.99999... = 1 !? Stop it!

[Hanspeter Schmid]

I've seen many proves and disproves, people are flaming each other,
but nobody seems to care about the basic question:

What is meant by the notation 0.9999... ?

If that is defined, it is immediately clear whether 0.999... = 1 or
not. If two people use two different definitions they may or may not
get the same result, and the discussion which result is correct is
meaningless. Both results may be correct, since they do not base on
the same premises. It should rather be discussed which definition of
0.999... should be preferred or whether there exists a definition of
notations like 0.999... that is generally accepted.

Cheerio, Hanspeter.
--
-===-=-====-=-===== Hanspeter == Schmid =====-=== hobby-musician classical-
Hanspeter Schmid | See first, think later, then test.
Signal & Information Processing Lab | But always see first. Otherwise you
phone: +41 1 632 35 46 | will only see what you were expecting.
e-mail: sch...@isi.ee.ethz.ch | Most scientists forget that. (D.Adams)
>ETH-Z<====-=-===== orienteering runner ===== pan-flute trombone =========-

[Jeffrey C Humpherys]

Well, different strokes for different folks. An analyst is going to
say .999.. = 1. An engineer will say .90000 = 1. A physisist will say
c=1. A logician will argue with everyone and prove nothing and/or
disprove everything. Mathematicians in general don't agree on anything.
Logicians don't agree with anyone on anything. Philisophers will ask if 1
exists or whether it is our perception of 1 that exists. Computer
programmers think that .9999995 = 1.

If you pick up the theory of Real Variables (e.g. Royden), .99999..=1
because the reals are complete. If you want to talk about what
algebraists or logicians think that .99999...is, then they will perhaps
disagree with the analysts.

Jeff

[John P DeMastri]

The point is that the sequence 0.999... is pretty well universally meant
mathematically as an infinite sequence of 9s to the right of a decimal
point. Can we agree on that? Good. Now what value does that
represent? Can we agree that that is that the question we are asking?
Good. Now, the value of any decimal notation is the sum of the value of
each of the digits in that notation, agreed? Good. Now, the value of
the summation of the infinite series described above is well known and
easily proven to be 1. Agreed? Good, now let's move on, already!!!

John DeMastri

[Eduardo Chappa L.]

In discrete metric 0.99999.... does not exist :)

[J. B. Rainsberger]

In article <5134vj$s...@elna.ethz.ch>,

sch...@isi.ee.ethz.ch (Hanspeter Schmid) wrote:
>I've seen many proves and disproves, people are flaming each other,
>but nobody seems to care about the basic question:
>
> What is meant by the notation 0.9999... ?

It is the limit of the sequence

(0.9, 0.99, 0.999, 0.9999, 0.99999, ...)

just as *any* decimal representation is the limit of a sequence of finite
approximations, whether the representation is finite or not.

Thank you for your sanity.

JBR.

J. B. Rainsberger, BA

Calvin says:
1. If you can't win by reason, go for volume.
2. The ends justify the means, but only for me.
3. The best way to find out if there are monsters under
the bed is to tell stories about devouring little kids -
sometimes they laugh.

[Jeffrey C Humpherys]

: John DeMastri

I was the first person to reply to the origional post and proved it the
same way. But then.... people got anal.

Jeff

[Jeffrey C Humpherys]

Eduardo Chappa L. (cha...@math.washington.edu) wrote:

: In discrete metric 0.99999.... does not exist :)

Another example of somebody trying to show off that they know something.

Jeff

[Eduardo Chappa L.]

On 12 Sep 1996, Jeffrey C Humpherys wrote:

> Eduardo Chappa L. (cha...@math.washington.edu) wrote:
>
> : In discrete metric 0.99999.... does not exist :)
>

> Actually you are wrong. You don't need any notion of a topology to get
> that .9999 = 1. This is an increasing sequence in a complete space, with
> addition well defined. Thus, 1 is the lub, and so you get that .999... = 1
> without any notion of distance.
>
I'm sorry i have to answer this mail, but i see some confusion in your
answer, I hope this mail will be useful to let you see where your mistake
is.

