In grade school and high school, I was taught a real number is a number with a decimal expansion--that is, a finite sequence of digits followed by a decimal point followed by an infinite sequence of digits.

When I moved on to studying analysis, I was introduced to the Dedekind cut construction of the real numbers, and then proved every real number could be expressed as a decimal expansion.

Now presumably, the real numbers could also be rigorously constructed as decimal expansions, very similarly to the Cauchy sequence construction.

Is there a particular reason why, when Dedekind performed his original construction of the reals, he chose to use cuts rather than decimal expansions (or base 2 expansions for that matter)? Is there some unforeseen difficulty in rigorously constructing the reals by means of decimal expansions?

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    $\begingroup$ Dedekind cuts are more natural, in a sense. If we construct the reals using a specific base (base $10$, for example), it's a bit arbitrary. $\endgroup$ – MathematicsStudent1122 Dec 2 '16 at 21:59

One problem is that multiple decimal representations correspond to the same real number. Of course, this is easily solved, though.

I think the real point was foundational: defining reals as decimal expansions is defining them as sequences of rationals (the Cauchy definition is via equivalence classes of sequences of rationals - the decimals approach picks out a "canonical" Cauchy sequence for a given real). By contrast, Dedekind cuts define a real as a set of rationals. On the philosophical side, sets are slightly simpler than sequences, and Dedekind was very interested in developing the foundations of mathematics.

Dedekind's definition is also more natural in that it doesn't fix a base: so it really defines the real numbers without making any arbitrary choices.

  • $\begingroup$ In the case of binary expansions, multiple representations can be viewed as the algebraic relations that must be enforced, so not a problem but part of the 'logic' to be embraced. Compare with en.wikipedia.org/wiki/Presentation_of_a_group The algebraic logic to handle "De Thiende ('the art of tenths') / Simon Stevin" would no doubt be intractable. $\endgroup$ – CopyPasteIt Dec 11 '18 at 14:51
  • $\begingroup$ @CopyPasteIt I'm not sure what your last sentence means (it's certainly not the case that the logic for handling decimals is intractable in any sense that I'm aware of), or what your distinction between the algebra and the logic is. Certainly it's a problem insofar as it leads to technical issues in certain arguments, although this is subjective. And nothing is substantively different between binary and decimal expansions (although binary is more natural than decimal). $\endgroup$ – Noah Schweber Dec 11 '18 at 14:53
  • $\begingroup$ My answer to this question can be chalked up to 'food for thought'. You can look at the (infinite) sigma notation and manipulations on it as an abstraction, nothing to do with a series as a limit. You are then using group presentation theory. I am not advancing this as something to be adopted or assimilated - but the logic is coherent. $\endgroup$ – CopyPasteIt Dec 11 '18 at 15:08

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