
Point nine recurring ... - RiderOfGiraffes
http://www.physforum.com/index.php?showtopic=13177
======
scott_s
The most intuitive explanation I've read is that if two numbers are distinct,
then you can find a real number inbetween them. No such number exists for
0.999... and 1.

~~~
RiderOfGiraffes
As posted below: (1.0+0.99999...)/2

~~~
ErrantX
can you solve that equation?

(the answer is no, again it's proveable but it is beyond me to do so. Someone
else will have to jump in)

~~~
frig
See, what's so awesome about that answer is it allows the idiocy to keep
rolling. Here goes:

Proof: assume .9... < 1 (b/c lol _I know this_ ); then: X = (1+.9...)/2 => 2X
= 1 + .9... =>

\- 2X < 1 + 1 = 2 x 1 (b/c 1 > .9...)

\- 2 x .9... = .9... + .9... < 2X (b/c .9... < 1)

=> 2 x .9... < 2X < 2 x 1

=> .9... < X < 1

=> proof there's a # between .9... and 1, assuming of course .9... < 1.

What's cute about this is it's employing a similar conceptual error: the above
proof is literally circular -- it's proving something equivalent to '.9... <
1' (that there exists a distinct # between .9... and 1) by assuming '.9... <
1'; it's just different-looking enough to hide the circularity / equivalence
if you don't want to see it (the analogy to the 'debate' as to whether .9...
== 1 should be apparent).

~~~
ErrantX
Indeed - or to put it more simply it's just rearranging the equation, not
solving it ;P

~~~
scott_s
That's not an equation, it's a number. There's nothing to solve.

~~~
frig
And if you did 'solve' X = (.9... + 1)/2 using the standard 'algorithms' for
doing it by hand you will wind up with X = .9... + .0...1 (more on this in a
sec); you get there by noticing 1.9... / 2 looks like it ought to be .9...,
but "of course" 2 x .9... == .9...8, ergo you can deduce you have .9... + a
remainder of .0...1.

If you don't believe .0...1 exists (infinite 0s followed by a 1) then this
ought to convince you that .9... == 1 (as (x+y)/2 == x => x == y) but since
we're (hypothetically) arguing with someone who believes .9... != 1 it's quite
plausible that the person is also going to believe in .0...1 existing, also.

What I suspect is actually going on (I've also met otherwise-savvy people who
will debate this issue also) is that the human numerical-cognition system uses
something like a crude form of nonstandard analysis internally; and, under
this theory, you see this internal system (and its intuitions) leak out in
some people smart enough to get that .9... really does have 'infinitely many'
9s but who also for whatever reason are both undereducated on this point of
mathematics and generally self-assured.

Briefly: you can extend the real #s with a new entity (call it delta) such
that:

\- delta > 0

\- if r is a nonzero # then delta < |r|

\- (for now) delta^2 = 0

(in effect: delta is an infinitesimal, it's nonzero but smaller than any other
positive #, and delta^2 is 0 is mainly an algebraic convenience).

Once you've extended things by including delta you can write extended #s in
the form r + s x delta (eg: 1 - delta).

If we re-define ".9..." to be 1 - delta, then the following all drop out of
that:

\- .9... != 1 (1 - .9... == delta != 0)

\- .9... < 1, (1 - .9... == delta > 0)

\- and we even get: (1 + .9...)/2 = 1 - (delta/2) ==> (1 + .9...)/2 > .9...

And more generally in my experience most of the spinning-in-circles around
people with beliefs about .9... != 1 is that the people who vociferously
believe .9... != 1 don't even know about non-standard analysis (or
infinitesimals, etc.) and thus can't articulate what their intuition is
telling them, but what their intuition is using is basically (a nonrigorous
form of) analysis with infinitesimals.

~~~
slackenerny
_(for now) delta^2 = 0_

Oh, that's another wonderful story — of Clifford algebras:
<http://en.wikipedia.org/wiki/Dual_number> , useful in automatic
differentiation.

As to your general point of what constitutes "internal representation",
someone below tries to convince people that definitions involved with Reals
are arbitrary and can be made differently thus forming Hyperreals. But no. One
first have to construct Reals _anyways_ , then one arrives at infinity, which
is not a number, hyperreal or otherwise, but a consequence of numbering _per
se_. Nothing ad hoc here. Hyperreals are that or another notation, but the
main thinking is still the same, just paradoxes pop in different places, the
only gain is increase in confusion as hyperreals, tuns out, do not conform to
that "internal representation" in many more places than the poor 0.(9) .

