I have been looking for various proofs on why the infinite repeating decimal .999....=1 and I came across an explanation using Dedekind cuts on Wikipedia's website: https://en.wikipedia.org/wiki/0.999...#cite_note-13

But the definition presented brought up other questions for me that I had some difficulty answering.

1) My understanding from the Wikipedia page is that real numbers are defined by Dedekind cuts, where a real number is then equal to the infinite set of rationals less than it. If real numbers are sets then how do the usual operations on real numbers translate? For example, is addition set union? What about division?

2) I would like to present the proof to some high school senior students, but would like to read a "friendly" introduction of Dedekind cuts first. Do you know of any good resources?

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    $\begingroup$ You can see Ethan Bloch, The real numbers and real analysis (2011) $\endgroup$ Commented Sep 13, 2015 at 18:57
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    $\begingroup$ Not necessarily the place to start, but definitely worth reading at some point: I'm a huge fan of (the translation of) Dedekind's original paper, "Essays on the theory of numbers" (gutenberg.org/files/21016/…). $\endgroup$ Commented Sep 13, 2015 at 19:01
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    $\begingroup$ I know this is not what you are asking, but for the record, Dedekind cuts are not a nice way to define real numbers: the "better" way to do it is to use Cauchy sequences of rationals, because this extends to other sets that have less structure than the real numbers. The best resource I've seen for the various definitions of real numbers is the Appendix chapters in Spivak's Calculus, if you can get hold of it, but it's probably not as friendly as you might want. It does have, in the exercises, a number of alternate ways of defining the reals. $\endgroup$
    – Chappers
    Commented Sep 13, 2015 at 19:08
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    $\begingroup$ If I may be so bold, if you are giving a mathematical presentation to high school students, you may want to pick a more flamboyant, immediately satisfying result. Students are likely to consider this result as just micromanaging definitions, and indeed you will spend a huge amount of time trying to give students some understanding of Dedekind cuts when they barely know what a set is. Plus you will have students who simply will not grasp the concept within an hour and will eat up your time with foundational objections, leaving those that do understand bored and frustrated. $\endgroup$
    – guest
    Commented Sep 13, 2015 at 19:09
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    $\begingroup$ If $0.999\ldots<1$ there would be some numbers in between. What would be their decimal expansion? The only way to resolve this is to declare that $0.999\ldots$ and $1$ are names for the same number, namely $1$. $\endgroup$ Commented Sep 13, 2015 at 19:29

1 Answer 1


Dedekind cuts are somewhat elementary from an analysis point of view, but understanding exactly why they give the (complete) real numbers with all arithmetic operations defined as expected with all the desired properties is a bit harder. For addition, if you have two upper bounded sets of rational numbers then you can form the set of all pairwise sums of those rational numbers, and then take the least upper bound and show that this satisfies axioms of addition. Multiplication is similar for non-negative upper bounded sets of rationals, and care has to be taken when signs between two reals are potentially one or both negative. Division is accomplished if you can establish that multiplicative inverses of positive reals exist.

I've seen a few presentations of Dedekind cuts and they're all basically the same. I don't think there's any "silver bullet" that would allow you to significantly more easily explain the concept (complete with operations defined on reals) to high school students.


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