So I know infinitesimility is suppose to = 0, but I was thinking, what does infinitesimility * infinity equal? I mean, I would think it would have to equal 1, which argues infinitesimility = 0.
What do you guys think?
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So I know infinitesimility is suppose to = 0, but I was thinking, what does infinitesimility * infinity equal? I mean, I would think it would have to equal 1, which argues infinitesimility = 0.
What do you guys think?
How would anything equal multiplied by 0 manage to be equal to 1?Quote:
I mean, I would think it would have to equal 1, which argues infinitesimility = 0.
The way you've set that up sounds wrong.
Soma
I seriously doubt that infinitesimal is considered anything resembling 0. While math is not my area, I think there is a huge difference between something infinitely small and nothing at all and this difference somehow has been expressed in maths. What I remember learning is that an infinitesimal is a quantity approaching zero but not reaching it. It's distance to zero is infinitesimally small.
Infinite quantities may be applied to maths, but you already realize that rules are different for them when for instance we learn that any operation on infinity returns infinity. So, you shouldn't just assume good old algebra is the way to solve your question. 0 * 1 is not what you should be looking for. Infinite quantities aren't numbers at all.
Instead here's three possibilities:
- If we were to represent infinitesimals differently than infinitely large quantities, it would probably make sense to establish a rule based on the position of them in the equation. That is to say an infinitesimally small number multiplied infinite times is still an infinitesimal number, while an infinitely large quantity multiplied by an infinitesimal amount is still an infinitely large quantity. This holds true for all operations.
- We can simply establish that infinitesimal quantities are infinite quantities in fact. Which they are. There's no direction for infinity. So an infinitesimally small quantity is an infinite quantity. Infinite quantities aren't big or small. They are simply infinite.
- Or we can remember that infinitesimal is not the same as 0, as such -- and because any operation on infinity results in infinity -- the answer to your question is infinity.
If you look closely, all the three possibilities are the same. :)
Infinitesimals don't equal 0. If they did, all of calculus would be nonexistent.
The problem is that infinity doesn't exist in the real numbers, so it doesn't apply to the same rules as real numbers. The only way of dealing with infinities, without using the extended reals, is to use infinite limits. You're conjecture that 1/∞ * ∞ = ∞/∞ = 1 can be disproven by showing that both limx->∞ 1/x = 0 and limx->∞ 1/x^2 = 0 and that limx->∞ x = ∞, then considering both limx->∞ 1/x * limx->∞ x = limx->∞ x/x = 1 and limx->∞ 1/x^2 * limx->∞ x = limx->∞ x/x^2 = limx->∞ 1/x = 0, to see that there are different possible answers depending on the infinity in question. So, in other words, depending on the "size" of the infinity, you get different results. Ratios of infinities, and a few other special cases like these are called indeterminate forms. There are a few good calculus resources out there, if you want to learn this type of stuff. I particularly like MIT's OCW Calculus, with Wikipedia to fill in the gaps. Unfortunately, they spend absolutely no time learning limits.
I suggest you learn limits. They can be a fun for people who like algebra. You finally get to *almost* divide by 0! :D
Oh, it wouldn't, I'm trying to imply that maybe infinitesimal != 0.
Yes this is as far as I know too, but I found some compelling arguments showing that because infinitesimal is so small, that is, if one were to search for it's 'end', they could never find it, thus making it nothing.
Well, I can't find the pages with the proofs anymore. But here's a good reference nonetheless: 0.999... - Wikipedia, the free encyclopedia...
Now, I hope my assuming that saying that 0.(999) = 1 is the same as saying that 0.(000)1 = 0, isn't wrong :)
True, and in real life calculations, they're not used. To me, this seems to lead to holes in the mathimatical system, though, for example the infamous divide by 0 problem. The idea is too see how infinity and numbers would co-exist in the same mathimatical system. Formal math seems to say they don't/can't.
We know that infinity is bigger than 1, and that infinitesimal is smaller than 1, so, when a difference in relationship to tangible real numbers can be shown, doesn't that prove that they aren't the same thing? On other words: Surely something can't be smaller and bigger than something else at the same time ;)
The problem with this is the assumption that any operation on infinity results in infinity, for example, 0 * infinity, is of course, 0.
Just as we have 1000, and 0.001, we have infinity, that is, 1(00)0, and infinitesimal, that is, 0.(00)1, I purposed 1 as an answer because if 1000 * 0.001 = 1, then why not 1(00)0 * 0.(00)1 = 1?
This breaks the rules, though.
No, 1 IS .999... and .999... IS 1, just as 1/3 IS 2/6 and 2/6 IS 1/3. They are two representations of the exact same number. That's why they are the same, not because the nines never end. In terms of math: 1-.999... = 0 not .(000)1.
They exist, you've just not been introduced to the formal math for dealing with infinities and real numbers. Extended real number line - Wikipedia, the free encyclopedia :p
Do you mean 0 or 1/∞? If the later, then you need to reread my last post. :)
We don't. Infinity isn't measurable. There may be a need to express infinity in two directions in mathematics. But it's probably because we shouldn't, that theorists have so much trouble defining what mathematical rules they will obey.Quote:
We know that infinity is bigger than 1, and that infinitesimal is smaller than 1
We can surely prove, by logic alone, that an infinitesimal is infinite. What you should probably be rid of is the notion that infinity is only something that is bigger than something else.Quote:
so, when a difference in relationship to tangible real numbers can be shown, doesn't that prove that they aren't the same thing? On other words: Surely something can't be smaller and bigger than something else at the same time
Oh, no it's not :)Quote:
The problem with this is the assumption that any operation on infinity results in infinity, for example, 0 * infinity, is of course, 0.
