This is an essential assumption of Bayes’s Calculus. If you doubt that this is common, then just take a cursory look at the mathematical community here, here, here, and here. And here and here. Do you know what this means? It means that Calculus, like probability (see my deconstruction of probability), is false. The argument goes something like this:
0.333… is 1/3, right? Well 1/3×3=1. But surely 0.333…x3=0.999…! Therefore, by one or another form of the transitive property, 0.999…=1!
In addition to being a near-blasphemous usage of the transitive property, it is just plain false. Think about it in the following manner. 0.1 is necessarily greater than 0.0X, where ‘X’ is any countable number. 0.1 is also necessarily greater than 0.0XX. And so on. No matter how many X’s you add to the series, it will never equal or be greater to 0.1. Therefore, by mathematical induction a la carte, no amount of repetition of 0.0XXX…. could ever equal 0.1, which is what is necessary to add to 0.9 in order to equal 1. Importantly, (0.9 x / x<0.1)≠1 Λ (0.9 x / x<0.1)<1. Therefore Calculus is false. A house built on sand cannot divide itself.
Notice that all I needed to disprove this foundation of calculus was mathematical induction.
Advice to all my readers: Don’t let “math wizards” intimidate you with technobabble. And note that I am not alone.
1 = 1
and
1/3 = 1/3
if you sum 1/3 +1/3 +1/3 = 1
i dont know where da hell did you learn math !!
A simple proof:
Obviously 1/3+1/3+1/3 = 1; any 4th-grader knows that. But that’s irrelevant — it is inaccurate to refer to 1/3 as 0.33333… because even a penultimate number of 3s after the decimal won’t reach 1/3. Thus the .333…*3 = .999… analogy doesn’t apply.
1/3 = 0.333333 to infinity
that mean
0.333 to infinity + 0.333 to infinity + 0.333 to infinity = 1
sample and stupid xD
Not only has the concept of “Infinity” been dealt with in a previous post, but the equivalence – without reference to hypofinite math! – has been refuted in my argument above. In fact, your comment there is just a restatement of precisely what I refuted.
A standard proof that 0.99… = 1
Let 0.99… be replaced by a variable:
x = 0.99…
Multiply both sides by 100:
100x = 99.99…
Subtract x from both sides:
100x – x = 99.99… – 0.99…
100x – x = 99
Factor and simplify:
(100 – 1)x = 99
99x = 99
Divide both sides by 99:
x = 1
Since x = 0.99… and x = 1, then 0.99… = 1
QED
Mathematics is a means to discover and invent ways of finding systematic patterns. It does not have to have any correspondence to reality. The fact that it so often does was referred to by Eugene Wigner in his paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” link here: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
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density Theorem in real analysis says
if x and y are real number with x<y, then exits a rational number r such that x<r<y
so, it’s clear there is no rational number between 1 and 0.99999999…, it’s mean 1 and 0.99999999… are the same number.
Gentle Molly Toff,
Nice try. Instead of proving that 0.999…=1, you have instead proven the age-old thesis that women are deceivers. Do you think I am so foolish as to allow in a question-begging assumption such as “99x = 99″? I am not.
Furthermore, you state that “[Math] does not have to have any correspondence to reality.” No further comment should be necessary.
And Eugene Wigner is not a credible source since he has a vested financial interest in (contemporary) mathematics being a legitimate field. On the economics of writing, please see the famous tome by intellectual Charles Beard, “An Economic Interpretation.”
Sweet Aria Turns,
Your post is very challenging. My first instinct is to say, “So much the worse for Chebotarev and his co-conspirators.” After all, the following is not a valid argument form:
1) Noted Scholar’s thesis says X
2) X is inconvenient for math
__________________________
3) X is false
However, because of the intuitive appeal of the Density Theorem, which you so parsimoniously explicate (see Molly’s egregious subterfuge above), I am tempted to try and salvage it. Unfortunately I am in no position to do so now, as I have piles of research yet to do. But I will point out this: The traditional approach of calculus, that 0.999…=1, fairs no better under Density Theorem. After all, there is some number (so the story goes) before 0.999…., but I ask you: What comes between *that* number and 1? Quod errata, my dear, quod errata.