It seems me very interesting the new definition of 0.9999.... that you
have given, according to what I understand from your mail you are defining
0.99999... = sup {0.9, 0.99, 0.999,....}
when defined in that way, every analyst would say : yes 0.9999... = 1.

Now if you quote the fact that the set of real numbers is complete and
that's why increasing sequences converge to its supremum then I understand
that you are using the usual metric in R (the set of real numbers).
Unfortunately this theorem is not true in discrete metric, since in this
metric the only convergent sequences are those that "are constant at
infinity".

Therefore we have a difference in general between limits of increasing
sequences, which might no exist, like the example above, and supremum that
might exist, also as in the example above.

The usual definition of 0.9999... is a limit of a geometric series,
that's why I said that this limit didn't exist in discrete metric. I hope
that you understand now what I meant.

Edo.

[Jeffrey C Humpherys]

Eduardo Chappa L. (cha...@math.washington.edu) wrote:

: In discrete metric 0.99999.... does not exist :)

Actually you are wrong. You don't need any notion of a topology to get
that .9999 = 1. This is an increasing sequence in a complete space, with
addition well defined. Thus, 1 is the lub, and so you get that .999... = 1
without any notion of distance.

Jeffrey Humpherys

[Jeffrey C Humpherys]

Eduardo Chappa L. (cha...@math.washington.edu) wrote:

You can sum numbers in a geometric series without a metric, if the
numbers are all positive. This is how.....Royden does it this way.

Let S be the collection of all finite subsets of (.9, .09, .009, ...)
We define the sum of the series .9 + .09 + .009 +... as follows.

Let F be the sum of each element in S. Note that each element in
S is a finite number of the (.9, .09, .009, ...), so {.9, .009} is in S,
and adds to .909 in F. Then by taking the sup of F, we get our geometric
series. Since the reals are complete, i.e., upper bound => lub, and F is
bounded above, then it has a lub, and it is easy to show that it is 1.
Thus, the geometric series sums to one.

It should make sense that the way we add things shouldn't depend on our
notion of distance. Although we can not convienently use the notion of
distance to prove convergence, we can still define cantor sets and other
decimal expansions without a distance and relate them to our usual notion
in R.

Also, it is worth noting that this approach to sums allows us to make
theorems about reordering series. i.e., the fact that a reordering of a
series doesn't change the sum if the series is absolutely convergent comes
directly from this.

Jeffrey Humpherys

[Jeffrey C Humpherys]

Eduardo Chappa L. (cha...@math.washington.edu) wrote:

: It seems me very interesting the new definition of 0.9999.... that you

: have given, according to what I understand from your mail you are defining
: 0.99999... = sup {0.9, 0.99, 0.999,....}
: when defined in that way, every analyst would say : yes 0.9999... = 1.

: Now if you quote the fact that the set of real numbers is complete and
: that's why increasing sequences converge to its supremum then I understand
: that you are using the usual metric in R (the set of real numbers).
: Unfortunately this theorem is not true in discrete metric, since in this
: metric the only convergent sequences are those that "are constant at
: infinity".

I am using no metric.

The sup still exists, metric or not. Completeness tells us this.
I am using the def of complete as upper bound => lub, not as every cauchy
sequence converges. Is that the problem....

The completeness of the Reals is an Axiom (According to say, Royden),
and doesn't involve any notion of topology. It is an ordered (totally)
set where ub => lub. For example the power set (although partially
ordered) is complete.

Jeffrey Humpherys

[David Ullrich]

Hanspeter Schmid wrote:
>
> I've seen many proves and disproves, people are flaming each other,
> but nobody seems to care about the basic question:
>
> What is meant by the notation 0.9999... ?
>

> If that is defined, it is immediately clear whether 0.999... = 1 or
> not. If two people use two different definitions they may or may not
> get the same result, and the discussion which result is correct is
> meaningless. Both results may be correct, since they do not base on
> the same premises. It should rather be discussed which definition of
> 0.999... should be preferred or whether there exists a definition of
> notations like 0.999... that is generally accepted.

Good luck.

--
David Ullrich

?his ?s ?avid ?llrich's ?ig ?ile
(Someone undeleted it for me...)

[Eduardo Chappa L.]

