False, 0.9999.... is a limit, though for all intents and purposes it can be substituted with 1, it is not the same thing. There is an iota of difference. Example, an object can accelerate infinitely towards the speed of light but never reaches it.
>>2
The Calculus taught me that as long as the function converges, the limit is a value that is the same thing as other values. Hence, since 0.999... converges, it is equal to 1, which is the limit of its convergence.
A line has no area, yet is exists on a plane. I see the two situations being fairly similar. There is zero difference between a plane, and a plane with a line subtracted. Hence, there is zero difference between 1, and 1 with an infinitesimal subracted.
Name:
Hump4us2007-10-07 2:33
>>2
Divide 1 by 3. You should get .333333... Multiply that by 3. You should get .999999... Under all circumstances, and I mean ALL circumstances, .9999... is 1.0. After all, zero in itself is the reverse of infinity. Where as infinity never ends, so does zero at being infintisimaly small. And for the record, it is possible to reach the speed of light. It is just not within human grasp, and very well may never be.
Name:
Anonymous2007-10-07 2:50
>>2
0.999... is a limit in the sense that EVERY real number is a limit (of a cauchy sequence of rational numbers), and 0.999... = 1.000...
Name:
Anonymous2007-10-07 2:54
>>5
1.) It requires an infinite amount of energy for a particle with mass to accelerate to the speed of light.
2.) The universe contains a finite amount of energy.
∴ It is impossible for any particle with mass to move at the speed of light.
>>6
0.999... isn't a number, 8 isn't a number, IV isn't a number, none of these things are numbers. I think people are arguing over what symbols mean, not numbers. Numbers are concepts unto themselves. Obviously the symbols 0.999... and 1 aren't the same thing, because christ, look at them. Not so obvious is what is meant by the symbol 0.999..., series or limit. While it is not intuitive, the symbology of decimal fraction digits has come to be intertwined with the definition of real numbers as limits, and so on that basis we say that 0.999... and 1 are symbols for the same number.
Name:
Anonymous2007-10-07 3:21
>>8
It's not what the numbers look like that's the issue. It's what the numbers represent.
Name:
Anonymous2007-10-07 3:26
>>7
In the creation of the universe, matter had in fact traveled faster that the speed of light. The matter had reached it's destination before it's own image. Interestingly enough, it's because of this that the universe was created in 3 min rather than near instantaneously. Time distortion caused weird side effects.
>>9
Yes, but people see what they see, and if they aren't operating under the same idea, or under the formal real number definition, they aren't going to agree because what the symbols look like leads to their idea of what is meant.
Name:
Anonymous2007-10-07 4:17
If they aren't operating under the formal definitions and axioms, they should make up new terminology for their attempts to construct a new system and not confuse everybody by using the same words to mean different things.
When something is understood in general to mean one thing, the onus is on the person using the word to use it correctly, not the person reading it to cover every single possible thing that it could mean if the user was an idiot.
>>12
Hardly anyone learns or gives a shit about formal definitions and axioms, hence the disconnect. When all is said, 0.999... is a shit way to represent the unitary quantity, as far as most people are concerned. Face it, the formal definition of real numbers is not even close to common knowledge.
There's a distinct topological difference between the x-y plane and the x-y plane with the line x=0 removed.
One's connected and the other isn't.
Name:
Anonymous2007-10-08 12:22
All of you, you fail. Asking if 0.999... equals 1 is pointless. There is no answer. Accept it, bitches. It is like asking "why does a spatial point have zero dimensions?"
This has something to do with the geometric sum to infinity.
0.9999999... = 0.9 + 0.09 + 0.009 + ...
r(ratio of geometric progression) = 0.09/0.9 = 0.1
and a(the first term)
Sum of a geometric series approaching infinity is given as S=a/(1-r) if r<1, and S=a/(r-1) if r>1.
Therefore, given that r is less than 1...
S= 0.9/(1-0.1)
= 1
Convergence of numbers is when an total infinite number of sums is a finite number. 0.9 + 0.09 + 0.009 + ... will be 1 as the number of the sum approaches infinity.
Name:
Pasis2007-10-09 11:40
Grah. Correction.
*This has something to do with the geometric sum of a series approaching infinity as well.