As I said before, you cannot think in algebra terms. I'd say it is infinite. But a search on google reveals I would be wrong; It's in fact undetermined. Which in my own view means we don't have a proper mathematical model to deal with infinity. In any case, no. It's not zero. Again infinity is not a number, it's no longer even a measurable concept. The fact we try to give it mathematical properties is probably something someone one day will give a condescending smile at, for our brave but futile attempts.
Indeterminate. See above. :p
Yeah, we do, but it only deals with the sure parts of it, and leaves you to fend for yourself when you get to the indeterminate ones. It does give you a few weapons to help you defend yourself though. I've slaughter many a indeterminate with l'Hôpital's Rule. For more detail, see above.
:)
Let me extend on this by giving you a concrete example why infinite is both "bigger" than 42 and "smaller" than 42. Or, in other words why infinity is both infinitely large and infinitely small. Or, to be more precise, why infinity isn't either.
The example is a derivation from Zeno's arrow paradox. If you fire an arrow at a target, that arrow has to travel half its way before it can start doing the other half. But then, it needs to travel half of that before it can travel the other half of the first half. And so on, ad aeternum.
You can suddenly reach the conclusion that the arrow path can be divided infinite times into infinitesimal sections. Note already the need to express "infinity" and "infinitesimal" in the same sentence to mean the same thing. One can attribute that to a trick of the tongue. But if you can divide space into infinitely small portions, the arrow need to take an infinite amount of travelling before it reaches its target. And that's the paradox. It shouldn't ever reach it.
Physics will have something to say about all this. But here I'm just demonstrating to you that infinity is both small and large. And it is both things at the same time. And exactly because it doesn't have any measurable properties, it should in fact not even be determined in terms of being small or big. It's just infinite.
This is all very interesting, but please bear with me while I don't get anything. :)
I don't understand why that would be so. Does this have to do with calculus equations often have the double mirror effect? Kind of like calculating PI? (Not sure what the real term is)
Your equations don't make sense, why do you say that limx->∞ 1/x * limx->∞ x must equal limx->∞ x/x?
Yep, when I said .(000)1 I meant 1/∞, I guess I should use that form instead, from now on.
So, if .999~ is 1 (which I'm not necessarily disagreeing with), what is 1 - 1/∞?
Why? Because we live such that we can't 'capture' infinity, everytime we try, it just acts like a number so big that it can't be counted - even when not restrained by time. I can't think of an exception to this.
(Yes I do realize that infinity isn't a real number)
Look at Well-definition - Wikipedia, the free encyclopedia
There's It shows that it kind of is.
Yeah probably, but I like to try anyway ;-)
42. That's good :D
Why do you say "infinity and infinitesimal in the same sentence to mean the same thing"? It seemed to me, that you meant different things. It seems like you said "infinite" in reference to "so many that the amount has no end, and "infinitesimal" in reference "so small their smallness has no end". I don't see the paradox your talking about. If the length is infinite of course the arrow would take an infinite amount of time to reach the target. :confused:
My thinking is, your right, infinity can't be measured. Let me give an illustration of what I'm thinking. Say, in space (by space I mean an imaginary void, not real space), there's a road, who's size is infinite (NOT infitesimal), remember, infinity means "boundless", so the road's size wouldn't be bound, and without bound, things keep going. So while on the road, we take a measuring tape and measure part of the road we're standing on, which we measure to be... 42 inches. Now, if we want to, we can walk a few yards and measure the same length out again. How does that not prove that 42 is smaller than infinity? Now, let's say we can pick up the road. Now matter which way we place it, we won't be able to put it within the 42 inches we have measured out. How does that not prove that infinity is bigger than 42?
I haven't even studied limits to a satisfactory degree, but they do have properties like other things, such as logs. lim x to a f(x) * lim x to a g(x) is the same as lim x to a f(x)g(x). Of course, you're asking people to explain calculus to you at this point which is a bit annoying I imagine.Quote:
Your equations don't make sense, why do you say that limx->∞ 1/x * limx->∞ x must equal limx->∞ x/x
My interpretation of what's been said is this: the road is really a number line, with signs posted. Start from an arbitrary number N not forty-two and try to go to 42. I'm willing to bet that there are infinite real numbers between N and N+1 that you won't even make it to the next sign. Now, is infinity bigger or smaller than forty-two? Well, it's damn meaningless to measure!Quote:
So while on the road, we take a measuring tape and measure part of the road we're standing on, which we measure to be... 42 inches. Now, if we want to, we can walk a few yards and measure the same length out again. How does that not prove that 42 is smaller than infinity?
Of course, you can deal with infinity by using limits. For example, one of the first things you do, studying limits is find definition of a function. Is that information useful? I dunno. For me math is more about application than anything else, but I can't tell you what limits are applied to yet. I'm still learning calculus myself....
So these algebraic rules are suppose to be transparent of limits? I guess that makes sense. I'll look closer at it again.
