0.999... = 1

Oh snap!

>>26
I wanted to know what you think the word "converge" means.  You just went through a discussion of this or that "converges" but you never once indicated what you think the act of converging means.  Your "intuitive" description indicates that you believe to converge means to approach zero.  Now what do you believe is meant by "approach"?
>>27
That's exactly what was how I described what the other guy said, contrasting it with what the other guy #2 said.  I gave two descriptions describing what two people said, to answer one of the two people, to aid seeing what each thinks is meant.
>>28
So it is a limit because it is a limit.  Pretty fuckin' circular.  Is the numeral a limit or the number?  What is a numeral to you?
>>32
You can't get away with doing arithmetic on a numeral without first assuming what is truly represented by the numeral.  And if you tried to multiply by hand, you'd never finish unless you made a specific assumption.

>>40
And still completely irrelevant to the issue at hand. The fact that you don't realise that makes me fear for our public education system.

It's amusing and sad how people try to get around the FACT that 0.999...=1 by taking the limit, by questioning the use and definition of the terms "limit" and "converge".

If a function converged infinitely on 1.3422390230234230239408, they have no problem with that.  But if another function converged infinitely from 0.999... to 1, OH BOY, they have a BIG PROBLEM with it.

The simple fact is that 0.999... is a converging limit, and that limit at infinity is ONE.

We use infinite convergences in the Calculus all the time, fuckwads.  We get answers that are perfectly correct -- practically and theoretically.  Anyone who doubts that is just a kook who probably flunked Alegbra for political reasons (i.e. they were too busy smoking pot and chasing pussy to ever study hard enough to UNDERSTAND MATH).

In other words, the simpletons who mistakenly believe they understand the problem, just don't understand what the "..." notation in the number 0.999... means.  The "..." indicates the operation of a LIMIT.  So, the simpletons MISS the point that 0.999... is not just a number as written, but it's a FUNCTION in disguise.  As we know, we operate functions in order to arrive at results.

Once you understand that, the next determination you need to make is DOES IT CONVERGE.  Yes, 0.999... does converge.  Then, the next thing is to determine which SINGLE VALUE the limit converges at, assuming infinity is reached (which for a number line, is always TRUE).  The single value is ONE.

gb2college, you fucking dipshits!

>>41
Numerals represent real numbers, real numbers are defined as the set of limits of Cauchy sequences of rational numbers. It's "pretty fuckin' circular" because it is a DEFINITION. This "argument", if you can call it that, is essentially equivalent to:
Me: Shampoo is something you use to clean your hair.
You: No it's not, but I'm not going to explain why I think it's not.
Me: But the word "shampoo" is defined as "something which is used to clean hair."
You: CIRCULAR LOGIC

>>42
The fact that you don't see a connection between the DEFINITION of the real numbers and the validity of a statement about the real numbers makes me fear you're a middle school student. Let me spell it out for you step by step:
0.999... is a real number.
{0.9, 0.99, 0.999, 0.9999, ... } is a Cauchy sequence of rational numbers which trivially converges to 0.999...
{0.9, 0.99, 0.999, 0.9999, ... } also converges to 1 in the rational numbers.
Ergo 0.999... = 1, because by the definition of convergence, a sequence can converge to only one value.

>>44
"So, the simpletons MISS the point that 0.999... is not just a number as written, but it's a FUNCTION in disguise."

No, it is the limit of a sequence. Numerals do not represent functions. gb2introanalysis, you fucking redcream.

>>45
And now you're contradicting yourself. gj.

>>45
0.999 = a number
0.999... = a sequence

A sequence is a function by definition.  You have to perform the sequenced operation in order to achieve the result.  That sure sounds like a fucking FUNCTION to me!

again >>6 where's OP?
i think he meant .(9), ie the periodic number.

Just in case someone forgot elementary operations for periodic numbers:

7/9 = .(7)
8/9 = .(8)
9/9 = .(9) oh wait... 9/9 it's... one !!!!!eleven!!!111!
10/9 = 1.(1)
etc..

>>47
No, 0.999... is a number. It is equal to the limit of a sequence; that does not mean it IS a sequence. 2 is also the limit of a sequence, as are pi and 5 and e. Do you call all of them sequences? If no, then you're an idiot for thinking that 0.999... is a sequence. If yes, then you're an idiot as it's trivial to see that each of the numbers listed above is the limit of not only one sequence, but infinitely many sequences. You can not identify a real number with "a" sequence; only with an equivalence class of Cauchy sequences of rational numbers.