~~~
frig
I'm not trying to claim the "internal representation" is somehow more correct
or even prior-to the real #s, etc.; I do think (though admit I haven't proven)
that a great many common examples of innumeracy can be explained by assuming
the innumerate are using a "reals + infinitesimal" representation...which
doesn't make the innumerate claims right, but possibly helps get at why they
make those particular mistakes and not others.

In the previous post I was hoping to make (only) the following claims(s):

\- in my experience, the theory "people who claim .9... != 1 are (unknowingly)
equating .9... to 1 - delta" has explanatory power (in that the results of
doing calculations with deltas are aligned with what these people claim)

\- I suspect that the internal human numerical representation is often a
cruder, less rigorous form of nonstandard analysis, and that the reason you
encounter otherwise-smart people who won't budge on .9... != 1 is that for
them the question goes straight to that internal representation

The just-so story for why I think the internal numerical representation is
approximately real #s + some infinitesimal would go like this:

\- small distances aren't reliably perceivable under natural conditions: if I
have two identical 8L pots and one has 4L and the other has 4.001L in it I
probably can't tell which is which visually; if I'm not in very controlled
conditions artifacts of ambient lighting or perspective or variations in the
pot material will overrule any perceivable visual difference between the two

\- actions with nonzero actual effect often have no perceivable effect: if i
have an 8L pot with 4L in it and I insert 1 ml from a pipette I probably can't
see a difference even after the ripples settle down, but of course I will know
that there's more water in the pot after I put some in than before

...and thus it'd make sense (at least for items with a continuous scale of
size) to keep track of both:

\- what size do I perceive this to be? (the real component)

\- if I know extra information (eg: that I added a .001L) about the size does
that information say that the actual size is going to be > or < the size I
perceive it to be (the "infinitesimal" component)

...and it's not unreasonable that for ad-hoc, intuitive reasoning the mind
would (roughly speaking) make the "correct" adjustments when combining size
measurements (eg: (x + delta) + y > x + y, (x + delta) + (y + delta) > x + y +
delta, (x + delta) + (y - delta) 'is a wash between' x + y, etc.)...after all
even babies and animals apparently can do simple sums
intuitively/automatically.

------
ErrantX
> 1/3 = 0.3333... remainder 0.0...1 (yes this is 0 recurring with a one at the
> end)

The main logic failure is there.

~~~
drcode
can you explain the logic you think is being failed? The alternative (that 1/3
is exactly 0.3333...) seems equally arbitrary to me.

It appears to me that the people on this thread arguing that 1!=0.9999... have
come up with a fully consistent, alternative way of defining equality when
involving infinite fractions that is no worse than the conventional
definition.

To expand his argument a bit: The argument is:

    
    
         0.3     *3+0.1    =1
         0.33    *3+0.01   =1
         0.333   *3+0.001  =1
         0.333...*3+0.0...1=1
    

...seems reasonable to me (again, it requires a slightly different definition
as to what "equality" truly means)

Kind of an interesting idea, actually. (Of course, that's after only looking
at it a couple minutes... the idea might fall apart under more scrutiny)

~~~
jerf
"can you explain the logic you think is being failed?"

You can not have an infinite series of something, followed by something else.
One definition of "infinite" is that it never ends, which is also why it isn't
a number (a number is intrinsically something that would have a definite end).
So to have "an infinite series of something, followed by a 1" is to have "a
series of something that never ends, followed by a 1 after the end". I can
type those words without crashing English, but it has no meaning.

"The alternative (that 1/3 is exactly 0.3333...) seems equally arbitrary to
me."

No. I'm not even sure what else to say. If you need convincing, start the long
division on paper and keep going until you're convinced.

No, seriously, keep doing it until you're convinced. Right now. Don't reply
with a counterargument until you've done that. There's nothing arbitrary about
it; to prove that there is, you need to show some point where you came to a
choice and you _chose_ to add the next 3, rather than some other hypothetical
alternative.

"have come up with a fully consistent, alternative way of defining equality
when involving infinite fractions that is no worse than the conventional
definition."

Well, no worse, other than also defining an internally-contradictory
definition of "infinity" for the sole purpose of winning an internet argument.
Other than that, no worse, no.

