Nate T.

My first concern to the set of solutions approach you give above is extraneous roots. Sometimes not all of the solutions arrived at via a solution method will actually satisfy the equation for various reasons-- you would have to take this into account somehow. My second concern is that you'd have to "flatten" expression involving nested radicals (or other multivalued functions). For instance, since we certainly want sqrt(sqrt(x)) = 4thrt(x), we'd have to have formally that {{sqrt( -sqrt(x)), sqrt(sqrt(x))}, {-sqrt(-sqrt(x)), -sqrt(sqrt(x))}} = {4thrt(x), -4thrt(x)}, which are sets of different levels, and wouldn't even be equal if flattened, since some of the roots on the left side are imaginary.

Well, it's not as though the "restriction" approach eliminates the other roots from contention. It's just that after one extracts a square root from real equations like x^2 = a, one really has two equations to solve: x = +root(a) and x = -root(a). The only reason one does a "restriction" is to let root(a) be a single number, so that one can break up the possibilities into separate equations. That's true; pedagogically, that kind of mistake does follow from a "restriction" approach. However, if they broke up their equations into two parts as above, there would be no problem.

That's interesting; you got a BS in math, and then pursued a PhD in physics? Was that difficult? Did you take a fair number of physics classes in your undergrad career? Well, if you do want to do a "multi-valued function" kind of approach, you would have to treat the sets that come out of functions like the square root function as numbers in order to make sense of equations involving radicals (and any other multivalued function like the inverse trig functions). Therefore, you'd have to extend the real number line to include every possible such multiply nested collection of sets of real numbers (in order to make sense of the arcsine of the square root of one, say). Then, you would have to find a way to denote a general member of this extension if you want to solve equations with this extension; how to do this isn't obvious, at least to me. It is something of a trade-off, but I do think that it's better to restrict (albeit awkwardly) the squaring function to a bijection in order to define a square root function instead of extending the real numbers in the way outlined above. After all, in the former case one can always describe the roots in terms of the one chosen by convention (this even holds true for complex roots). To be honest, I haven't thought out a way to extend the usual algebraic operations to relations, but I'm guessing you'd encounter messes fairly quickly. Do you have a specific kind of task in mind for which the multi-valued function approach would be better?

Since you're using terminology out of Category Theory, I'll ask something I should have asked up front: exactly how much mathematical knowledge do you have? I think the concept you're looking for is that of a relation. This generalizes the idea of a function by allowing that one point in the domain may be "related" to more than one point in the range, and that points in the domain need not be paired to any pints in the range at all. As for "inverses" versus "inverse functions," relations always have inverse relations, so that wouldn't be an issue. Right, but how do you overcome the algebraic closure problem I give above?

Depending on where you look in mathematical literature, a "mapping" is either a function or a continuous function. In either case, by the definition of a function, you cannot have a function which maps one number into "several numbers." Such a "function" mapping one number into many is called a relation. You could do this, but in practice no one ever does. If you'd like to do it, I suppose that's okay, but you yourself would have to extend the arithmetical rules operating on these pairs of real numbers, since I don't know of anyone else who has done it. One problem with this is that if you wanted to extract roots of roots, you would have pairs of pairs of real numbers, and so on. Since you have to account for every possible operation in this way (i.e., you want the algebraic operation to be closed), you would have to extend the real numbers to a set which is notationally nightmarish.

Extracting square roots was originally invoked as a way to solve equations which look like "x^2 = a," for x. The solution set of this equation is the set of all x which satisfy this equation. Hence, if you want "the square root of a" to mean a number and not a solution set, you have to specify (through convention) which solution to the equation is meant. Mathematicians choose the positive root and express other roots in terms of that one. Given that the squaring function is a two-to-one function (except at zero), it has no inverse, and so there is no "square root" function unless one restricts the domain of the squaring function so that it is one-to-one. People usually do this by restricting the domain of the squaring function to nonnegative values (which amounts to choosing the positive solution above). So no more of this "numbers are equal to sets" business.

We also have things like Drug Addict + Cocaine = Happy, too. Do you think that a life of addiction to hard drugs is a happy life? Do you think it's a good life? You're promoting Hedonism here. Also, in what sense are you "justified in believing whatever you want ... as long as it doesn't affect me"? You may be legally within your rights to do whatever you want without violating another's rights, but you are not morally justified if it means you are diminishing your own life.