On 12 Sep 1996, Jeffrey C Humpherys wrote:

> Eduardo Chappa L. (cha...@math.washington.edu) wrote:
>

I don't need to read Royden's book to realise that all his arguments
depend on the fact that R is complete with the usual metric. Try to prove
that the same hold in the rationals and you'll see the difference.

For example the sequence 1 , 1.4, 1,41, 1.414, 1.4142, etc has a limit
in R, if you want also, a supremum, but not a limit or a supremum in the
rationals. You are using that R is complete with the usual metric (this is
equivalent to the supremum axiom). You need topology, it is inherent,
although you can construct a theory without mentioning the word "metric"
or "topology".

Now it is not clear what is you definition of 0.9999....

Edo.

[Eduardo Chappa L.]

On 12 Sep 1996, Jeffrey C Humpherys wrote:

> Eduardo Chappa L. (cha...@math.washington.edu) wrote:
>

I guess you should know that completeness is equivalent to the fact that
every bounded above set has a (unique) supremum.

[Jeffrey C Humpherys]

Eduardo Chappa L. (cha...@math.washington.edu) wrote:

: I don't need to read Royden's book to realise that all his arguments

: depend on the fact that R is complete with the usual metric. Try to prove
: that the same hold in the rationals and you'll see the difference.

: For example the sequence 1 , 1.4, 1,41, 1.414, 1.4142, etc has a limit
: in R, if you want also, a supremum, but not a limit or a supremum in the
: rationals. You are using that R is complete with the usual metric (this is
: equivalent to the supremum axiom). You need topology, it is inherent,
: although you can construct a theory without mentioning the word "metric"
: or "topology".

Dude, you just don't get it! Let me try again. Completeness says that
every set which is bounded above has a lub, agreed. That is the
definition.

Now, the set (not sequence), {.9,.99,.999,...} has an upper bound, say 1.
Thus it has a lub. This lub, is a real number. It is less than or equal
to 1. We can show that it can't be strictly less than one, by supposing
it is, .... (I won't patronize you). Thus the lub of the set = 1. Agreed.
Now, remember how nonnegative sums are defined. This is in my last
response. The sup of that set, is the sum, is the same as the lub of the
above set, and so .9 + .09 + .009 +... = 1.

Now, nowhere did we write for all e > 0, there exists a N, such that
d( , ) < e when n > N. No where is the notion of distance used. We have
completeness and a total ordering on R, as well as the regular axioms on
R, i.e., the field axioms. That is it! Think intuatively about this.
Why should our notion of distance effect how we add?

Now, I agree that in the discrete topology, we cant prove convergence.
There is no notion of convergence! But, we can define sups, because we
have an ordering. Thus, .99999... = 1, regardless of the topology put on
R, but instead of saying that the sequence {.9, .99, ...} converges (a
notion which needs a topology), we say, .9 + .09 + .009 ... = 1.

Therefore, your statement about .999... not = to 1 in the descrete metric
is false.

Jeff

[RCarfrey]

> What is meant by the notation 0.9999... ?

Actually, the correct notation that I learned in grade school is 0.9 with
a bar over the "9". This is read "zero point nine repetend nine" It is
of course not possible to write the bar over the 9 so we write 0.999...
-RCar...@aol.com

[Terry Moore]

In article <51nui5$6...@newsbf02.news.aol.com>, rcar...@aol.com (RCarfrey)
wrote:

Define "correct".

I was taught to put a dot over the first and last of a group of
repeating digits. Were my teachers ignorant?

There is no such thing as "correct".

Terry Moore, Statistics Department, Massey University, New Zealand.

Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan

[Jeffrey C Humpherys]

Terry Moore (T.M...@massey.ac.nz) wrote:
: I was taught to put a dot over the first and last of a group of

: repeating digits. Were my teachers ignorant?

Agreed. Did god come down and say, "Bar over is the only thing that is
accepted". The nice thing about notation, is that one can put a little
personallity in to his work. I like using real big smiley faces and
hearts and Lucky Charms cereal shapes.

I had a proff once who said that convolution * looked like an anus, and
so when he wrote f * g, he said f sphinctor g. As usual the crowd saw
dead, and didn't get it. Personality is hard to come by in Math.

Jeff