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 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