0.9999... never happens anyway. There's no fraction that gives you that decimal unless you fuck up the long division. So don't even bother with this shit.
>>7
Actually, it happens every time you eg divide a number by itself, since it is *gasp* the exact same thing as 1. The '1' notation is just more convenient to write.
Name:
Anonymous2006-05-25 21:35
If you rewrite .9999999... as .9 + .09 + .009 + .0009 + ..., and then rewrite those numbers as fractions, you get 9/10 + 9/100 + 9/1000 + 9/10000 + ... So therefore .9999... is really just an infinite geometric series with the first term = 9/10 and the rate =1/10. Using the formula for the sum of an infinite series, which is a/(1-r), you'll get
(9/10)/(1-1/10)
= (9/10)/(9/10)
= 1
Q.E.D.
I thought this up myself at two in the morning one time. Other people have probably done it this way too, since it's the only really solid proof I've ever seen of .999999... = 1.
Name:
Anonymous2006-05-25 23:16
>>9
No need to go through all that rubbish. You just have to show that there is no real number between .99999999... and 1.
Why would you ever want to write 0.99999... instead of 1 in the first place? It's not like you will ever encounter it in arithmetic. It's just a curiousity, a strange construct.
Name:
Anonymous2006-05-26 3:00
>>8
Nonsense, it never happens. I'm not talking value, I'm talking the representation of the value. You never see 0.99999... when you divide an integer by itself, tough guy.
It can be even simpler.
1/3 = .33333...
3 * 1/3 = 1
3 * .333333... = .99999...
∴ .9999... = 1
Name:
Anonymous2006-05-28 0:07
>>1
>What the fuck. Why is that true. They got different numbers in them.
Lets list a few different ways we can express the value 1 without using that number, shall we?
3-2=1
2/2=1
cos(0)=1
8935^0=1
|√(0.5)+√(0.5)i|=1
i^4=1
ln(e) = 1
So why in the world would the fact that .999... doesn't include the number 1 prevent them from being equal?
(my intention was not to prove .999...=1, but rather to dissuade the assumption that since they include different numbers, they cannot be equal)
Name:
Anonymous2006-05-28 2:00
|√(0.5)+√(0.5)i| = 1
Could you prove this? I'm not so good with imaginary numbers and would like to see if this is true... Unless you meant to write
||√(0.5)-√(0.5i)||
Which I can see how it would work
Name:
Anonymous2006-05-28 20:13
No, it's correct, |√(0.5)+(0.5i)|=|(√2+√-2)/2| right.
Take √2 out of the bracket, you get |(√2(1+i))/2|
Cancel the √2 you get |(1+i)/√2|
Square the fraction you get |(1+2i-1)/2|
Collect like terms |2i/2|
Simplify, end up with |i| which we know = 1.
Name:
Anonymous2006-05-28 20:20
it depends on what norm you are using lol.
Name:
Anonymous2006-05-28 21:27
>>18
Or you could use a calculator.
Or you could defer to the complex number plane, plot the point, and then realize that the point is exactly π/4 on the unit circle in such a plane, and then realize that absolute value means distance from zero, which in this case is 1 because its the fucking unit circle.
In other words, you shouldn't have had to do all that algebra.
Yes, but he did say prove and while plotting the point on the complex plane is simpler, I can't easily do that on a text board. I could've described it like you did but I wasn't sure if that would be satisfactory. Besides, it's not that much algebra.
It's a flaw in the Number System since you can't apply these laws into reality and are impossible to co-erce with realistic physics. Since, in realistic physics, there are no fractions and all matter/energy exists in integers.
So my point is, such numbers are non-applicable to reality, which is where the confusion exists.
Name:
Anonymous2006-05-30 21:47 (sage)
wen eye git a nerection, my peenis is so long an hard yu cud poke it wit a needle and it would burst and the exploshun wud be enuff to destroy teh world!!!!!