Yep, it's basically a number line. Sure, there's an infinite real numbers between N and N+1, but that holds true to even finite number lines, and that doesn't stop one from traversing them by 42.
Differentials(denoted with the d in front of them) are not zero, if they were, the derivative would be meaningless, the integral also, by inclusion. df/dx=limh->0 (f(x+h)-f(x))/h This says that the ratio of two differentials(infinitesimals), is finite. If they were zero, then the ratio would 0/0 , and therefore undefined. I'm not going into detail on what they really are, it takes more explaining than I really want to give, since there are many resources already available. (MIT OCW + Wikipedia = everything I know)
A law of limits is that, iff limx->c f(x)*g(x) = L, then (limx->c f(x))*(limx->c g(x)) = L I used the inverse.
1/∞ is indeterminate, you can't give it a single representation. Neither can it be represented with a real number(.(000)1 is not a number). To prove .(000)1 is not a number, AFAIK, requires a real analysis concept called Cauchy Sequences. Basically, it means that any real number can be expressed as a convergent sequence of differences. There is not sequence that can converge to .(000)1.
Remember that ∞ can't be used without limits(within standard analysis, in nonstandard, they use rules that give the same results as if you were to take a limit, which is why nonstandard analysis is largely considered extraneous). So, you take the limit limx->∞ 1-1/x = 1. In general, any number minus an infinitesimal is that same number. Counterintuitive, maybe, but true, definitely.
To help Mario, what he means by infinities and infinitesimals being the same, is that the are the same in their immeasurableness. On is always, unconditionally larger, and the other is always, unconditionally, smaller.
The derivative of f(x) is defined as f'(x)=limh->0 (f(x+h)-f(x))/h, notice the fact that at x=0, it is 0/0, an indeterminate form. The derivative, intuitively, is how much the function rises, as it the amount it runs approaches zero. Derivatives, an application of limits, have innumerable applications. And integration, the most important application of derivatives, have even more.
How long have you been studying? I did for a few months before I actually started this semester at my local community college. It will definitely help you later. Many have issues when they first encounter a new paradigm of math, and being pre-exposed is like a vaccine against failing. Anyway, good luck, and if you ever get stuck on something, I have my PMs turned on, *unlike some people* cough cough...
Assume a rectangle with a width of 1 and a height of pi. The area of the rectangle is width x height = 1 x pi = pi.
Divide the width of the rectangle by two, and multiply the height of the rectangle by two. The area of the rectangle is width x height = 1/2 x 2pi = pi.
Repeat ad infinitum and one arrives at the case of inifinitesimally small x infinitely large. The result is ... pi.
Then again, were the original rectangle of width 1 and height e, then infinitesimally small x infinitely large would equal e.
However useful, infinity and infinitesimality are concepts, not numbers. You cannot just multiply them and expect to get a defined answer in return.
I think you pretty much answered yourself in the last sentence. What's an infinitesimal quantity if nothing else than an infinite quantity? Can you bound it? More on that below, where I revisit 42.
The paradox is achieved the moment you realize that yet still the arrow always reaches the target. Strengthening the idea that infinity boundless quality is only possible due to it's dimensionless attribute. And "dimensionless" doesn't fit with "big" or "small".Quote:
I don't see the paradox your talking about. If the length is infinite of course the arrow would take an infinite amount of time to reach the target. :confused:
I didn't say it wasn't. I said it is also smaller than 42. It is both things at the same time in fact. Or more probably, neither one of them. It does not respect our puny attempts at giving it a dimensional property. If I tell you that I measured the distance between the arrow and the target as being 42 paces, won't you immediately reach the realization that infinity is after all also smaller than 42 paces?Quote:
My thinking is, your right, infinity can't be measured. Let me give an illustration of what I'm thinking. Say, in space (by space I mean an imaginary void, not real space), there's a road, who's size is infinite (NOT infitesimal), remember, infinity means "boundless", so the road's size wouldn't be bound, and without bound, things keep going. So while on the road, we take a measuring tape and measure part of the road we're standing on, which we measure to be... 42 inches. Now, if we want to, we can walk a few yards and measure the same length out again. How does that not prove that 42 is smaller than infinity? Now, let's say we can pick up the road. Now matter which way we place it, we won't be able to put it within the 42 inches we have measured out. How does that not prove that infinity is bigger than 42?
Infinity is boundless, dimensionless, nor greater or smaller, unless you get a specific need to bring it down to our level of comprehension, in which case you will use math to try and define a more or less formalized set of rules to try to explain/comprehend/use it. Problem is you are probably always going to find these paradoxes and you are going always to have to resort to mathematical fallguys like indeterminates... and accept the practical consequences of your audacity (NaNs). I don't think you can represent infinity within a dimensional field without that removing its properties. The moment you explain it in terms of size, you know you missed the point.
Here's a BBC documentary on infinity, with some of it's apparent paradoxes and peculiarities, if you have an hour to kill.
YouTube - BBC Horizon (2010) - To Infinity and Beyond (complete, uncut)
Fascinating! You stole one hour from my workday. Well done :)
Interesting prospect the idea of an infinite number of universes, each of them infinite. I was thinking this same issue yesterday. How could one describe this concept, given that an universe being infinite, because its boundless, would certainly have to preclude the existence of any other universe.
