For the remainder of this post, the statement 0.99999... = 1 will be denoted by S.
Over the course of my math education and my professional teaching career, S has occupied considerable time and provoked much thought on my part and reflection among my students, countless mathematicians and, now, the math blogosphere (see Polymathematic's famous series of posts!). Sane individuals (aka, non-mathematicians) remain skeptical about S, unwilling or unable to grasp the equality in the statement.
They argue: "0.99999... gets closer and closer to 1 but how can you say it EQUALS 1. There's always a gap!" Ah, the mystery of limits!
For many years now, I have been posting my 'proof' of S on various listservs, discussion groups (including MathShare, the one I moderate) and blogs. Here's the reaction I 've generally received: ____________________
That's right - silence. Because I like to put a positive spin on things, I take that to mean no has found a way to refute it! I've even occasionally heard a student say that this convinced her/him.
I don't want to bore the veterans out there who've heard and read all of the well-known arguments, most of which have 'holes' in them (or should I say, discontinuities!). Even using the basic formula for the sum of an infinite geometric series doesn't necessarily satisfy the Odd Thomases (sorry, I'm a Dean Koontz addict) who will continue to question the validity of the statement.
Any attempt to justify S necessarily requires (to paraphrase Liping Ma) a profound understanding of fundamental principles regarding the real number system and my argument is no different.
Enough already -- Here it is:
Non-Rigorous Explanation: If 0.9999... is less than one, then there must be a decimal between it and 1. But this is impossible!
Rigorous Explanation:
Step 1: Consider the sequence: 0.9, 0.99, 0.999,...
Since this is an increasing sequence of real numbers bounded above by 1, this sequence has a limit, L, namely its least upper bound. As many of you know, I am using the Completeness Axiom for the Reals (known by other names). An excellent reference for the axiomatic structure of the real number system can be found here.
This demonstrates that 0.99999... does exist (i.e., it is a real number). Thus,
0.99999... is the limit L of the above sequence. Verification of the existence of 0.99999... is what is often lacking in other demonstrations of S.
Step 2: L is either greater than 1, equal to 1 or less than 1. We need only consider the last 2 cases.
Step 3: Reasoning indirectly, assume that L<1. By the density property of the real numbers, there must exist at least one real, x, between L and 1. Since L is different from x, it must differ from it in some decimal place. The tenths place? No! Since x is less than 1 and greater than L, it must have 9 in the tenths place. The hundredths place? No, again for the same reason. Need I continue or do you see we've reached a contradiction? Therefore, our assumption that L is less than 1 is false. Thus, L = 1 or, equivalently, 0.99999... = 1. QED!
Ok, your turn! Feel free to critique the proof or present your own favorite argument for or against S. Also, would you consider using this type of argument when teaching this topic?
Saturday, December 15, 2007
0.99999.. equals 1: Oh no, not another 'Proof!'
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14 comments:
i just thought of it like:
1/9 = .11111111
2/9 = .222222222
so why can't 9/9 = .9999999 = 1?
there was also another proof that my 6th grade math teacher showed me that involved multiplying the .99999 by 10 then subtracting something. unfortunately, i've forgotten. Sorry, Mr. Tully!
Along the lines of what d rhee said, the same people who are freaked out about S, do not seem nearly so concerned about the statement (shall we call it T) that 0.33333.... = 1/3 exactly.
I like your argument. As to whether I would use it with students, um, not middle schoolers. :)
I use the "standard" one that Mr. Tully probably used. Let N = 0.9999....
Then 10N = 9.9999....
10N - N = 9.9999... - 0.9999...
9N = 9
N = 1
Less rigorous, but a little more accessible to middle schoolers. Are they convinced, I don't know. But perhaps via early exposure, we will be able to gain the same level of acceptance for S as we seem to already have for T.
Hi D--
Thanks for that contribution!
The 1/9,2/9,..pattern is always a favorite of students and adults alike. I know I enjoyed it whenever I saw it! One could also do this with thirds:
1/3 = 0.33333...
2/3 = 0.66666...
3/3 = 0.99999...
However, as wonderful as these patterns are, they do not provide a rigorous mathematical proof. By the way, some students usually asked me how we can divide 9 by 9 and obtain 0.99999...
If we alter the usual division algorithm, we can actually come up with that decimal (the formatting will be off and I'll use **'s to denote blanks for spacing)
**0.99_______
9)9.00000
**8*1
****90
****81
*****9 etc
I'm guessing that Mr. Tully used the following famous algorithm for converting repeating decimals to fraction form:
(*) N = 0.99999...
