It depends on what set of numbers you're considering. In the reals, we can prove very immediately that 0.999... = 1: Least upper-bound axiom => Axiom of Archimedes => Non-existence of infinitesimal reals => 0.999999... = 1 The only thing you can question here is the least upper-bound axiom, which is true because it just is. If you reject the least upper-bound axiom, there are sets of numbers (such as the surreals) for which 1-(1/omega) exists and has a decimal representation of 0.999999.... However, unlike the reals, a decimal representation does not determine a unique surreal. Sincerely, Adam P. Goucher http://cp4space.wordpress.com

----- Original Message ----- From: Thane Plambeck Sent: 11/14/12 05:39 AM To: math-fun Subject: Re: [math-fun] when and why was it agreed that 0.999...=1?

Vi Hart dealt with all these flawed proofs that 1= 0.99999....in a video

http://www.youtube.com/watch?v=wsOXvQn3JuE

On Tue, Nov 13, 2012 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:

...

I'm pretty sure the shift in thinking -- at least among mathematicians -- occurred when Cauchy came up with the modern epsilon-delta definition of limit. Which Wikipedia says occurred in 1821.

I've seen way too many non-mathematically trained people insisting (e.g., on sci.math) that 0.9999... definitely does not, can not, equal 1. So I guess the shift among non-mathematicians is still in progress.

--Dan

On 2012-11-13, at 8:33 PM, Gary Antonick wrote:

...

Hi all,

I'm wondering if anyone knows when and why it was agreed that 0.999...=1.

That is, the expression meant its limit of 1 and not something that got closer and closer to 1.

I've asked several people and have gotten some big pieces but not quite the whole story: Keith Devlin (last week): before Cantor mathematicians considered "0.999..." to mean a *growing sequence* of 9's after the decimal. After Cantor it was decided that the expression meant the *limit* of this sequence: an infinite number of 9's after the decimal all at once. Steven Strogatz (this afternoon) suggested talking to John Stillwell John Stillwell (a couple hours ago) seemed to indicate the shift in perspective happened gradually, with Zeno arguing for the growing sequence (which would never get to 1) and everyone after the Axiom of Choice agreeing that the expression was referring to its limit.

-400 Zeno: 1/2 + 1/4 + 1/8 etc will never get to 1 -350 Aristotle: 1/2 + 1/4 + 1/8 etc eventually gets to 1 -300 Euclid: 1/4 + 1/4^2 + 1/4^3 + etc eventually gets to 1/3 1585 Stevin: 0.999... = 1 1671 Newton: 0.999... = 1 1858 Dedekind cuts 1880? Cantor 1904 Zermelo: Axiom of Choice

Does anyone have more precise timing for this shift from thinking?

All the best,

Gary _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun