Refutations

(note:  this post has been closed to comments;  comments about it on other pages will be deleted!)

THE FINAL UPDATE!

Okay, so I'm still on vacation, but it's clearly time for me to step back into this.  Thanks to the people at scienceblogs, namely Goodmath/Badmath and the EvolutionBlog for taking up my cause.  And thanks very much to the people in the comments of my other posts for doing their best to explain.  Some of my explanations here use some of their ideas.

Let's do this by classifying the problems people have with my posts, from the most ridiculous to the least (roughly):

You must be a public school teacher, and I fear for your students.  You don't know enough math to teach it.  Stop filling their heads with nonsense.

Wow.  I actually teach at a private school.  Some of my former students have gone on to get perfect scores on SAT's, study advanced math at top-level universities, and place well at the national level in math competitions.  I have B.A. in math from a top-20 school.  All the information online (you can start here and here) agrees with me.  In addition, while I know that there are public high school teachers who are not very good with math, all of the ones I know who teach at, say, the pre-calculus level or higher also agree with me.  Please don't insult the entire profession just to discredit me.  If you're that worried, go to graduate school and study math, and teach it yourself.

.999... is clearly less than 1, but mathematics isn't advanced enough to handle infinity, so you can't prove it.  My intuition isn't flawed, math is.

This is another one that's so wrong that I barely know where to start.  Historically speaking, this debate is quite old.  In the 19th century, this apparent paradox (.999... is clearly less than 1 vs. .999... equals 1) was addressed by some great mathematicians (Cauchy and Dedekind, among others).  In particular, they formalized the notion that the real numbers are infinitely finely divisible, and in their formulations, all arithmetic operations worked the way they seemed like they should.  Using their formulations, all proofs result in .999...=1.  Those formulations are discussed more at the wikipedia site linked above.  Mathematics has been handling infinity well (using definitions that don't even require the use of the word 'infinity') for at least 100 years now.  Until you've studied that work, you might want to be careful about saying that math can't handle infinity.

Variations on:  1 - .666... = .444..., but .999... - .666... = .333..., so 1 and .999... can't be equal.

You'd think that people trying to argue with mathematicians would at least check their work:

1 - .666... = .444...?  If that's true then to check it, I should be able to add .666... and .444... and get 1.  But .6 + .4 is already 1.  So the sum of two larger numbers can't also equal 1.  The check fails.

Variations on:

10x = 9.5
- x =    .5
9x = 9

So x = 1 and also x = .5.  SEE!! I can use your stupid method to prove that 1 = .5!

If x is .5, then 10x is 5, not 9.5.  Your equations are unrelated, so of course you can prove something false.

(A lot of birds with one stone):

10 * .999... isn't 9.999..., it's 9.999...0
1 - .999... is .000...1
2 * .999... is 1.999...8

This is a common mistake among my students.  You're mistaking notation for mathematics.  Notation is not mathematics.  Mathematics is the study of ideas about patterns and numbers, and we have to invent notations to communicate those ideas.  Just because you can write something that looks mathematical, that doesn't imply that what you wrote has meaning.  The number written as .999... has meaning:

9*(1/10) + 9*(1/100) + 9*(1/1000) + ...

Every 9 in the list means 9*(1/something).  But .000...1, for example, is an abuse of notation.  It doesn't correspond with any meaning, so it doesn't communicate anything.  If you think it means something (and putting 1 at the END of an ENDLESS list of zeros shouldn't mean anything), you're going to have to tell me what's in the denominator of the fraction represented by that decimal place.  If you can't tell me that denominator, you're not using the notation right.  If you tell me the denominator is 'infinity', then see the next entry.

1 - .999... is 1/infinity, which is a number bigger than 0.

Mathematicians don't really use the noun 'infinity' very much, and when they do, it's usually as a shorthand for an idea that is relatively easy to define without the concept of infinity.  They do use the adjective 'infinite' to describe lists and sets, and the adverb 'infinitely' to describe how the real numbers are divisible.  While some intuitive ideas can be captured by using the idea of 'infinity' as a kind of number, you have to be very careful with it, and standard arithmetic doesn't usually work.  But as an intuitive idea, anything that might be written as 1/infinity never behaves differently from the number 0.  I can't prove that, however, since 1/infinity doesn't really mean anything.  Using infinity as a number creates fallacies that even the doubters of .999... = 1 would disagree with.

.999... effectively equals 1, but it doesn't actually equal 1.  It rounds off to 1.  You can never really .999... going on forever because you can't live long enough to write it.