Name:
Anonymous2006-05-31 3:48
gad
Name:
Anonymous2006-05-31 3:48
adg
Name:
Anonymous2006-05-31 3:48
adf
Name:
Anonymous2006-05-31 3:49
.9999999... to my knowledge, does not equal to one. It is always less then one, always.
but, it is infinitly close to one, just never exactly one
>>34 It is true that all matter and energy is quantised but time and probability and other factors exist as real numbers.
Name:
Anonymous2006-05-31 11:29 (sage)
>>41
That implies that the computing power of the universe is beyond that of a Universal Turing Machine. That's extremely counter-intuitive; it makes much more sense for space and time to be quantized, and I see no reason for it not to be.
Name:
Anonymous2006-05-31 13:48
Is 0.9999999... = 1?
Name:
Anonymous2006-05-31 13:59
Don't believe the stupid lies. How could 0.999... be 1 when we all know that it isn't? Think!
Name:
Anonymous2006-05-31 14:38
assume .999~ = 1
subtract 1 from both sides
-.000~1 = 0
multiply by 2
-.000~2 = 0
thus
.000~1 = .000~2
divide both sides by .000~1
1 = 2
Name:
Anonymous2006-05-31 14:59
.000~1
We are beyond math now. We are in the realm of Meta Math!
Why do you fucking retards return every time this thread occurs? I can only hope that each time I post this, one person is convinced, that way, assuming there is only a finite number of idiots here, eventually you will all learn.
Let:
x = 0.999~
Multiply both sides by 10.
10x = 9.999~
Subtract 0.999~ (which is equal to x) from both sides.
9x = 9
Divide both sides by 9.
x = 1
We began by stating that x = 0.999~, therefore.
0.999~ = 1
A proof that requires nothing more than middle-school algebra. Satisfied?
Name:
Anonymous2006-05-31 23:51
>>45 Uh, no. There is no last digit in the infinite expansion of 0.999... So you can't do operations like that. Analog of using infinities and ordinary arithmatic to show that 1=2...
(And if there were, which there isn't, it would be a 9, not a 1)
Name:
Anonymous2006-06-01 1:11
>>39 >>40
What the fuck? People are posting all sorts of mathematical proofs, and you're saying "to your knowledge"? Jesus fucking christ. Learn to yield to someone who has a clue.
You know that link 49 and 5 posted up? You might want to look at that, I think you'll find it jolly interesting.
Name:
Anonymous2006-06-01 7:13
>>49,52
There exist people who are fundamentally unable to comprehend the concept of infinity. Don't worry too much about it. Instead, we should revel in our superiority.
Name:
Anonymous2006-06-01 8:02
that's funny because if you understood infinity you would understand that 0.999... isn't 1
Name:
Anonymous2006-06-01 12:35
Remember what infinity means. Alright, instead of using mathematical rigour (which is self-evidently reliable anyway) I'll try and use common reasoning. 0.9999... is infinitely close to 1 right? The seat of my trousers is infintely close to the chair I'm sitting on, that means it's touching it. In the same way 0.9999... touches 1, it therefore equals 1.
Name:
Anonymous2006-06-01 14:48
Nice try, but no. It's kinda like the integers 1 and 2 are touching, but they are not equal. That's just how mathematics works, sorry.
Name:
Anonymous2006-06-01 15:58 (sage)
The reason why the 1=2 proof works is because you're dividing by zero during the proof.
As for The "10-.999~"=9, I'd say that it'd equal 9.000~ with a one at the end (of infinity). However, for anything non-purely theoretical, yes, you can round it and make it one.
I didn't mean that, read it again. 'Touch' was used to illustrate the fact that you can imagine something being infinitely close to something means it touches it. As in, as the difference between 0.999... and 1 becomes infinitely closer you cannot possibly measure and state it. They are therefore not consecutive, but the same number; they are touching rather than the shortest of distances apart.
So, taken in this sense, 1 and 2 are consecutive, but not touching.
...Just look at the wiki, that'll make sense.
Name:
Anonymous2006-06-01 19:14 (sage)
>>58
Two possibilities:
1. These people are trolling. You are wasting your time.
2. They are like Christians; no amount of logic will make them see they are wrong. You are wasting your time.
Infinitely close = touching is a "real world approximation", and doesn't really stand up to mathematical rigour. 0.999... is as much 1 as 22/7 is Pi. That is, not at all.