Induction does not prove your point. Induction proves that your fact is true for any specific number of digits X that you name–that’s different from proving it true for infinitely many X’s. Furthermore, the wikipedia section you site is not a set of actual, mathematical counterarguments, it’s a set of observed counterarguments that are easily refutable, but are cited as demonstrations that students (like you, apparently) often cannot understand why .9 repeating equals 1.
You are under no obligation to believe the centuries of mathematical theory that claims that .9 repeating equals 1. However, you are under a moral obligation to do so with a little humility, which ought to make you think twice about saying that those centuries of mathematical theory by some of the smartest minds in history completely missed a simple induction argument. It’s not that they missed it; it’s that it’s incorrect.
Chebotarev’s density theorem is a deep theorem from number theory and has nothing to do with Aria Turns’s ‘density’ theorem. It appears that you typed ‘density theorem’ into Google or Wikipedia and name-dropped the first item that you saw without inquiry into its relevance to this farcical debate. This action should speak for itself.
Ms Turns was referring to the fact that the rationals are topologically dense in the reals – that is, they ‘get everywhere’ and if you paint all of them black it will look as if you have painted the whole number line – and I don’t know a construction of the reals that doesn’t yield this as an immediate property.
I ask you this question: Suppose 0.9999… is different from 1. What, then, is half of 1.99999… ?
oh hang on. sorry mate, I repent. You’re absolutely right.
Dreamer,
Thank you for your concession.
How come you didn’t give your comments on Todd and Vishal’s blog about your conclusion being misleading since Inf is not an element within the set of all Natural numbers?
oh by the way, in terms of the words you used in this entry, let me rephrase it: Infinity is NOT a countable number. X can only be a countable number as you said.
Dear hi second-rate dumbass,
Insofar as I can even discern what you are trying to say through the drivel, it is still not substantial. Please take your negative attitude elsewhere.
I wandered here from another blog and saw your math, and I had to comment…
Your argument is this, in your own words:
(Premise) 0.1 is necessarily greater than 0.0X. 0.1 is also necessarily greater than 0.0XX. And so on. No matter how many X’s you add to the series, it will never equal or be greater to 0.1.
(Conclusion) Therefore, no amount of repetition of 0.0XXX…. could ever equal 0.1.
Note that your conclusion is identical to your premise.
And your premise is false, because 0.0999… is equal to 0.1, by the same mathematics used by Molly Toff above.
If you disagree, then please show exactly where Molly Toff has erred in her math.
Are we sure this whole site is not a spoof?
Hazel,
Just because you don’t agree with something, doesn’t mean it’s not even serious. That’s pretty arrogant if you ask me.
And anyway, I’m not the only one who has doubts about this particular part of calculus. Some of my other posts are way more controversial!!
That’s my point. I don’t see anything on your site that is reasonable. For instance your post on the fact that the speed of light is constant shows virtually no understanding of the experimental work that has verified Einstein. Your arguments are simple and uneducated. Furthermore, all the proofs people have shown here that 0.99.. = 1 are solid. The fact that you think you are bringing down such basic math and science seems so sophomoric that my first thought was that your site is a spoof. If it’s not it seems to me you should concentrate on your education more.
Hazel,
Are you running for President of Patronizing?
These are blog posts. You can’t expect long tracts like Principia Mathematica. These arguments will develop over time. See: Kuhn’s Structure of Scientific Revolutions. Furthermore, the “proofs” you refer to are frequently disputed on this very thread.
notedscholar: Your dispute of Molly Toff’s proof consisted of, “You have proven the age-old thesis that women are deceivers. Do you think I am so foolish as to allow in a question-begging assumption such as “99x = 99″? I am not.” I concur with Hazel. I call shenanigans.
Hazel: I think I love you.
Kudos for standing up to the mob over on UD.
There is no meaningful difference between 0.99999 with infinite repeating 9s and 1. What don’t you get about that? I think you use your calculator too much and your brain too little.
I’ve decided notedscholar needs tagging with this meme
Is this blog a parody, or is its author seriously espousing crank mathematics?
For those of you convinced that .999… is not the same as 1, in spite of the numerous mathematical proofs of said fact: if that’s the case, then what is the number halfway between the two?