Then the thought occurred to me as I was rereading about the arrow paradox. And it starts with the premise, "Is infinity boundless? Think again". Let's look at the following bounded interval of real numbers; [0, 42]. It's perfectly bounded. Yet, there's an infinite number of elements inside. The universe of elements inside this closed interval is infinite. It's an infinite bounded universe -- another delicious paradox on what possibility the biggest mass producer of those in cosmology; Infinity.
ed: well, by definition infinity is bound to produce an infinite number of paradoxes... big deal :p
That's a different kind of infinity. The measure of members in a set is called cardinality. There are infinitely many infinities(cardinal numbers) used for denoting which infinite sets are bigger than others. (That was purposefully overcomplicated, it's actually pretty simple.) Like, for example, that the set of real is bigger than that of naturals.
Cardinality - Wikipedia, the free encyclopedia
And, yes, it is boundless, by definition. :p Wikipedia says that the word itself is from the Latin for "unboundedness."
You don't argue in math. You need to have a clear definition of the concepts you are using.
As for your example, if f(x) -> oo and g(x) -> oo , we still cannot say much about f(x)/g(x). The limit can be 0, it can be oo, or something in between.
EDIT:
And no, 0.(000)1 is not a number. What would it mean? 0.(9) means lim(N->oo) sum(1...N) 9/10^N.
Suppose that 0.999... != 1.
Okay, what's the average of the two? For that matter give me ANY number that exists between them.
If there's one thing I've learned about math, it's that Sang-drax is always right.
What's the average of infinity and 12? You can't plug concepts like that into normal math equations. The idea of 0.999... is different from a specific number.
And certainly 0.999... is not equal to 1. Just because you can't find a number in between them doesn't make them the same. The inequality 0.999... < 1 still holds for any number of nines.
Prove it. (The qualifier "any number of nines" sounds strange to me, given the context: we're talking about the 9s repeating without end.)Quote:
Originally Posted by Perspective
Isn't it self-evident that an irrational number composed entirely 9s is smaller than its closest higher integer? Maybe you math folks have some way to formalize this proof. I do wonder though if it is required when we have already been expressing such things in other ways. Like with an interval, for instance ( [0, 1[ ).
Incidentally, Sang-drax expression above has a quality that is, I think, not being properly acknowledged: N->oo meaning isn't really that N grows towards infinity. Although that certainly can be safely implied, that description can hide one important piece of information for this purpose. What N->oo truly means is that N grows unbounded. And given the accountability properties of Real Numbers I think a simple Cantor's diagonal is enough to prove that 0.(9) is not 1.
0.999 ... is 1 for the same reason 0.333... is 1/3, it's just a different expression for the same number.
I can't proof it, but I can proof the reverse, that 0.9999..... = 1. I haven't read the entire thread, but I couldn't find the actual proof in it glancing over it.
10*0.99999.... = 9.99999....
Subtract 0.99999.... from 9.99999... As there are an infinite number of 9's in both, each 9 in 0.999... has a matching 9 in 9.9999... it "removes". Hence:
9.9999... - 0.9999... = 9
So let's say a = 0.99999...
I just showed that 10*a - a = 9.
9*a = 9
a = 1
Not my own proof, but I've always particularly liked this proof...
The problem here is there is in quantifying infinity. I think what you're trying to say is that as the number of nines approaches infinity, the value approaches 1. (?)
Been a while since I've done any real math, but I'm much more comfortable with that than saying 0.999... = 1.
edit: btw, I like the identity proof.
You're right saying that: as the number of nines approaches infinity, the value approaches one. However, if there is an INFINITE amount of 9's, as indicated by 0.999..., then the proof I presented holds.
There's the difference between the number of 9's approaching infinity, or there actually being an infinite number. They're not the same thing.
That is true for any finite number of nines. You can prove it with nested interval theorem. EDIT: Actually, I'm not sure I can :). However, you can infinitely nest intervals with 1 and .999... always in them. Nested interval theorem states there is exactly one number in every infinitely nested interval.
Of course I can. It's 0.999...
Real numbers are uncountable. The "you cannot find a number" is simply an exploitation of our inability to represent these numbers to their infinite precision. It does not mean however such a number can't exist. Many theorems have proven the infinite set of rational numbers. The same way we use oo to represent infinity, for the purposes of this debate we can simply stipulate that 0.(9) is not a number, but the representation of a number that has an infinite number of 9s.
As such, 0.(9) stands between 0.(9) and 1. Or, if you want a more complete answer, oo stands between 0.(9) and 1.
Where is this "infinity" thing coming from? The fact that the number of digits is infinite is irrelevant.
How do you express the value 1/10 in decimal? It's 0.1. How do you express it in binary? It requires an unending number of digits. Thus, the infinitude of the representation is an artifact of the chosen base, not anything to do with the number itself.
It's a tough concept and there are many ways of looking at it, but 0.999... is indisputably equal to 1. Period.Quote:
And certainly 0.999... is not equal to 1. Just because you can't find a number in between them doesn't make them the same. The inequality 0.999... < 1 still holds for any number of nines.
The real numbers are dense. This means that between any two unequal real numbers there are an infinite number of real numbers. Thus if 0.999... and 1 are not equal, there are an infinite number (uncountably infinite, actually) of values between the two. Name just one of them.