Except that 22/7 is only equal to pi when you're looking at the first few decimal places, yet .999... is infinitely close to one when there are an infinite number of nines (that is, the decimal is written out fully), so that argument is pretty stupid.
There is no real number between .999... and 1, so therefore .999...=1.
Name:
Anonymous2006-06-02 1:17
>>57
"with a one at the end (of infinity)"
There is no end of infinity! That's the whole point. There isn't a one at the end because THERE IS NO END.
For purely theoretical math, yes, 0.999... is EXACTLY equal to 1. It's no different than writing 2/2 or cos(0) or 3*5/15. It's exactly equal to 1.
>>60
You're talking about mathematical rigour? That's laughable. Find me a mathematician on the planet who thinks 0.999... is not 1, and I'll give you my first born son.
Name:
Anonymous2006-06-02 1:19
>>50
Uhh, the real number line is continuous, so the limit of x at any point is equal to x. Since the limit of 0.999... is 1, then 0.999... = 1.
In other words, you just proved yourself wrong. Congratulations.
Name:
Anonymous2006-06-02 1:48
>>34
There are no fractions in physics? Are you high?
Example 1 of 10983758302098373856563924: electrons have spin 1/2.
Name:
Anonymous2006-06-02 2:05
This is way totally something worth getting yr panties all in a bunch over because it has so many real world practical applications.
Name:
Anonymous2006-06-02 2:21
You all fail. 0.999... isn't 1 and no amount of whining and false analogies will make it so.
I expect that child to be airmailed to me ASAP. And don't forget to insure it. On the off chance that it's not white, only ship me its heart (sacrificial reasons).
Name:
Anonymous2006-06-02 2:31
n = 0.9999...
10n = 9.9999...
9n = 9
n = 1
Name:
Anonymous2006-06-02 4:05
i think your math is flawed
n= .999999...
10n=9.999999....
9n=8.99999999....and infinity of nines ....9999991
n= .99999999...
n does not = 1
Name:
Anonymous2006-06-02 6:24 (sage)
>>67 Abraham Robinson
lolol
I think you're getting your systems mixed up here, genius!
>>69
Read this again: There is no end of infinity! That's the whole point. There isn't a one at the end because THERE IS NO END.
You are a bad loser. At least admit defeat with a shred of dignity.
Name:
Anonymous2006-06-02 6:51
>>71 You wouldn't know dignity if it beat Super Mario Bros. in 5 minutes and 47 seconds.
Name:
Anonymous2006-06-02 6:58 (sage)
>>71
I'm not even the guy who he'd been talking to. I just found his bringing up of nonstandard analysis hilarious.
Name:
Anonymous2006-06-02 9:26
>>65
Let's not forget
a) pi, e, and every other mathematical constant
b) h, c, and every other physical constant
lollerwaffles
Name:
Anonymous2006-06-02 9:36
>>67
Way to fail. Abraham Robinson was not dealing with numbers you fucking tard. He was performing analysis on hyperreals; he defined things in sequences. Guess what, if you convert a hyperreal to ACTUAL NUMBERS, you get zero, which means 0.999... = 1. Another tard who proved himself wrong. Congratulations.
Learn some fucking a) basic calculus, b) set theory, and c) group theory, and then stop trying to apply random concepts to the real number line.
I can't believe how fucking stupid everyone is. This is like an argument on religion. You can throw away pages and pages of mathematical proofs, but you read a random piece of nonstandard math that doesn't even apply to real numbers, and you fucking cling to it like a security blanket.
Name:
Anonymous2006-06-02 9:39
>>69
>9n=8.99999999....and infinity of nines ....9999991
I think YOUR math is flawed. You just put a last number on your decimal expansion there pal. The number has to end in "...", that's the whole point. What is so complicated here?
Name:
Anonymous2006-06-02 9:45
>>74
Not to mention the fact that trancendentals (due to radicals) and imaginary numbers (anything in wave analysis, such as electrodynamics or quantum mechanics) are also everywhere in physics.
Immesureable fail on the integer physics there bub.