Just because different people may choose to do different things in a moral situation doesn't mean one of them was wrong, or that each person's choice is subjective. Both people can be in the same situation and decide what is in their best interests and do different things. It just means that they have different circumstances, or operate in different contexts. Hence, ethics is personal (in that valuing presupposes a valuer), but not subjective (in the sense that you can choose options which make your life better as opposed to worse).

Yes, but a post isn't a work of art-- you're trying to communicate with people. Communication is best done concisely, trying to convey your idea using the simplest possible language.

Why the fancy verbiage? You won't be taken seriously around here if you can't express yourself in plain language.

I had assumed that the Lebanon Revolution had more to do with the Revolutions in the Ukrane and Georgia, since they seemed to be similarly modeled.

ItOE, 2nd ed., pp. 78, 94

Bryan, I should explain. When mathematicians write a sum from i = 1 to i = infinity, that expression is shorthand for the limit as N goes to infinity of the sum i = 1 to i = N. This is nothing new, it's what I've been saying this whole time, that the sum of a series is the limit of the sequence of partial sums. Why the distinction between an approximation of an actual sum you can never calculate and the limit of the sums? What possible use could there be in distinguishing between them? First, it isn't the smallest conceivable number, not if you want to usual operations of addition and multiplication to apply to it. Just take half of "1/infinity." There, something smaller. Second, we aren't using the same concept of infinity. I can say things like lim(x to infinity) f(x) = M, because I know it really means: for all e > 0, there exists an N > 0 such that |x| >= N implies |f(x) - M| < e. When you use it, you can't fall back on the epsilon-delta definition of the limit. You're really injecting infinity into the number line as an actual number, not as the property of a sequence. There are systems in which that is valid, but I prefer to keep my number line more tied to the perceptual level. Just make sure to mention before you say that 0.999~ < 1 that you're a non-standard analyst. It'll be great for parties. You know what-- okay. That's fine. I'll leave it to you to figure out what 0.999~ means.

Hal, Yuck! No Platonist, me. I should have used the term "constructed." Hmm, that's another good point. I don't quite know what to think of the Axiom of Choice. But from a pedagogical standpoint, Zorn's Lemma takes a lot of terminology. "Every nonempty partially ordered set having the property that every chain is bounded from above has a maximal member." You'd have to teach them all of those terms, and never use any of them ever again in the calculus course. Not exactly tough for a college course, but I can see the high school kids' eyes glazing over right now. They're equivalent constructions of the real line. The Dedekind cut method is awkward when it comes to multiplication, so it's generally liked less than the Cauchy Sequence construction. Although it does depend on who you ask. Well, don't get me wrong-- infinitesimals are a lot easier to understand from the get-go. And I'm certainly not attacking non-standard analysis as being logically flawed or something. I just think that, (1) pedagogically, the introduction of infinitesimals will raise more questions than it answers. ... hmm, that's actually a pretty good point. Except the motivation between the construction of the complex numbers and infinitesimal numbers are totally different. No one wanted to recognize the validity of imaginary numbers. Even Cardano, the man who first took them seriously, called them "false roots" and dismissed them as nonsensical, until he realised that he needed them in order to get real answers. Infinitesimals, on the other hand, are backward from this. People wanted a structure that could do something for them, and they just made one up for the sake of introducing a new kind of number into the number system (albeit with some pedagogical justification). Complex numbers were the result of closing the algebraic structure of the reals under root extraction. They also simply and unify many other branches of mathematics, such as the Fundamental Theorem of Algebra, which states that any polynomial of degree n has exactly n complex roots, and integral evaluation in the complex plane using analytic continuation. Such theorems are not possible without complex numbers. They have an elegant geometric strucure in the plane, and have been used to prove construction theorems in Euclidean Geometry. Considering infinitesimal roots of polynomials and solving infinitesimal equations can be done, but isn't really fruitful. Constructing this grand systems of infinitesimals to make the calculus work is fine, but then all evidence of the method used is wiped away in our final answer as irrelevant. Infinitesimals don't apply in any other branch of mathematics that I know of-- their sole purpose is to make taking limits conceptually easier. They don't have any beautiful geometric interpretation-- in fact, as far as I know, they behave just like formal variables. I guess it comes down to not constructing anything more than what you need to get the job done.