Dear Digdug:
Thanks but no thanks for addressing the least important part of my response.
NS
Brad D:
I have never in my life been accused of using a CALCULATOR. You need to check your assumptions my friend.
NS
Dear Davis:
I call false dichotomy. It’s only “crank mathematics” if you presuppose against it. As for “parody,” a few people have responded to me with words like that. As I understand parody, maybe I sometimes do it (like having my name be notedscholar – a parody of academia). Similarly my blog’s subtitle is humorous. BUT if you mean my fundamental opinions, then it’s not parody. It’s polemics, and it’s also minority opinion (which leads some people to say “troll”, and idiotic term), but not parody.
Okay. First of all it’s a nonsense question. That’s like asking “where is the fifth corner of a square?” But I’ll humor you. Your very question can be turned around on traditional mathematics. Observe:
What is the difference between (0.999…-0.000…1) and 1? The same supposed “paradox” applies. Does it bother anyone? No, it doesn’t bother anyone.
But in any case – and I’ve already answered this over at Topological Musings (http://topologicalmusings.wordpress.com/2008/11/12/using-mathematical-induction-to-disprove-the-foundation-of-calculus/#comments) – the answer to your question can be given symbolically. The difference between the two is precisely 0.000…1, where the notation “…” indicates Penultimate (P) decimal places. If you need background on P, see my much earlier post on Infinity.
Hope this clears things up.
NS
Dear notedscholar:
Thank you for not addressing my response at all.
I ask again: Where, exactly, has Molly Toff erred in her math?
Dear Digdug:
Gentle Molly Toff erred on three logical accounts, one of which was mathematical in nature. You can feel free to reread my response to her. Until then, it’s been fun.
NS
I’m 99.999999999999999% sure this guy is a fake. On the off chance that I’m wrong, he’s the dumbest mf’er that ever lived.
@ Notedscholar
It might not make much sense to you that there should be a number bewteen every two distinct real numbers, but is is actually one of the axioms for the real numbers (in most axiomatisations). This is one of the deeper properties of real numbers. It makes my head spin just thinking about how many real numbers there are. I just can’t wait until next semester when I will take a nice real analysis course and get to understand how they really are constructed.
See axiom 2 of: http://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals
or http://en.wikipedia.org/wiki/Dedekind-complete for more information.
As for your counter argument, 0.000…1 is not a real number. It is one of the characteristic things about infinite sequences, that they actually are infinite, and has no end. Thus there cannot be a 1 “after” all the others.
To be a bit more precise there is exactly one decimal in every real number, for every natural number (ofcourse infinitly many might be 0). For instance the 4th decimal of pi is 1 as (pi = 3.1415…). So in 0.00…1 what natural number gives you 1? The answer is that none of them do. So 1 could not be a part of the decimal expansion. So your counter argument falls short on the fact that is is not talking about real numbers. Perhaps you could fit it into the theory of hyperreals, but I doubt it.
Peace!
Dear Epsilon,
I try not to delete other people’s comments, because I support discussion.
But try a little less of the sarcatic ad hominem and a little more of addressing actual arguments!
NS
Dear “Logician”,
I doubt you are really a logician. I’ve known several logicians!
Simply appealing to “axioms” doesn’t help. We can make just anything an axiom. And citing old dead people for the already subjective judgment doesn’t help your case.
You say,
But, this results in paradoxes about infinity itself. What is between the Penultimate and Infinity? Even worse, what is between 0.999… and 1? What is between the last “infinite ordinal number” and one? At some point, real numbers stop (I have called these “countable numbers”), and fake numbers begin. You know, there’s a reason we have the term “irrational numbers.” But the sense of these terms has obviously been lost on some people.
NS
In your response to Davis (on November 26), you say that asking what is halfway between .999… and 1 is a nonsense question.
So, let’s put it differently: what then is (.999… + 1)/2? If you still consider the question nonsense, then either you believe that some pairs of numbers in your system cannot be added, or that some numbers cannot be divided by 2.
Dear Todd Trimble,
First, I would like to paraphrase David Lewis: Nonsense is nonsense even when you talk it about mathematics.
To address your point, it is clear that 0.999…+1 = 1.999… where “…” means “penultimate decimal places.” Now keep in mind that the “Penultimate” is axiomatically the last number in a series. This axiom does not preclude there being – for the purposes of representation only – “more” decimal places in order to produce half of 0.999… than there are places in 0.999… itself. However, this difference is clearly illusory, since “more decimal places” very clearly does not necessarily yield larger numbers. But what is of mathematical interest in my theory is that 0.999… could not have decimal places added to it. So the formulation 0.999… + 0.000…1, for example, is nonsensical, because there is no decimal place after P decimal places. Of course to represent the division you ask for we may “pack in” more decimal places before we hit P decimal places, but this is surely a limitation on mathematical notation and not my theory.
NS
(1) In a comment on my blog, I asked you what is the difference between 1 and .999… and you responded, “.000…1″. I took that as your asserting that
1 – .999… = .000…1
(whatever that might mean exactly). Now you tell me that the “formulation”
.999… + .000…1
is nonsensical. Why would the first equation be sensible but this formulation not? What prevents you from saying it equals 1, according to the first equation?
(2) “Of course to represent the division you ask for we may “pack in” more decimal places before we hit P decimal places, but this is surely a limitation on mathematical notation and not my theory.”
I asked you a simple question, which you avoided answering directly. What in “your theory” is 1.999…/2? Vague references to “packing in” more decimal places doesn’t cut it.
In order to have a mathematical theory, you need to have a precise language in which to express it. So “limitation on mathematical notation” necessarily means a limitation of “your theory” — how are we to make sense of your theory if you can’t or won’t supply adequate notation in which to express it?
“First, I would like to paraphrase David Lewis: Nonsense is nonsense even when you talk it about mathematics.”
You said it!
By your logic, a person could never cross a room, because he would first have to traverse nine-tenths of the distance across the room (0.9), then nine-tenths of the distance that remains (0.09), and so on.
Dear Todd,
I’m afraid you’re losing me – and I can’t help suspect that you’re trying to.
I’m also afraid that you and I, in our cross-discussions across blog, have been equivocating on the decimal notation. This should clear up some of your troubles. I may have assumed traditional notation once or twice on your blog. Some signals got crossed.
“Packing in” is not a vague reference. What I mean is that the decimal places are more or less illusory since they do not add to the size of the sum, and still “fit” into the larger number. Hope this is clear.
This is just snobbish. Are you telling me that Newton had to have already formed his theory before forming his theory? Ludicrous. In any case, Wittgenstein showed that there can be such a thing as “wordless” knowledge and experience.
NS
Dear A.A.,
Nice try, but I also took intro philosophy class. You’re not using my logic. You’re using Zeno’s logic.
Snap.
NS
Nicely spotted. Now refute what I said.
No, not trying to lose you; just trying to get straight answers to the straightforward questions I asked above in (1) and (2). I’m still waiting.
No, it’s not. But it might help if you would just answer the concrete questions posed in (1) and (2).
And better still if you would not use insert made-up stuff between the blockquote tags: I never said
and you know it. To use that as the basis of an ad hominem attack against me (calling me “snobbish”) is dishonest, to say the least.
As you know, NS, Newton was not out to defeat math and science. The burden of proof that science and math are defeated rests on you, and I’d advise you to get used to the fact that any such “proofs” had better be precise. Otherwise they convince no one.
Poe’s Law
You idiots. Think about it for one second. 1-0.1=.9, yes? extend the number of decimal places to infinite, and the difference between two becomes zero. Therefore they are equal. That is the short greatly abridged answer, you are concerned to greatly with the decimal representation of a rational number, which is actually an approximation of 0.999…
The person who stated that the example of 3*1/3=1 does not prove this has made this mistake. 1/3=/= 0.333 it equals 0.333… if one can appreciate the difference. An irrational number cannot be completely described in a decimal and hence trying to approach this problem from that angle is doomed to a conclusion as misguided as the approach that formed it.
http://en.wikipedia.org/wiki/Geometric_series#Sum
Dear Noam,
Thank you for PROVING THE POINT.
This isn’t, by any chance, Noam Chomsky is it? Because he wouldn’t be so careless with his citations.
NS
I stumbled upon this today and I just want to say something:
If we assume that 1 is an absolute value that is nothing else but 1, then nothing else can be 1, than what already is.
You see two apples. They are identical. They = 1
You bring in a third apple that is as close to identical to these two as possible, with the most meaningless smallest possible thing left out. You could say it was identical and none would really care since it made no difference in the world and it could be used as it was…
Except the knowledge that there is a flaw there, makes it not the same.
It is totally irrelevant to anyone in any possible situation in real life, whether it was exactly the same, or the closest to the same as anything can possibly be. But the truth is that if it’s not the same, it’s not the same. And why even bother trying to argue that something what is not something, is something, when it’s clearly obvious that it cannot be.
You don’t need fancy equations to understand this. You just need to admit to yourself, that if two things are different, they are not the same. Even if it made no difference in the world, it’s obsolete to claim so.
I know someone will pop up and say: “Well if it makes no difference in the world, then it is the same, as there is nothing that the other apple can that the other cannot”
The only thing they cannot do, is be the same.
The problem with this lies in semantics. And if you claim that two different things are the exact same, then you could just say that everything is the same at the same time. Because it just takes the ground from under the possibility of actually handling matters as individuals with differences. And it also makes it obsolete, since if everything is the same, there is no point in trying to differentiate things from one another.
I took this too far and almost confused myself to insanity, but prove me wrong and I’ll give you a cookie!
Consider this:
In regards to .999… (point nine recurring), I hope we can say that .999… > 1 is a false statement without much argument. Therefore, the converse must be true: .999… 1. No matter how many zeros we put in between the decimal and our nonzero digit, .999… will always have more nines beyond it. If A plus any positive P is greater than B, we must conclude that A cannot be less than B. Thus, .999… cannot be less than one. .999… >= 1.
We’ve already established that .999… cannot be greater than 1, and since we now know that it cannot be less than 1, that leaves only one option:
.999… = 1
Consider this:
In regards to .999… (point nine recurring), I hope we can say that .999… > 1 is a false statement without much argument. Therefore, the converse must be true: .999… <= 1.
For any positive P, .999… + P > 1. No matter how many zeros we put in between the decimal and our nonzero digit, .999… will always have more nines beyond it. If A plus any positive P is greater than B, we must conclude that A cannot be less than B. Thus, .999… cannot be less than one. .999… >= 1.
We’ve already established that .999… cannot be greater than 1, and since we now know that it cannot be less than 1, that leaves only one option:
.999… = 1
So , if 0.(9)=1 can you tell me why…
1.0000..-0.7777…=0.33333…,but 0.9999…-0.7777…=0.22222..
0.33333….0.22222…
sorry for the mistake
1.00…-0.7777..=0.2222..223
0.2222..2223 is not 0.22222…
.777…=7/9
1-(7/9)=2/9=.222…, NOT .222….222…..222…..2223!!!
Which is an example of why people shouldn’t try to subtract INFINTELY REPEATING decimals without converting them to their fractional form first. Also, calculators round and should not be taken literally.
Funny, as one of the top contributors on the early inquiry site WHquestion, I held the same position—thas ·’9′ ≠ 1|1·’0′. Thus their differend is ·’0’1, a quasicomplex—true, it’s not mootly “real”—infinitesimal or differential, the iota in surreal algebra. Its half is either indefinite, or ·’0’05—it’s all arbitrary.
My dear Math Chick,
You make an interesting point, that calculators should not be taken literally. I wonder if electrical engineers would agree with you? In what way do calculators produce metaphors? Compare the sentence “.777…=7/9″ with the sentence “Life is a house.” What can we learn from this? Do you really wish to insult the achievements of engineering in this fashion?
NS
A just, clear, and excellent response, sweet Autumn D.C.
I frankly have nothing to add.
NS
notedscholar:
Do you know what Cauchy sequences are? Defining real numbers in terms of equivalence classes of Cauchy sequences of rational numbers makes this equality really, really simple to understand. Read a book on analysis to learn more. Good luck!
I guess I’m beating a dead horse here, but let’s follow our dead Noam’s link a bit further: http://en.wikipedia.org/wiki/Geometric_series#Repeating_decimals
0.(9) = 9/10 + 9/100 + 9/1000 + 9/10.000 + [...]
