0.999...= 1 - Hackdot

Re:0.999...= 1 by obastani on Tuesday April 28, 2020 @01:04PM (#23009066)

This is a good point, but an even more basic issue is that the question "what is a number" is a matter of definition. There isn't a "correct" definition of numbers; only one that we've accepted as standard. The accepted definition of a "real number" is actually quite complicated [1], and it's certainly not easy to convey why this complexity is necessary. Other definitions are also possible [2], but nonstandard.

The simplest definition is: a finite decimal ak ... a1.b1 ... bh is defined to be a fraction and an infinite decimal is defined to be a limit. You'd still have to define what a limit is, but that is somewhat more intuitive.

[1] https://en.wikipedia.org/wiki/Dedekind_cut

[2] https://en.wikipedia.org/wiki/Hyperreal_number

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Re:0.999...= 1 by btilly on Tuesday April 28, 2020 @03:53PM (#23011111)

Sorry, no.

There are TWO standard definitions of the real numbers. Namely Dedekind cuts and Cauchy sequences. (They are completely equivalent.) The usual decimal representation of a number turns out to be a Cauchy sequence.

The "simplest definition" that you provide turns out to be rather non-simple in practice. Try proving that multiplication is commutative to see the difficulty.

There are plenty of other number systems out there. Try https://en.wikipedia.org/wiki/Surreal_number or https://en.wikipedia.org/wiki/P-adic_number or the complex numbers.

https://en.wikipedia.org/wiki/Surreal_number

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