Bryan, There isn't one. If you have a natural number N, you can always add one to it to get a bigger natural number. So was I. That is the definition of a real number. You're right about that, but that's beside the point. You're claiming that because every partial sum 9/10 + 9/100 + ... + 9/10^N is less than one, the number 0.999~ must also be less than one, which is a non sequitur. The problem with this is that you can't 'literally' add up an infinite list of numbers. If you try, you will never be done. What you can do, however, is determine what that sum is getting very close to. That value, which is defined to be the sum of all of the numbers in the last, is the limit of the sequence of those sums. Once again, what is this "infinity", and why are you dividing by it like it were a number? Even supposing that "1/infinity" makes sense, you haven't answered what it means for a quanitity to be infinitely small, you've only given an example. Okay, let's distinguish between two things. The final answer of each of the partial sums 9/10 9/10 + 9/100 9/10 + 9/100 + 9/1000 ... and the limit, which is the value to which these final answers tend. If you mean that each of the partial sums is less than one, then that's right. But 0.999~ isn't any of these partial sums. It's their limit. You don't need to literally sit down forever to calculate 9/10 + 9/100 + ... in order to show it equal one. All you have to do (and this is the epsilon-delta definition of the limit) is show that no matter how close to 1 you specify, you can add up enough terms of the series to get even closer. That's what it means for that limit to be equal to one. I'm not sure you understand what my proof was claiming, so I'll do it in more detail. PROPOSITION. There is no greatest real number less than one. Proof. Suppose that there is a greatest real number less than one. Call it 'a'. Consider the number b = (1 + a)/2. First it is true that a < b. Second, it is true that b < 1. But a is defined to be the greatest number less than one. Since b is less than one, it follows that a is greater than it. Therefore, a is strictly less than itself, which is a contradiction. Hence no such number a exists. 0.999~ is the limit of the sequence (0.9, 0.99, 0.999, ...). I'm a senior math major.

Hal, Ocam's Razor cuts both ways. You can object to the definition of the limit as being overcomplicated and that of infinitesimals being much simpler, but it also postulates the existence of a huge number of entities to create the "superreal numbers," the vast majority of which you throw out of your final answer anyway when you take the "Standard Part" of your answer at the end, effectively ignoring the infinitesimals in your answer. Think of the philosophical conundrums that'll come up when students ask what these infinitesimals are, and you cannot answer them because their existence relies on Zorn's Lemma and other such high-level construction techniques. I'd rather teach my students some first order logic than tell them how to construct an abstract field extension with the Axiom of Choice. Not to mention, for those that go on in math, Analysis uses the idea behind limits to construct the real numbers, so it's helpful to have a background in limits. The seminal idea in topology is also based upon this definition. I'll stick with the epsilon-delta definition because I believe it integrates better into the hierarchy of knowledge. With the aid of limits, infinity is defined in terms of finite quantities and first order logic, which everyone understands. Postulating the existence of elements of an enlarged real line with certain preconstructed properties which are then "interpreted" as infinitesimals strikes me as formalistic, regardless of how intuitive infinitesimals might be. To me, it's the equivalent of saying: "Well, trying to construct calculus based on solely the real numbers is too much work. That infinitesimal idea almost worked, except for that small detail that there are no least positive numbers. But I want there to be a least positive number! Why not just say there's a least positive number and see what happens?" Finally, regarding the "intuitiveness" of infinitesimals, I don't regard the concept of an infinitesimal as being coherent. I regard it as a hand-wave, as a first attempt at getting a real idea of what the concept of infinity ought to be if we're going to base it our previous understanding of numbers.