0.(9) = a/(1-r) = (9/10)/(1-1/10) = 9/9 = 1
0.(9) = 1
QED
Further info here: http://en.wikipedia.org/wiki/0.9_recurring
PD: Extra points to whoever figures what does ⑨ mean.
Str,
I don’t know what str stands for, or what you are trying to say in your comment. If Professor Cauchy has made “this” easy to understand, great! But ease of understanding does not yield truth.
Cheers!
NS
El Suscriptor Justiciero,
First of all, Noam Chomsky is not dead. Are you confusing him with his wife? She recently passed away, which is very sad.
Second, I agree that the link proved the point, so with whom are you arguing?
NS
Oops, typo. I meant “our dear Noam”.
But that would make nonsense out of your horse metaphor.
1/3 is not equal to 0.3333…. in your world. And I think you know why. You stipulate that you do not believe infinity to be an actual concept. You are satisfied that you have disproved the concept of infinity. Unfortunately to say that 1/3 = 0.3333…. you have to be willing to say that 1/3 = 0.3333(followed by an INFINITE number of 3s). As a scholar you should know, that you can’t have it both ways, either infinity exists or it doesn’t. And as long as it doesn’t exist you simply need to refer to 1/3 as one third. And I don’t think you’ll disagree that 1/3 + 1/3 + 1/3 equals 1.
It is completely incorrect to say that 1/3 = 0.33333… with a FINITE number of 3′s following. If you cannot see that I strongly encourage you to sit down and do this problem, long division style. 1 divided by 3. Now when you’re done come back and give us your answer. Ohhhhhh wait, you’d die before you finish, but yeah, that’s totally not infinite.
Good day sir.
Next to Last:
Noted Sholar is like those modern geocentrists who say “Relativity means that any given object may be considered the center of the universe. Therefore geocentrism isn’t wrong. Therefore the Earth is the center of the universe, everything orbits around it, and Galileo and relativity are wrong.”
Noted Scholar (I misspelled last time) said:
That’s not a “question-begging assumption”, it follows from the premises. “X” doesn’t even have to come into it, if you don’t like it. It is obviously true that if
(100 * .999…) = 99.999…
then
(100 * .999…) – (1 * .999…) = 99.99… – (1 * .999…)
and therefore (here comes the factoring on the left side, is this the part that’s actually disputed?):
(100 – 1) * .999… = 99.99… – (1 * .999…)
or 99 * .999… = 99
QED
Another error I feel the need to correct. plamen said:
The result “1.00…-0.7777..=0.2222..223″ is wrong. I’m guessing it was mistakenly derived from induction, probably with a calculator.
It’s true that 1-.777 = .223, and that however many (finite) 7s you put in there, you will get a bunch of 2s followed by a 3. But it’s still not the case that 1-.(7) contains an infinite number of 2s “followed by a 3″; it’s simply an infinite number of 2s, or, more simply, two-ninths.
Dear Notedscholar,
In what field are you noted, and whom has noted you, precisely?
If you are so noted, why are you not using your real name and degree and in which field is this degree?
As an aside note, degrees in theological studies are not relevant to making one a scholar in mathematics, and so make you nothing more than an ignorant fool in what you try to disprove.
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Your proof attempts to use the inductive property, but that can only apply to finite numbers. An arbitrarily large finite number of 9s following 0.0 will always be less than 0.1, but that isn’t the same as an infinite number of 9s. Technically, an infinite number of 0s is a theoretical construct which would be analogous to the limit of the sum of X*10^-n from 1 to n as n approaches infinity. Your inductive argument shows that any finite sum will be less than 1, but that says nothing about the limit.
Consider 1/x where x>0. By the archimedian property, there is always a number between 1/x and 0 (for instance, 1/2x). What you’re essentially arguing is that the limit of 1/x as x approaches infinity isn’t 0 because any finite value of x will yield a 1/x greater than 0. It’s a stupid claim, and betrays a lack of understanding of what a limit is, and of calculus in general.