Multiply both sides by 10:
(**) 10N = 9.9999...
Subtract (*) from (**) to obtain:
9N = 9 and the magical result appears!
All of these methods, no matter how convincing or impressive, do not address the essential nature of an infinite repeating decimal, namely a limit of a sequence:
0.9,0.99,0.999, ...
Students often attempt to apply ordinary arithmetic operations (like merely subtracting the infinite decimal part), to infinite sequences, but such operations have to be shown to be valid. Justification normally requires theorems on limits and one has to demonstrate convergence of each limit involved.
Sorry to be so evasive about the logic flaws in these other arguments. So much more needs to be said here for clarity.
D--
I am interested in your opinion of my argument. Does it make sense? Did it convince you or were you already convinced?
Isn't treating it as a geometric series:
0.9(1+ 1/10+1/100+.....) and finding the sum a rigorous argument that can constitute a mathematical proof?
Or, is it that you first have to show that the limit of the geometric sum exists, and that it is equal to blah blah blah before you can arrive at the result?
TC
a little confused about:
"Students often attempt to apply ordinary arithmetic operations (like merely subtracting the infinite decimal part), to infinite sequences, but such operations have to be shown to be valid."
did you mean "invalid"?
if so: why? and how? is it based on aristotle's argument that infinity cannot exist in the real world (also taught by Mr. Tully). For instance, if we had an infinite line of coffee cups, and i drank a hundred of them. there would still be an infinite amount left, creating a paradox. but i don't see a problem if i drink an infinite number of cups.
i'm also curious how we could prove 9(1+.1+.01...)=1 as TC suggested. i'm not sure what the next step would be.
i'm also confused about the "limits of the sequence."
i fear that if anyone tries to answer all of these questions at once i'll be in worse shape.
regarding the statement itself, i had accepted it to be true, mainly because i have not thought much about it. but now i have my doubts.
a side note: does .0000...1 = 0?
D, Mathmom, TC et al--
Where do I begin to tell the story of 0.99999... (Anyone who remembers the first few words of that lyric is at least as old as me!)??
I don't claim to be an expert in all things infinite, but I can share my understandings from years of research, dialogue and thought:
(1) Before one can apply the ordinary arithmetic operations on infinite repeating decimals as if they are rational, one must first establish that they are real, then rational. Thus, Step 1 of my argument using the Least Upper Bound Axiom.
(2) Applying the rules of arithmetic to these numbers is not transparent. The operations must first be DEFINED! Thus, although, students fully accept that
0.33333... + 0.66666... = 0.99999..., mathematicians needed to make such statements rigorous, which means they can be deduced logically from underlying definitions, axioms and/or theorems.
The equality of the addition statement above follows from the Basic Limit Theorem that the limit of a sum of two infinite sequences equals the sum of their respective limits. One must first
show that 0.33333..., which is by definition a LIMIT, actually exists and, in fact, equals 1/3, etc.
Mathmom, while I agree that my argument seems to be way over the heads of middle schoolers and they are more likely to appreciate the algorithm you described, I strongly urge you to try the intuitive non-rigorous form of my argument:
:If 0.99999... is less than one, boys and girls, then produce a decimal between IT and ONE!" By middle school, they should already know or will learn that between any rationals there is another rational (in fact, infinitely many), and we can allow them to assume 0.99999... exists and is rational. My rigorous proof was not intended for them! By the way, I've used this non-rigorous approach with real middle-schoolers and some get it immediately.
Getting back to the algorithm of letting N = 0.99999... and allowing that N exists for the moment, remember you are telling children that the infinite repeating part obviously cancels out! When dealing with infinity and infinite processes, what is really obvious!
D--
Inspired question about 0.0000...1! Thinking individuals have asked about that 'number' for, perhaps, centuries! But is it a number. Mathematicians would argue that it is not well-defined and in fact does not exist! WHY? Well, what you're trying to do is create the SMALLEST POSITIVE NUMBER! By the axioms of the real number system, such an 'infinitesimal' does not exist! This is equivalent to trying to create a number between 0.99999... and 1, the whole point of my proof. I've actually heard students say to me that 0.99999... is THE LARGEST NUMBER LESS THAN 1! Again, this is the essence of the entire question. By the density property of the reals or the rationals, THERE IS NO LARGEST REAL (or rational)LESS THAN ONE. Flaming in uppercase here is not a rant, it's purely for emphasis!