Bryan, If you want to admit a literal infinity into the natural numbers, that's your business. But you'd better be ready to supply an infinite collection of objects on demand to prove that your concept belongs in the same genus as the natural numbers 1, 2, 3, ... No, it isn't. It makes no reference to "infintely small amounts" nor "least positive numbers." But since you agree to this definition, this discussion is over: 0.999~ = 1. I refer you to any Real Analysis text for the proof, if you're interested. This is an arbitrary assertion. How do you know that the limit of a sequence of terms which are less than one is less than one? Do you have a proof of this? You have to define 0.9 + 0.09 + 0.009 + ... to be the limit of the sequence (0.9, 0.99, 0.999, ...). There is no other consistent way to describe what you're talking about. You misunderstand what I'm asking for. I accept the unit we are working with is one. You say that an "infinitely small amount" is a number that is really small, but still not zero. I asked you how small that was. You evaded the question. I ask again: how small is 'infinitely' small? And you'd better not answer "smaller than any positive number", since no such quantity exists. Every interation you get is less than one, that's right. But none of those is the final answer, since you can always stick that back into f again. Hence, your observation that all of the finite iterations are less than one is irrelevant. You must describe this 'final' answer in terms of a limit, and the limit here is also 1. To which proof does this refer? I was referring to my proof that there is no largest number less than one. There are no "infinite number of steps" in that proof. In fact, none of the proofs that I've presented so far contain an infinite number of steps. I'm beginning to think that you think that the 'x' in that proof isn't a constant, but a variable. You have demonstrated nothing but the fact that you don't understand what 0.999~ means. In any case, we are beating a dead horse here. If you do accept the definition of real numbers that I proposed, the ball game is over, and 0.999~ = 1. If you want to use infinitesimals and whatever god-awful number system you can dream up to justify your assertion that 0.999~ and 1 are not identical, be my guest; I'll have nothing to do with it.

Bryan, You haven't even told me what 0.999~ is yet, let alone that it's a number not equal to one. In fact, if you assume that 0.999~ < 1, you can get a pretty big contradiction. Suppose that 0.999~ < 1. Then e = 1 - 0.999~ is a positive number. Hence, there exists an integer N such that 1/10^N < e. Now it should be clear that 0.99...9 (N + 1 9's) < 0.999~. But this implies that e = 1 - 0.999~ < 1 - 0.99...9 (N + 1 9's) < 1/10^N <= e. Once again, we get e is not itself, which is a contradiction. Okay, that's a good step, and you're right-- the concept 1 is far less abstract than 0.999~. If you insist... The way the real numbers are constructed from the rational are as equivalence classes of Cauchy sequences of rational numbers, equipped with termwise addition and multiplication. It's a result in Analysis to show that this structure is an algebraic field and has the familiar order properties of the real line, as well as being complete. Essentially, this means that every real number can always be identified with a sequence of rational numbers (a, b, c, ...). In our case, each of the sequences are equivalent to the real number 1: (1, 1, 1, ...) (0.9, 0.99, 0.999, ...) (1.1, 1.01, 1.001, ...) (0.9, 1.1, 0.99, 1.01, ...) You're right, to this extent: if you accept axiomatically the existence of an infinitely large number, you can build a system in which infinitesimal numbers exist, and in such a system the series 0.9 +0.09 + ... will converge to a number less than one by an infintiesimal. You can even say which infintiesimal. But if you'd like to stay in the real number system, where there are no 'infintiely large numbers', just sequences of numbers increasing without bound, you cannot have a smallest positive number. If you think you have such a number, then half of it is always both positive and strictly smaller, which is a contradiction. But what is the 'smallness' exactly? Presumably it's some number-- you appear to be postulating this 'something' as existence without identity here. If a number is larger than zero it has to be larger than zero by some amount. What amount is it? Okay, well let's work with this. Let's define f(x) = (1 + x)/2, as you've done above, and consider the sequence of numbers f(x) f(f(x)) f(f(f(x))) ... You claim that the "final answer" a = f(f(f(...))) is equal to 0.999~, right? Okay, well clearly by the definition of a, we have f(a) = a. Therefore, (1 + a)/2 = a, whence 1 + a = 2a, so a = 1. QED. Such a number doesn't exist. I gave you the proof of that already. If you can find an incorrect step in that proof, please show me.