>>46
And now you're pretending to see something wrong with my post without actually mentioning what it is. gj.

These threads are worse than retards.

>>49
No, 0.999... is a number. It is equal to the limit of a sequence;

Wrong, it's an equivalence class of Cauchy sequences that get close to one another.

>>51
Any Cauchy sequence of rational numbers is also a Cauchy sequence of real numbers (trivially), and any Cauchy sequence of real numbers converges. So yes, it is a limit of a sequence. If you would try reading my entire post before replying, it would have been quite clear to you that I am familiar with the definition of real numbers as equivalence classes of cauchy sequences of rational numbers.

>>52

Wrong, it's a Dedekind cut.

>>53
Actually, same thing. I think you are able to form real numbers from Cauchy sequences of rationals.

You need to define decimals first.

.a1a2a3... (base n) is equal to the infinite sum of ai/n^i for i=1  upwards. Thus, .999999... is equal to 9/10+9/100+... which is a geometric sequence (9/10)*(1+1/10+1/100+...)=(9/10)*(1/(1-.1))=(9/10)*(1/.9)=(9/10)*(10/9)=1

>>47
"0.999 = a number"

Firstly, you mean REAL number. What these "numbers" you speak of? What is a number?

Secondly, I can look at this semantically. This is not a number, but rather a DEFINED SYMBOL. Thus, 0.999... is not a number, but a symbol representing such a number.

>>53
There are approximately 18,000,000,000 different ways of defining the real numbers which are equivalent to the equivalence classes of cauchy sequences way, and dedekind cuts is one of those 18,000,000,000.

>>56
"Secondly, I can look at this semantically."

You could, but linguistics has very, very little place in math.  You can't call a mathematician wrong on a linguistic basis if he's got the concept right.  If the proof's wording sucks, come on in.  Else, gtfo.

ABCDEFGHIJKLMNOPQRSTUVWXY and Z are all also defined symbols which hold meaning for you.  But are they really letters?  According to you, these would be defined symbols that represent what we call "letters."  Hiragana are letters, too.  But Hiragana can be represented by combinations of two or more of the 26 symbols I just named.

>>58
Wrong again. ABCDEFGHIJKLMNOPQRSTUVWXY, while it is not a symbol itself, is an expression. An expression is a finite sequence of symbols. A symbol is anything within the set of a formal language called its alphabet. Whereas all of those letters are symbol

The reason why semantics has much place in mathematics is because the symbols of formal languages form a finitary system, and makes concrete representations of abstract things. The syntatic part of mathematics also has its place, but semantic implication is more important by far than syntatic implication. A defined symbol is a representation of an expression or another symbol, but is not itself a part of the alphabet, and by consequence is not a symbol itself.

You would know all this if you had taken a course on mathematical logic. All this comes straight from my mathematical logic textbook.

What I was saying was that .999... is not a real number, but rather a finite string of defined symbols of "." followed by "9," "9," "9," "." "." and finally, "."

Also a source for information:
http://en.wikipedia.org/wiki/Formal_language

>>59
No one cares about the symbolic representation, just the value. 

But hey, let's all write out the grammar and look at the parse tree. With the power of a regular language will nip this problem in the butt!

For someone with such a hard-on for formal languages and semantics, you sure play loose and fast with your definitions. You have given, by far, the worst explanation of a formal language I have ever seen.

Please don't ever post here again.

>>60
Hey, I'm not a logician. I can only offer rudimentary knowledge on it. Yes, I did mix up semantic and syntatic parts.

.999... is accepted as short-hand for the limit as n approaches infinity of the sum from k=1 to n of 9*10^(-k).  everything converges, exists, is finite, etc.  so it is equivalent to a (finite) real number, which can be proven to be 1

>>60
The correct phrase is "nip it in the bud".

>>45
So if it's by definition, stop confusing the kids with this "proof" shit.

Let's call it "Dedekind cunt".
And no one in this whole thread has a clue what the word "converge" means; they just remember when to use the word when discussing the topic.

>>64
It's not by definition. It follows from the definition. There's quite a difference there.

Everything in math follows directly from definitions. That's what makes it math.

>>67
this is true

by the definition of truth.

Q.E.D.

The Ministry of Truth
You guys are fucking commies

pi =~ 4