“At some point, real numbers stop (I have called these “countable numbers”), and fake numbers begin. You know, there’s a reason we have the term “irrational numbers.” ”
Do you have a similarly fun description of, say, imaginary numbers? Or hyperreals? I’d love to see those!
One of the best jokes I’ve heard in a loooong time. Thank you!f
Next to Last,
I am sorry that you are disappointed! Hopefully you can be more optimistic and satisfied now.
Okay, I see what you are trying to do. You are trying to link my critique of calculus to my denial of infinity. Nice try. But you know, even though my refutation of infinity corroborates my critique of calculus, it is independent. But I guess you are right, that the convergence of evidence supports my position!
Also, from “you will die before you finish” it does not follow that “the task is infinite.” This inference is completely squirrilous.
Yours,
NS
Lenoxus,
I was not aware that geocentrists make this argument.
Thanks,
NS
Lenoxus,
You write, “(100 * .999…) – (1 * .999…)”. But one of the paradoxes of infinity is that you cannot subtract one from the other. For example, Hilbert’s Hotel.
Best wishes,
NS
Dear Sane Individual,
I am noted in the sciences and maths [sick].
I do use my real name in social networks, but I don’t want to associate with my home institution… after all it has been bad for Behe who has done so.
Thank you,
NS
Dear Greg,
First, despite your claim that I have stupidity, thank you for at least admitting that my argument proves something. Now, you make the same mistake as Lenoxus above, by assuming that infinity does not involve paradoxes. Yet you also admit that your view is nothing more than a “construct,” like gender or race. How can you hope to refute my argument based on this?
As for calculus, I am criticizing calculus! As for limits, I of course am pulling the limit back to the penultimate in general, as you can see in my famous post on negative numbers.
Next time,
NS
Persilja,
You are correct that imaginary numbers and irrational numbers are a joke! In fact, that is why they were invented! I think I have a post on imaginary numbers that you can find. Try Google!
bye,
NS
Wow… just wow.
I refuse to believe this is a real argument you’re making. This, and every other post you’ve made, is riddled with mistakes and mathematical face-palms.
But congrats, you just got a bunch of pageviews out of me
Hi Brett. Welcome to the blog. I’m confused about which argument you have in mind. Like many of my posts, this one has generated a long stream of serious discussion. Also, I do not know what a “face-palm” is (at least not in this context).
For what it’s worth, thank you for the pageviews…
Cheers,
NS
The proof that mathematicians use is x=0.999… and 10x=9.999… However, if you replace 10x with ANY OTHER NUMBER times x, x is still equal to 0.999… and not 1. The proof only works for that specific situation, not the entire set. x=0.999…=1 is an extraneous answer.
the actual proof goes
0.999…=x
10x=9.999…
10x-x (9.999…-0.999….)=9, therefore 9x=9;
Simplify 9x=9; x=1
Also, in response to what you said, you can’t just separate it. The fact is that in that case, it=1, but it otherwise it=0.999. Therefore, they are equal. Just because two different methods of thinking lead to two different answers doesn’t make one wrong. It just makes them equal. For instance, 1/2 is equal to one half, but it’s also 10 tenths divided by two, which is 0.5. That doesn’t mean one is wrong. It means they’re equal.
Congratulations for rediscovering hyperreal numbers! Unfortunately, they are already well-known.
Hint: your “countable numbers” are what are commonly known as “real numbers”, and your “real numbers” are in fact the hyperreals. Hyperreals include reals, positive and negative infinity (what you call Penultimate, or P), as well as infinitesimals (which you usually write as “0.000…1″ – the common notation is “0.(0);1(0)”). The full notation is a.b(c);d(e), which corresponds to a.b, followed by infinitely many repetitions of c, followed by d, followed by infinitely many repetitions of e. In your notation, that would be “a.bc…de…”.
You are perfectly correct that 0.(9);(0) != 1.(0);(0) using hyperreals. However, 0.(9);(9) = 1.(0);(0).
0.999… is equal to the limit of 1-1/10^n as n increases withouth bound. When n is raised without bound, 1/10^n approaches 0, and thus the limit, and 0.999… are equal to 1!!!