Bryan, I appreciate your candor in trying to defend the idea that there is no metaphysical infinity. But don't misunderstand what it means-- in mathematics it's perfectly valid to talk about infinite sets, infinite series, etc., because mathematics is a discipline involving concepts of method. All numbers also 'only' exist as abstract concepts. In what way is the infinite series 0.999~ less real or less valid than the concept of 1? She also mentions that mathematics is a concept of method, which includes the valid concept of infinity to help aid the ultimate goal of mathematics to measure things. First, both reality and limits are necessary to make a theory of calculus. Second, infinite sums and limits are calculus concepts! One uses the concepts of calculus to solve problems in reality, yes. ...? This isn't how real numbers are defined. It merely steals the concept of 'complex numbers', which are defined in terms of the real numbers. First of all, all numbers 'only' exist as concepts. You did say "infinitely small amount away", which is what an infinitesimal is. Care to elaborate on what an "infinitely small amount" is? The reason I brought up that expression was because it helped to prove that there is no greatest number less than one. The argument goes as follows: Suppose there is a largest value less than one. Call it 'x'. Then (x + 1)/2 is strictly greater than x, but strictly less than one. Therefore, by the definition of x, it follows that (1 + x)/2 <= x. But then x < (x + 1)/2 <= x. This means that x is something other than itself. But A is A, so contradictions do not exist; one of our premises is wrong. The wrong premise is the beginning one, that x is the largest number strictly less than one-- no such number can exist. Let's get something straight right now. A sequence is a countable list of numbers. A series is the limit of a sequence of partial sums of numbers. The above isn't a sequence, it's a series. And anyway, if you really accept this, you've already admitted that .999~ = 1. If you open ItOE to the section "Definitions", Rand defines a definition to be "a statement that identifies the nature of the units subsumed under a concept." Moreover, it is the statement which condenses and explains all the other knowledge of the concept in question under the context of one's current knowledge. If you try to do this for the various 'definitions' for 0.999~ you present above, you'll find little more than an anti-concept, since most of those definitions don't even describe the same objects and are contradictory. What else is the 'actual number' .999~ besides the limit of the sequence (0.9, 0.99, 0.999, ...)? I agree that just because the limit of a sequence is one does not mean the sequence is one. Sequences aren't numbers, and I never claimed that they were. How close is it? Blank out.

Bryan, 0.999~ doesn't refer to anything in reality? You'd better be careful about statements like that. Does -1 refer to anything in reality? How about i = sqrt(-1)? If these numbers are concepts of method, then why isn't 0.999~? After all, just as negative and complex numbers have many applications, so do summing series and limits (in fact, they make all of calculus possible, which is the only thing that makes modern physics possible). Also, I wasn't able to get a straight answer out of you about the status of 0.999~. Is it a real number? An 'abstract' number (abstract in what sense)? One minus an infinitesimal (what is an infinitesimal)? The greatest number less than one (which can't happen, btw)? A sequence of rational numbers? You seem to be using all of these definitions interchangably. I contend that the only way to make any sense whatsoever of the expression 0.999~ is for it to represent the limit of the sequence (0.9, 0.99, 0.999, ...). Since one can show that this sequence converges to 1, it follows that 0.999~ = 1. I agree that the sequence never does reach one--that's fine. But the limit of the sequence is what 0.999~ denotes, and that is equal to 1.

Bryan, I'm confused by these statements. Is .999~ a number, or not? If so, how do you define this number, what is it? What do you think it means for a number to exist in reality? What does it mean for a number to 'approach' another number? You know that there is no greatest number less than one, right? If you think you have such a number x, then the (1 + x)/2 is still less than one, but greater than x. If 0.999~ doesn't exist in reality, what do you think about any other irrational number like pi or the square root of two? Since we can only give finite decimal approximations of these numbers, do they also not exist in reality?

The reason you need to define .999... as the limit of the sequence (.9, .99, .999, ...) is because there is no meaningful way to subtract 9.999... from 0.999... without this definition, and the proof given originally in this thread uses this as a step. Normally, if you want to subtract two decimals which terminate, you use an algorithm that everyone learned in grade school, involving subtracting along columns, with borrowing from the next column sometimes being necessary. The important thing is that this procedure is a right-to-left procedure, i.e., it begins at the rightmost decimal place and works its way backwards. However, you cannot do that with 9.999... and 0.999..., since there is no rightmost column. You have to define 9.999... - 0.999... to be that number to which the sequence 9 - 0 9.9 - 0.9 9.99 - 0.99 9.999 - 0.999 ... gets infinitely close to. Since each of the numbers above is just 9, it's clear that this sequence apporaches the number 9, and that's why 9.999... - 0.999... = 9.

How do you know that 9(.999...) = 8.999...? That seems just as dubious a step as saying that 9.999... - .999... = 9.

Free Capitalist, In what way is this an error?

Moose, That's something different. That .999... = 1 isn't a fallacy, it's just counterintuitive, kind of like some of Xeno's paradoxes.