> "Thomas McGahee" <

tom_mcgaheeRemoveMERemoveMESIGMAIS.COM> wrote, over and over and

> over:

>

> > Whether you like it or not, mathematically the mumber 1 exactly

> > equals .9 repeating.

>

> and David VanHorn <

.....PICLISTspam_OUT@spam@MITVMA.MIT.EDU> replied:

>

> > That's as nonsensical as saying that 2=3 for large values of 2.

> > ....

> >

> > 0.99(followed by any finite or infinite number of nines) is by

> > definition, not equal to 1.0

> >

> > The value of 0.99(inf..) is less than 1.0 by an infinitely small (but

> > non-zero) amount.

>

> [Sorry to single you out, Dave; this applies not only to you,

> but to everyone else who's posted essentially the same thing.]

>

> It's been a long time since I've written a "God DAMN it, you peopole

> are pissing me off" email to the PICLIST, and since James Newton has

> imposed a "no profanity" rule, I guess I won't write one now.

>

> Still, though, you people ARE pissing me off. 0.9 repeating IS

> exactly equal to 1. Someone posted a simple algebraic explanation

> earlier; was that TOO simple? Here, for your edification, is a more

> complex explanation, courtesy of the sci.math FAQ (where the

> "0.999... = 1" question used to be #1 on the list):

>

> 11Q: Why is 0.9999... = 1?

>

> A: In modern mathematics, the string of symbols "0.9999..." is

> understood to be a shorthand for "the infinite sum 9/10 + 9/100

> + 9/1000 + ...." This in turn is shorthand for "the limit of the

> sequence of real numbers 9/10, 9/10 + 9/100, 9/10 + 9/100 +

> 9/1000, ..." Using the well-known epsilon-delta definition of

> limit, one can easily show that this limit is 1. The statement

> that 0.9999... = 1 is simply an abbreviation of this fact.

>

> oo m

> --- 9 --- 9

> 0.999... = > ---- = lim > ----

> --- 10^n m->oo --- 10^n

> n=1 n=1

>

>

> Choose epsilon > 0. Suppose delta = 1/-log_10 epsilon, thus

> epsilon = 10^(-1/delta). For every m>1/delta we have that

>

> | m |

> | --- 9 | 1 1

> | > ---- - 1 | = ---- < ------------ = epsilon

> | --- 10^n | 10^m 10^(1/delta)

> | n=1 |

>

> So by the (epsilon-delta) definition of the limit we have

>

> m

> --- 9

> lim > ---- = 1

> m->oo --- 10^n

> n=1

>

> Does that make it clearer?

>

> It's a basic mathematical FACT that 0.9 repeating is equal to 1. If

> you don't "get" this, ask questions. Say, "Gee, that sure seems

> counter-intuitive." Find your old schoolbooks and see if it's

> explained there. Post a message to the sci.math newsgroup if you

> want a thousand people to tell you to read the FAQ... But for God's