A proof that 0.999~ <> 1

[capsulecorpjx]

http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/

I thought this was interesting.

Any mathematicians can comment on the proof above that 0.999~ <> 1?

[zaza]

No! Stop it now! You've already been told about this.

To comment on this Hackenstrings link you keep peddling, I quote:

It should not be thought that the Hackenstrings approach in any way shows that conventional mathematics is `wrong'. Rather, the axioms of Hackenstrings arithmetic are different from those of the real numbers, one consequence being that in Hackenstrings 0.999... < 1. So any proof that 0.999... = 1 must fail when applied to Hackenstrings, which in turn must mean that one of the `different' axioms has been used.

To rephrase this - the Hackenstrings system uses a different set of arithmetic axioms from real numbers. When you use a different set of axioms then you can have whatever you like. If you want to say "0.999...<>1 according to the axioms specified for Hackenstrings", then I'm prepared to believe you, on the grounds that I'm not sufficiently interested in Hackenstrings to research it and try to prove incorrect an expert in the field. However, in the land of mathematics in which the majority of us dwell and in which presumably we are intended to contemplate this question, given that you have not specified otherwise, then 0.9 recurring is equal to 1.
I'm sorry that you don't like it, and you can argue with us as long as you like, but you can't argue with nature I'm afraid.

I would also like to quote, from the same webpage:

A Frequently Asked Question (`FAQ') in the sci.math Internet newsgroup concerns the relationship between 0.999 ... and 1; indeed, this question is FAQ #1 for that group. Usually, some student or amateur cannot quite believe that these two numbers are the same, and asks for reassurance or proof. Of course, there is a satisfactory analytic proof...


Please, I beg you. No more.


zaza

[capsulecorpjx]

I justed wanted a professional mathematician's view point on this.

And the whole concept is an issue of language.
But I think the claim that 0.999~ = 1 is applied for any number system in that 0.xxx~ = 1 Where x is the top number of a number system (9 for base 10, 1 for base 2).

Therefore, if you can disprove this theorum in a different number system that has the language for it, then you can also disprove it in the base ten system.