Ok, I'm far from done here but I will leave you with the following famous paradox:
1 - 1 + 1 - 1 - 1 +... = ??
ZERO? ONE? 1/2? None of these? Does not exist??
This proof loks mightily familiar.
I would offer along the same lines, if anyone is interested, the proof that 2=1.
This will be simple enough for everyone here, but it's a great prof to use for students when you want them to be able to identify errors in proof logic.
Correction to last line of the paradox:
1 - 1 + 1 - 1 + 1 - 1...
[It's supposed to alternate!]
This question may seem trivial but some of you know it's far from obvious and really led to a rigorous development of infinite series and the need for precise definitions. Change the definition of the sum of an infinite series, allow rearrangements of the terms of certain types of series, and one can make any sum you please!
Justin,
Yes, I would like to see your version of such a proof! I've seen many and they're all powerful instructional tools. Finding logic flaws or fallacies requires an understanding of mathematics beyond algorithms.
One of the things that could be coming up here is the idea of Surreal Numbers. I believe that these were initially invented by John Conway, and there is a nice book by D. Knuth about them. Instead of all this business about how to define a real or a rational number from integers which are defined from whole numbers which come from Peano's Axioms (Whoo, that was a lot of derivation). Surreal numbers are formed from a just a few simple rules.
Rule 1: Every number corresponds to a set to two sets of previously created numbers, such that no member of the left is greater than or equal to any member of the right set.
Rule 2: One number is less than or equal to another number if an only if no member of the first number's left set is greater than or equal to the second number, and no member of the second number's right set is less than or equal to the first number.
That second rule is rough, but the beauty part is that out nothing you get numbers. Why out of nothing? Because we need "sets" of numbers, and we can start out with the empty set of numbers. So the two sets are { empty | empty } The "|" divides (might I even say cuts) the two sets apart.
It turns out after some work that that { empty | empty } = 0. This allows us to make new numbers like:
{0 | empty } = 1
{empty | 0 } =-1
and the whole numbers follow quite nicely. The fractions that have denominators powers of two come quickly, and after an infinite amount of time the "Surreal Number" infinity and 1/infinity come up.
It is really a rather amazing thing. That I encourage everyone to check out.
Matt--
Fascinating stuff! Leave it to the ingenuity of the human mind to create a new kind of number to address the limitations of the reals, that is, the issue of "there's no smallest positive real"! I started reading the Wikipedia article and it's a (sur)real screen-scroller!
Thanks for sharing this. I may have mentioned on my blog, awhile ago (Alec Klein interview) that I had a student who could not accept the notion of limit and tried to devise a new entity similar to surreals. John (not Conway!), if you're out there, are you one of the creators of this idea? After all, Leibnitz invented infinitesimals for similar reasons.
Lets presume you somwhow knew that you where going to die for certain when you where 100 years old. Are you living when you are 99.99999999...?
If you're a mathematician you've reached your expiration date (aka,limit) at 99.9999... !!!
As I indicated in the 'proof' in the original post, assuming that 99.9999... is less than 100 leads to a contradiction since by the density property of the rationals (or reals) there must be at least one rational (or real) between 99.999... and 100. But this is impossible unless you postulate there is a new kind of 'number' like surreal numbers.
Of course, I have no way if you were using irony in your comment or trying to disprove this fact!
Why .999... does not equal 1.
Gotta invoke the algebraic argument to disprove the S theory, or at least show it as questionable.
First, it is assumed 1/3 equals .333... and I submit that it is infinitely unequal. Again, this is algebra, so the assumption of limits is an assumption. I am yet to observe a prove where 1/3 actual equals a particular number and have demonstrated (to myself) that the fraction / division results in 'number' that is forever not equal to 1/3.
Second, x = .999... is a bit illogical or inaccurate, as x is presumed finite. Would be more accurate to say xxx... equals .999... Therefore 10xxx... would equal 10 times 0.999... in which case the resulting number is (absolutely must be) a number that has a zero on the end. Where / when does that zero appear. Either you admit that it never appears, or you are left with a number that looks a lot like 9.999...0. Accept that logic that your mind must be enabling to see 10x as equal to 9.999... and you can accept the number 0.000...1. It really isn't hard to accept that second number as existing, but most see it as incoherent. It is as coherent as 10 times .999... when (algebraic) logic is applied to the zero that (absolutely) must appear on the end of the .999...
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