*Convergence of numbers is when the total of an infinite number of sums is a finite number. Well, at least that is how I think of it.
First point of contention: 0.99999... is approximately equal to one.
Draw a number line and place exactly where 0.99999... is. If its approximately equal to 1, how close would you place it to the number 1? One may think that the number moves closer and closer to 1 but never reaches there; however this is not the case. Since 0.99999... is a fixed number and not a sequence/series, it will not move on the number line, which brings me to my next point.
Second point of contention: 0.99999... as a geometric sequence/series.
You're not adding (0.9)/10^n where n is 1,2,3... to this number an infinite number of times. That is nonsensical. The number 0.99999... is a concrete and fixed quantity, not a sequence. Since the number extends infinitely, it is EQUAL to its limit, not "approaching" it. The limit in this case happens to be 1.
>>35
ah so, that is the question then, what does 0.999... mean to the reader? (We already know what it means to the fairly advanced student of mathematics, so everybody shut the fuck up for a second.) The reader's meaning will shape the reader's perception. A series or a number?
Name:
Pasis2007-10-10 12:50
I think there will be other perceptions on how to view the subject, one way or another.
Looking at it as a form of a series is one of those views. It is a form of series, anyways. ._.
Huh, whut? On number two. What is with the adding 0.9/10^n where n is a positive integer approaching infinity thingy? Is this what you meant?(but if you want it to work in there, the value of n must be a zero to represent the first term and going towards a positive integer when going up to infinity): 0.999... = 0.9+0.09+0.009+...+0.9/10^n, but the last part that you wanted to add is really not necessary. Anybody with a mathematical inclination can see that it's a fixed series/sequence of numbers without the extra appendage.
0.99999... is not a solid and concrete quantity, in my opinion. Can you write the number down in its entirety? I certainly can't. It's a geometric sum too. A sum that works its way up geometrically in sequence where the number of terms is approaching infinity. 0.999... = 0.9+0.09+0.009+...+0.9/10^n(Really, I don't think it's necessary).
I don't know how to modify post, if there is a function like that in here anyways... so, yes. When the number of the geometric terms(or however you may call it) reaches infinity(which is the limit, as you said), it'll be 1 when summed together(a finite number), not approach 1. My apologies, I was too lazy to clarify myself, and I suck at explaining what I mean. I will get my terminology wrong many times.
Name:
Anonymous2007-10-10 13:10
>>37
If 0.9999... is a series, then so is e, and π, and 1 and every damn number in existence. Just because two things have the same value doesn't mean they can't be differentiated and classified based on their composition. All series aren't fourier series simply because they can be rewritten to be one.
There are no 'consecutive terms' in 0.9999...
Name:
Anonymous2007-10-10 13:31
These "proofs" are all flawed.
1/3 is not the same as 0.333...
0.333... only tends to a limit of 1/3 but will never reach it.
This is where one gets the meaning of sequences and series wrong. Let's just say I give you 0.888, I bet another form to put it is 0.8+0.08+0.008, no? It is a series where the ratio for each consecutive term is 0.1 and the sum of these three terms is 0.888. If I give you 0.929292, I can place it as 0.92+0.0092+0.000092, where the ratio is 0.01. Add them together, you get 0.929292. It is the same as 0.99999999999999.... where I can keep placing it as 0.9+0.09+0.009+0.0009+0.00009+... and henceforth until the number of terms reaches infinity and the geometric ratio of each consecutive term is 0.1.
e and Pi are both constants. They have a fixed value in which both of their values are not in a definite series. 3.1419... has no definite area where one can put a series into it because there is no definite pattern in the constant itself. It is a number as a whole. But I can place an easily definable value like 0.999 as 0.9+0.09+0.009. It is true when you sum the three terms all together, no? This will be true for 0.99999.... as 0.9+0.09+0.009+...
I guess one has yet to learn about geometric progression, yes?
I do not think one can put a decimal point 1 at infinite decimal's end. ._.
Seem logical, but it is wrong in my opinion. One cannot define a point in infinity, therefore no number could be placed there. Maybe an unknown, but not a number.
You can classify the cauchy sequences representing 1 as much as you want, but that doesn't change the fact that 0.999... and 1 are one and the same object. The LIMIT of these sequences.
