE INFINITY CONCEPT IN THE SOLUTION.”
No, they do not. Limits are simply the rigorous formalism by which we deal with infinities. You not understanding how a limit works or how decimal digital strings work does not imply that limits remove infinities. They do not, by construction, and the equivalence between a limit and a hyperreal-valued function is further proof of this.
“Therefore the limit should not have a representation of infinity in the answer (limit) itself!”
Your claim is nonsense. All it demonstrates is your failure to understand limits. This failure I already indicated earlier, and you denying the failure means nothing if you cannot show there is no failure, which you cannot, by this argument.
“(Logically, a number cannot be a never ending addition of other numbers. It’s senseless.)”
You are wrong. Numbers necessarily must be several infinite sums. Otherwise, no number could ever exist. 1. Every number is the summation of some finite number and infinitely many zeroes or vanishing summands. For instance, 1 + (1 – 1) + 0 + (2 – 2) + 0 + ••• = 1 since adding 0 leaves the number unchanged by the axioms of summation, and iterating this process indefinitely yields the same result. 2. Non-trivially, all irrational numbers are necessarily the sum of infinitely many non-ZeRo summands. This can be proven from the definition of irrational numbers. By your argument, no number exists, or else, addition does not exist. Both claims are false. Hence your assumption is false by proof by contradiction.
“Yes, if you accept that the symbols 0.999… are a number,…”
They are, by definition. Every valid digital decimal representation can be parametrized as a summation, which is a number. In fact, by definition, the decimal representation of a number is a notation of abbreviation for the partition of the number into sums of the exponential order basis with respect to a chosen radix.
“…but it represents a sequence/infinite sum, ie, {0.9, 0.99, … } or (0.9 + 0.09 + ….).”
No, you are wrong. It does represent an infinite sum, it does NOT represent a sequence. You keep using the phrases “infinite sum” and “sequence” as if they were interchangeable, but they are not interchangeable. An infinite sum is a number, the limit of the sequence. Hence, you are correct in that, by definition, 0.(9) = 0.9 + 0.09 + 0.009 + •••, which in turn is lim {0.9, 0.99, 0.999, …}. It is the limit of a sequence, not a sequence. Digital strings never are sequences. Not that this matters: you can define infinite summation without appealing to the limit of sequences, especially in the surreal numbers, but addition in the surreal numbers will still give that 0.(9) = 1.
“I haven’t failed to notice anything other than “0.999…” being called a number.”
Yes, which means you have failed to understand the definitions of infinite summation, sequence, limit, digital strings, and number, since if you understood the definitions of these, you would not fail to understand that 0.(9) is a number, nor would you claim that limits remove infinity.
“I have easily completed advanced and engineering calculus, doing these kinds of problems.”
You may be good engineer, but a horrible mathematician. If you passed every advanced math course in your educational institution easily, as you claim, and yet you have such a terrible understanding of these rather basic definitions, then all it implies is that your institution did an an absolutely terrible and hopeless job with their mathematics education department.
“I “understand” what you want me to do, and I understand what you WANT me to believe. Just because I reject certain things doesn’t make it less understandable. Those are just plain ol’ empty attacks. Am I attacking you?”
This is not a question of me wanting you to believe anything, because this is not a question of opinion, this is not a subjective topic. You can either accept the definitions, and hence be correct, or reject them, and inevitably be incorrect. You cannot reject them and still be correct, because of the problem I mentioned very early on in this response. There is no attack here. There is just me telling you what the facts are. You refusing to believe the facts does not make them not facts, it makes you wrong. That is how beliefs concerning objective topics work, by the way.
“Sure it does, in a very logical sense.”
It really does not. You want to talk “logical sense”. Okay then. We are using classical deduction and the axioms of arithmetic as outlined by classical mathematics. You stated the premise that the limit is a statement about convergence, not a statement about being there. Then you concluded that the limit of a sequence should not be associated with the sequence itself, and therefore, not with 0.(9). But your premise does not imply your conclusion, because your conclusion has nothing to do with “being there”. Nowhere in the definition of decimal representation is it required that one be there, nor is this assumed with a sequence either. Hence, in a very logical sense, it really does not prove your point. Learn some formal logic next time, maybe.
“I do not. They are not numbers since they are really a function or representation of a sequence…not a number.”
They are not. Infinitesimals are absolutely not functions, since a function is defined as a map from a set to the other such that the output for every input is unique. Infinitesimals are not maps, so they are not functions. They are also not sequences. They are defined as numbers e with the property that if r is a nonzero real number, then |e| < r, & 0 < |e|.