You're saying 0.(9) is a number which is unequal to itself?
EDIT: Okay, let's start the long process of going through each of the thousands of explanation of why 0.999... == 1.
Let x = 0.999...
Thus 10 * x = 9.999...
Thus 10 * x - x = 9.999... - 0.999...
All the nines on the right of the decimal subtract away:
10 * x - x = 9
9 * x = 9
x = 1
The only way the proof fails is if 0.999... != 0.999...
It follows on the same concept that oo does not equal oo, brewbuck. It's another of infinity paradoxes; also somewhat explored by Hippasus when he proved that the square root of 2 is both odd and even.
Conceptually you know very well that for every precision, no mater how large, that you can think for 0.(9), you need always to add another decimal place. You'll do this indefinitely. However, for 0.(9) to become 1 you need to add 0.(0)1. You are required an infinitesimal to reach 1. You are thus required another infinity.
But I also believe your axiom to be wrong:
And here you will make your first mistake:Quote:
Let x = 0.999...
Thus 10 * x = 9.999...
Thus 10 * x - x = 9.999... - 0.999...
They don't really. Not for the "last" decimal place, represented by an infinitesimal. If we replace the above for a finite quantity it's easy to see why:Quote:
All the nines on the right of the decimal subtract away:
10 * x - x = 9
Let X = 0.999
Thus 10 * x = 9.99.
Thus 10 * x - x = 9.99 - 0.999
Oops!
Next you make your second mistake, I believe:
But oo-oo != 0. It's undefined. It's both 0 and 1, or 12 and 13. Or any other number you can imagine. One of the properties of infinity is that it does not equal itself. Mathematically your expression becomes undefined at this point. The thought is that we know there is a number for your expression, but we can never reach it. As such it's undefined.Quote:
10 * x - x = 9
So, to finalize your axiom:
10 * x - x = undefined
I suppose I can be corrected somewhere since my math skills are tremendously limited. But more than my gross attempt at correcting you, I think it stands that 0.(9) is the representation of a number that never reaches 1. On the right side of an interval, this number would be represented as "1[" and this is a crucial hint to the nature of this number and the fact it does not equal 1.
I'm probably being pants on head retarded about this but like I said before, 0.333... is 1/3.
Therefore
1/3 + 1/3 + 1/3 = 1
Iff 1/3 = 0.333... then 0.333... + 0.333... + 0.333... = 0.999...
QED.
There are real numbers that result in a repeated sequence in any base. 0.2 doesn't stop being 0.2 when you express it in binary. But I think 1 is special because it's a unit. I think that in many bases (above binary perhaps, unless you entertain fractional bits) 1 has alternate expressions.
But again... pants on head retarded here.
Ah, I see your point now. Trouble is it has been long established that irrational numbers cannot be represented as fractions. So, 0.333... cannot be really represented as 1/3. Why? Well, exactly because of the example you gave. Many other examples exist where the irrationality of irrational numbers is evident every time you try to represent them as fractional numbers, like the aforementioned Hippasus proof.
We may do so on a daily-basis for convenience sake because the margins of error we impose become acceptable. But when it comes down to discussing the nature of 0.333..., or 0.999... for that matter, bringing in fractional numbers will not do because it will introduce an error margin that simply isn't acceptable.
1/3 is not irrational on the basis that I just expressed it in rational form. (e.g. sqrt 2 is irrational. ) But I see you're point: 0.333... is an approximation of 1/3, just as 0.999... would be of 1. As an approximation, it couldn't be exactly equal, but the difference is infinitesimally small, so by all means of proof it would be equal.
EDIT: By that I mean 0.999... or 0.333... carried out to some finite place would be an approximation but the difference between the actual numbers and 1 and 1/3 respectively are so small.
Infinity is not a number. .999... is. Therefore, this is irrelevant. I'm not sure how sqrt(2) relates, but I'll let it pass.
You're considering finite expansions. Finite amounts are irrelevant, because .999... is an infinite expansion. The fact that the distance(amount you must add) between them is an infinitesimal means they are equal. Since, apparently, brewbuck's statement isn't enough, here's a link:
http://www.calvin.edu/~rpruim/courses/m361/F03/overheads/real-axioms-print-pp4.pdf
Name ONE rational between .999... and 1. That's all I want. Anything else is intuition, and math is not based on intuition.Quote:
Theorem 0.22: [The rationals are dense in the reals.]
Between any two distinct real numbers there is a rational number. (In fact,
there are infinitely many such rational numbers.)
"Axioms" are true by definition. I would use "proof" or "statement" in this case, IMHO.
Yeah, oops, we've reached the end of an infinite expansion and shifted it. Big oops.
"indeterminate" is the word you're looking for, and 1-.999... isn't indeterminate. Again, .999... is a number, not infinity.
It never reaches anything, it's a constant. 1 doesn't reach 1, it is 1. Refer to the quote of the theorem above for a rigorous proof.
I tried to prove that 1 = 0.999... a different way after that post you replied to, and Mario responded to me with something about how 1/3 is irrational. That's how sqrt 2 came into the discussion, because in saying that 1/3 is rational, I wanted to give an example of an actually irrational number. Just something for Mario to think about because he couldn't think of any integer fractions a/b to express it. As quickly as it entered the discussion, sqrt 2 leaves it.Quote:
Infinity is not a number. .999... is. Therefore, this is irrelevant. I'm not sure how sqrt(2) relates, but I'll let it pass.
