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Re: a .9999rep = 1 debate, can someone comment please?
Posted:
Nov 6, 2000 4:44 PM
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Steve,
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.
Guillermo
"Steve Lord" <slord@wpi.edu> 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
>
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