Recall that introducing one contradictory premise into a logical system allows
you to prove any statement. If you are not bothered by a "small" contradiction
to prove a dubious point, you don't understand math. Thus, making up
definitions to prove a point must be analyzed in the _full_ context of a
mathematical system, not just analyzed on some other abstract measurement
system (that doesn't matter). Breaking infinity to win an argument doesn't
work.

...

Most people go about proving this the wrong way. Proving something about the
string "1" vs. ".999..." is wrong. In order to show the two are distinct
entities, you need to show a situation in which they behave differently. No
such situation exists. Until you do that, "1" = ".9999..." is no more
surprising that "2/2" = "1". Or (ahem) "9/9" = "1". Numbers are what they do,
they have no existence beyond what they do.

On that note, when teaching I think the point that "=" means "is
bidirectionally fully substitutable by" is not taught properly. "2 + 2 =
[underline]" is a flawed question; oh, we all know what it means, but "2 + 2 =
2 + 2" is a perfectly correct answer to the question as written. We ought to
define a separate operator for what we really mean, "simplification", so we
can do something like put "2 + 2 => [underline]" on a test, and save the
equality operator for actual equality.

That is also why, for instance, the definition of the square root function
must include the stipulation that it takes the _positive_ root, because
otherwise "sqrt(4) = 2" is _not_ a true statement; if sqrt(4) means "both"
roots somehow it is _not_ fully substitutable with 2. (You can't say that both
"sqrt(4) = 2" and "sqrt(4) = -2", either, because then "2 = sqrt(4) = -2".
Oops.)

That's the background for my statement that .9999... = 1. It means that
there's no way to tease the two apart with any (valid!) mathematical
operation. Here's another way to say it: On the real number line, if you have
real number A and real number B, and there are no points between A and B, then
A and B must be equal. To show they are two different real numbers, you must
show a number C between 1 and .9999.... You can not do this... well, again,
you can't do this _validly_.

(Also, per RiderOfGiraffe's point, yes, there's some simplification here, but
I think it's fair to run on the theory that intrinsic to this argument is that
we're on the real number line. People who know enough to talk about
transfinite ordinals don't usually get into these arguments; by then, they've
learned the secret of math lies in the definitions chosen and understand the
ultimately contingent nature of any answer. :) )

~~~
drcode
Again, I don't think this is necessarily that fruitful a discussion, since the
whole point of the guy's argument, as I see it, is that you can slightly
adjust the meaning of "equality" as it relates to infinite fractions and
maintain a completely self-consistent mathematical system.

    
    
      > You can not have an infinite series of something,
      > followed by something else.
    

Why is that? How about the number 19...91? why are you denying me that number?
:)

    
    
      > If you need convincing, start the long division
      > on paper and keep going until you're convinced.
    

When I do that, I constantly am left with a remainder. If I accept that
1/3=0.3333..., as you're arguing, then the remainder disappears, for
unexplained reasons. An alternative argument is that the remainder persists in
the form 0.0...1 and then you don't have to "hand wave away" the remainder.

    
    
      > Recall that introducing one contradictory premise
      > into a logical system allows you to prove any statement
    

I agree with that completely, but I haven't seen the "contradictory premise"
yet... all I see (at this point) is some alternative premises that seem fully
internally consistent.

    
    
      > In order to show the two are distinct entities, you
      > need to show a situation in which they behave
      > differently.
    

I think that's actually the best argument you've put forward so far.
Basically, an argument in favor of choosing the more pragmatic way of handling
these situations.

~~~
jerf
"When I do that, I constantly am left with a remainder. If I accept that
1/3=0.3333..., as you're arguing, then the remainder disappears, for
unexplained reasons."

No, it doesn't. You stopped finitely soon into an infinite process. You
apparently didn't take it far enough.

No disrespect intended, but a HN comment is not a place to post a full
description of infinity. Hie thee hence to a textbook. And read it like a
_textbook_ , not an internet post. You know, where you go over it with a fine-
tooth comb looking for opportunities to leap up and call your opponent a Nazi.
Yeah, I know that style. Without that, counterarguments are getting mangled on
their way into your brain, because you don't actually understand them in Math.
That can't be corrected by people batting down various ill-posed English-based
objections.