I knew what Mario wanted to say, so I responded to that as well.
Whether my proof is rigorous or not is another thing, but I work as well as I can.
BTW one of the benefits of being friends with me User Name is that PMs open up, so you can do that now.
I'm not sure I understand how there can be a last digit in an infinite string of digits. I also have heard that this is a common sort of misunderstanding when thinking about this problem (imagining that there can somehow be a "last" digit even if arbitrarily chosen).
How about a proof that 1 + 2 + 4 + 8 + 16 + ... is equal to -1?
X = 1 + 2 + 4 + 8 + 16 + ...
X = 1 + 2 * (1 + 2 + 4 + 8 + 16 + ...)
hey, the thing in parentheses is X again
X = 1 + 2 * X
X - 2 * X = 1
-X = 1
X = -1
There are such proofs for absurdities like "0 = 1" but they mostly involve on steps which are obviously problematic such as dividing by zero. There's nothing obviously wrong with the above... Or is there? ;)
I can't actually find anything wrong in your proof that ISN'T done in the 0.9999... proof. However, I've worked on your proof, I find this one slightly more interesting, as it's easier and the result is even stranger:
Same concept as your proof...Code:X = 1 + 1 + 1 + 1 + ...
X = 1 + (1 + 1 + 1 + 1 + ...)
X = 1 + X
0 = 1
Because it means the world is square ;-).
Okay brewbuck, I've been working on my OWN proof that 0.999... = 1 (though there are many on wikipedia, but I'm sure about this one). Not using the same method that you used. Try to find something wrong with this:
Is that a better proof for you?Code:0.9999... is, per definition:
0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
(that's the definition of ..., agreed?)
If so:
1 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 1 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 1*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 1*10^-2 + 9*10^-3 + ... + 9*10^-x )
...
= lim(x -> inf) ( 1*10^-x )
= 0
So:
1 - 0.9999... = 0
1 = 0.9999...
It is not an approximation; it is another representation. 0.333333333333, on the other hand, is an approximation of 1/3.Quote:
Originally Posted by whiteflags
How can I possibly name a rational, when the interval for which you require me to do so is bounded by irrationals? Explain to me how can I name a rational in [0.(9), 1[ and, more importantly even, how my physical inability to do so does in any way mean that 0.(9) is the same as 1. Do you deny irrationals? Why don't you ask me for an irrational when you know perfectly well that's the only number representation I can give you?
As for the axiom you introduced, I'm afraid at this point you are possibly taking advantage of my basic knowledge of mathematics. I'm pretty sure that it may be valid for some sort of calculations where the infinite string of decimals required by 0.(9) does not need to be factored in. But it cannot be valid here where the proposition 0.(9) == 1 is being discussed to an absolute certainty. And because we are discussing the infinite nature of rational numbers, you cannot honestly say 0.(9) equals one without introducing an error margin.
So, it's my turn to play cat and mouse with you: Name me a rational representation of 0.(9) where it becomes equal to 1 and the error margin is 0. Dishonest question, I know. But you made me one too.
EDIT: BTW, I took a long and closer look at the axioms you linked me to. And nowhere I can see where anything even resembling your proposed proof that 0.(9) == 1 is being discussed, particularly where it relates to this statement of yours: "The fact that the distance(amount you must add) between them is an infinitesimal means they are equal". This is a very strong thing to say. Nowhere it is being discussed there and never before I heard such a statement. It's particularly confusing, because those axioms reach a point where it is established that "Between any two distinct real numbers there is a rational number. (In fact, there are infinitely many such rational numbers.)". This pertains also to Cantor's proof that shown there are more irrationals in the Real set of numbers, than there are Rationals (another issue of differently sized infinities).
The question I pose you is then, given there's an infinite amount of rational numbers between any two real numbers, how does one represent one such rational number between irrational limits? Our inability to name such a number really means there isn't any? If there isn't (and then 0.(9) equals 1), how come there's in fact an infinite number of rationals between these two limits?
Hence why I was careful by placing last between quotes. I was hoping you understood it wasn't meant to be talking literally. Instead, if you wish, "for an infinite number of digits there will always be one next digit that doesn't allow you to zero out the 9s like you did in your example".
This is all to do with the proportions of infinity. Let me give you an example with natural numbers:
Let's look at an infinite series of natural numbers; 1, 2, 3, 4, ....
Let's remove from that series all even numbers; 2, 4, 6, ...
We now have two infinite series. One with all odd numbers and one with all even numbers. It could be said that these two infinities are smaller than the infinity of natural numbers. But all are infinite. We will also have an hard time explaining which of the odd or even infinities is larger. One should be larger than the other because we took each number alternatively from an infinite series. For a given natural number placed on the upper limit, we can immediately gauge at the proportions. But without that possibility, we cannot say which 3 infinities are larger or smaller. Infinity does not have a proportion. And as such you cannot do what you proposed to do. For every 9 you zero out in 10*x-x, there will always be another 9 next in line needing to be zeroed out. 10*x and X are both infinities in this context. But they aren't of the same size.
