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0.999999... = 1?

Name: Anonymous 2006-05-25 9:53

What the fuck. Why is that true. They got different numbers in them.

Name: Anonymous 2006-07-12 13:17

lol at axiomatic proofs.

also, 0.000...1  can exist and not mess with .999~ = 1, it would be lim n->inf 1*10^-n
= 0

so 0.999~ - .000...1 = 0.999...8 = 1
since 0.999...8 = [sum i=1 to inf 9*10^-i] - [lim n->inf 1*10^-n] = 1 - 0 = 1

.999~ = .999...8 = .999...7
you never get any farther from 1 doing that though, since theres still an infinite number of 9's before that non-existent last digit.

Name: Anonymous 2006-07-12 13:57

THIS THREAD IS ULTIMATE FAIL YOU FUCKING CLUELESS IDIOTS!!!

Name: Styrofoam 2006-07-13 1:33

>>161

Wow, you're fucking retarded.

0.000...1  does not fucking exist.  Explain how you can have a 1 after an infinite number of zeros.  "Why, the one goes here!"  No it fucking doesn't, it has to go after an INFINITE NUMBER OF ZEROS YOU FUCKING DICKWEED.  HAVING THAT LAST 1 IMPLIES THAT YOU TERMINATE THE FUCKING DECIMAL, AND THEN IT'S NOT INFINITE ANYMORE, DICKFACE.  GO BACK TO MIDDLE SCHOOL MATH.

Name: Styrofoam 2006-07-13 1:34 (sage)

Fuck, I meant to sage.  Why won't this thread die?

Name: Anonymous 2006-07-13 15:00 (sage)

>>164
Because you just revived it

Name: Anonymous 2006-07-13 15:56

>>163
Dear Dumbass,
I understand that 0.000...1 can't have a final digit and thus could not truly have a 1 'at the end' since there is no end.  If you wish to, you could infer this understanding from the statement I made regarding a similar representation, "since theres still an infinite number of 9's before that non-existent last digit" in reference to .999...8
Without regard to your idiocy, my point stands.  That point is that such a theoretical representation of the value (lim n->inf 1*10^-n)  would be equal to 0.  Since, as you so helpfully noted, there's an infinite number of 0s.  If your intent is to project the facade of a mathematician, please try to keep an open mind when dealing with representation of an abstract concept, and worry more about the properties and implications of a concept rather than its presentation.
Yours truly,
>>161

Name: Anonymous 2006-07-13 16:57

1<1

Name: Anonymous 2006-07-13 18:58

I think its time for socratic logic

lol @ symbolic logic

Name: Anonymous 2006-07-13 19:10

1. 0.999999... does not exist
2. 1 does exist
3. existence does NOT equal non existence
4. therefore, 0.999999... is not 1

Name: Anonymous 2006-07-13 19:23

>>150


That's exactly right. Thread over.

Name: Anonymous 2006-07-13 21:58

>>169
0.9999~ exists

0.9999~<=1

Name: Anonymous 2006-07-13 22:22

>>170
No it isn't. He's proof is circular. lol let's say 1/3 = 0.33333...

Name: Styrofoam 2006-07-13 23:57 (sage)

>>166

So what you're saying is, you are acknowledging that 0.000...1 has no actual meaning, and neither does 0.999...8.  I note that in >>161 you say >>you never get any farther from 1 doing that though
So you admit that 0.999...8 (which is a nonsensical number) = 0.999~ = 1.

Really, I think we agree that 0.999~ = 1, but I'm arguing that 0.000...1 is nonsensical while you're arguing that it's a valid concept.  Come to think of it, I would agree that 0.000...1 is a valid concept, as the infinitesimal.  But that's not how the infinitesimal is defined, and I still say 0.000...1 is a stupid way of writing it.

>>170

0.999... and 0.999~ both imply an infinitely repeating decimal, numbskull.