Whoops. I did not see that it is a sum of a series. My apologies on that. But wait. Yes, it is a sum of a series, but not the sum of a *geometric* series, where the ratio of any two consecutive terms must be constant. Euler's number does not have a definite geometric series. Which is why it generates such a number to begin with.
0.9999... is a geometric series. It has a definite ratio, unlike the two other constants you mentioned(Yes, they are the sum of a series, but they are not geometric in nature).
Try finding a *geometric* series for Pi, if I were to give the number as a whole, there is no definite way one can put it.
That's alright. It sounds like you're foreign, so I don't blame you for not being completely rigorous in putting across what you intended to mean. There isn't a geometric series for pi or e. I'm guessing you don't mean Euler's constant by "Euler's number", but e. Euler's constant we know almost nothing about.
Also, <<One cannot define a point in infinity, therefore no number could be placed there. Maybe an unknown, but not a number.>>
Complex analysis (along with many topological spaces) frequently makes use of a "point at infinity" which helps immeasurably in theoretical proofs and so on. It might not be as intuitive a notion, but it's treated in the same way as other complex numbers.
Well, I guess if people from all over the world could have access to it... Unless I am wrong - only selected countries get to access 4chan. Don't know, man. I just arrived here.
Eh? 4chan seems like an international board to me...
>>69
Well, it was a minor error, and few people cared anyway ... largely since you're such a gaping asshole of a fagswallower.
Still, 0.999... = 1 and there's nothing you can do about it anyway. Algebraically you can arrive at that conclusion, and the concept of limits supports it, and that's good enough for me.
Name:
Anonymous2007-10-13 7:35
c = 0.999...
10c = 9.999...
10c - c = 9.999... - 0.999...
9c = 9
c = 1
If you still don't believe 0.999... = 1, don't worry. It's because of your poor education and not because you're stupid.
Name:
Anonymous2007-10-13 9:09
1 - 0.999999 = 1/10^n -> n -> infinite
Thus (1-0.9999)* 10^n = 1
According to you 0 * 10^n = 1 thus 0=1
AMIRITE?
I'm probably not up to your standard of Science, but surely if photons move at the speed of light then it must be possible for other particles to move at that speed also, given that we are all made of the same fundamental strings.
Name:
Anonymous2007-10-13 17:11
>>78
Photons have zero rest mass. Nothing else has rest mass.
>>82
The problem here is idiots not grasping the concept that 0.999... = 1 has been proven in so many different ways, by so many different people, for so many different years, that we're getting bored trying to convince you. You're like Young-Earth Creationists, and we have better things to do.
Name:
Anonymous2007-10-14 13:51
>>85
Oh, please! You don't seriously believe that the earth is several millions years old, do you? Just think, 40 tons of cosmic dust falls on Earth every day. If the Earth really was that old, it'd be covered in dust now!
All you have to rely on for your spurious claims are highly inaccurate carbon datings, which mistakenly assume that radioactive particles thousands of years ago behaved exactly as they do now.
Sheesh, gullible sheep like you are the reason so many people still believe that we actually landed on the moon.
>>85
It's proven that the limiting value is 1, but no proof will convince everyone that 0.999... actually means or implies the limiting value in the first place, because not everyone has taken a post-secondary course about defining real numbers as limits.
The irritating thing is that people keep confusing 0.999... (meaning the decimal place is followed by infinitely many nines) with some number where the nines stop. 0.999... is just another decimal expression of the number 1. It's just a matter of notation. Yes, if the nines stopped at any point, that number would be less than one. But every number has two expressions for their decimal expansions. (Ex. 3.00000 = 2.99999.....) It's just the nature of our number system.
>>93
no, they aren't confusing that at all. it's that they see the nines as an ordered set that perpetually grows but of course does not finish growing, ever.
>>19
That's a shitty proof, and it's thrown around everywhere.
>>20 >>14 is being rigorous; RedCream was offering handwavy intuition.
>>48
As a general note, instead of all this bullshit about whether or not ".99..." is a series or not, why can't we just define real numbers as equivalence classes of cauchy sequences with the same limit? This definition of real numbers is consistent with the original, and we'd kill half this non sense.
>>52
He makes too many of them for us to care about. In this case, the connectedness of the topology is irrelevant.
