0.999…. and the Debate that Repeats Forever

In Calculus class, it was often hard to resist distractions. I’d taught those kids for four years, and there was too much fondness built up between us. We swapped movie suggestions, debated Beatles albums. I’m surprised we got anything done.

But one day’s digression stayed surprisingly on-topic. (Well, at least it had to do with math.) The students got curious about repeating decimals.

Adrian led a pack of disbelievers in the claim that 0.999… = 1. (By 0.999…, I mean the decimal number in which the 9’s go on forever and ever.) “You’re crazy,” Adrian said. “You’ve lost your grip on reality, Orlin. Snap out of it.”

With my sanity under challenge (for neither the first nor the last time), I pushed back by offering the standard proof of the fact:

At this point, dissent began to flow among Adrian’s ranks. Some surrendered to my argument, but Adrian held firm. “Witchcraft,” he said.

“Fine then,” I said. “Give me a proof that they’re different.”

“No proof needed,” Adrian reported. “It’s by definition.”

“Ah,” I said, “but you wouldn’t want your definition to lead to inconsistencies, would you?”

“It doesn’t. Besides,” Adrian huffed, folding his arms, “you don’t tell me what I want.”

“Well, consider this,” I said, and quickly offered an alternative justification – not quite a proof – of the same fact, one to highlight the absurdity of Adrian’s stance:

By now, the other students had begun to turn on Adrian, who still held firm. “He must have clicked a YouTube video about 9/11 being an inside job,” Angel suggested. “He’s grown paranoid and delusional.”

You’re delusional,” Adrian replied. “I’m just standing up for what’s right. And true. It’s the American way.”

I could have left it there – Adrian had delivered his performance, landed some good-one liners (even Kevin was cracking smiles), and instigated a debate that led to a cool proof. It was probably time to steer the lesson out of the wilderness and back onto dry pavement.

But the Socratic in me (I grew up with a family cat named Socrates) couldn’t resist another attempt. “Suppose you’re right, Adrian,” I said.

“Thank you.” Adrian bowed to the class.

I pressed on. “If 0.999… isn’t equal to 1, then what is it?”

“It’s the number right next to 1.”

“What do you mean?”

“It’s so close to 1 that there’s no number in between. It’s the biggest number that’s still smaller than 1.”

“I see,” I said. “So what’s the distance between 1 and 0.999…?”

“It’s a tiny little number, of course,” Adrian replied.

“How tiny?”

“The tiniest number of them all.”

“Isn’t that just 0?” I asked.

“There you go again, with your crazy talk,” Adrian scolded. “It’s not 0. That’s the smallest number. This is the second-smallest number.”

“And this second-smallest number,” I said, “what happens if you divide it in half?”

Adrian shook his head. “That’s dangerous. Like splitting the atom. Creates an explosion.”

“C’mon,” I said. “What if you take your little number, and divide it by two? Dividing it by two should make it smaller. But you said it’s already the smallest.”

“That’s why you shouldn’t divide it,” Adrian said. “You’re being reckless.”

“But can’t any number be divided by two? Isn’t there always a smaller number?”

He shook his head. “Nope.”

I turned to the rest of the class. “Does this remind you guys of anything?”

They nodded. “Limits.”

It was a teacher’s sweetest dream: an oddball digression that winds up shedding new light on the central themes of the course. I gestured triumphantly. “You see, Adrian? If there were a ‘smallest’ number, the concept of a limit wouldn’t make any sense. Calculus would fall apart.”

“No calculus?” Annie exclaimed. “You’re saying if we agree with Adrian, then calculus goes away?”

“All right!” Stephany said.

“Okay,” I said, sensing trouble. “I think it’s time we get back to Newton’s Method.”

“Why should we bother?” Jose said. “You just admitted that calculus doesn’t exist.”

Adrian was seizing up with giggles. “Is this funny for you, Adrian?” I asked.

He was too busy laughing to reply.

Note: As usual, I’ve used imagination to plug the holes in memory – though Adrian is every bit as lovable and mischievous as described. Good luck at SLO!

For what it’s worth, the notion of a “smallest positive number” (call it ε) wreaks havoc on the real number system. ε winds up behaving like a bizarre zero (bizero?). Specifically, the only way to salvage the existence of division is to say Aε = ε for any A (just like zero), and yet A + ε ≠ A (unlike zero). It creates a version of the reals where every number has precisely two “neighbors” – one directly above, and one directly below. And the distance to these neighbors is incomparably smaller than the distance to any other number.

Mathematicians out there: While this obviously destroys many of the reals’ nicest properties (ε, for example, has no multiplicative inverse, so we’ve lost field structure), are there any internal inconsistencies?

