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Usually school children are taught fractions and decimal representations way before the notion of limit. So they must come to the idea that infinite decimal sequences like 0.999... are the same as an infinite sequence of partial sums 0.9 + 0.09 + 0.0009 + ... And when they are told that 0.999... = 1 they unavoidably get the wrong impression that the sequence has a fixed value.

I know from my own bad experience that many students are not aware of the fact that a strictly increasing sequence has no fixed numerical value since it increases without end. Usually mathematicians know that the limit has to be applied but from time to time I have even heard mathematicians claim: Every finite sequence fails but the infinite sequence is 1 - as if the infinite sequence of partial sums could contain an infinite partial sum or could somehow reach 1! Therefore it is very important to clear up these things early enough in order to prevent such mistakes.

My question: When is the appropriate time to correct this wrong picture and to tell them the truth about the meaning of the decimal representation?

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When is the appropriate time to correct this wrong picture and to tell them the truth about the meaning of the decimal representation?

I am not entirely clear yet on what the wrong impression here is, but a reasonable answer is: The appropriate time to discuss the precise meaning of decimal representation is when limits of sequences are discussed, and the level of rigour can be the same. Once infinite sums are also introduced, one can take a second look at the subject.

A rigorous approach to limits and sequences seems to happen at around the first or second year of university mathematics studies in the universities with which I am familiar.

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  • $\begingroup$ I agree completely. But I would like to add: Before this time the pupils should not be told that 0.999... = 1, because without this provision the equation is false. $\endgroup$ – Wilhelm Mar 9 '18 at 14:37
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    $\begingroup$ There are many things one learns about before knowing all the details. I for one am glad I do not have to know all about internal combustion engines and engineering and such to drive to the grocery store, and once there I do not have to know economic theory that might lie behind the prices I pay for the food I buy. $\endgroup$ – Dave L Renfro Mar 9 '18 at 18:28
  • $\begingroup$ @ Dave L Renfro: But don't you think that students of mathematics and sciences should know about these principle things? If however they first learn that 0.999... = 1 without knowing about the background it is hard to convert them. Certainly this lack of education is the reason why so many quarreling about 0.999... swamps the internet. $\endgroup$ – Wilhelm Mar 9 '18 at 19:37
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    $\begingroup$ @Wilhelm This debate only swamps the internet when you are around. Most of the time it doesn't come up, because almost who cares about this issue already understands it. I think that students can have an informal understanding of limits already in elementary school, and talking about repeating decimals is a good place to start wrestling with these issues. $\endgroup$ – Steven Gubkin Mar 10 '18 at 20:39
  • $\begingroup$ @StevenGubkin Why not write an answer? $\endgroup$ – Tommi Mar 11 '18 at 7:11

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