One does equal 3 minus 2… There is no way around it, except that where a number cannot be shown, fixed, and finite; there it is difficult to say what it equals…You cannot say what 1/3 of anything is…It is not the answer, but the question 1 divided by 3…Put that in decimal form, and mulitply it by three, and you know the answer: 3/3 which is one… Keeping it in decimal form only reproduces the nonsense endlessly…It does not inform, and it does not illustrate, and it does not ease which is the purpose of all forms… We do not have forms like math because they muddy reality beyond comprehension, but because the give us knowledge… .999… is garbage for hogs…
Hey Wonderer,
Still campaigning this thread, eh? It’s got to be one of the ILP classics by now.
How 'bout this idea: we can define an infinitesimal as follows:
infinitesimal = x - y_max where x > y. That is, an infinitesimal is the difference between x and the maximum possible value for y when x > y.
That definition works for me, but I’m still a bit skeptical whether it is subject to arithmetic in the same way as all other numbers.
I seem to recall from my courses in calculus the difference between an open limit and a closed one. I forget which is which but one is a limit that can be attained whereas the other is one that can’t. So in the equation x^2 = y, for example, the limit of y as x approaches 2 (say) is 4, and it is an attainable limit (open/closed - I forget which) because x can equal 2. On the other hand, the limit of y as x approaches infinity is likewise infinity, and it is an unattainable limit because x can never reach infinity.
So the question that lies before us is this: in the equation sum(9/10^x) = y, is the limit of y as x approaches infinity an open or closed limit? In either case, the limit of y is 1. But is it an attainable limit?
You seem to argue that y_lim is unattaible, which tells us that y_lim as x approaches infinity - 1 = infinitesimal, but I say y_lim is attainable, which tells us that y_lim as x approaches infinity - 1 = 0 and so y = 1 as x approaches infinity.
Who’s right?
Well, I would say that I’m not sure, but the proof given in the OP seems to settle the matter for me - the limit is attainable.
you think infinities are grabage and do not give knowledge? tell that to all of calculus, which, given its history of leading to the mathematical knowledge-base for scientific theories spanning all of physics (along with all of the real-world material technology these theories have generated) and beyond, might beg to differ.
symbolic representations of infinities or unending sums/sets are indeed very practically useful.
fractions (i.e. percentages or relations) are no different fundamentally than decimal numbers. they are symbolic-language ideas meant to represent a QUANTITY of/and/or relative value. you can indeed have a quantity of 1/3 of X, and this DOES indeed give us added meaning: we learn that the total quantity of X is 1/3, or 33% of its maximum (full) possible value.
fractions represent ideas of probability as well as quantity; these are two distinct aspects where we are given knowledge by these symbolic representations… and by this measure, you might even consider a fraction (i.e. think of it also as a RELATION between TWO numbers) as providing more information than a single number by itself… 1/3 gives more information that 1 or 3, because the concepts of the quantities 1 and 3 are both contained within 1/3.
Almost our entire social reality is made up of infinites that are alternately useful and useless… God is as much an infinite as justice or liberty which are moral concepts, which is to say: spiritual forms… I think it would be great if we only had finites to deal with, but then, we would not be human, and certainly not what we are… So if you find mathamatical infinites useful; then by all means, enjoy them…It does not matter to me, but it should matter to you when the basic identity behind all math, that one is one should be supplanted by an infinite representation of one which is hardly useful, and so, is pointless… Ultimately, all our concepts are not useful because they are truthful, but are tuthful because they are useful… I would ask what use has the representation of one as .999…??? If it has no use it has no truth… No matter with what form we represent reality, it must ease our lives to tell us truth…The truth as we have it, that life is difficult, and complicated is not the one we desire… We want a new truth… .999… is just reinventing the wheel out of a log…It does not us get closer to a desired reality…
Juggernaut, if you ever take advanced math or physics or any hard science you’ll see that, far from useless, infinite decimals like 0.9[bar], and identities like 0.9[bar] = 1, are essential for analyzing all sorts of practical problems. They help us make predictions about the world. Without them, many concepts in math would cease to exist, and the applications of those concepts in the sciences would become impossible. You owe the computer you’re typing this on to a human mind that designed it, and that mind used infinite decimals and infinite series identities to design it. So in a way, you owe the computer you’re typing on to 0.9[bar] = 1. What could be more useful or real than that?