>>56
GTFO Apartheid fag. What convinces you that Pasis is from sub Saharan Africa? He sounds Indian... oops, I got pwnt by >>65.
Uhh... what? I do not understand what are you talking about. May I have an example or two please? Ai kanot spikkin in 'saintipik' ingwish. Sou manny beeg n hado wards. Layman terms, if you may. I is a simple village ricepicker. =_=
Eh, racism and its associated topics comes from the human ability to differentiate, in my personal opinion. It simply cannot be avoided or eliminated. Things like this is bound to surface, so I cannot be bothered to get mad about this.
Cauchy sequences are special in some way, though I fail to appreciate its importance. It suffices to say that there is a theorem that says that all convergent sequences of real and rational numbers are cauchy (and vice versa).
I make reference to equivalence classes because that would eliminate all the hassle about ".999..." being a (geometric) series, number, blah blah blah. By establishing all these things as elements of the same equivalence class, there is no ambiguity that all are equivalent to 1.
Thus .999...95, 1.000...1, .999... are all convergent sequences with a limit of 1, and hence are all equal to one.
Seeing differences is normal, but hating is much less so. This is especially true for a /sci/ board. I'm glad you're neutral.
I can barely read/write myself. Sorry for assuming that you could. You admitted that your English wasn't great, so I assumed that you were better at Chinese.
What kind of lag are you referring to? Internet lag?
Name:
Anonymous2007-10-16 2:59
>>102
FYI, Cauchy sequences in rationals are not (necessarily) convergent. Eg, {3, 3.1, 3.14, 3.141, ...} is a Cauchy sequence of rational numbers which does not converge to any rational number. However, every Cauchy sequence of real numbers is convergent, as the real numbers are complete.
Cauchy sequences whose elements are rational numbers converge to something (real numbers).
Cauchy sequences whose elements are real numbers converge to something (real numbers).
>>104
Thank you for pointing out the exact theorem. What I said is not a contradiction, so all is well.
>>106
I fail to see how a country can be a major contributor to lag. Is there a lot of spam coming out of your country?
Ai. I just went in and bathe, and I realized something while I was in there.
I was wrong on the limit of 0.555... /facepalm
Why am I so stupid?
The limit is... 5/9. ._.
Otherwise given as 0.555... again /heh
Sorry, I wasn't really thinking when I saw 107. /swt
This sucks when everything has to be pulled out straight from the head. I should refer to the articles and notes more and think about it first. ._.
Thanks for the responses. I think my point is made. Why is it that people accept 0.555...=5/9, yet they can't apply the same logic to conclude that 0.999...=1/1?
The number {0.000...1} doesn't exist as an infinite expression. The actual expression is {0.000...}, which converges to ZERO. Hence, your stupid fucking formula should be:
what your doing is rounding up, otherwise no it doesnt come out to be one, its a decimal and stays as such
-but then again what would i know all i do is build cc's and repair battle crusiers for deep space flight, theres no mathematics in my job.
Name:
Anonymous2007-10-18 10:16
0.000...1 denotes a finite (but unspecified) string of 0s followed by a 1.
e.g. 0.000....1 might mean 0.00000000000000000000001 or 0.0000000000000000000000000000000000000000000000001
Name:
Anonymous2007-10-18 19:43
funny to think that there is an actual proof out there for this...
It is impossible for any particle to ACCELERATE to the speed of light. Einstein's theory doesn't say anything about particles already travelling at the speed of light.
Name:
Anonymous2007-10-23 6:26
>>130
that's the whole idea of infinite energy: you can't. Except for massless particles, which always travel at the speed of light.
Name:
Anonymous2007-10-23 7:20
What is with the hate? Calling those particles massless?
It does not equal 1, it gets infinitely close though.
0.999... is not a variable, it is fixed. It cannot get infinitely close. It is already infinitely close (i.e. any positive difference is greater than the actual difference => the difference is zero => they are the same) or it is not infinitely close (i.e. different so there is a number in between them).
Name:
Anonymous2007-10-24 9:55
hrm, if we're not being picky, why not say 1=2... I mean, it's pretty close...
If it was >>76 then yes, you can. Arithmetic isn't about working digit by digit. These are numbers, so we consider them as ideas, not physical, actual objects. Therefore, there is no "time" needed to do arithmetic. The equations exist per se - they can be proved true or false at any given moment. These ones are, by the elementary laws of arithmetics, true.
replace x with 5 and do similar calculations, it will prove 5=1, I shit you not... SO, IT MUST BE TRUE...
