0.99999~ = 1 ?

Fair enough. Is there somewhere I can see the mathematical proof of that? Well is infinity + 1 > infinity. I guess not. But even in my fairly limited context of undergraduate level physics (as opposed to pure or higher level maths) the concept of more rapidly approaching infinity and thus "degrees of infinity" has arisen. At least in some sort of conceptual principle.So even if I am taking the role of the idiotic punchbag for the more mathematically literate here I do so on the basis of asking whether things are as obvious as you seem to be claiming?

Put like that - I guess so.But does that mean that infinity squared is the same as infinity to the power of 10? For example. Well I had to try. Yes. But once we start invoking infinity all sorts of mad sounding stuff is mathematically true in ways that mathematicians I have spoken to never sound wholly convinced by themselves in ways that are well beyond me to know why they have such arguments.Maybe I am extrapoloting too far. Maybe every mathematician would say that 1=0.999R exactly and without any reservation of any sort. Without any debatable notion of infinity being involved.I am just not knowledgeable enough to say. But it still strikes me that once you need to invoke infinity nothing is as cut and dried as seems to be being asserted here.Edited by Straggler, : No reason given.

That they are equal? Look through this thread, I'd say. Otherwise, look it up on wiki, there are some nice proofs there as well. More than infinity? How's that even possible? Indeed. Infinity + 1 = infinity.Have you heard of the Hotel with infinite rooms analogy? It goes something like this:There is a Hotel that has an infinte number of rooms and an infinite number of guests. Now, one more guests checks into the hotel. How do they fit that guest into the hotel? Simple, they move all the guests one room, and the free room can now be occupied by the new guest.Now, how many rooms does the hotel have? Still an infinite amount. Ah yes, there are more then one infinity.Try this for a fun experiment. Take a circle, now, draw lines of infinitely small width from its center to its edge. How many lines are tehre? An infinite amount. Now, draw a bigger circle around it, and extend your lines outward to that circles edge. Suddenly there are spaces between the lines! But how can this be, if the lines are infinite? Apparently, to math guys, they are.

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I hunt for the truthI am the one Orgasmatron, the outstretched grasping hand
My image is of agony, my servants rape the land
Obsequious and arrogant, clandestine and vain
Two thousand years of misery, of torture in my name
Hypocrisy made paramount, paranoia the law
My name is called religion, sadistic, sacred whore.
-Lyrics by Lemmy Kilmister of Motorhead

A very nice video related to this subject. Link here

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I hunt for the truthI am the one Orgasmatron, the outstretched grasping hand
My image is of agony, my servants rape the land
Obsequious and arrogant, clandestine and vain
Two thousand years of misery, of torture in my name
Hypocrisy made paramount, paranoia the law
My name is called religion, sadistic, sacred whore.
-Lyrics by Lemmy Kilmister of Motorhead

Screwed if I know. But then degrees of infinity is no less mad and that is a genuine mathematical concept. Well once we invoke infinity we can make all sorts of stuff sound insane. Wich is kinda the problem with invoking infinities. They are not very practically helpful. So I am told. In my limited experience maths guys can argue about the nature and application of infinity infinitely.

Hi HuntardWow, infinity ("the abyss") really drove Cantor crazy hehe.Thanks for the link, I was looking for something to watch tonight (as opposed to watching "The Wire" for the third time).Thanks for the link.

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But does that mean that infinity squared is the same as infinity to the power of 10?

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There are degrees of infinity. For instance the infinite number of real numbers is greater than the infinite number of integers (as shown by Cantor's diagonalisation argument).Off hand I don't remember exactly how they relate mathematically, it's been a while since I read up on this stuff.

Hi Straggler First off, 0.999~ is a product of using base 10. I don't think you would have any difficulty with saying: Or with saying: Put them together and you have: 0.999... - Wikipedia

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One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using long division, a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333, in which the digits repeat without end. This decimal yields a quick proof for 0.999 = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 0.333 equals 0.999. And 3 1⁄3 equals 1, so 0.999 = 1.[1]

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Of course, this also means that 6.999R ≡ 7.0 ... Not necessarily, as there are many instances where you can do Limit analysis where the result is and each limit on it's own is infinite, yet the result is a finite number.Enjoy.

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One of the problems with the term infinity is that, infinity is not a number and the basic math functions don't work in the same way as with a finite number.For example there is an infinite set of numbers called integers.
There is another infinite set called rational numbers, all integers are contained in the set of rational numbers. Both sets are infinite so if one subtracted the integers from the rational numbers there would still be an infinite set of numbers left. The infinite set of integers added to the infinite set of algebraic irrational numbers & the infinite set of transcendental numbers gives another infinite set, the real numbers. This infinite set when added to the set of imaginary numbers (all real numbers multiplied by i (the square root of -1) gives part of the infinite set of complex numbers (numbers written in the form a + bi where a and b are real and i is the square root of -1Edited by bluescat48, : missing line + typos

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There is no better love between 2 people than mutual respect for each other WT Young, 2002Who gave anyone the authority to call me an authority on anything. WT Young, 1969Since Evolution is only ~90% correct it should be thrown out and replaced by Creation which has even a lower % of correctness. W T Young, 2008

How about "a probability of 1 - ε"? False as false can be.If the former was smaller than the latter, there would have to be some quantity by which it was smaller. But there isn't.Edited by Dr Adequate, : No reason given.