In all seriousness; can you give me an example of .9[bar] used in any equasion essential to physics or advanced math??? I mean, I am sensitive to words like like, and while I have read a lot of physics, and always pay attention to the formulas and equasions, I do not recall ever having come across .9[bar] used anywhere in place of one…
Let me clarify. I can’t give you an example where 0.9[bar] = 1 is used verbatim in a formula, because it is not powerful enough to help you do anything. However, 0.9[bar] = 1 logically follows from more powerful results which DO show up everywhere in math and its applications. Specifically it’s a corollary of the geometric series formula, which is a corollary of Taylor’s theorem, an essential result of calculus which is constantly and universally applied in the sciences. I don’t think I should go into the details because they require a lot of math background, but here’s an example from physics where the geometric series formula is used.
In this example, the geometric series formula is used to verify that a physical model makes reasonable predictions. Specifically, we expect that if a bunch of atoms are undergoing radioactive decay, then eventually all the atoms should decay, and we want our model to be consistent with that. In the model, that prediction is backed up by the geometric series formula given in “Check Your Understanding” Problem 1, and 0.9[bar] = 1 is a corollary of that formula (just substitute P = 0.1, A = 1 into the formula given, and you get the statement 0.9[bar] = 1 exactly).
The model for radioactive decay discussed in the page is simplified and doesn’t represent reality exactly. But that is why physics is so powerful. By simplifying and abstracting reality, physicists can derive models that are simple enough to solve exactly, and these exact solutions make approximately correct predictions about the real world. To solve the simplified models, a variety of advanced math formulas are often needed. Those formulas often imply 0.9[bar] = 1 as a trivial consequence, as in the example above.
If you want to do physics, you need these formulas, and if you want to be logical and consistent in using them, you have to accept all of their logical implications, including 0.9[bar] = 1.
The fact that you have not seen these formulas in your reading of physics is probably because you are most likely reading books written for laymen, not textbooks which teach you how to actually solve problems in physics. These laymen’s explanations can give you a sense of the excitement and general aims of science, but they will not show you how physicists do what they do, or what tools are necessary. These books are a sort of museum for the layman to come in and admire the displayed achievements of science. However, you will not learn how a museum-displayed rocket works from looking at it; neither will you learn physicists do what they do by reading general interest books by (for example) Hawking or Greene.
Thanks for your honest reply…I have nothing againt utilitarian math…I put it right up there with creative finance…The fact that I see is that having an infinite answer to one divided by three, we multiply that infinite by three…We know the answer is one…I don’t know what happens when we try to multiply an infinite, or try to add infinites… You can tell me, but if you could actually demonstrate an infinite, it would not be an infinite…
Here’s how it works:
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Suppose you are told there is this mystery number, call it x. We don’t know its value yet but we know a few things about it. We know it’s greater than or equal to 0. We know it’s less than 1/10. We know it’s less than 1/100, which is 1/10^2. We know it’s less than 1/1000 = 1/10^3. And so on. For each number n, we know that it’s less than 1/10^n. What can this number possibly be? Can it be greater than 0? No, because if it were, then it would be bigger than 1/10^n for some n. (Convince yourself of this. Think of a number, any number, and find a number n such that 1/10^n is smaller than it.) Therefore, since x is not less than 0 and not greater than 0, it must either be 0 or not a number. It is reasonable (and very useful down the line, as discussed in my previous post) to say that it is a number, so we are forced to say that x = 0.
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Now suppose you are told there is this mystery number, call it “the difference between 1 and 0.9[bar]”. We don’t know its value yet but we know a few things about it. We know it’s greater than or equal to 0. We know it’s less than 1/10. We know it’s less than 1/100, which is 1/10^2. We know it’s less than 1/1000 = 1/10^3. And so on. For each n, we know that it’s less than 1/10^n. What can this number possibly be? By the argument above it must be 0. So the difference between 1 and 0.9[bar] is 0. Hence 1 = 0.9[bar].
i would say 1/10^(n+1)
imagine .9[bar] shown on a graph where each point on the x axis represents an additional 9 in the sequence, making the total closer and closer to 1. The curve would plateau approaching what appears to be 1, as more and more 9’s are added to the sum of the number represented by the axis.
For each 9, we have a corresponding 1 which makes up the true difference from the graph points to 1.
imagine superimposed on the same graph with the .9bar curve, a curve which begins at .1 then moves to .01 at the 2 mark on the x axis, and continues in that fashion.
when we think of the function in these terms, as the .9 function approaches .999[bar] or “1” as it appears, so too does this .1 function approach .000[bar]1
.999[bar] does not approach a number just as the .1 function does not approach a number
1 does not appear anywhere in the .9 function on the graph shown below, and niether will 0 appear in the .1 function as a y value.