You can pretend to treat them the same, but for the purpose of proving that 0.(9) equals 1, you failed for this reason: For every single rational x, up to infinity, your calculations will fail. So how do you propose your proof to be valid when it fails every single time, all the way to infinity? Is this a case of many wrongs do a right? ;)
Is it? Or is 1/3 actually 0.(3) + 0.(0)1. (That is, it simply can't be accurately represented as a single decimal).
Just as 1/3 * 3 = 1 != 0.(9)... 0.(3) + 0.(0)1 = 1 != 0.(9).
How can 0.(9) be a number any more than infinity or 0.(0)1? I understand that infinity really can't be treated as a number without big problems, but the same seems to be so for 0.(9) as well. 1 - 0.(9) = 0.(0)1 and 1 / 0.(0)1 = oo.
If one were to accept the proofs as valid (at least, EVOEx's proof, even after reading up on limits I still don't understand them enough to get what I've seen of them here), then one should have to also accept the proof: oo + 197 = oo = oo + 2.(9), therefor 1 = 0.(9) and 197 = 2.(9). Which we know doesn't work.
But it can be represented in decimal. It just takes infinite places.
0.0... is zero. 0.0... * 1 = 0, so I dunno what you mean to say but 0.333... + 0 is still 0.333...
It doesn't matter how many places you carry zero out to, it's going to be zero still. Unless I totally misunderstand what you mean by 0.(0)1 -- but I think if you think it means something else, then you are probably wrong.
Well then write it that way. But 0.333... means 1/3 carried to infinite places, so the infinitesimal is taken into account. Plus, lim x -> infinity ( 1 + 1/x ) = 1. There is no difference here.
I know. It was a joke. I mean, my "proof" "showed" that 1 = 0 (you think I didn't realize that wasn't true?) and if 1 is indeed 0 then one can proof AND disproof ANYTHING. That the world is square, round, but not square, and not round, and that I exist but do not.
Yes, EVERYTHING becomes true.
Tell me what's wrong about my proof then. You can use exactly the same proof for 1/3 = 0.3333
Calculators show you 0.33333333334. That's because they simply don't have enough 3's to represent it, because calculators, unfortunately, can not display an infinite amount of 3's.
In fact, if a 4 ever does appear, then 3*(1/3)rd is GREATER than 1.
Let me show you: 0.3333333334... Multiply that by 3. The 4 becomes 2 and generates a carry: everything to the left of it becomes 0 and generates a carry.
Hence EVER adding the 0.(0)1 would mean that 1*(1/3) > 1, which is obviously false.
It's an informal proof. Still not convinced? I can make it formal if you didn't understand it.
Read my last proof, and tell me what you think of it. Tell me where you think I was wrong.
I propose this to anyone who does not believe that 0.9999... = 1. Read my proof and point me where my "error" is. You can't, because there isn't any.
It may be though that you misunderstand "...". It doesn't mean, repeated a lot of times, it means repeated infinitely, so it is about limits.
Yeah, an infinitesimal must be named. Can't be represented by something like 0.(0)1. In fact our numeric system capabilities to handle any sort of infinite numerations (uncountable sets) is perhaps at the basis of the misconceptions here. The attempt to represent 0.999... or 0.333... and even perform arithmetic operations on them is the type of error that allows one to reach conclusions such as 0.999... equal 1.
There's an error introduced on such calculations that is simply left unexplained in the eagerness to show the radical idea that irrational numbers can amount in fact to integers by some mathematical operation. A proposition that -- and forgive me my rudeness -- I find strange coming from folks dealing every day with the problems of floating point arithmetic.
But if nothing else, at least the thought that 0.999... represents a number that is close to, but never reaches, 1 is an universally accepted idea. That this is being disputed is a complete surprise to me. Maybe someone is playing devil's advocate. I can accept that since I don't have the math skills many of you do in here and haven't be able to formalize a proof. But I'm not dumb either. And can read proofs. I just haven't seen any yet that shows me 0.999... equals 1. And that's the type of claim that requires proof.
Of course we can represent 0.9999..., as you and I both just did. "0.999..." is the representation mathematicians agreed upon, just as "1" is what we agreed upon for using the number 1.
Another representation, according to the definition, would be in a formula:
I could have used the sigma sign for sums, but I can't write formula's properly here, but you get the point.Code:lim(x -> infinity) 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x
So as I asked you; what's wrong with MY proof. And is it really "universally accepted"?
Not according to wikipedia:
0.999... - Wikipedia, the free encyclopedia...
And any other actual mathematical resource.
The textbook answer for non-mathematicians like me is that those infinite sets are of the same size since there is a one-one correspondence between the set of natural numbers and that set of odd numbers, and a one-one correspondence between the set of natural numbers and that set of even numbers, and of course there is a one-one correspondence between the set of natural numbers and the set of natural numbers.Quote:
Originally Posted by Mario F.
