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Possible Duplicate:
Does .99999… = 1?

At supper today my daughter was discussing her maths (she's 13) - she had been studying putting decimal numbers into what she called standard form $A*10^n$, and what were the possible values for $A$.

She proudly said that all her classmates had said that $A$ should be between 1 and 10, while she had said that it should be between 1 and 9.9999 ... (recurring)

What interested me was that this is the first realistic attempt to distinguish between the two that I've ever really encountered. It led to an interesting discussion about open and closed intervals.

I've reformulated the next bit to clarify what I was trying to get at

So my question is whether there are any other places where a distinction between the two might have the vestige of a mathematical rationale - in the sense that there was, in my daughter's formulation, an interestingly intelligent attempt to get at the idea of the openness of the (end of) interval by distinguishing two different ways of writing the same number.

Our mathematical notation and ideas do not follow this intuition - indeed there can be a strong resistance to it, not least because it is such hard work to teach some students that the two numbers really are the same.

But I am interested in whether there are other naïve ways of looking at the distinction between the two formulations which are grasping at real mathematical content (rather than confusion) and which could help my daughter to explore the mathematical concepts she is on the edge of understanding.

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Mark Bennet
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    There is no difference between $10$ and $9.9999\ldots$; both decimal representations correspond to the same number. – Arturo Magidin Jan 11 '12 at 20:32
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    @ArturoMagidin: There's no difference between the real numbers they represent (they represent the same real number), but they are different representations of that number. – Isaac Jan 11 '12 at 20:35
  • @0.999999999999...9999999999998 , do you watch Dexter on Showtime? Admittedly, not really a plan for world domination... – Will Jagy Jan 11 '12 at 20:46
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    Can someone identify in a comment the question this duplicates - I did try to find one and evidently failed. What I was trying to do btw, and why I tagged intuition, was not to identify that the numbers were the same (I know that), but to identify why people (thinking about intelligent 13-year-old children, for example) might think that they were different. I want to teach my daughter good maths, and I was looking for ways of exploring the same idea with her. – Mark Bennet Jan 11 '12 at 20:51
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    @MarkBennet: I'm not sure why a comment with the link wasn't generated upon the first close-as-duplicate vote, but the link is now at the top of the question. – Isaac Jan 11 '12 at 20:54
  • @MarkBennet I think the most useful thing for you is rational numbers and repeats; as $7 | 10^6 - 1$ you get $1/7 = 0.142857142857...$ The number theory aspect is, with $a/b$ and $\gcd (a,b) = 1,$ the smallest $n$ such that $b | 10^n - 1$ is needed to find the period of the repeat decimal, and may always be the period, I will need to look that up. A high-school contest book I have calls that the order of 10 modulo $b.$ Titu Andreescu and Dorin Andrica, "Number Theory" Birkhauser about 2009. – Will Jagy Jan 11 '12 at 21:04
  • @MarkBennet: The key is to try to understand what the decimal representation means (exactly what is being discussed in the class, if I understand you correctly), and then translate what it is supposed to mean for the purpose of $0.9999\ldots$. You may want to start with the fact that $(1/2) + (1/2)^2 + \cdots + (1/2)^n+\cdots$ equals $1$, and go to the fact that $(9/10) + (9/10^2) + \cdots + (9/10^n) + \cdots = 1$, so there is no difference between the numbers represented by the two notations. – Arturo Magidin Jan 11 '12 at 21:22
  • @Isaac I originally posted the link in a comment manually, then later removed the comment. The presence of a comment with the link may have suppressed the automatic posting. – Austin Mohr Jan 11 '12 at 21:25
  • @WillJagy: See http://math.stackexchange.com/a/445/72 – Isaac Jan 11 '12 at 21:29
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    @MarkBennet See the earlier answer by Isaac. I expect your daughter would enjoy this: http://store.doverpublications.com/0486210960.html by Albert Beiler. Repeating decimals are chapter 10, pages 73-82, especially for $1/p$ for $p$ prime. The simplest property is when 10 is a primitive root $\pmod p,$ you get the repeating part of $a/p$ a cyclic permutation of that for $1/p.$ So, ( 10^18 - 1 ) / 19 = 52631578947368421 and 2 * ( 10^18 - 1 ) / 19 = 105263157894736842 and 3 * ( 10^18 - 1 ) / 19 = 157894736842105263, and you can see where the extra 0 goes in the actual decimal 1/19. – Will Jagy Jan 11 '12 at 21:47
  • I have to say that I think the question I wanted to ask is different from the identified duplicate - certainly the answers there are not quite what I was looking for. I'm not voting for a reopen, because I can see from responses so far that I'll likely get the same kind of answers. But I do think it is interesting how young people learn maths and that such intuitions are worth noting. – Mark Bennet Jan 11 '12 at 21:50
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    @WillJagy Thanks, great spot - I have a copy to hand. – Mark Bennet Jan 11 '12 at 21:51
  • See also this answer on the general case of periodic decimal expansions. – Bill Dubuque Jan 11 '12 at 22:33
  • @MarkBennet You're welcome. – Will Jagy Jan 11 '12 at 22:47
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    Voted to reopen as the question is about teaching subtle concepts such as the quoted example of open, closed and half-open intervals. Another example is that while fractions with a denominator with prime factors only of 2 and/or 5 have two decimal representations, different rules apply in other bases (in the so-called factorial number system, all non-zero rationals have this property). – Henry Jan 12 '12 at 02:54
  • @BillDubuque Thanks. Hardy and Wright has a chapter on decimal expansions too. – Mark Bennet Jan 12 '12 at 06:27
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    @MarkBennet: I'd be more comfortable voting to reopen if the question were edited to make it clearer what kind of answer you're looking for. – Isaac Jan 12 '12 at 21:53
  • @Isaac: My daughter, not knowing any convention, used the two different ways of expressing the same number to distinguish between the "open" and "closed" ends of an interval. The possibility had never occurred to me before. Having noted this quite unexpected and unconventional way of looking at things, does this suggest any other similar possibilities? – Mark Bennet Jan 12 '12 at 22:08
  • @MarkBennet: similar possibilities like other ways in which one might think of 10 and 9.9999... as distinct to serve a particular purpose? – Isaac Jan 12 '12 at 22:11
  • @Isaac: basically, yes. But my daughter is not initiated into "mathematics" as it is known by mathematicians. So she has identified a possible way of looking at things which is not the route mathematics chooses, but is almost "what might have been" except that there is a good reason for the way things are. But my intention was to explore naïve possibilities, rather than sophisticated ones. I think most of us have left such possibilities behind, but I think that for teaching, understanding what people might be thinking about what we think is obvious can be very helpful. – Mark Bennet Jan 12 '12 at 22:24
  • @MarkBennet: If you edit some of what you've said in your last two comments into the question itself (I'd think in or near that last paragraph), I'll gladly add my vote to reopen the question. – Isaac Jan 12 '12 at 22:32
  • Why is there a vote for deletion of this question? – t.b. Jul 02 '12 at 00:21
  • @MarkBennet There might be a reasons to "tag" numbers to distinguish between open and closed ends of intervals (tagged stops of games in CGT is very similar), but I think it's more misleading than helpful to try to use equivalent decimal representations for that purpose. – Mark S. Dec 10 '13 at 18:28

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