“The limit of the infinitesimal sequence is 0.”
There is no such a thing as the infinitesimal sequence. A sequence is either made of real elements, or infinitesimal elements, or infinite elements, or surreal non-standard elements, but there are infinitely many of each type. Infinitesimal elements are numbers by definition, and again, if you reject the definition, you reject mathematics. Now I leave you the choice to be rational or irrational.
“It is false because 0.999… is not a number.”
It is indeed a number. You have yet to prove it is not a number. To prove it you must show that the summation semantically implicit in the digital string is invalid or impossible, or that the digits themselves are not numbers. Yet you already disproved both in a previous comment by stating that 0.(9) = 0.9 + 0.09 + 0.009 + •••.
“Horse – 1 = 0 is a false statement just the same.
0.999… – 1 does not equal a “number”.
This would be true because horse is not a number, and according to you, neither is 0.(9), but the problem is that you are wrong, because we have already shown 0.(9) must in fact be a number.
“1 – 0.999… also does not equal a number. I suppose it equals: 1 – (0.9 + 0.09 + …)
which equals: 1 – 0.9 – 0.09 – … which would give a sequence: {1, 0.1, 0.01, …}
Maybe you want to represent that as 0.000…1.”
No, digital representations are defined as the abbreviation of a summation, not as a sequence. It is strictly incorrect to represent a sequence with a number, which is what digital representations are.
“It’s not a number though, just the same 0.999… isn’t. They are on the same playing field. Either both not numbers, or both numbers.”
You have yet to prove that they are in the same field. There is no possible summation which gives 0.000…1. There does exist such a summation for 0.999…, and you proved this yourself. Hence you are wrong.
Your entire argument relies on establishing false, unprovable premises, and on not understanding basic definitions in mathematics. You still have yet to present a valid argument.
“Wait, what? “OUR NUMBER” ? 0.999… is a collection of (infinite) operands (numbers). There lies the problem right off the get-go.”
The operands are finite. Every operand is finite and small because they are all less than 1. There is no collection, though. Collections in mathematics are not operations. What there is a summation, and the operands are the summands. 0.(9) is a summation of infinitely many finite and small terms. There are many ways to calculate this sum, but regardless of the way you do so, using limits or not, the total is 1. Sums are numbers. This is undeniable. Whether you can complete the summation if perform every step individually at a fixed pace is not relevant because the status of the operation is not determined by the status of an algorithm designed to evaluate the operation, it only depends on its definition. Thus, the idea that we can never get there is a red-herring. Regardless, 0.(9) is not a sequence.
Engel, I for one just wanted to say thanks for that great refutation of Mathman’s (the irony of that name!) comments. You’ve been logically consistent, and explained it in a simple enough manner that even I understood it.
I know that comment must have taken a long time to write, and I just wanted you to know the effort wasn’t wasted.
“Thank you for contradicting yourself and proving what I had already proven.”
Lol. These high and mighty comments are tiresome.
You can’t sum infinite operands. Which is why we need to take the limit. Then we define what is impossible, to make it possible. I am allowed to reject the impossible. An infinite sum is something we can analyze and say something about, for instance, what the limit is as we APPROACH infinity, ie, as we let some variable become arbitrarily large. (We don’t every approach infinity, as there are infinite ‘objects’ beyond what we even imagine as ‘close’ to, or approaching, infinity).
Of course I am rejecting that concept. What do you think I’ve been doing all along?
” They do not, by construction, and the equivalence between a limit and a hyperreal-valued function is further proof of this.”
We are not talking about hyper reals. Maybe I’m mistaken, but I’ve NEVER seen a limit result, with infinity in it. The whole point is to set up a limit with a variable→∞ to see if it has a result. If the limit exists, then obviously no ∞ involved.
“You are wrong.” No I am not. You cannot have a precise number, when you endlessly add more numbers, in every single case. What you have is some sort of function, ie, infinite sum if you will, which is not a sum, but we call it one. Infinite sum is oxymoron in itself. You never have the sum. A sequence of sums, yes. Never a final sum. As close as you want to imagine, I imagine the sum being infinitely far from that. Again, the LIMIT of the infinite sum is not the sum itself. The sum doesn’t exist.
“They are, by definition.” Which is what I am arguing. 1 is the number, formed from taking the limit of 0.999… , where 0.999… is a representation of SUM n=1 to ∞ (9/10ⁿ).
