An infinite question: Why doesn't .999~ = 1?

Cecil's Columns/Staff Reports
301

The Great Unwashed, he doesn’t claim that they flip-flop in-and-out of existence. He is simply making the equally absurd claim that, so long as you are trying to represent a number with an infinitely long sequence of digits, that numerical sequence (whatever it is) cannot properly be used to represent the number it theoretically equates, because the sequence never actually “reaches” the equivalent point on the number line. So 0.999… never reaches 1, and cannot be said to be equal to it. The mistake has to do with how he treats doing something infinitely often. As long as you and he don’t agree on the meaning of doing something infinitely often, you will not resolve this silly debate he is insisting upon.

And you will notice that engjr fled this debate long ago…

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Imposing an unnecessary infinity and then refusing to agree that 1/3=.333… reminds me of the Tortoise’s cleverer refusal to accept Achilles’ birthday.

[QUOTE=the Tortoise and Achilles, as related by Douglas Hofstadter]

One fine May day, the Tortoise and Achilles meet, wandering in the woods. The latter, all decked out handsomely, is doing a jiggish sort of thing to a tune which he himself is humming. On his vest he is wearing a great big button with the words “Today is my Birthday!”

Tortoise: Hello there, .Achilles. What makes you so joyful today? Is it your birthday, by any chance?

Achilles: Yes, yes! Yes it is, today is my birthday!

Tortoise: That is what I had suspected, on account of that button which you are wearing, and also because unless I am mistaken, you are singing a tune from a Birthday Cantata by Bach, one written in 1727 for the fifty-seventh birthday of Augustus, King of Saxony.

Achilles: You’re right. And Augustus’ birthday coincides with mine, so THIS Birthday Cantata has double meaning. However, I shan’t tell you my age.

Tortoise: Oh, that’s perfectly all right. However, I would like to know one other thing. From what you have told me so far, would it be correct to conclude that today is your birthday?

Achilles: Yes, yes, it would be. Today is my birthday.

Tortoise: Excellent. That’s just as I suspected. So now, I WILL conclude it is your birthday, unless …

Achilles: Yes-unless what?

Tortoise: Unless that would be a premature or hasty conclusion to draw, you know. Tortoises don’t like to jump to conclusions, after all. (We don’t like to jump at all, but especially not to conclusions.) So let me just ask you, knowing full well of your fondness for logical thought, whether it would be reasonable to deduce logically from the foregoing sentences, that today is in fact your birthday.

Achilles: I do believe I detect a pattern to your questions, Mr. T. But rather than jump to conclusions myself, I shall take your question at face value, and answer it straightforwardly. The answer is: YES.

Tortoise: Fine! Fine! Then there is only one more thing I need to know, to be quite certain that today is

Achilles: Yes, yes, yes, yes … I can already see the line of your questioning, Mr. T. I’ll have you know that I am not so gullible as I was when we discussed Euclid’s proof, a while back.

Tortoise: Why, who would ever have thought you to be gullible? Quite to the contrary, I regard you as an expert in the forms of logical thought, an authority in the science of valid deductions, a fount of knowledge about certain correct methods of reasoning. . . To tell the truth, Achilles, you are, in my opinion, a veritable titan in the art of rational cogitation.
And it is only for that reason that I would ask you, "Do the foregoing sentences present enough evidence that I should conclude without further puzzlement that today is your birthday

Achilles: You flatten me with your weighty praise, Mr. T-FLATTER, I mean. But I am struck by the repetitive nature of your questioning and in my estimation, you, just as well as I, could have answered ‘yes’ each time.

Tortoise: Of course I could have, Achilles. But you see, to do so would have been to make a Wild Guess-and Tortoises abhor Wild Guesses. Tortoises formulate only Educated Guesses. Ah, yes-the power of the Educated Guess. You have no idea how many people fail to take into account all the Relevant Factors when they’re guessing.

Achilles: It seems to me that there was only one relevant factor in this rigmarole, and that was my first statement.

Tortoise: Oh, to be sure, it’s at least ONE of the factors to take into account, I’d say-but would you have me neglect Logic, that venerated science of the ancients? Logic is always a Relevant Factor in making Educated Guesses, and since I have with me a renowned expert in Logic, I thought it only Logical to take advantage of that fact, and confirm my hunches, by directly asking him whether my intuitions were correct. So let me finally come out and ask you point blank: “Do the preceding sentences allow me to conclude, with no room for doubt, that Today is your Birthday?”

Achilles: For one more time, YES. But frankly speaking, I have the distinct impression that you could have supplied that answer-as well as all the previous ones-yourself.

Tortoise: How your words sting! Would I were so wise as your insinuation suggests! But as merely a mortal Tortoise, profoundly ignorant and longing to take into account all the Relevant Factors, I needed to know the answers to all those questions.

Achilles: Well then, let me clear the matter up for once and for all. The answer to all the previous questions, and to all the succeeding ones which you will ask along the same line, is just this: YES.

