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0.999... = 1

Name: Anonymous 2007-11-17 12:16

Oh snap!

Name: Anonymous 2007-11-17 13:25

troll

Name: Anonymous 2007-11-17 13:40

>>2
Aren't we all trolls on this forum?

Name: Anonymous 2007-11-17 14:06

fapfapfapfapfap

come on baby, another 100 posts!

Name: Anonymous 2007-11-17 14:08

>>4
Why sage?

Name: Anonymous 2007-11-17 14:21

x= 0.9999
10x = 9.9999
10x - x = 9.9999 - 0.9999
9x = 9
x=1
way old.

Name: Anonymous 2007-11-17 14:27

>>6
This guy tells the truth.

Name: Anonymous 2007-11-17 15:51

That's enough of that.

Name: Anonymous 2007-11-17 23:33

>>1
you cunt

Name: Anonymous 2007-11-17 23:35

>>6
circular reasoning, see the 6 previous threads on the topic

Name: Anonymous 2007-11-18 0:33

>>5
fine, so sage this time
fapfapfapfapfap

Name: sage 2007-11-18 0:56

sequence can be shown to converge to 1
this has been beaten to death a dozen times already

Name: Anonymous 2007-11-18 10:39

>>10
Nothing circular about it. The fact that you don't understand 0.(9) * 10 = 9.(9) doesn't mean it's not true.

Name: Anonymous 2007-11-18 15:16

>>10
Actually, what >>6 said is pure, airtight proof.  It establishes a definition of x being the same as .999... and going from there with only valid arithmetic and algebraic operations.

Name: Anonymous 2007-11-18 18:24

>>13
>>14
Incorrect morons.  To accept the procedure, you must first accept that the numbers are the respective implied limits in the first place.  If you don't take that as an assumption, you aren't able to ever complete the arithmetic operations necessary to solve the equation.  Ergo, you are attempting to prove that the expression 0.9999... is the limit 1 by first assuming that 0.9999... is the limit 1.  Hence, circular reasoning.

Name: Anonymous 2007-11-18 18:34

>>12
what does that mean to you, "converge"

Name: Anonymous 2007-11-18 18:36

>>15
Your post is nonsense, I hope you realise this.

Name: Anonymous 2007-11-18 18:38

>>17
Learn to read, jackass

Name: Anonymous 2007-11-18 18:58

>>18
Learn math, jackass.

Name: RedCream 2007-11-18 20:57

0.999... expressed as a limit, converges.  The result of the convergence is 1.

All else is jackoffery.

Name: Anonymous 2007-11-18 21:04

>>20
All numbers are limits. You're embarrassing yourself more than usual.

Name: Anonymous 2007-11-18 22:18

>>15
i don't understand

Name: Anonymous 2007-11-18 22:41

>>16
The basic definition of convergence of a sequence over a metric space (in this case, R with d(x,y)=|x-y| is given by:

{x_n} is said to converge to x iff for every epsilon > 0, there exists an natural number N such that for all n > N, d(x, x_n) < epsilon.

Basically you take x_1 = 0.9, x_n = x_(n-1) + 9/10^n as your sequence, which clearly becomes 0.999... as n approaches infinity and show that it converges to 1.

Name: RedCream 2007-11-18 22:50

>>21
Only the former is true.  0.999... expressed as the usual limit is 1.  You made no point whatsoever except to affirm the premise by which I made my conclusion.

What's the embarrassment in that?  In fact, the embarrassment here should be YOURS, since you failed epicly in pointing out the flaw in this chain of reasoning:

1. 0.999... has a limit.
2. That limit converges at infinity.
3. That convergence value is 1.

Name: Anonymous 2007-11-18 23:16

>>19
Stop trolling, you can't read can you
>>24
he unfortunately isn't claiming that 0.999... has a limit, he is claiming that 0.999... is it's own limit, ie circular reasoning, therefore no need to prove that 0.999... is 1 because he simply defined that it is 1 in the first place.
>>23
you never once said what the word "converge" means in all of that.

Name: Anonymous 2007-11-18 23:35

>>25
Although I gave you the definition of convergence of a sequence, I guess you want the more intuitive explanation of what convergence means. You beautiful layman, you.

Essentially, as n->inf the amount that x_n may differ from x approaches 0. You could think of this as the diameter of the set of points x_n from some n going to infinity monotonically decreasing as n increases. If the diameter of a set is 0, then it is a set of one element (in this case, the value the sequence converges to).

Name: Anonymous 2007-11-19 1:43

>>25
Oh, wow. Mathtards in my /sci/. 0.999... doesn't have a limit because it's not a function or a sequence. It's a goddamn number. A number cannot be used to express a sequence of values. It's a value. One value.

Name: Anonymous 2007-11-19 1:52

>>24
Numbers don't HAVE limits, numbers ARE limits. Of Cauchy sequences of rational numbers.

>>25
It's not circular reasoning, it is the DEFINITION OF THE REAL NUMBERS.

Name: Anonymous 2007-11-19 11:36

>>27

"0.999..." is not a number.  It is a numeral, a written expression that represents a number.