Obviously, according to you, no one is allowed to challenge or have their own intellectual thoughts. To ‘solve’ the infinite 9 issue in 0.999…, we take the LIMIT, and we end of up with 1. It’s the same reason something like 1/10^∞ is poor symbology. However, the limit 1/10ⁿ as n→∞ = 0. This does not mean 1/10^∞ = 0.
“Yes, which means you have failed to understand the definitions of infinite summation”
Then how in the hell was I doing engineering calculus in high school? sheeesh. You and your condescending attitude again. I understand the definitions fine. The only failure of understanding here is you seeing my point. A few of you all have that same attitude. “If you don’t accept everything our way, you are wrong, and you don’t understand anything!!”. It’s pretty sad.
“yet you have such a terrible understanding of these rather basic definitions”
Once again, I understand what YOU are saying about the definitions and concepts just fine. I understand what you want to me understand just fine. You just want me to “shut up and just do it how we tell you to”.
” You can either accept the definitions, and hence be correct, or reject them, and inevitably be incorrect.”
So according to you, being incorrect that 0.999… not equaling anything, and the limit 0f 0.999… equaling 1, has what consequence?
I asked if I was attacking you, and you didn’t answer. You are STILL attacking me.
“Then you concluded that the limit of a sequence should not be associated with the sequence itself, and therefore, not with 0.(9). ”
You changed what I said. Because I didn’t say that.
The “infinite sum” (or string of digits), 0.999… forms the partials sums sequence {0.9, 0.99, 0.999, …}
The LIMIT of the sequence is equal to the LIMIT of the infinite sum, both being 1. 1 = 1.
“because we have already shown 0.(9) must in fact be a number.”
You haven’t. If you had, I wouldn’t be responding with all this right now.
The LIMIT of the infinite sum 0.999… = 1.
We defined 1. Not 0.999…
“No, digital representations are defined as the abbreviation of a summation”
1 – SUM 9/10ⁿ (n→∞)
“No, digital representations are defined as the abbreviation of a summation, not as a sequence. It is strictly incorrect to represent a sequence with a number, which is what digital representations are.”
Ok, if 1 and 0.999… are both digital representations of the same number ‘one’, then what are the 2 digital representations of the result 1 – 0.999… ?
There is no possible summation which gives 0.000…1
1 – SUM 9/10ⁿ (n→∞)
“The operands are finite”
What I meant was, there were “infinite amount” of finite operands. There are ‘infinite’ 9’s, thus, there is never a sum in the first place. There is only a limit. There are partial sums, because you have a sum; it’s done. You can’t have infinite and sum for every case. Maybe zero. And what the hell is “infinitely many” ?? How many is that exactly?
” There are many ways to calculate this sum,”
Wrong. There are many ways to calculate the LIMIT of the infinite sum, or the convergence of the sequence of partial sums. I have no issue with either of those. I have an issue with saying “zero decimal infinite 9’s is equal to the limit of the infinite sum of them” The “having infinite 9’s” is just nonsense. You can’t have that to even start with. We can have arbitrarily large amount of 9’s if you like. But this argument about “having all infinite 9’s there at once” is just assertion.
Another way to word it is as follows: 0.999… TENDS TO 1, using limit concept wording. The limit EQUALS 1.
You also run into other problems if you “allow” infinite in the representation of a number.
If you mean 0.999… to be ‘zero decimal 9 infinite times’, then we can show:
0.9 = 9/10; or, ‘zero decimal 9, one time equal one 9 over 1(one zero)
0.99 = 99/100 (two 9s over 1 with two zeros)
0.9(n times) = 9(n times) / 1[0(n times)]
0.999… = 9…/10…
BOOM!
0.9(infinite times) = 9(infinite times) / 1[0 infinite times]
This shows one particular issue with allowing ‘infinity’ to be part of a number.
However, using the limit on both sides would result in 1 = 1. Thank you.
It’s impossible for Infinite/Recurring 0.9 (0.999…) to ever equal 1
Because of the Infinite/Recurring 0.1 (0.001…) Difference!
Only by the using the + Calculation can it be possible (0.999…) + (0.001…) = 1
0.999… Can NEVER! equal 1 or become 1 on it’s own! or there will be a contradiction to the definition of Infinite/Recurring 🙂
Anthony.R.Brown
y’all need to chill
if anyone doesn’t think 0.999…=1 then check https://en.wikipedia.org/wiki/0.999… and its reference section at the bottom if you’re still unconvinced
yeesh