Name:
Anonymous2006-06-02 12:18
>>65
Don't forget, quarks have a charge of either -1/3 or 2/3.
Name:
Styrofoam2006-06-03 2:43
I think the guy who said there are no fractions in physics was confused by quanta. Still, it's pretty stupid.
Also, I'm going to say it again because people don't seem to be getting it: You can't have "an infinite number of zeros (or nines) and then a one." There is no fucking end of infinity. If you want to stick a one after an infinite strings of zeros, start running to the end of this infinite line. When you get there, you'll have proven me wrong. And you had better start now; infinity is a long way to run.
Name:
Anonymous2006-06-03 2:53
1/3 != .33333... it is just the closest representation that this stupid arab number system can give us. down with it i say
You cannot represent 1/3 as a finite number of decimal places, but you *CAN* represent it as an *infinite* number of decimal places, that's the big step to realise :]
Name:
Anonymous2006-06-03 18:43
Well you can't really, because you'd never finish writing them down.
Name:
Anonymous2006-06-03 20:01 (sage)
>>82
Sure you can, just write every 3 half as wide as the one before it.
Name:
Anonymous2006-06-03 22:32
>>82 It's not about being able to write it down, though
Correct, but 0.999... is 1, so long as the ...'s are there.
Name:
Anonymous2006-06-04 13:54
stop trolling. 0.999... isn't 1
Name:
Anonymous2006-06-05 1:34
You people are fucking retarded. There have been multiple proofs posted and you still don't understand that .999... = 1. You idiots obviously have NO concept of infinity.
Name:
Anonymous2006-06-05 1:50
>>84
It is if you use the word "represent", which is what I was responding to. You'd never really show all of the digits, because you can't.
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Anonymous2006-06-05 1:53
Anyway, chew on this: According to the same argument, in binary, 0.0000000000000... equals 1. WTF?!
>>90 Yes, but represent does not mean write down. I could represent something with an infinite number of 9s, even though I can't write that down. Anyway this is kind of getting pointless, even by 4chan standards (4chan has standards?)
Name:
Anonymous2006-06-05 10:30 (sage)
>>91
No. If the base is 10, the repeating digit is 9. If the base is 2 (binary), the repeating digit is 1:
0.111... = 1.
Name:
Anonymous2006-06-05 14:52
>>95
Yeah, I figured that out myself when I was in bed for the night, but didn't feel like getting up to correct myself.
Name:
Anonymous2006-06-05 14:54
>>94
I'd like to see you try to represent it with an infinite number of 9s. Better get started, you have much work to do.
Name:
Anonymous2006-06-05 15:11
0.999... = lim(n -> inf, 1-(1/10^n))
Name:
Anonymous2006-06-05 16:12 (sage)
>>80,97
You can actually extend the 'decimal point'-system for representing non-integers in several ways. One such extension is to append an infinitely repeating group of digits with '...' instead. In that system, 1/3 can be exactly represented by '0.333...'.
(Another (more logical) method is to introduce a second decimal point, preceding a group of repeating digits. 1/3 would then be represented as 0..3. In that system, eg 7/23 = 0.3.18 (and 1 = 0..9, etc).)
Name:
Anonymous2006-06-05 16:25
>>98 >>99
All you are doing is pretending that there are infinitely many digits by using a symbolic notation for the idea. You never accomplish showing the infinite number of digits, because you can't.
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Anonymous2006-06-05 16:41 (sage)
by using a symbolic notation for the idea.
lol, that's pretty much the only way you can communicate ideas, no?
Name:
Anonymous2006-06-05 16:57
>>101
There is a difference between actually showing a set of digits and indicating through symbols that a larger set of digits is implied. You can only imply the infinite set, you can't actually show all of the digits.
Name:
Anonymous2006-06-05 17:05 (sage)
>>102
Obviously ideas involving infinity force you to find economical representations of them, but how the hell do you go from that to saying "1/3 != 0.333..."? Those are just different ways of representing the same thing.
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Anonymous2006-06-05 17:28
Well, the symbolism of 0.333... implies a process that can't be finished. I'd prefer to say that the limiting value of that process is what equals 1/3.