Name: Anonymous 2006-07-14 12:06

we do agree, so lets be friends. <3

Name: Anonymous 2006-07-14 21:17

(1/3)*3 = 0.99999... < 1

Name: DoubleAW 2006-07-15 0:52

>>160

That was enough.

All of you, quiet now. -_-

Name: Anonymous 2006-07-15 1:05

1/3  approximatley equals .333...

.333... is not a number, because it has infinite number of threes after the decimal place, and infinity cannot be applied to a number

Name: Styrofoam 2006-07-15 1:28 (sage)

>>177

How did you pass middle school math?  Seriously.

Name: Lumen 2006-07-15 16:11

>>177

0.333... is in |K. This number exist.
1/3 * 3 = 1

0.999... = 1 (proof here 100 times)

if .000...1 exists, it will be ... 0. Because .000...1 * 10 = .000...1. So .000...1 * 9 = 0. And .000...1 = 0.

We have .9999... + .000...1 = 1 like you want :D But I am note sure .000...1 exists... But IF it exists, it's 0

Name: Anonymous 2006-07-15 16:17

>>173
>Really, I think we agree that 0.999~ = 1, but I'm arguing that 0.000...1 is nonsensical while you're arguing that it's a valid concept.  Come to think of it, I would agree that 0.000...1 is a valid concept, as the infinitesimal.  But that's not how the infinitesimal is defined, and I still say 0.000...1 is a stupid way of writing it.

No, god no. That would make no sense as the definition of an infinitecimal because it's zero.

An infinitecimal is a FINITE, NON-ZERO value; it's defined as being small enough to satisfy whatever arbitrary precision (error) you need, no matter how small.


I can't believe this fucking thread is still going on. I can't believe >>157 thinks there's actually some debate about whether 0.999~ equals 1. Drop out of school and go bag my fucking groceries.

Name: Anonymous 2006-07-15 16:57 (sage)

>>180
Of course the real line doesn't include infinitesimals, so 0.999~ = 1 when you're using that, but introducing infinity, infinitesimals, or other weird numbers tends to mess things up, as is the case here. With infinitesimals, 0.999~ < 1.

Name: Anonymous 2006-07-15 19:20

>>181

What the fuck are you talking about! Do you have any idea what an infinitecimal is? It is a FINITE NON-ZERO value. It has nothing to do with the fact that 0.999~ = 1, and it does not "change" the real number line. Learn some fucking calculus, then kill yourself.

Name: Anonymous 2006-07-15 19:48

>>181
The mere fact that you refer to infinity and infinitecimals as "wierd numbers" shows how clueless you really are.

Name: Anonymous 2006-07-15 20:01

>>182,183
The mere fact that you keep spelling the word with a 'c' shows how clueless you are. They are 'weird' insofar as that they don't belong to the reals, and operations on them are not straightforward.

On the real number line, the proof that 0.999~ = 1 rests on the fact that if two numbers are infinitely close together, they must be the same. If you add infinitesimals to that however, two numbers can be infinitely close together but still separate. Therefore, 0.999~ will not be the same as 1 anymore.

Name: Styrofoam 2006-07-15 23:35

>>184

Infinitesimals have nothing to do with 0.999~ = 1, dumbshit, because the number line has nothing to do with it.  Go look at the two proofs I have provided in this thread, the algebraic one and the one using an infinite series.  They both concretely prove that 0.999~ = 1 without regard to the fucking retarded number line.

Seriously, fuck the number line.

>>180

I was just arguing that 0.000...1 could potentially be a representation of the infinitesimal.  I said it would be a retarded representation, but one nevertheless.  If you don't think it's the infinitesimal, then what the fuck kind of value do you think 0.000...1 has as a concept?  The very way it's written is nonsensical!

Name: Anonymous 2006-07-16 6:19

>>185
Your proofs are meaningless, because you assume infinite summation and operations on infinite representations work on the hyperreals exactly as they do on the reals.