These are all really arguments that claim that .999... isn't really a number, and that you therefore have to stop writing it or round it off at some point.  Look, either you allow the possibility that there could be infinitely many (note the use as an adverb!) decimal places or you don't.  If you don't allow it, you'll have a lot of trouble with proofs that pi or square-root-of-2 can't be written using a number that has finitely many decimal places.  If you do allow it, you have to be prepared to discuss what happens if they all equal 9 at the same time, and you have to discuss it without rounding or talking about when they end.

.999... = 1 if you allow limits, but not if you're just talking about numbers.  The limit of the series isn't the same as adding up the numbers.

The evolution blog linked at the beginning of this post has an excellent discussion about this.  The upshot is that once you admit that there's an infinite geometric series here (which you have admitted as soon as you merely write .999...), there is no difference between the limit and what the thing actually equals.  They have to be the same by any defintion that is internally consistent.

Your fraction argument only works if I admit that 1/3 equals .333....  I don't think it does, so I don't think your arguement works.  1/3 can't be precisely expressed with decimal numbers.

Well, at least the people who argue this are not abusing notation, and they're not attacking me personally, and they understand that the assumptions have to be correct in order for the argument to be correct.  So I'm giving some credit to this one.  But unfortunately, .333... really does equal 1/3.  If you think 1/3 equals some other decimal, you're going to have to tell me what it is.  If you think that you can't express it with decimals, then remember that the very word 'decimal' itself comes from our base 10 number system, and that's a biological coincidence due to our 10 fingers.  In a different base, 1/3 might be no problem, but 1/2 might be.  Any problem results from notation, not from the concept of 1/3.  Remember, notation is not math, notation just communicates math.

Okay, I really have to go now...I'M ON VACATION, PEOPLE!!!

Comments

[Ross Stratman]

Euclidean geometry also suggests tha .999(repeating) = 1. Here goes.

1) For every distinct point on a line (number line included) There MUST exist at least one point between them.

2) If .999 repeating is actually LESS than 1 then there MUST exist a distinct point on the number line separate from 1.

3) Since there must exist a number between the two (see Theorum 1) then... WHAT, pray tell, is that number?

[Flavio]

.99999... = 1 ONLY in standard analysis.

In non-standard analysis that IS NOT true.

YOU should know that axioms matter, and since 1900s mathematicians don't believe there is anything are the "right" axioms!!!! Haven't you ever read about the parallel postulate debacle...

So learn something yourself, and not necessarily .999... = 1. That depends on axioms.

BTW for computable reals, that is also not true, 0.9999 != 1.

[Polymath]

Yes, it's true...I do understand that axioms matter, and that with other axiom sets, there are mathematically consistent ways of proving that the numbers aren't equal. But I do make explicit (not in the first post, but in subsequent ones) that I'm talking about the standard real number set. Furthermore, the people who disbelieve can't even argue logically in the standard real numbers, let alone the hyperreals or other sets.

[craig]

Proof 1/3=0.33333rec

a=0.99999rec
10a=9.999999rec
9a=9
a=1
a/3=1/3
a/3=0.3333333rec
as a/3 is 1/3
1/3=0.333333rec
QED

[no name]

So I am just going to put my two cents in.

I like you algebraic proof. It was nice to terminate the repeating decimal that way. However, let's first look at 1/3
10x = 3.3333333...
- x = .3333333...
9x = 3
x = 1/3

It works. Yay. If we were to stick 1/3 back in for either equation, it would work. Let's do pi next. I will just do 3.14
10x = 31.4
- x = 3.14
9x = 28.26
x = 28.26/9 = 3.14
That worked just as well, but only up to the two digits we used. If x = 3.14, then 10x = 31.4. Hey look, it checked out. For the final test.

10x = 9.9999999...
- x = .9999999...
9x = 9
x = 1
what? .999999... = 1? But that's impossible? .999999...<1. Well, yes that's correct. if you plug in .999999... back, you get 10x = 9.999999... and x = .999999...

Wow, it worked for an irrational and repeating number, so it doesn't really matter what type of number it is, the algebraic proof works. I also don't want to hear about how 3.14 is not irrational. Do it to however many digits you want, it still works.) Surprisingly, but yes, it works.

[Guilherme]

What about this? .9999... is a fraction (since it periodically repeats 9). So it is a racional. A racional arbitrally close to 1 is 1.

[Bradley]

Very sorry, and I regret doing this but after lots of reading and thought on the issue, I question the validity of the basis of the whole argument that .9r = 1.

Time and time again in these comments multiple posters on both sides of the argument have acknowledged that the fundamental issue of the conflict is that the mathematics aren't flawed - rather that the symbolic representation of the idea is flawed.

I'm inclined to believe that the importance of the entire argument is lacking if this factor cannot be eliminated. I believe it is impossible to exhaustively convey the existence of any entity, object, or concept through notation (which also makes me question my ability to convey my very thoughts to you all now through language).