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Anonymous2006-06-05 17:39 (sage)
No, the '…' represents the "limiting value" directly. Otherwise the whole notation would be rather useless, wouldn't it?
Name:
Anonymous2006-06-05 20:05
>>Fraction proof
>>The standard method used to convert the fraction 1⁄3 to decimal form is long division, and the well-known result is 0.3333…, with the digit 3 repeating. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…; but 3 × 1⁄3 equals 1, so it must be the case that 0.9999… = 1.
If we have a fraction for 0.3333... which is 1/3,
then we MUST have a fraction that = 0.9999...
Oh wait, we don't have a fraction that = 0.9999..., but we do have a similar one that = 1 !!!!
Amirite or am i wrong? Is there a fraction that = 0.9999... ???
106 asked a dumb question, how can you expect a good answer? I fraction that's equal to 0.999... is 3*(1/3)=3/3. I could say 9/9 = 0.999... too, or pi/pi = 0.999... for that matter.
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Anonymous2006-06-09 1:52
>>105
The ellipsis ... doesn't have a formal definition in mathematics. It merely stands for "and so forth". Their meaning is derived from conventional use. I recognise that in practice, a limiting value is understood by agreement among certain academics, but it is not necessarily so.
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Anonymous2006-06-09 2:08
I should revise the above statement to include the fact that the more official ways to show the repetition of digits (bars, dots) are widely understood to mean merely that: the continued repetition of digits without end. The interpretation that they must symbolize a limiting value of that repetition is an interpretation after all, and those that do not share the interpretion will not readily accept the assertions of equality.
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Anonymous2006-06-10 14:12
>>110,111
This is irrelevant. The problem is not in the interpretation of the ellipses, it's on the definition of the number system you're using. On the real number line, the '...'s don't have to mean limiting value; 0.999... is still equal to 1.
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Anonymous2006-06-10 14:46 (sage)
the '...'s don't have to mean limiting value
What else could they mean?
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Anonymous2006-06-11 20:03
>>113
They merely mean that the 9s are repeated forever. The fact that the ellipses imply convergence to the limiting value is a consequence of the continuity of the real number line.
Name:
funny ringtones2006-07-01 8:10
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Name:
Anonymous2006-07-01 8:16
>>39
But...but .99999 is .33333 X 3
.333333 is 1/3, thus .99999=1
Not infinitely close, SAME
OR ELSE PARADOX
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Anonymous2006-07-01 10:26 (sage)
As a result of not having infinitesimals, it cannot be 'infinitely close', thus must be equal.
However .99---> infinite places will never be equivelently equal to 1 because of the extra 0.0(infintite)0001 needed to make it one, so therefore is will always be approaching 1
It can be described as a limit LIM(n -> infinity)[.99~~]-> 1 where n is the decimal places
Haha, I just realized how much of an idiot >>119 really is.
>>It can be described as a limit LIM(n -> infinity)[.99~~]-> 1 where n is the decimal places
That's actually correct, but he thinks that's an argument against .999~ = 1.
Guess what? "LIM(n -> infinity)[.99~~]-> 1 where n is the decimal places" is exactly the same as .999~. This equals 1. Congratulations, now go back to high school math.
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Anonymous2006-07-04 5:09
1 is better than 0.999... because 0.999... has to converge to be 1, but 1 doesn't have to converge to be itself.
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Anonymous2006-07-04 5:49 (sage)
>>124
0.999... doesnt have to converge to be 1. It IS 1 already.
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Anonymous2006-07-04 6:21
>>125
Exactly. 0.999... IS 1, there's no fucking limits or convergence involved here. So stop saying that, you goddamn idiots. The convergence of a limit is a tool to prove that this decimal expansion is correct, but it doesn't actually have anything to do with the expansion.
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Anonymous2006-07-04 17:24
Well then what you are doing is starting with 1 and then partitioning it indefinitely, not starting with 0.9 then adding 0.09 then adding 0.009 and so on to magically create 1 without benefit of limit. You cheater.