But 1/10^(n+1) isn’t a single number. It becomes a single number when you substitute in a number for n, say 1000. But 1/10^(1002) < 10^(1001), so 1/10^1001 can’t be the desired single number because it must be smaller than 1/10^k for ALL k, including k = 1002. If you put in 100000000 for n, same thing will happen; 1/10^1000000001 cannot be the desired number. No matter what you put in, it won’t work. There is no number greater than 0 that is smaller than all of the infinitely many numbers {1/10, 1/100, 1/1000, …}. Unless you want to say that .0[bar]1 is a positive number different from 0, but this creates its own problems as I discuss below.
These two statements seem to contradict each other. Is .000[bar]1 a number or not? Does the .1 function approach it or not?
If .000[bar]1 is a number which is not equal to 0, then its reciprocal 1/.0[bar]1 should be a number too. However its reciprocal is bigger than any number. It’s infinite. Do you want to posit that infinity is an ordinary number? If so, let’s give it a name, call it H. What is H + H? is it the same as H or is it more? If H + H = H then H = 0, a contradiction, so we must have that H + H is not equal to H. Similarly 3*H is different from H and H+H, and the result is there’s all these infinite numbers floating around. That’s kind of cool but it also makes arithmetic very complicated (for example, what’s H^3/(H^2 + 1)? how can we assign it a value that creates no contradictions?)
Wouldn’t it be easier if our number system had no infinite numbers in it? If you’d prefer such a system, the only consistent choice is to make .000[bar]1 = 0.
The number .0[bar]1 can be whatever you want it to be in math, so long as you’re consistent. If you say it’s 0, you get a number system with no infinite numbers. If you say it’s not 0, you get a number system with infinite numbers. A number system with no infinite numbers is easier for most mathematicians to understand, so they choose to say that .0[bar]1 has the value 0. There is another system that has infinite numbers, called the hyperreal numbers, but the introduction of infinite numbers makes it very complex and hard to work with. So we say .0[bar]1 = 0 because it’s possible to do so consistently, and it’s a lot easier than the alternative.
That’s why we say “approaches”. You can approach a city by walking closer and closer to it, without ever reaching it. Your distance to the city is a function of time that approaches 0 but never reaches it.
Aporia…One is a single number…100 is 100 single numbers… All you math minds will have to correct me if I am wrong, and trust me, I struggle with sand box arithmatic; but the concept: Number, is built upon a single number, a unit; upon which the concept of numbers is built… As Aristotle said: all numbers are in ratio to one, excepting zero, and that is not so much a concept as an anti concept…
As one English jurist once said: there are no imaginary cases… Numbers must represent realities to be useful… I understand that a lot of philosophical space is given to infinities… I also understand where mathimatical infinities can have a use… If I do not see this as such a case, and I am wrong it would not be the first time… I have been wrong so often in my life that it is a wonder I can trust myself with toast in the morning…In any event, there has to be some pooint where our concepts reflect reality, and even while we know that our realities are not fixed in time, and that our concepts are not fixed in time, that to have some use we must hold the illusion for at least a moment that both concept and reality are fixed, and if we cannot say .9[bar] is any particular quantity, that is fixed in time, then it cannot be true because it cannot be proved for veracity… I know it is a simple objection, but I think it is valid…We can count cows on our fingers… Really??? How many cows have you had on your fingers??? As a manor of speaking, we can, since one equals one…
No numbers are equal to zero…Zero is not a number…Zero is not anything…
I agree with you that a number should not be a process or change over time, the way .9[bar] appears to do. But like I said, it’s useful to make .9[bar] a number. So logically, if we wish to make .9[bar] a number, we should not think of it as changing “as the 9’s increase to infinity”. Instead .9[bar] must be thought of as the destination being approached by the sequence of “partial decimals” (0.9, 0.99, 0.999, …). Imagine a grasshopper which starts 1 foot from a castle wall. He hops 9/10 of the way to the wall, then hops 9/10 of the remaining distance, then 9/10 again, and so on. His distance from the wall goes like (0.9, 0.99, 0.999…) as his hops increase over time. He never gets to the wall, but he is always approaching the wall. 0.9[bar] is a representation of where he would be after infinitely many jumps. But we know that that’s the wall, so 0.9[bar] = 1.
Forms do not work because they are imaginary, but work because we can think of them as real… The work is what we do having the form set in our minds…The work should not be the form itself…I can thread a needle as a form with a thread as a form, but if the thread was always changing its form, why would I, and how could I???I first must nail down what the form is, and for that reason all forms are conserved…That is the meaning of identity…
I repeat, the number 0.9[bar] is not changing. It is always the same single number. It always holds the same single place on the number line. It is the number representing the unchanging singular destination of the grasshopper. Even though the grasshopper moves, his destination is always the same. And if you apply some logical thinking as illustrated in my previous post, the only possible destination of the grasshopper is 1. Hence 0.9[bar] = 1.