The "..." notation in this numeral means nothing more nor less than that you are taking a limit.  It is an abbreviation for a more complicated expression involving an explicit limit.

You fail.

Utterly.


Name: Anonymous 2007-11-19 13:28

>>29
The "..." notation in this numeral means nothing more nor less than that you are taking a limit.
No, it's not. It's a shorthand notation for an infinite number of 9s, just like 1 is a shorthand notation for "1." followed by an infinite number of 0s.
Since you're the first to mention the difference between numerals and numbers in this thread, it's pretty ironic that you'd get it that wrong.

Name: Anonymous 2007-11-19 14:16

>>29
You can't just make up your own system of mathematics then pretend like it's the one we've been using all along. Get with the program or get out.

Name: Anonymous 2007-11-19 15:06

>>14 here.

>>15
The initial assumption in >>6 isn't that .999...=1.  It's that x = .999 (which we are allowed to define as true).  From there, we use operations that keep the original equation intact, until we come to 9x = 9.  We divide both sides by 9, and viola!  .999...=x=9/9=1.  lrn2math, fool.

Name: Anonymous 2007-11-19 17:43

>>6 here

guys i really didn't want to mess things up. I just revised it again for uni short time ago so i was able to prove it.

As far as i know that proof is mathematically right and logically too.

Just think if it's wrong, by contradiction you should prove that it exists a number between .(9) and 1. Do it, then I'll say i'm wrong. No problem.

in b4 (.(9) + 1) / 2  :)

Name: Anonymous 2007-11-19 18:05

In fact if they are different then you can write a terminating decimal that lies between them.

>>30
And what is the meaning of the written symbol 0.9999999 (with an infinite number of 9s)?  Oh, that's right, a limit!

Name: Anonymous 2007-11-19 18:43

>>33
How about you prove there's no number between .(9) and 1 instead of shifting the burden of proof.

Name: Anonymous 2007-11-19 18:57

The fact is you need to define arithmetic on recurring decimals, before you can use elementary operations on them, and show that it is a consistent model.

The way you do that is in the construction of the real numbers, and thus it's pointless to use that sort of reasoning, because if you bother to spend the time proving that your proof is valid, you essentially prove the equality of 0.999.... and 1 along the way

Name: Anonymous 2007-11-19 19:23

>>34
Right, and "1" is shorthand for "1." followed by infinitely many 0's, which is a limit. EVERY real number is a limit, because that is how they are DEFINED.

>>35
It's been proven countless times. Or rather, it has been proven that 0.(9) = 1, which in turn implies that there is no number strictly between 0.(9) and 1.

Name: Anonymous 2007-11-19 19:59

FAPFAPFAPFAPFAPFAPFAP


>>35
by the way, you fail furiously

Name: Anonymous 2007-11-19 21:00

>>37
If that how you want to define your "limits", then they have no bearing whatsoever on this discussion. GTFO.

Name: Anonymous 2007-11-19 23:07

>>39
No, moron, that is not how I define my limits. That is how I define the real numbers. This has been explained before in the thread.

Name: Anonymous 2007-11-19 23:07

>>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.

Name: Anonymous 2007-11-19 23:36

>>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.

Name: RedCream 2007-11-20 0:07

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).

Name: RedCream 2007-11-20 0:12

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!

Name: Anonymous 2007-11-20 1:05

>>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.

Name: Anonymous 2007-11-20 1:11

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

Name: RedCream 2007-11-20 3:19

>>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!

Name: Anonymous 2007-11-20 5:43

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..

Name: Anonymous 2007-11-20 17:25

>>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.

Name: Anonymous 2007-11-20 17:30

These threads are worse than retards.

Name: Anonymous 2007-11-20 17:32

>>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.

Name: Anonymous 2007-11-20 19:12

>>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.

Name: Anonymous 2007-11-20 21:07

>>52

Wrong, it's a Dedekind cut.

Name: Anonymous 2007-11-20 21:18

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

Name: Anonymous 2007-11-20 21:22

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

Name: Anonymous 2007-11-20 21:30

>>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.

Name: Anonymous 2007-11-20 22:04

>>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.

Name: Anonymous 2007-11-21 14:03

>>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.

Name: Anonymous 2007-11-21 18:24

>>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

Name: Anonymous 2007-11-21 21:42

>>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.

Name: Anonymous 2007-11-21 21:58

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

Name: Anonymous 2007-11-22 0:28

.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

Name: Anonymous 2007-11-22 0:43

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

Name: Anonymous 2007-11-22 0:46

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

Name: Anonymous 2007-11-22 0:50

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.

Name: Anonymous 2007-11-22 2:39

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

Name: Anonymous 2007-11-22 4:26

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

Name: Anonymous 2007-11-22 12:27

>>67
this is true

by the definition of truth.

Q.E.D.

Name: Anonymous 2007-11-24 4:02

The Ministry of Truth
You guys are fucking commies

Name: Anonymous 2008-09-08 6:18

pi =~ 4
Don't change these.
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