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Anonymous2006-07-04 22:22
I can't believe this thread is still going on. 0.9999... = 1
The End
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Anonymous2006-07-07 12:53
now this is very simple:
0.999999999....=0.9999999999...
1=1
simple mathematics
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Anonymous2006-07-08 3:41
if 1 = .999...999
then .999...999 = .999...998
then .999..998 = .999..997
...
then .999...067 = .999...066
then 1 = .999...066
then 1 = .999...XXXXXXXXXXXXXXXXXX
etc.
so if 1/infinity doesn't matter at all, like the notation 1 = .999... indicates, then an infinite number of infinities wouldn't matter. Thus, if 1 = .999..., then 1 = .5, or anything else.
CORRECT ANSWER AHEAD:
1 does not equal .999...
But in all practical mathematics useage, they both work the same, and can be interchanged. But they are NOT the same.
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Anonymous2006-07-08 4:02
It's not cause it's getting rounded off to a higher number?
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Anonymous2006-07-08 17:14
It's not that complicated.
.9999~ = .9999~ while 1=1.
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Anonymous2006-07-08 18:05
Ok, let's get serious here. 0.999... = 1. An "easy" way to see it, that does not involve any of the proofs shown above, is this:
Theorem: 0.999... = 1
Proof: Let R be the set of real numbers. R is know to be an Dedekind-complete ordered field. Since R is a field, addition and multiplication are defined with their usual properties; since a field is a group under addition, it follows that R is an ordered group, and therefore it defines an uniform structure. An uniform structure is complete, and therefore for every two members a,b in R, there exist infinite members (denoted by s) s such that a < s < b. Since there is no real number s such that 0.999... < s < 1, it follows that 0.999... = 1. QED
The idea behind the math: if two numbers are different, one can always find a number in between them. You cannot find a number in between 0.999... and 1. Therefore, 0.999... = 1
PS: (1 + 0.999...)/2 does NOT does the trick; if you evaluate the division correctly, you find 0.999... once again.
Following that logic
1 = .999...
and
.999...9 = .999...8
therefore 1 = .999...8
But the number .999...9 is between 1 and .999...8
So 1 can't = .999...8
But if A = B and B = C, but A does not = C, then there has to be a problem with the logic system, no?
Well, the reason that there's nothing between 0.999... and 1 is because 0.999... gets infinitely close to 1, ie the limiting value is 1. It doesn't have to follow that 0.999... means the same thing as 1.
Name:
Anonymous2006-07-09 4:18
>>135
Fuck you're stupid. Writing .999...8 makes no sense. You can't tack an 8 on the end, THERE IS NO END. STAB YOURSELF IN THE FUCKING FACE PLEASE KTHX, GOD YOU IDIOTS PISS ME OFF.
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Anonymous2006-07-09 15:43
First of all, just a quick note: the transitivity of equality (A = B and B = C -> A = C) DOES hold in the reals, but it does NOT in many other sets.
>>138
No, that's the WHOLE difference between the rationals and the reals. In the reals, "be infinitely close" is the same as "be the same". That follows from the completeness of the uniform structure.
>>135
Explain me exactly HOW do you intend to equate 0.999... to 0.999...8. in crude terms, you can't really put something "after an infinity of 9's". However, I'm willing to give you the chance: how do you prove that 0.999... = 0.999...8 ?
Really guys, I'm make it a little clearer now: in the reals, there is NO SUCH thing as the "last number before a number". Between any two different numbers, there is an infinity of other numbers. Think a little about the following related question:
Which is the smallest number greater than 0?
Remember that 0 = 0.000... Is any of the guys talking about "0.999...8" willing to say that the smallest number greater than 0.000... is 0.000...1?
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Anonymous2006-07-09 16:50
what the fuck are you talking about? equality is transitive by definition
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Anonymous2006-07-09 18:21
>>140
Remember that 0 = 0.000... Is any of the guys talking about "0.999...8" willing to say that the smallest number greater than 0.000... is 0.000...1?
Well, I guess I would. Although I might argue that 0.000...1 isn't even an actual number, and is more of a concept, if even that.