The work we do with 0.9[bar] = 1 set in our minds is useful scientific modeling work, as I discussed and illustrated earlier with the radioactive decay example. Certainly nothing in the ordinary world will be usefully represented directly by 0.9[bar]. For anything like that, we’d just use 1 as you have said many times. But 0.9[bar] = 1 comes about as a consequence of certain natural, simple models of common physical processes. In these cases, 0.9[bar] represents a result from an idealized model that approximates a real-world event or process.
.999… is not even a number… fractions are not numbers…Ones are numbers…Two ones are two…Three ones are three…You might represent fractions, but you can hardly show what they represent, so 1/3 is the problem, and not the solution; and 1/2 is the problem, and not the solution… The more these problems are changed in form the less the reveal any truth, that is: As a concept that can be compared to a reality…It is easy to represent one unit as one on a page, and notch in a gun, or a finger on a hand…In that instance we are not making a statement of equality, but of identity…One is one…If .9[bar]could represent one, then the world would spare me, and write one in place of it… We reduce our fractions… We would not try to say 3/3, or 5/5 were one, or put them needlessly in any equasion, let alone as an answer…We know it is one,and write it so… Don’t try to represent reality as so many fractions…First of all, it is not… It adds complexity without end…How many ones do your have in your pocket if you count every coin as one??? If you count every coin as two halves, or as three thirds or as ten tenths, does it show any better the reality of the situation, or does it work only to confuse reality by making an unnatural division of nature??? We simplify our nature, our reality with forms/concepts, of which math is only one example… Cut everything with Occam’s razor…
If the only thing you ever want to do with numbers is count coins in your pocket, then I agree with you – there’d be no point in using 0.9[bar]. But there’s more to numbers than counting things! Even in the grocery store things are measured in finite decimals, like 1.73 pounds, and 1.73 is just a way of writing the fraction 173/100. If 1.73 isn’t a number, then what is it, and why can we add and subtract and multiply and compare with it just like a number?
1/3 cup of water is an amount of water such that, when you put 3 of those amounts together, you get a cup of water. I can do that in my kitchen. I think that shows exactly what 1/3 represents. It’s a little more complicated to represent than 1, and the representation depends on the idea of 1 cup for its existence; but the representation of 1/3 cup is just as precise and just as connected to reality as that of 1 cup of water.
I agree that we would not put them in an equation needlessly. The point is that sometimes there is a need. For example, suppose we wish to add 1/3 + 1/5. To do this we write
1/3 + 1/5 = 11/3 + 11/5 = (5/5)(1/3) + (3/3)(1/5) = 5/15 + 3/15 = 8/15.
In the middle steps of this solution, we represent 1 as both 3/3 and 5/5 because doing so allows us to create a common denominator for the fraction. The “reduced” form of 1 will not help us solve this problem – a more complex representation of 1 is needed. Also, “answers” have no special status or special need to be reduced in math, because answers don’t always answer all questions. Sometimes your answer becomes a tool to solve a new problem. But solving the new problem may require representing your old answer in a more complicated way. For example, our answer from the previous problem is 8/15. Suppose we want to sum 1/3 + 1/5 + 1/30. It will save us some work to use our answer from the previous problem: 1/3 + 1/5 = 8/15. But we need to turn our answer into 16/30 so that we can get it into a common denominator with 1/30 and get the final result, 17/30.
Therefore, the “simplest” representation of a number may not be the most useful or valuable one. It depends on what you’re using the number for.
It’s the same way with 0.9[bar]. For example, suppose you need to subtract 0.173 from 1. Is it easier to do the subtraction as 0.9999… - 0.173, or as 1.000… - 0.173? Most people would find the first subtraction easier to do, because the second would require several carries. And it would be much worse if you had to subtract an infinite decimal from 1. For example, suppose you wanted to subtract 0.12356[bar] from 1. Is that easier as 0.9[bar] - 0.12356[bar] or as 1 - 0.12356[bar]? Again, the first way requires no carries.
Numbers have many equivalent representations in math, and each representation has its own uses. Often (as with 1/3 or .9[bar]) each of those uses are good for different problems. But those problems are all real and important. They come from physics, finance, or even basic cooking. I agree that it’s good to keep things simple, but we will not oversimplify to the point where all we can do with our math is count coins.
It took 35 pages of debate, but I think it’s safe to say, we finally settled the argument! 