The string of nines is infinite. I take the string of nines, let’s call it string 1, and duplicate it to String 2. I can correspond the nth 9 in String 1 with the (n+1)th 9 in String 2, giving us a one-to-one correspondence, but we fill the first slot with a 9. Now we multiply String 2 by 10, subtract String 1 from String 2, and everything cancels out, giving us 9x=9, thus x=1.
“a near-blaphemous use of the transitive propertey”
When applied to numbers, a mathmatical property is always true.There is no way to use it ‘blaphemously’ , only incorrectly. However, since all the steps in this chain are true, the entire path is true.
Noted Scholar wrote:
First of all, .9… is not an infinite quantity, so this objection doesn’t really apply. You don’t have a basis for arguing that .9… is a number which somehow cannot be subtracted from itself.
Secondly, you apparently accept that the “rules” can be different when it comes to infinite sequences. Well, your original argument against the equality was inductive: 1 is greater than .9, .99, and so forth. This argument takes for granted that the same principle must apply even if the number of nines written down is infinite. But of course, the “rules” are different for infinities, in ways that are entirely self-consistent. For example, as the number of nines in the sequence increases, the difference between the written number and 1 becomes smaller and smaller. When the number of nines is infinite, the difference is “infinitely small”, which is necessarily the same thing as 0 in our number system.
0.333… =/= 1/3
/thread
You know, I have a similar problem with limits and calculus. It’s one thing to toy with the theory, but in physical application, I dunno, it seems problem ridden. For example, the idea of “instantaneous velocity”. O, we just ram a limit in there, as delta-t approaches zero, call it calculus, and claim that we have an equality. But we don’t REALLY. Because OBVIOUSLY if there is no duration of time, stuff doesn’t move, an therefore, can not have a velocity. It has always really bothered me that we use limits in this way, to tease out a quantity that we can use for further calculation and experimental verification, and we just sort of ignore the fact that what we are claiming is, actually, physically impossible. I have brought this up with one of my profs, and was sort of patronized about it, so I just keep it to myself now. But I feel like, I dunno, we are sort of missing some profound, and we keep passing over the clue to the revelation because it is inconvenient.
the simplest proof i’ve ever seen.
0.999…+0.999…-0.999…=1.999…-0.999…=1
Unnoticed,
Your theory OBVIOUSLY builds in the loaded notions of “limit” and “bound.” But these notions, as any student of maths knows, are intertwined with the very notions under dispute.
Pedagogically,
NS
John Doe,
This is completely insane. If the string of nines is infinite, then you can’t duplicate it. Let’s keep in mind that I am working in a non-Kantian framework here, so I reject the notion of multiple infinities.
NS
Thomegemcity,
You write, “When applied to numbers, a mathmatical property is always true.”
I can’t understand this sentences. Properties are not true or false, they are had or un-had.
Please come again,
NS
Lenoxus,
Thank you for your stimulating comment. It is better than many other comments here.
I’m not sure I’ve accepted that the rules can differ for infinity, since I argue there is no such thing! There can’t be rules for, e.g., unicorns. I say there is a “paradox” of infinity, but that is a paradox for my enemy, not for me.
As for the “infinitely small,” I think this is really just an argument for infinitesimals, which are not the same thing as zero (notwithstanding absurd formalisms to the contrary).
Friendship,
NS
.1 > .0X for any number X;
We’ll take X = 9 because that’s the most we can fit into a single digit.
.1 is more than .09 by .01; and greater than .0XX (with n X’s) by .001 (with n 0′s)
With an infinite number of 9′s, the 1 at the end of the amount that .1 is greater by is endlessly far from the decimal; we can’t get it there by adding digits one at a time. (Because you can’t count to infinity.)
This means that if we start with an endless number of 9′s (and therefore a difference between .1 and .09999… of 0.000…00001) Then the 1 has to be at the end of the endless decimal; such a thing simply doesn’t exist. There stops being a 1 at the end as soon as there stops being an end, which is why it becomes irretrievable. The difference between .1 and .0999… with an endless sequence of 9′s is 0.
Greg,
I can’t focus on this right now, because it’s too late where I am. I will reply soon. If I forget, please repost it.
Best,
NS
1-0.(9)=0.(0)1?
The parentheses indicate that there is no end. If there is no end, where does the one go?