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Anonymous2006-07-09 19:17
>>142
Why, you could call that concept the 'infinitesimal'. Ingenious!
Perhaps, if you added these to the real line, you could formulate a sort of 'non-standard' analysis...
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Anonymous2006-07-09 20:51
0.999~<1
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Anonymous2006-07-09 21:21
0.999~>1
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Anonymous2006-07-10 3:20
>>143
Just because you can name a concept doesn't mean it can actually be concieved
Hay guys
This guestion has no answer. It is both = 1 and < 1, depending on what you are trying to do with it
Name:
Styrofoam2006-07-11 2:43
God damn it, you stupid fucks are all god damn retarded. You know what? Shut the god damn fuck up about infinitesimals, the number line, and analysis, even though those are all valid ways of figuring out that 0.999~ = 1. Shut the fuck up, just for a second, and read this. Read it slowly.
Let:
x = 0.999~
Multiply both sides by 10.
10x = 9.999~
Subtract 0.999~ (which is equal to x) from both sides.
9x = 9
Divide both sides by 9.
x = 1
We began by stating that x = 0.999~, therefore
0.999~ = 1
There. Motherfucking algebra. No analysis, no calculus. Just motherfucking algebra. Go home and cut yourself.
.9999999 does not equal 1. It equals .9999999.
0.9(Dash above the 9) on the other hand, DOES equal 1. 1/3=0.3(Dash above the 3) times that by 3 and you get 0.9(Dash above the 9) And 1/3x3 equals 1. Think of the same rules as infinity, except with decimal places. But, .9999999999999(Put as many 9s as you want) does not equal 1, because .999+0.001 equals 1. It needs the dash ontop of the 9 to indicate that it has infinite decimals.
Hey, re-re. Algebra laws fall apart when you are dealing with infinities. Pretty much all laws fall apart when you are dealing with infinities. Don't they teach this shit in middle school anymore? What the hell. This is why the problem has no real answer, because you are trying to compare a simple number to an abstract concept. Just give it up already.
But it is debated. Why? Because there are really no definate laws when dealing with infinities. 1 * 2 = 2 is definate. It is not debated. I propose that the reason .999... = 1 is accepted is because there are no math problems where a 1/infinity amount is going to make a difference.
But conceptually, .999... and 1 are very different things. And in at least that respect, they are not the same number.
If you read the fucking proof, it doesn't matter that it's an infinitely long decimal. The first operation performed is multiplying 0.999~ by 10. This clearly produces 9.999~ Then, you subtract off the quantity 0.999~. That's it.
It's debated? That's fucking news to me. Oh wait, you mean it's debated by little 14 year old shits on the internet who know nothing about mathematics. Oh, and there are plenty of definite(gb2/spelling/) laws about infinities. Learn Calculus.
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Anonymous2006-07-12 2:20
>>156
Again, you are trying to solve a problem with infinite places with algebra. I'll say this again, you simply can not do that. Algrebra breaks down when dealing with infinity.
Let me put it this way. We are told in our first math class that infinity is not a number. If you don't accept this, you are a fool. Need more? A 1/0 slope is undefined. Why? Because you can't define an infinite slope. Because normal rules don't fucking work with infinity.
Since the .999 is followed by an undefinable number of 9s, there is no way you can apply math operations to it. This includes .999... * 10, or anything else you can think of.
Thinking about calculus? Well, you have a point. Or would, I suppose, if it weren't for the fact that calculus is, by it's very nature, an estimation (who the hell doesn't know this?). Why do you think they word it like "the LIMIT of f(x) as x APPROACHES c is L"? Because you can't prove that f(x) = L for any x. Why? Because they never actually meet.
Thanks for proving you know nothing about mathematics. If you actually managed to take a Calculus course like you imply, I suggest you take another. Either you are a fool, or your teacher was, and either way you have lost all credibility in this conversation. Leave now
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Anonymous2006-07-12 2:41
>>157
0.99999… is finite not infinite so whatever the fuck you are talking about fails.
Calculus an estimate? Go back to learning more basic maths dumbass.