Tortoise: Wonderful! In one fell swoop, you have circumvented the whole mess, in your characteristically inventive manner. I hope you won’t mind if I call this ingenious trick an ANSWER SCHEMA. It rolls up yes-answers numbers 1, 2, 3, etc., into one single ball. In fact, coming as it does at the end of the line, it deserves the title “Answer Schema Omega”, V being the last letter of the Greek alphabet-as if YOU needed to be told THAT!

Achilles: I don’t care what you call it. I am just very relieved that you finally agree that it is my birthday, and we can go on to some other topic-such as what you are going to give me as a present.

Tortoise: Hold on … not so fast. I WILL agree it is your birthday, provided one thing …

Achilles: What? That I Ask for no present?

Tortoise: Not at all. In fact, Achilles, I am looking forward to treating you to a fine birthday dinner, provided merely that I am convinced that knowledge of all those yes- answers at once (as supplied by Answer Schema w) allows me to proceed directly and without any further detours to the conclusion that today is your birthday. That’s the case, isn’t it?

Achilles: Yes, of course it is.

Tortoise: Good. And now I have yes-answer ro + 1. Armed with it, I can proceed to accept the hypothesis that today is your birthday, if it is valid to do so. Would you be so kind as to counsel me on that matter, Achilles?

Achilles: What is this? I thought I had seen through your infinite plot. Now doesn’t yes- answer (o+l satisfy you? All right. I’ll give you not only yes-answer co + 2, but also yes-answers ro + 3, ro + 4, and so on.

Tortoise: How generous of you, Achilles. And here it is your birthday, when I should be giving YOU presents instead of the reverse. Or rather, I SUSPECT it is your birthday. I guess I can conclude that it IS your birthday, now, armed with the new Answer Schema, which I will call "Answer Schema 2ro ". But tell me, Achilles: Does Answer Schema 2ro REALLY allow me to make that enormous leap, or am I missing something?

Achilles: You won’t trick me any more, Mr. T. I’ve seen the way to end this silly game. I hereby shall present you with an Answer Schema to end all Answer Schemas! That is, I present you simultaneously with Answer Schemas co, 2 to, 3 to, 4 a, 5 ro, etc. With this Meta-Answer-Schema, I have JUMPED OUT of the whole system, kit and caboodle, transcended this silly game you thought you had me trapped in-and now we are DONE!

Tortoise: Good grief! I feel honored, Achilles, to be the recipient of such a powerful Answer Schema. I feel that seldom has anything so gigantic been devised by the mind of man, and I am awestruck by its power. Would you mind if I give a name to your gift?

Achilles: Not at all.

Tortoise: Then I shall call it "Answer Schema ro 2 ". And we can shortly proceed to other matters-as soon as you tell me whether the possession of Answer Schema ro 2 allows me to deduce that today is your birthday.

…

Achilles: All right, though I’m beginning to doubt whether it will do any good.

… [I’ll truncate this conversation on this pessimistic note by Achilles]
[/QUOTE]

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The rest of us have clearly and repeatedly defined (0.333…) and (0.999…) on the number line … for which your only answer is “no, it’s not”.

The reason you give is that we’re including all the elements of the summation, that if we leave off even just one element, we’ll get a different answer. This kinda violates the rules of basic addition, we just can’t leave out a part and expect to get the correct answer.

As soon as you can define a point 0.333~ on the number line you will be able to define what number can be expressed by 0.333~. I think we’d all appreciate it if you left us out of the paradox, this seems yours and Zeno’s alone …

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Then the problem is solved. The point is 1/3.

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No one has shown (even RitterSport has not) how we reach point 1/3 by getting closer to 1/3 infinitely many times and reaching 1/3 not even one time!

I have never claimed that there must be a decimal expansion of 1/3. Where does that come from?

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It exists by definition in our mathematics. If it doesn’t in yours, you’re doing it wrong.

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No one is leaving out a part. We do consider all the infinitely many steps/digits - each and every n-th step/digit for each and every n ∈ ℕ.

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Of course. It’s the same mathematics that sets 0.999… = 1 without showing how the limit comes into play.

0.999… simply means infinitely many attempts to reach 1 - for each attempt is valid that it doesn’t succeed. There is no “limit step” or whatever.

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It does reach 1, you’re just doing it wrong …

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I never claimed that you had claimed that there must be a decimal expansion of 1/3.

No, I simply asked you to state what the decimal expansion of 1/3 is.

From your less than direct reply (why do you do that?) would I be correct in thinking that you believe the decimal expansion of 1/3 does not exist.

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That’s not true. No one has shown how to get there in a way that satisfies your flawed rules.

Any finite number of digits in the decimal system will differ from 1/3.