It's manifestly sane.Suppose we have a set S. We can consider the set of all subsets of S --- call this P(S).So for example if S = {1,2} then P(S) = {{},{1},{2},{1,2}}.Now, the size of P(S) must be strictly greater than the size of S. This is obviously true in the finite case. Less obviously, it must also be true if S is infinite.This means that there must be lots of different sizes of infinity. If S is the set of natural numbers, which is infinite, then P(S) must be bigger than that, and P(P(S)) bigger than that, and P(P(P(S))) bigger than that, and so forth.---I shall give the proof of the statement above, because it's one of my favorite proofs in mathematics. Proof that P(S) always has more members than S:Note that P(S) always has at least as many members as S, because for every member s of S, P(S) contains the element {s}. So if they're different sizes, P(S) must be the larger of the two.This we shall prove by contradiction. Suppose that they were the same size. Then we could put the members of S and the members of P(S) into one-to-one correspondence, since that's what it means for two sets to be the same size. Hence there would be a function f from S to P(S) such that for every y in P(S) there is an x in S such that f(x) = y.Now, for every z in S, z either is or is not a member of the set f(z). So, under our assumption that the function f exists, we can form the set T = {All z in S such that z is not in f(z)}.Now, T is a subset of S. Therefore T is a member of P(S). Therefore, there is some member t of S such that f(t) = T.Now, here's the clever bit. Is t a member of T?. Well, if it is, then it isn't, and if it isn't, then it is --- by definition of T.So we have produced an absurdity, therefore the function f can't exist, therefore P(S) is larger than S.Edited by Dr Adequate, : No reason given.Edited by Dr Adequate, : No reason given.

Well I accept what you say. And when you guys explain it does all kinda make sense. But there is still something that seems intuitively wrong about the whole thing. For example is it true to say that 0.999R is a whole number? Thanks for this. That does actually make sense. Even to me.

I get all the arguments for 0.999R being entirely equalt to 1. But it still seems "wrong" that 0.999R is a whole number. Surely I am not alone in this intuitive feeling? Otherwise it wouldn't even be a topic worth highlighting. Oh God. Please. Let's not go there. You know what I mean about the inherent impossibility of certainty in evidence based arguments just as well as I do.

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But it still seems "wrong" that 0.999R is a whole number.

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Isn't that just a matter of presentation ? You see something that doesn't look like a whole number, but in fact it's just an odd and impractical way of writing 1 (or just wrong if you go with the finitists).

Yes, things aren't really totally clear cut or intuitive. In truth it all comes down to the nature of the real numbers and some historical issues.If I have a number like 0.99, everybody will understand what I mean. It is simply 9/10 + 9/100, no problem. However if I write down something like:
0.999....This is supposed to represent:
9/10 + 9/100 + 9/1000 + 9/10000 + .........So 0.999... is an infinite sum. The problem is (and this was a big problem historically) that it isn't obvious what I mean by an infinite sum or how I define what it sums to. So without some extra mathematical machinery to give a definition to 0.999..., it means absolutely nothing.This is a serious problem. Very early in their history the Greeks discovered numbers like the square root of 2, which can only be represented using an infinite sum. Numbers like the square root of 2 and Pi, don't immediately need infinite sums to have a definition, since we have other ways to define them. However some numbers are only known as infinite sums. Also although Pi has another another definition, it's completely geometric (ratio of diameter to circumference). The only purely numerical definition of Pi is as an infinite sum, so within a numerical context you would probably want a good definition.It was only in the 19th century (although there was some earlier work in Europe and India and possibly ancient Greece) that people began the hard work of actually giving a definition to infinite sums.It turned out that the only sensible definition is to define an infinite sum to be the number that it is approaching. In the case of 0.999..., you can see that with the addition of each 9, it approaches 1. Thus it is 1, by definition.It may be better to think of 0.9999... to be the number constructed by a new mathematical operation called "infinite summation" and under the rules of infinite summation it is 1.
The intuitive confusion isn't actually incorrect. 0.999.... is not the same type of thing as 0.99, except with an infinite number of 9s. It's actually a different mathematical beast, belonging more in the realm of analysis than arithmetic. When presented with 0.999... you can either:
(a)Reject it because it requires the use of the infinite to construct it. In which case it simply has no meaning, like writing down "0.9car9" or something. This is the position of finitists.
(b)Accept that the use of the infinite in its construction is valid and attempt to give a meaning to that construction. The only meaning that makes sense and agrees with the mathematics we already know is the one I've given above and under that meaning it is 1.History has shown option (b) to be more fruitful.