ONOZ IT HAS INFINITE DECIMAL PLACES LOOKS LIKE WE CAN'T DO SHIT ABOUT IT, SORRY
Except as a repeating decimal, it is a rational number. Get that, shithead? 0.999~ IS A RATIONAL NUMBER. CHRIST, YOU ARE AN IDIOT. OOPS, CAN'T DO OPERATIONS ON RATIONAL NUMBERS! Can't fucking multiply a rational number by 10, NO FUCKING SIR!
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Styrofoam2006-07-12 3:51
You know, if this idiot won't shut up about not doing operations on an infinitely repeating decimal, I whipped up a proof using an infinite series. Maybe that will shut him up.
0.999~ = SUM(n=1,inf,9/(10^n)) [The series is 0.9 + 0.09 + 0.009 + ...]
Sorry about the notation, but that's the best way to express it on a BBS.
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Anonymous2006-07-12 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.
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Anonymous2006-07-12 13:57
THIS THREAD IS ULTIMATE FAIL YOU FUCKING CLUELESS IDIOTS!!!
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.
>>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
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Anonymous2006-07-13 16:57
1<1
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Anonymous2006-07-13 18:58
I think its time for socratic logic
lol @ symbolic logic
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Anonymous2006-07-13 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
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.
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
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Anonymous2006-07-15 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.
>>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.
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.
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Anonymous2006-07-15 19:48
>>181
The mere fact that you refer to infinity and infinitecimals as "wierd numbers" shows how clueless you really are.
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Anonymous2006-07-15 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.
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.
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!
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Anonymous2006-07-16 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.
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.
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.
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Anonymous2006-07-16 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
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Anonymous2006-07-16 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).
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.
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.
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.
>>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.
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Anonymous2006-07-17 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.
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Anonymous2006-07-17 13:10
>>199
Rather, I should say it's due to the completeness of the real number line.
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Anonymous2006-07-17 13:25
>>188 >>189
Yes, any recurring decimal number is rational. You can convert any recurring decimal x with period length n to a rational number just by taking 10^n x - x:
x = 0.18181818...
100x = 18.181818...
99x = 18
x = 18/99 = 2/11
And i am not a math studiant. i am a math teacher. And this fucking .99~8 or 0.000...1 DONT EXIST! I'ts easy to prove than 10^-x with < -> infinite = 0. Not 0.00...1
>>202
They're letting twelve year olds teach math know?
Well, at least he's right.
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Anonymous2006-07-17 13:34
*now. Fuck.
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Anonymous2006-07-18 0:48
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Anonymous2006-07-18 5:10
0<1=FAGGOTS
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Anonymous2006-07-18 21:37
The best I can say is that lim(n -> +infinity- [infinity greater than zero, approached from the lesser side]) { sigma(x=1; x <= n) {9/(10**n)} }== 1.
That is to say, as you increase the number of 9 that follow 0. in the expression 0.999999, the value of that expression approaches 1. 0.999... == 1 only in the case that you have iterated an infinite number of 9 after the decimal.
uhhhh someone else finish what i'm saying if i'm going in the right direction. or am i just blowing air out of my ass
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Anonymous2006-07-18 22:07
oh and also
.9~ + epsilon == 1
defining epsilon, the infinitesimal, is an exercise left to the reader
>>210
Hey guys in non-standard analysis 0.999... != 1. Now STFU.
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Anonymous2006-07-19 5:41
>>209
Epsilon is not the infinitesimal here. That epsilon is just zero. If you were trying to refer to epsilon-delta limit proofs, that epsilon is NEVER an infinitesimal.
Why do people keep saying infinitesimal when they have no idea what it means?
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Anonymous2007-05-07 3:46 ID:mS9qBufE
I AM A HARVARD MATH PROFESSOR AND I THOUGHT I SHOULD RESPOND TO THIS TO SET THE RECORD STRAIGHT. 0.9~ IS NOT EQUAL TO 1 BECAUSE YOU CAN NEVER FINISH WRITING DOWN ALL THE 9'S. IT'S THAT SIMPLE.
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Anonymous2007-05-07 3:49 ID:YvAmVJNg
I am sure that you left some behind and the next guy that sits down will get HIV.