0.3 differs by 1/30
0.33 differs by 1/300
0.333 differs by 1/3000

We see that when dealing with finites in this manner we “can’t get there”. Zeno’s paradoxes deal with how this affects what seems like a valid description of real world situations, and the resolution wasn’t “this math can’t be used to describe this situation”, it was “infinities are tricky and need to be dealt with not as a very big finite”.

If we continue the exercise above to ridiculous lengths we get a very small distance from 1/3, but no finite gets there. The n’th step always differs from 1/3 by 1/(310^n). Now we can’t “get there” by performing the discrete infinite steps as described and you insist that this means that 0.333… repeating belongs to the interval [0.3, 1/3>, (or [0.333333, 1/3> or whatever), and you’re free to do so. You give up the ability to do math with non-terminating decimal representations, but that’s no big loss. You haven’t produced any great insight into math though. Your peculiar non-definition of 0.999… and its cousins isn’t useful in any way. It’s looking at Zeno’s paradoxes and saying “this math doesn’t work here”. The rest of us see however that the “infinities are tricky and need to be dealt with not as a very big finite” gives us something more, special and universally useful. By the rules of infinities we get that “at infinity”, a conceptual and non-intuitive place, 1/310^n is zero, and if it is zero then 0.333… is 1/3.

There’s nothing magical about your step-by-step approach that makes our choice of rules to deal with infinities invalid. All you’re doing is describing a different set of rules.

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We understand that you don’t get the limit. We also understand that you have no interest in learning despite the many incredibly patient attempts to get you over that hump. What we don’t understand is what this gets you.

You can’t do math with your approach. Zeno understood this. He knew that this approach was limited (pun, yes). He also knew that his version of 0.9999~ must equal 1.0. That was his paradox. It was frustrating. Without the tools to handle that problem he could not advance and go further and deeper into math. Which is exactly where you are.

So you can’t do math for yourself. Nor can you do math with anything else, for as soon as you do you get told you’re doing it wrong, and not understanding what every other mathematician in the world acknowledges. You’ve throw up a gigantic unscalable wall for yourself even though there are huge glowing arrows pointing to a doorway through you can quickly and easily walk.

Why? You’re only hurting yourself. You’re not showing a hole or flaw in math. You’re not smarter than all the mathematicians of the past 150 years. You’re an Internet crank. If you look at the dozens of threads on this subject you’ll see that all the cranks who dispute it have the tell of posting the same single point of “logic” over and over and over. Real mathematicians can tell you in fifty different ways why something is true, and then go on to show why this must be true because of the way it affects every other part of math. You are obviously not doing that. Anyone familiar with the way cranks argue would note that you are wrong even if they knew no math at all.

What’s your next step?

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I cannot see what the decimal expansion of 1/3 could be.

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1/3*10^n is zero if n = infinity. But this mathematics is erroneous, because n ∈ ℕ. There is not infinity in ℕ.

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It’s not me who has to do the next step.

We complete 0.333… in infinitely many steps, but we don’t complete reaching 1/3 in infinitely many steps. That’s why they are different. It is well known in set theory that the union of { [0, 0.3], [0.3, 0.33], [0.33, 0.333], … } is [0, 1/3) and not [0, 1/3].

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Not your next step in continuing to make this false case. Your next step in actually using your false belief to do math. What math do you do with it? Do you do math at all? If not, what difference does this make to you? If so, give us a concrete example of your math solving a real-world problem.

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We’ve noticed … try long division … do they even teach that in grammar school anymore?

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This is a somewhat puzzling response to the question. It could mean:

  1. You believe there is no decimal expansion of 1/3
  2. You suspect there may be one, but you are not knowledgeable enough to say what it is
  3. You wish to avoid taking a position on whether one exists
  4. Something else?

Can you help by saying which of these is correct?
Also: Is 1/3 unique in this regard? Or are there other fractions whose decimal expansions you’re unable to see?

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Folks, why is this “discussion” still ongoing? All it does is prove his point (in a weird, not so very helpful way).

He refuses to accept that the repetition of something “infinitely often” produces a result that is identical to the defined limit. So 0.999… != 1 because the first is a repeating sequence of digits that never “reaches infinity” in terms of how many '9’s there are. As long as he persists in this fundamentally different concept of how to treat something repeating infinitely often, no argument, discussion, reasoning, etc. is going to change his viewpoint. The same would be true of ANY repeating decimal expansion of a fraction (as well as, obviously, irrational numbers). So 1/7, 23/346, etc. are all going to have the same trouble.

Indeed, if you will, we can understand his viewpoint best by considering irrationals. Take π. (3.1415926…). Since no matter how far we extend the decimal expansion, we can always extend it farther (infinitely far!), and since we cannot EVER know exactly how far off our finite expansion is, any decimal expansion of π is not equal to π. The assertion here is that ANY decimal expansion of 0.999… is not 1, because no matter how far you go along the infinitely long path, you always are just that little bit short. As Exapno Mapcase points out, if you view it that way, what use is that math?

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Well … it’s taking longer than we thought …