Our very idea of symbolic notation and mathematics is specific to our existence. Likewise, multiple mathematical concepts and operations utilized in the 'proofs' that have been thrown around in this article are also limited to the bounds of our system of symbolic representation. Who is to say we are allowed to define all these items through our various systems of notation when even the entirety our own existence throughout history is as minute in the grand scale of the known universe as the infinitesimally small amounts this argument seeks to explain.

We can prove conclusively that, based on our conceptual representation of these unquantifiable amounts that the numbers do make sense and lead us to the conclusion that .9r really does equal 1. However, we cannot hope prove this exhaustively. There are so many possibilities in the universe (which, by the way, we don't even know the true expanse of) for explaining the argument against .9r=1 that it would be impossible to fathom them all.

I'm not dismissing the blood, sweat, and tears of generations upon generations of renowned mathematicians - I'm merely encouraging us all to look ahead with an open mind free of the shackles of our traditional schools of thought because we may soon find ourselves on the brink of an entirely different dimension - one in which the rules and symbology we've used so conclusively to describe processes and concepts in the world as we know now will not apply, and will have their foundations shaken from under them.

[Rich]

I have been thinking about this for quite some time myself, and I find this discussion quite interesting to say the least.

I think the major issue in the line of reasoning that 0.999... with the 9's repeating off to infinity is equal to 1 is that, each of your proofs add or subtract "infinitesimals," and so some information is lost in these results. Let me explain, however. I do not disagree in any way that 0.9999... is another real-number representation for the real number 1. This is because to treat 0.999... as a real number we come to the conclusion that 1 - 0.999... is equal to some number closer to zero than any other real number, and thus we might then say that 1 - 0.999... = 0 since they differ by an infinitesimal quantity.

First, consider your proof using simple algebra. I personally like this proof, it is quite neat. Anyway, what you do is something like:

0.999... = x; thus clearly
9.999... = 10x; hence by subtraction of these
9.999.... - 0.999... = 10x - x = 9x, thus x = 1 = 0.999....

What I want to analyse is when you say:
9.999... - 0.999... = 9.
Here you are subtracting two numbers which are indeed infinite geometric series. So it might be suitable to turn to that discussion to see the crux of my problem so far:

When you consider the geometric series, say 0.999... = 0.9 + (1/10)9 + (1/100)9 + ..., mathematicians have discovered through some simple multiplication and subtraction of finite geometric series that the sum of the first n-terms is:

Sn = a + ar + ... + ar^(n-1)
= (a - ar^(n))/(1-r)

Which can be simply obtained. Now, when we say that the sum of an infinite series equals a finite number, what we are saying is that
lim[n->inf] Sn = S in the reals

Provided that 0inf] Sn = lim[n->inf] (a - ar^(n))/(1-r)

And the right-hand side obviously *equals* a/(1-r) since we say that r^(n) = 0 as n -> infinity.

OR DOES IT? This is the crux of the matter, as this EQUALITY has to hold if your subtraction is to be true. I mean, if there is some ambiguous undefined component in these results, then we won't be able to say even that 0.999... = 0.999..., since first we would have to establish that this component is the same for both infinite geometric series. But consider:

lim[n->inf]r^n, with 0 0 as n -> inf, but we cannot say that r^(n) = 0 as n -> inf, because it doesn't!

So this limit introduces a term in each geometric series that is infinitesimal--it is closer to zero than any other number. And the important part is that we have no clue how to add and subtract infinitesimals, just like infinity - infinity makes little sense to us. Which means that 9.999... - 0.999 = a number closer to 9 than any other real number, maybe we might write 9 +/- I, for some extremely small I. Now of course if we just want to deal with real numbers, we can ignore infinitely small numbers.

This topic is quite vital, in my opinion at least, and I like your explanations. But one might witness that in Calc II, for example, we do stuff like:

dy/dx = h(x) =>
dy = h(x) * dx

Thus we can substitute into an integral and change the limits of integration, and ultimately solve this integral. But we must recognise that dy and dx are numbers closer to zero than any other real numbers! We got them via a limiting process; therefore, infinitely small numbers are not "pointless" things to be thrown away all the time.

For the record, however, the argument that 0.999... = 1 is quite appealing in that it demonstrates how infinitely small values are handled typically in mathematics--namely, we cast them away as "really" zeros when we take the limits.

[Supernova426]

It seems that all of the arguments against ".9999... = 1" are stating that representations of numerical values are different from the numerical values themselves. You must use your brain to truly understand the true numerical values of repeating decimals, as Garthnak has so wisely described.

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