Yes I do indeed mean an infinite number of 9s and 0s.
I think there is some mileage in representing numbers this way :
1. It 'feels' like a natural extension to the way numbers are ordinarily represented: if you can have an infinite number of places to the right, then there should correspondingly be an infinite number of places to the left of the point.
2. Doing arithmetic this way causes no logical inconsistencies (as far as arithmetic is concerned - try it).
3. There are two 'pleasing' things which come out of this notation a. If we define ....999.000... = -1 then ....999.999.... must be zero. Dividing this number by 9 we would get .....111.111.... which must also be zero. In fact any infinitely repeating finite sequence of numbers (from the left of the point to the right) would represent zero. This allows you to find the negative of any rational number which contains a repeating sequence eg -(1/7) = ....142857142857.000000....
b. If we represent things in binary, then ....11111.0000.... = -1. This turns out to be just twos complement arithmetic using an infinite number of bits. This nice thing about it is that all you would have to do to find the negative is to flip all 0s to 1s and vice versa : ie 1 = NOT(-1) OR ...0000.1111....= NOT(....1111.0000....). This works for any number - which is neat.
So it may seem ridiculous to have an infinitely large number represent a negative finite number. But by the same token, isn't just as ridiculous to represent a positive finite number, by the addition of an ever decreasing number of smaller numbers?
You can't prove it as such, fundamentally you need a definition somewhere.
"Steve Lord" <firstname.lastname@example.org> wrote in message news://Pine.OSF.4.21.0011061038040.31349-100000@mathlab.WPI.EDU... > On Sun, 5 Nov 2000, Guillermo Phillips wrote: > > > Can anyone prove that "....99999.00000....." equals -1? Absurd you might > > Define "....99999.00000....." If you're going to say "An infinite number > of 9's before the decimal place and an infinite number of 0's after", then > you have to ask yourself: does this make sense? > > > say! But try adding 1. So isn't this the same question - ie one of > > definition? > > It's a question of "Does ....99999.00000.... make sense?" Since you > brought it up, _assuming that ....99999.00000.... is defined_, what is 1 + > ....99999.00000.....? > > Steve L >