EDIT 9/7/2013: Check out James Tanton’s great take on this question.

35 thoughts on “0.999…. and the Debate that Repeats Forever

  1. Wouldn’t 0 * ε equal both 0 and ε in that case? You probably could define that particular product to equal 0 – but I’m not sure if that introduces any new problems.

    1. Yeah… I feel like the 0 rule should supersede the ε rule, but I’m not sure that’s necessary, or if it’s an additional assumption. Perhaps the answer lies in the “non-standard analysis” Wiki page…

    1. Thanks. That sets off some bells in the back of my head. I’ll read up.

      How’s life, by the way? I remember hearing that you were teaching?

      1. Life’s good! I was teaching (calculus and geometry) for a couple years, but moved on to grad school last fall. You’re out in California now, right? Enjoying the other coast?

        1. Yeah! One more year in the Bay Area, then who-knows-where for my wife’s postdoc. Let me know if you ever hit up Oakland/SF.

  2. Adrian:A bullet cannot hit a man. It’s fired and the man retreats, it covers the new distance and the man retreats again. Orlin; The lead is successively reduced until it ceases to exist. Adrian; You cannot subdivide a fraction again and again to get zero. The bullet cannot hit the man because it cannot get to him.

    1. Apply Zeno’s Paradox to the time class ends. Your students will converge to the existence of calculus quickly enough. 😉

  3. I prefer the number line based argument myself. The multiply by ten argument always strikes some of my students as some sort of magic trick. However, the argument that if the numbers are unequal then there is something between them on the number line appeals to a sense of completeness on the number line. Although my students don’t have the real analysis background or the college Advanced Calculus ideas, there is something inherently appealing about this completeness idea.

    1. I totally agree. The multiply-by-ten argument comes across as tricksy, and the “1/3 times 3” argument is only a little better. Appealing to the real line itself works better. And if they STILL don’t believe it, now you’re talking about the properties of the reals, which is a much more interesting and useful conversation than quibbling over the technicalities of a magic-looking proof.

  4. There’s a link between Goedel’s proof and nonstandard analysis: if you assume that the Goedel sentence is true (add it as an axiom) you get standard analysis, and if you assume it’s false, you get nonstandard analysis (its infinitesimals are the result of assuming that omega-inconsistency is not inconsistency and therefore there are infinitely large numbers past the naturals, of which the infinitesimals are just the reciprocals).

    But what’s truly weird is the possibility that there is a finite proof that the sentence is false! Goedel’s theorem does not exclude this, and although most people assume it isn’t so, it could be — in which case nonstandard analysis is the only analysis. Deeply bizarre.

    1. Thanks. That’s really cool. It’s not surprising, I guess that Goedel offers some insight on the deep axiomatic stuff, but I’m impressed that the Goedel sentence alone could cleave standard analysis from nonstandard. (Also funny – and a little scary – to think that the standard approach could, in an unlikely but conceivable scenario, be all wrong.)

      Your comment and others make me want to read up a little on nonstandard analysis. I don’t know what “omega-inconsistency” is, but I like the idea of infintesimals as reciprocals of larger-than-natural numbers.

      1. ω-inconsistency is this: Suppose that in a certain first-order theory of the natural numbers, for a certain function f,

        P0: f(0) is true and has a proof;
        P1: f(1) is true and has a proof;
        P2: f(2) is true and has a proof;

        yet there is no proof either that Pω = ∀x f(x) or that ~∀x f(x). Since it is independent of the rest of the theory, we can add either Pω or ~Pω as an axiom. If we add Pω, we are both omega-consistent and consistent; if we add ~Pω, we are still consistent, but we are ω-inconsistent, which feels paradoxical but actually is not.

        The Gödel sentence G is of the form ∀x f(x), and if we add G as an axiom, we have standard number theory, which is ω-consistent. But if we add ~G as an axiom, we have non-standard number theory, which is ω-inconsistent (but not inconsistent). In non-standard number theory, the range of x is necessarily extended: there are not only natural numbers but also supernatural numbers (distinct from negative numbers, irrational numbers, imaginary numbers, etc.) In either case, we can now take our theory and construct a new Gödel sentence G′ which is independent of the new theory, which is how we know that number theory is radically incomplete and can never be made complete. Unless …

        Unless there is a finite proof of ~G. That is truly bizarre, and most number theorists reject it out of hand, since what it says meta-mathematically is “G has a proof”. But it is possible that such a proof could be found. If so, number theory would not be inconsistent, because we can never construct this proof of G: it is necessarily infinite in length (handwave, handwave). But if such a proof existed, as I said, non-standard number theory would be the only number theory, and consequently there would be no radical incompleteness. We would just have to accept ω-inconsistency as a fact of life.