(If it could, I would have seen it happen by now.)

~~~
drcode
I agree this discussion can't be taken much further on an HN thread for the
reasons you give :)

------
jrockway
People are still talking about this? I remember being convinced of the right
answer ("yes") back in elementary school. If 0.9 repeating is not equal to
one, name the number between them.

~~~
RiderOfGiraffes
(1+0.9999...)/2

~~~
jibiki
Sigh. Here we go, a definition of the real numbers. I'm only doing this once.

Let's look at the set of all cauchy sequences of rational numbers. A sequence
of rational numbers is just a map N->Q. We write a_0, a_1, a_2, ... for the
sequence which maps 0|->a_0, 1|->a_1, 2|->a_2, ... Such a sequence is called
"cauchy" if it has the property that:

    
    
      For all epsilon > 0 there exists an N such that n,m > N => |a_n-a_m|<epsilon.
    

Basically, this means that the tail of the sequence bunches together.

Now, we define an equivalence relation ~ on the space of cauchy sequences. We
say:

    
    
      a_0, a_1, a_2... ~ b_0, b_1, b_2... if:
      a_0-b_0, a_1-b_1 ,a_2-b_2... converges to zero
    

(A sequence a_0,a_1,... converges to zero if, for every epsilon > 0, there
exists N such that n>N => a_n < epsilon.)

Now, the real numbers are the set of equivalence classes of rational cauchy
sequences under ~, as defined above.

Clearly, .999... is (the equivalence class of):

    
    
      0.9, 0.99, 0.999, ...
    

And 1 is the equivalence class of:

    
    
      1, 1, 1, ...
    

So are these the same? Well, we have to check if our representatives are
equivalent. Does this sequence converge to 0?

    
    
      1-0.9, 1-0.99, 1-0.999, ...
    

Yes, it does. So 1 = .999...

~~~
RiderOfGiraffes
Sweet, I've not seen this approach before.

It still doesn't convince my colleague, because he says that the sequence
never gets to zero, but now I know where I stand. He can't deal with the
formalisms of convergence and limits.

It's also interesting to try to prove this via the Dedekind Cut construction,
rather than the equivalence classes of Cauchy Sequences. I'll have a go at
that later tonight.

------
rms
The naive perspective on this issue is because people intuitively perceive the
hyperreal numbers. In the hyperreal number system, .9999999999999999999999...
and 1 are different. But in almost all phrasings of this question it is
implied to be talking about the real number system.

Specifying it gets rid of all ambiguity. The real numbers .999... and 1 are
equal. The hyperreal numbers .999... and 1 are not.

<http://en.wikipedia.org/wiki/Hyperreal_number>

reply

~~~
MaysonL
This seems to be the only comment here written with a little mathematical
sophistication.

Whether 1 and .999… are equivalent depends on how you define 1 and .999…

~~~
rms
I find these threads deeply ironic (the 400 page ones, anyways) because both
sides arguing are ultimately wrong because both perspectives are correct.
People with the naive perspective aren't wrong for not specifying they aren't
talking about the hyper real numbers, because the other side doesn't specify
they are talking about real numbers.

At a philosophical level, the answer is that .999999999999999999999999999...
is different from 1. The difference is real enough to let you rigorously
define calculus.

This is one of those subjects that makes an online community go insane (with
400 page threads), because it's possible for the math people and the
philosophical people to go on and on without figuring out the right answer.

------
frig
Someone once trolled an objectivist form with something like this: how do you
square 'A==A' with '.9999...=1'?

------
dnaquin
power series.

0.999... = sigma 9/10^n n=1 -> inf = 9 * sigma 1/10^n n=1 -> inf = 9 * ((sigma
1/10^n n=0 -> inf) - 1) =

power series sigma 1/10^n n=0 -> inf = 1/(1-1/10)

= 9 * (1/(1-1/10) - 1) = 1

qed.

------
RiderOfGiraffes
See also the poll at <http://news.ycombinator.com/item?id=692383>

------
wlievens
I like this one:

    
    
      1/9 = 0.1111...
      2/9 = 0.2222...
      3/9 = 0.3333...
      ...
      8/9 = 0.8888...
      9/9 = 0.9999...
    

yessir

------
slackenerny
_Ad. OP's link:_

OMIGOD, just noticed, its a 389-page thread!

And I thought 1st page was already pretty epic..

------
bhiggins
some people seem to have a problem with the idea that two different looking
representations of something can represent the same thing.