I have read that article before Evoex. I find it curious however that quotes like the following are ignored though:
"Varying degrees of mathematical rigour" being the key expression here. It is ok to attribute the equality between those two numbers (or two ways to represent one, whatever you wish) within varying contexts. But not as an universal truth. The very nature of infinity stops math on its tracks and starts producing non-numbers as final results.Quote:
Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience
You may claim all sorts of proofs. But for every single one of them, there's a counterpoint that destroys them. I did that already, with no math skills of my own to speak off, to two of so-called "proofs" in here. Both also shown on that article. You cannot name one single proof in that set of "proofs" that remains true for an infinite requirement for mathematical rigour. And that is the requirement when you are dealing with infinity.
no, the universally accepted idea is that 0.(9) is 1
if you want to do some reading, 0.999... - Wikipedia, the free encyclopedia... has quite a few equations that show this
edit- looks like I'm a bit late with this one
Are you sure?
I'm confident the mistake in your proof is here:
Now you are saying that an infinitesimal equals 0 without first proving it. You ostensively reduce accuarcy to an integer without even flinching ;) But let's play ball and see what happens if I replace 1 by 2?Quote:
= lim(x -> inf) ( 1*10^-x )
= 0
Please read the posts above before replying. I already answered that 30 minutes before you posted.Code:0.9999... is, per definition:
0.9999... = lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
(that's the definition of ..., agreed?)
If so:
2 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 2 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 2*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 2*10^-2 + 9*10^-3 + ... + 9*10^-x )
...
= lim(x -> inf) ( 2*10^-x )
= still 0? Really useful it would be 1. But it isn't, is it?
You have an error: 2 - 9*10^-1 != 2*10^-1. Rather, you should get:Quote:
Originally Posted by Mario F.
Code:2 - lim(x -> inf) ( 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 2 - 9*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 1 + 1*10^-1 + 9*10^-2 + 9*10^-3 + ... + 9*10^-x )
= lim(x -> inf) ( 1 + 1*10^-2 + 9*10^-3 + ... + 9*10^-x )
...
= lim(x -> inf) ( 1 + 1*10^-x )
= 1
Ah thanks for the correction laserlight.
We still have the problem on how we get from lim(x -> inf) ( 1*10^-x ) to 0. Certainly this requires evidence of its own. We are supporting the proof of one idea by using itself. That can't do.
But we can do the limit and then by induction the proof would be true. Notice that the limit you mentioned involves an infinitesimal. I'd say that lim x->infinity (1/x) = 0 on the basis that for any x as x grows the quotient will be closer to zero. The limit is zero. Now you can evaluate 10^-x as -x gets smaller and get answers close to zero. That limit is also zero
But you cannot do that here whiteflags. It is exactly this closeness between an infinitesimal and 0, or 0.999... and 1 that we are discussing. It may be true for certain types of operations where limits are used to facilitate calculations. But it is wrong to propose a proof that 0.999... is 1 by using a method in which we simplify 0.999... to 1. Or an infinitesimal to 0. What use can that be to anyone?
Here is something a lot more interesting. Why, I think, a certain level of rigor and a more critic attitude to seemingly established concepts serves us better:
Is 0.999... = 1?
I don't propose to understand all that is being discussed. But clearly here the author understands that under certain scenarios 0.999... may be equal to 1, if we allow ourselves to introduce a margin of error into our calculations. But as an universal truth, we cannot. We are then forced to introduce other concepts such as, and I quote:
The acceptance that we musty introduce new concepts such as 0¯ or 1¯ and that our mathematical operations are powerless to deal with these quantities (real numbers) are fundamental to accept certain proofs into our lives without that meaning we should defend them as universal in the domain of Real numbers.Quote:
Clearly 0.9* = 1 + 0¯, so 0¯ is a sort of negative infinitesimal. On the other hand, you can't solve the equation 0.9* + X = 1 because, in cut D, the sum of a traditional real with any real is a traditional real.
In other words: A little more humility serves better anyone pretending they can prove 0.999... is equal to 1.
We didn't 'round' anything. It was an exact answer. You have been corrected with your "point" in my proof that was wrong. Look it up, it is known that:
lim[x -> infinity] 1/x = 0
That doesn't say that a very tiny thing is 0. It means that as x goes to infinity, 1/x will go to 0 and will get extremely close and even keep getting closer as x grows even bigger.
But in my proof I never used 1/infinity = 0. Just the limit.
Irrational limits? You're making up concepts and arguing completely from intuition.
How does the fact that .999... is irrational [, you incorrectly assume,] affect the fact that that if .999... != 1, you could name a rational between them? pi is irrational, yet you can name a rational in [pi/2, pi], or can you? It wouldn't surprise me.
I'm not taking advantage of your ignorance, you're just too stubborn(or otherwise) to understand the plain theorems I show you. Ever heard of the Dunning–Kruger effect - Wikipedia, the free encyclopedia, you need to read it.
Um, certainly not. There is no equality without an infinite number of digits. That's where the infinity comes from. See my last post, the one you quoted was before I really understood what was going on here (ok ok, I didn't read the whole thread, busted!). As the number of digits approaches infinity, the value of the expression approaches 1. I prefer that understanding. It is the same equality, but the wording makes it easier to understand I think.
The problem is that people often use infinity as if it were a number (I think this is the basis of Mario's beef with some of the explanations). There are some number of nines, call it "n". "n" is not infinity (that doesn't make sense), but as "n" approaches infinity the value of the expression approaches 1.
Yes you're absolutely right here, and that's kind of the point: if the number of nines approaches infinity then the number approaches 1.
However, if you write "0.9999..." that means an infinite amount of nines, meaning that according to the limits it approached 1.