.999 = 1?

Factual Questions
2051

trust me: Not “everyone” does. Also, that was a very bold statement for you to make, because I am sure you did not confirm with “everyone else”.

People are interpreting 9/10 + 9/100 + … where the […] mean “and so on”, which equals (by their definition) : 1.

They are then referring to 0.999… as a number which is made up of only 9’s, where the […] mean “infinite 9’s”. They are not encorporating “limit” as the meaning of […], but rather, “infinity” where by some magic, and endless string of 9’s *becomes *1.

Again, definitely not everyone thinks this. They are certainly saying 0.999… is its own “number” just as 1 is, and they are different representations of the same thing. Then, they go on to say that 0.999… = limit 0.999…
Which according to you is redundant because it already implies the limit.
0.999… = limit (1 - 1/10[sup]n[/sup]) = 1

But again, why does […] have to mean limit in the first place?
We can certainly have 0.999… as an endless string of 9’s without ever mentioning the limit. It would be useless as a “number” but could still be used as an infinite series concept, separate from the limit of that series.

Again, I sense you are calling “zero decimal nine nine nice (to infinity)” the number.
cos(0) = 1

Is “cos(0)” a number, as written? the real number 1 is its value.

Just as 2+2 = 4. 2+2 is an expression which *computes * to the number 4. There is an operation involved.

I see the point the was made that “there are different *representations *of the same value” but these representations are ways of representing the number in a non-numeric form, involving functions, or other concepts. “Ways of representing a number” doesn’t mean they *are *numbers.

2052

Do you accept that with the definition 99% of the posters in this thread use, where 0.999… is defined to imply the limit, 0.999… = 1 ? Because that’s what this thread is about, and your quibbling about how people who don’t know common definitions may misinterpret them is utterly uninteresting.

2053

What is the limit of 1 ?

The limit of 1 is 0.99999…

What is the limit of 0.99999…?

The limit of 0.99999…is 1
The limit of 1 is not equal to 1. The limit of 1 is equal to 0.99999…
The limit of 1 is not equal to 1, therefore 0.99999… is not equal to 1

The limit of 0.99999…is not equal to 0.99999…
The limit of 0.99999…is equal to 1.
The limit of 0.99999… is not equal to 0.99999…, therefore 1 is not equal to 0.99999…

A number is equal to itself and therefore 1 = 1
The limit of a number is equal to itself too therefore 0.9999…= 0.9999…

Because the number and the limit of the number are not the same, the numbers
1 and 0.9999…are not two representations of the same numbers.
They are two different numbers.

2054

You aren’t using the usual meaning of “limit” here, because usually a limit is a limit of a function or of a sequence. Single numbers do not have limits. So, in your version of mathematics, what is a “limit”?

2055

Yes, they do have limits.

Number 0.99999…is a single number.

The limit of 0.99999…is 1.

2056

Mathematical limits must be taken with respect to a parameter approaching something. Just saying “limit” is kind of nonsense.

2057

As above. The word “limit” on its own is meaningless. You have not defined what is being limited, nor to what value it is being limited. Further, you are implicitly assuming that your “limit” function produces a unique result. Given you have not even defined your “limit” in any form, it seems a tiny bit presumptuous to start to apply proofs based upon unstated assumptions.

Without clearly and unambiguously stating what you mean, and further proving such things as uniqueness, everything is little more than gibberish. An assemblage of terms without actual meaning.

2058

No, the notation 0.99999… is shorthand for a limit of a sequence, the sequence being
0.9, 0.99, 0.999, …
with the nth term of the sequence being
1 - 10[sup]-n[/sup]

The trouble with saying “The limit of 0.99999…is 1” is that the concept of a limit is already contained in “0.99999…”. In effect you are saying, “The limit of the limit of the sequence 0.9, 0.99, 0.999, …”.

2059

Except that you haven’t been saying this all along. This is an orthodox statement that you have been arguing against.

What seems to be apparent is that you use the term, limit is a way that is different from orthodoxy. You use the word infinite in a manner that is inconsistent. you use the ellipsis in a manner that differs from normal usage.
None of this would matter – if you were clear. But you are not.

What is also apparent is that you do not comprehend the concept of continuity as it applies to the Reals. Of the twelve axioms that are used to define the Real number set, you have confusion over the one that distinguishes R from Q. The following confusions are exhibited:
[ul][li]A failure to adequately distinguish a number from the representation of a number[/li][li]The erroneous concept that the properties of a number depend on which base it is presented in[/li][li]the insistence that infinitesimal numbers, which do not belong to R,be treated as a non zero difference between two real numbers[/li][li]referencing a source that claims sqrt(2) is not a number – patently a false statement. The Pythagoreans figured this one out.[/li][li]general confusion between the properties of R and Q[/ul][/li]This is set against a backdrop of muddled language. … is a number / … is a limit / … is a process. Implying n+1>1. “Cardinality of n”. And so forth. Numerous times you have been called to clarify a typo or otherwise nonsensical statement and your response to me was to read between the lines.

As has been stated several times, there are number sets other than the Real numbers that do what you want – Hyperreals and surreals have been mentioned in this thread before. But you are not displaying the rigour necessary to engage in a discussion of that sort.

How are we supposed to proceed from here?

2060

Ummm… What?

2061

Well, just look. The limit of 1 has 14 characters and starts with a T. 1 has 1 character and starts with a 1. Clearly unequal.

2062

I admire your tenacity, j_sum1, but you’re trying to have a discussion with someone who believes that existence of certain real numbers depends on what base you’re using to write out their digits. You’re wasting your time.

2063

You are right. I am doing this for my own benefit and not for his.
He is a lost cause. i am learning.

2064

You’re welcome to do whatever you like, of course, but you might have better luck spinning off a separate thread and asking questions about analysis, field axioms, etc. directly. You’d get a higher signal-to-noise ratio, if nothing else.

2065

Moderator Warning

I missed this earlier. schooner26, as I noted before insults are not permitted in General Questions. This is an official warning. Do not do this again.

Colibri
General Questions Moderator

2066

I believe Schooner was responding directly to my statement:
http://boards.straightdope.com/sdmb/showthread.php?p=17631538#post17631538

FWIW, it was not my intent to insult here but just to draw attention to a communication issue.
Likewise, I did not interpret schooner’s comment as an insult but more as an unwillingness to engage.
My apologies for raising the temperature in GQ.

2067

In view of this I’ll rescind the warning and reduce it to a note. However, let’s not cast aspersions on other poster’s intellectual abilities. We have an ongoing Pit thread for that.:wink:

Colibri
General Questions Moderator

2068

Ummm… what ??

Limit C = C
x → a

It is one of the first things you learn in basic analysis when limits are introduced.

So yeah, “single numbers” do so have limits.

2069

Thanks for manning-up. I owe an apology as well. It wasn’t the first comment, nor second, that I interpreted as an insult, so I felt I had to start defending myself.
Sorry for retaliating.

Let’s keep it clean and back on track

2070

So, lim 0.999… = 0.999…

What has all the fuss been about?