Name: Anonymous 2006-07-16 16:12

>>186

How are you this retarded?  Did you just learn the world hyperreal and decide to use it without knowing what it means?  0.999~ is a fucking rational number.  I'll write it with cruise control again: 0.999~ IS A RATIONAL NUMBER.  Christ, it's middle school math!  REPEATING DECIMALS ARE RATIONAL NUMBERS.  Now hust the fuck up about mathematical concepts you don't understand.

Name: Anonymous 2006-07-16 16:17

>>187
Express it as a quotient then.

Name: Anonymous 2006-07-16 16:28

>>188

9/9.  You know, 1/9 = 0.111~, 2/9 = 0.222~

I'm sure you won't accept that answer, so let me just say again that it's known to middle schoolers that repeating decimals are rational numbers.  1/3 = 0.333~.  Everyone who has ever taken a math course knows that.  I'm sure if I scrounge around the internet I can find a proof that all repeating decimals are rational numbers.

Name: Anonymous 2006-07-16 16:33

Right. I'm a maths student, I can whip out muh axioms and muh topology or prove it from the Dedekind cuts construction. It just seems that you're using circular reasoning by saying that 0.999... is 1 because it's rational and it's rational because 0.999... is 9/9 = 1. But then excuse me if I'm not following this thread closely lmao

Name: Anonymous 2006-07-16 17:04

>>187
What. 0.999~ is a representation for the infinite limit mentioned earlier. Which number that stands for depends on what you're working with (ie. 1 if it's the reals).

Name: Styrofoam 2006-07-16 17:21

>>190

That's not how I'm proving it.  I'm proving it with the well known proofs I already posted in this thread, the simple algebraic one with 10x-x and the infinite series proof.  But then retards like >>186 come along and complain that you can't do algebra on an repeating decimal.  Wait, you are >>186, aren't you?  If you're a "maths student" and you don't know that repeating decimals are rational, you need to start over.

>>191

So what you're saying is, 0.999~ = 1, unless we're not dealing with the reals.  You figured it out, sir, through this whole thread we were dealing with the god damn fucking reals.  You win.

Name: Anonymous 2006-07-16 18:26

Lame.. So you lose then.

Name: Anonymous 2006-07-16 19:44 (sage)

numbers are merely ratios of quantized values.

Name: Anonymous 2006-07-16 23:24

but 1 - 0.9999~ = 0.000~1.

All you guys fail at math.

Name: Ouzo 2006-07-16 23:40

It was said earlier that 0.999~=1 because there are no other real numbers between it and 1.
in that case 0.999~8=0.999~=1
then 0.999~7=0.999~8=0.999~=1

however we know that there are real numbers between 0.999~7 and 1, hell i wrote 2 of them down.
the fact that there are no real numbers between 0.999~ and 1 does not mean they are the same.

Name: Anonymous 2006-07-17 5:50

>>196
finally, someone who gets it.

Name: Anonymous 2006-07-17 8:09 (sage)

>>196 lol
however we know that there are real numbers between 0.999~7 and 1
I'm going to disregard the incredible abuse of notation, and focus on your logic: There are NO real numbers between 0.999~7 and 1, because you just proved all of them to be EQUAL to 1.

0.999~ = 1, therefore it CANNOT be that 0.999~7 < 0.999~ < 1. This works because the seven is 7*10^-inf = 7*0 = 0. (With infinitesimals, 10^-inf wouldn't be zero.)

Again, I know this is an incredible abuse of notation, and it should be stated with limits and such, but I'm afraid >>196 wouldn't understand, and I'm just pointing out a logic error anyway.

Name: Anonymous 2006-07-17 13:07

>>196
You fucking idiot. 0.999~7 and 0.999~8 are not numbers. They make no fucking sense. You can't stick a 7 on the end, THERE IS NO GODDAMN END.

And yes, if you can't find a number between two real numbers, then they're the same number. That's a property of the real number line known as completeness.

Name: Anonymous 2006-07-17 13:10

>>199
Rather, I should say it's due to the completeness of the real number line.
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