    1. Not stupid at all – I think that’s exactly the question to ask.

      I need to read up on my nonstandard analysis, so others might be able to answer better. But it sounds like there’s no natural number of ε’s could ever add up to 1.

      Put another way: if you want enough ε’s to add up to 1, it sounds like it’s possible, but “enough” means “more than any natural number.”

      1. “Enough” in this situation also seems to mean a number as close to infinity (yes,I know it’s a concept) as makes no odds , which would be strange, as infinite zeroes make zero but the aforementioned number is not quite zero, so that would make an infinite number if it was actually infinity you were multiplying by but it’s not, and that all circles right back to that number and zero’s fascinating natures.

  5. These are some really interesting concepts. As some of the other commenters point out, some of the math done seems more like math tricks than actual proofs. However, when it comes to concepts of infinities and the like, you need someone smarter than me to explain it. Good post though! I really enjoyed it.

  6. I wonder about $A + \epsilon + \epsilon$ for some ordinary real number A (say, 1). I think you’re saying this should equal $A + \epsilon$.

    What do you get if you subtract $A + \epsilon$ from both sides, then?

    I think you’ll need a whole hierarchy of epsilons, then, to make this work (as in the surreal numbers, where there exists a number strictly between any two sets of numbers as long as all the elements of one set are less than all the elements of the other). So your $\epsilon$ is what you get in between 0 on the one hand and all the (ordinary) positive numbers on the other. But then you need the number where epsilon is on the small set and all the ordinary positives on the other, so there’s another number there, and another, and another, and …

  7. A better version of the fractions argument is to go with 1/9, 2/9, 3/9…9/9. That way it becomes much more obvious, since 1/9=0.11…1, 2/9=0.22…2, all the way to 9/9=0.99…9.

  8. There’s a chapter on nonstandard analysis (exactly this, in which epsilon is considered an infinitesimal, which can be justified as a legitimate mathematical thing with the properties you’ve explained) in the back of the new edition of “What is Mathematics?” by Courant and Robbins. I’m sure there are other treatments of it, as well, but that’s where I’ve seen it.

  9. I want to complain about your first proof: while (in my experience) it and versions of it are satisfying for many students, it is really misleading in that it fails to explain at all what’s going on. The point is that a decimal expansion represents a particular infinite series — in this case, \sum_{n\geq1}9/10^n — and so any questions like this should ultimately be explained by an argument involving infinite series.

    The reason for the “should” in the previous sentence is that without going back to this meaning, decimals end up being some mystical object that obey certain rules but not others for no obvious reason (see e.g. the Kauffman decimal post). But “limit of a sequence of partial sums” is an object that we (=students in the second half of a calculus sequence) should be able to understand and manipulate without trouble. (OK, I’ve taught calculus before, so “without trouble” is a little bit of a joke, but hopefully the point is ok.)

    In the case of 0.99999…, we have an infinite decimal. None of the finite partial sums is equal to 1, but *in the limit* they approach 1, and this is what the “…” actually means we should do.

    1. I think you’re right. What that first proof does is brush the arithmetic of infinite series under the rug. It’s true, but not necessarily explanatory.

      Of course, infinite series are a pretty tricky concept for students not yet in calculus (and, as you say, for lots of students in calculus, too). Replacing 0.999… with 0.9 + 0.09 + 0.009 + 0.0009 + …. is somewhat illuminating, but most students will run into the same confusion about what that “…” really means.

      BTW, I found a great dialogue another teacher wrote dealing with some of those issues–it’s listed at the bottom of the post now.

      1. After rewriting it as that sum is the point where you go straight to Zeno: “Before class is over today and you can go on to your next class, first you have to get through the first nine-tenths of the hour we have. Then, we have to get through the first nine-tenths of the remaining tenth. But then we’ll still have a hundredth of a class to get through the first tenth of. And clearly this process will go on forever. Since you have decided that this infinite sum does not add up to 1 but something less than 1, you have decided that we will never get through a whole class. Thus, lucky you, we’ll get to be together here forever!”

        I should think the repulsion this argument inspires will persuade even the staunchest nonbeliever. (This is essentially a formalization of Kelvin’s comment above.)

        1. I like that. I usually present Zeno as a kid in the back seat asking “Are we there yet?” on the way to Disneyland. But trapping my students in class sounds a lot more fun.

  10. Look up the hyper real number system. In it, you can have infinitely tiny numbers. You can also have a number with an infinite number of 9’s that is less than 1 (there are actually multiple such infinite 9’s constructions, some closer or farther from 1.)

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