Re: Problems I have with 1.999...=2
- From: The Ghost In The Machine <ewill@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Tue, 10 May 2005 15:00:15 GMT
In sci.math, Kirby Cook
<kwmcook@xxxxxxxxxxx>
wrote
on Tue, 10 May 2005 06:17:22 GMT
<SlYfe.16681$_g1.9495@trnddc04>:
> Problems I have with 1.999...=2
>
> Commonly, the first "proof" or demonstration offered involves setting
> .999... =x, multiplying .999... by 10, subtracting x from 10x, and
> asserting that 9x=9. Taking that step by step, if x=.9 and 10x=9,
> 9x=8.1. When x=.99 and 10x=9.9, 9x=8.91. Lining up the first few examples,
> 10x 9 9.9 9.99 9.999
> -x -.9 -.99 -.999 -.9999
> 9x 8.1 8.91 8.991 8.9991
The main problem with this proof is that 9.9999... - .9999... can
be construed to be either 9.0000.... or 8.9999... , depending
on whether one can tolerate an "infinite borrow" or not.
>
> So, even if you claim leave to ignore the convention that multiplying a
> number by ten always adds a zero to the end of the number by arguing
> that there is no "end" to .999... (a dubious dodge in itself, IMO),
Infinities are very weird. Of course what I call "Gary Denke" math
is even weirder.
His notation is along the following lines.
x = .999...
10x = 9.999...990
10x-x = 8.999...991
This is a largely meaningless notation at this point, but it gets
stranger.
x = .999...9
x^2 = .999...998...001
x^3 = .999...997...002...999
x^4 = .999...996...003...999...001
x^2 - x^3 = ...???
x^3 - x^4 = ...???
> if
> you let those two numbers, x and 10x march out to "without limit" in
> lockstep, the upper 9 will always be one step to the left of the lower,
> and 10x-x will never equal 9.
Correct. The best method of approaching this sort of
problem is by formal limits. Of course, one might give
that the name "I give up and go home" strategy (this after
the two participants play "delta-epsilon" for long enough;
the epsilon-picker takes his expression and surrenders,
as the delta-picker sticks his tongue out), but there
aren't many others, really; one can't sit there and count
to infinity.
>
> The flaw in the oft-cited second "proof", or demostration, which tries
> to show that there is no number between 1.999... and 2, may be found in
> the law of the reciprocal, by analogy and inference. Say, for instance,
> that there is no number between the infinite sum (difference) 1-.999...
> and zero.Taking it step by step, 1-.9 has a reciprocal of 1/(1-.9) or
> 10;
Be *extremely* careful here.
The long division can be represented as follows:
9.999...
..1 | 1.00000...
9
1
9
1
9
1
9
1
...
which means of course that 1/(1-.9) = 9.999... . Since in standard
math 10 = 9.999... no one pays this much mind, but it is an issue
if one explores alternative paths.
> 1-.99 has a recirpocal of 100, and so on. So, if you assert that
> there is no number between 1-.999... and zero, then you are also
> asserting that there is no number between 1000... and a point often
> termed the point at infinity. And how many of you will buy that?
1000... isn't an integer. I for one don't see a problem here.
>
> It should be obvious, I hope, that I am directing my remarks not to
> those who carefully assert that 1.999... =2 is merely sloppy shorthand
> for the idea that 1.999... *approaches* 2. No, my remarks are for those
> hopelessly superstitious minds that claim that "equals means equals" in
> this case.
So how do we ascertain when '=' means equals and when '=' means
"as one takes the limit of an unspecified independent variable"?
An interesting problem.
>
> I use the words "hopelessly superstitious" advisedly, as it seems to me
> that the common invocation of the words "without limit" and all similar
> such are used with no more understanding than a twelve year old's
> fervent incantation to charm warts. The frank translation of "without
> limit" seems to be "the magic happens here", the magic being, most
> often, that something becomes nothing. The fact that we can trace this
> impenetrably ignorant superstition back to such a luminary as Leonhard
> Euler, who said in one of his most quoted dissertations, "There is no
> doubt that any quantity can be diminished until it vanishes and is
> transformed into nothing", is really no excuse for its perpetuation. At
> any rate, *I* certainly doubt his assertion! That Euler was a brilliant
> mathematician is true, but that he was something of a crackpot is also
> true, as evidenced by the above quote and the accompanying exposition.
> Consider: If *at any point* applying division to a finite quantity
> results in zero, that point is as much where you are as, well, as this
> point is where I am. So if I can say "Alakazam!" and make a small bit
> vanish, why can't I say it (maybe louder?) and make a big bit vanish in
> exactly the same way?
>
> Or, if you persist in asserting that Euler was right, what *exactly* is
> it about the words and concept, "without limit", that makes the magic
> happen, that transports us from the familiar world of 2+2=4 to a world
> where four divided by something equals ("Alakazam!") nothing? Just as
> though that four never was.
This may be an issue of symbolic versus standard representation.
For example, pi can be specified in a number of ways.
- 3.14159... - we'll be here all night, next morning, the next
millennium, and the heat death of the Universe, before this
is finished. Therefore most people use the ellipsis and
truncate it.
- ratio of an ideal unit circle's circumference over its diameter
(or an arbitrary ideal circle, if one prefers)
- 1 - 1/3 + 1/5 - 1/7 + ...
- sum(i=1 to +oo) ((-1)^i * (1 /(2*i-1)))
- lim(n->+oo) sum(i=1 to +) ((-1)^i * (1 /(2*i-1)))
All of these are finite (by necessity) although the last is
the most rigorous specification. [*]
..999... is an infinite, nonrigorous specification of the concept
lim(n->+oo) (sum(i=1 to n)(9 * 10^(-i)))
This concept is a specialization of the more general concept
lim(n->+oo) (sum(i=1 to n)(a * r^i))
which, as it turns out, is equal to
lim(n->+oo) (a * (1 - r^(n+1)) / (1 - r))
(one can prove the subexpression by induction; e.g.
r^3 + r^2 + r^1 + 1 = (1 - r^4) / (1-r)).
If abs(r) < 1, this is equal to a / (1 - r). However, as you've
probably already noticed, the limit made the infinitesimal r^+oo vanish.
Perhaps we'll need two definitions of equality to properly deal
with this problem (although for most mathematicians vanishing
infinitesimals are par for the course).
a =fin b for those equalities that can be proven without resort to
limits. For example 2 + 2 =fin 4.
a =lim b for those equalities which are inherently limit-based.
For example, 0.999... =lim 1.000... .
=fin implies =lim but not the other way around.
A slightly simpler notation might be '->' instead of '=lim'.
Most uses of '=' are in fact '->'. Does anyone really
care? Not as far as I can tell; this sort of question
is about as useful as counting the angels on the head
of a pin, and pi = 3.1415926536 (*sans* ellipsis) will
give the circumference of an (assumed perfectly circular)
orbit around the Sun at 1 AU to within about 15-20 cm,
and most reals will fit into IEEE754 (with a bit of
squishing), which stipulates about 1.845 * 10^19 (rational
approximations to) numbers to play with, some of them more
useful than others.
[*] there are far better ways of computing pi than this extremely
slow-converging series.
--
#191, ewill3@xxxxxxxxxxxxx
It's still legal to go .sigless.
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