I think what one group of people is trying to get to is that 0.9999 (repeat infinitely) is for all intents and purposes 1...
that however does not mean that it equals 1.
You see, there's a little thing called the equals sign, or "="...
One of the first things you learn in math is that whatever is on either side of the equals sign has to amount to *exactly* the same as on the other side, or the equation or whatever is just wrong. Plain wrong. Thus, any one single number cannot equal any other number than itself. 1=1... 1!=1.1... same idea...
Now, however, if you argue that because 0.999... is repeated infinitely and there by cannot be treated as a normal number but more like some sort of idea or concept, then yes that may be the case, but THEN you cannot equal a concept or an idea, something that is NOT a normal number, with something that IS a number... that would be like saying: cherry toppings=1...
but then again, this whole thread is just one big troll, but it's a fun troll...
Also, the sum to infinity of a sequence is: S=a/(1-r), where a is the first term and r is the common ratio:
Sequence is: 0.9+0.09+0.009...
a=0.9. r=0.1
S=0.9/(1-0.1)
S=0.9/0.9
S=1
Is three proofs enough?
Name:
Anonymous2007-10-26 13:17
>>149 That's true, but it's caused by imperfectness of the symbols we are using. The numbers in themselves are, as you said, equal.
Name:
Anonymous2007-10-26 15:07
the fact is that this 0.999... (or usually written 0.(9), or some other ways not possible on a keyboard) doesn't actually exist as a number... whenever 0.9 has few enough decimals not to be infinite, it does not equal 1... however, whenever it "steps over the infinite boundary" it is no longer 0.999, it is instead 1... same goes with 1.(1) (becomes 1.2) and any other number you can think of...
Name:
Anonymous2007-10-26 15:18
To 152:
x=1.1111111...
0.1x=0.11111...
0.1x=x-1
0.9x=1
x=10/9
x=1.111111...
So, 'fraid not. It would work with 1.1999999 though.
Name:
Anonymous2007-10-26 17:48
instead of trying to prove it using mathematical formulas, which may or may not be true due to any number of factors in different scenarious, logically reason your way to a conclusion. Math is logical and can be proved logically very easily, but is harder to prove mathematically, because proving something "with itself" is like trying to say "that's the sun, because it's the sun".
It is the fact that when a number "crosses the border" between having a set number of decimals, i.e. being finite, and having an infinite number of decimals, it is no longer the beforementioned number, it instead "becomes" (or more accurately, it already is) the 'next' number in a 'logical series'.
Though truth to tell I am damn tired right now, and I can't really think straight. What I just said might not apply to series of decimals below .(5)... or maybe even below .(9)...
But I don't really care since this is just 4chan and you'll just troll anyway... so if someone wants to pick up where I left, go ahead.
Yawn, different ways of doing the same thing. True, dedekind bypasses the axiom of choice, but cauchy sequences are a lot more...intuitive, in my opinion.
>>160
you are thinking backwards. >>161
on the contrary, I am separating symbols from meaning, something you, as a cultist, can not.
Name:
Anonymous2007-10-27 14:04
OP here: I suck cock
Name:
Anonymous2007-10-27 15:41
>>164 (the real number, not any other sort of number, or anything else)
Yes, you're right, I'm a cultist. I've seen the light. From now on, every time I say, write, or even think about a number, I will append the statement "(the real number, not any other sort of number, or anything else)." No longer will I be so foolish as to believe that any human with a functional brain could interpret "0.999..." on its own to mean "the real number 0.999..., and not any other sort of number, or anything else."
>>166
you moron, you are still confusing what you see with what you believe it to mean.
Name:
Anonymous2007-10-28 1:35
This thread makes me hard.
Name:
Anonymous2007-10-28 2:49
>>169
So to clarify, you really do agree 2/3 != 4/6. Thanks for playing.
Name:
Anonymous2007-10-28 4:16
I have the sneaking suspicion that the same trolls acting stupid are also posting some of the proofs. Leave the trolls and idiots to natural selection and move on. Come on, /sci/ has some interesting content sometimes. This isn't it.