1=.999999...?

[Pnoom]

1.000[bar] is a whole.
0.999[bar] is not.

1.000[bar] = 1
0.999[bar] does not.

What does 1/3 equal in decimals?

0.333[bar] is the accepted way we can represent it in decimal form. The number is derived from 1 divided by 3.

And 3*1/3 = 3/3 = 1, correct?

So 3*0.333[bar] = 3*1/3 = 1.

Now multiply 0.333[bar] by 3 the long way. Multiply every digit by 3 (carrying ones where necessary, etc).

You get 0.9999[bar].

So 0.999[bar]=3*0.333[bar]=3*1/3=3/3=1

[Sidhe]

heres an example....

say we want to travel 4 miles east in an electric car.

the problem is that each time you can only travel half the distance that you just traveled.

the first distance you travel is 2 miles, the next is 1 then .5, .25, .125... and so on.

you will for eternity try to travel a total of 4 miles but it will never happen.

the sequence of numbers converges to 4, but still will never equal four.

similarily, if .999[bar] really exists than it is not converging to 1, it is converging to 1-(1/infinity)

i have already demonstrated why infintesimals cannot be quantified, and if .999]bar] exists (the 9's can go on forever) than an infintesimal must exist.

once again i have shown why infintesimals are not quantified in decimal values...

don't just say im wrong... prove me wrong...

don't be a slave to symbols and numbers, use reason to prove me wrong, not the same old proof i have already objected to

Xeno's paradox has been proven to be a faulty thought experiment about 1,000,000 times the only problem is that to understand why you'd need to understand some calculus. But let's just say that the relation between distance and time is not as Xeno thought. And that as a ball bounces it will eventually approach the floor and remain still and leave it at that. If you're interested Google Xeno's paradox and there are hundreds of examples that explain it how it is pretty much a solved problem in maths and has been for thousands of years, Archimedes for example was the first to solve it, using something approximating calculus or Taylor series. Also take a brief look at the logarithmic decay function (mostly used in radioactive decay) explains Xeno's paradox is not a paradox at all just sloppy maths. Put it this way if it was true an amount of radioactive caesium would never decay to 0, nor would half lives be half.

Put simply time and motion is continuous not in the way described by Xeno but by that described by integration and differentiation, since those models work with stupendous accuracy and Xeno's paradox doesn't I would go with the maths of today.

[Rocket]

And 3*1/3 = 3/3 = 1, correct?

So 3*0.333[bar] = 3*1/3 = 1.

Now multiply 0.333[bar] by 3 the long way. Multiply every digit by 3 (carrying ones where necessary, etc).

You get 0.9999[bar].

So 0.999[bar]=3*0.333[bar]=3*1/3=3/3=1

No. 0.333[bar] does not actually equal 1/3. It is just a representation.

[Sidhe]

And 3*1/3 = 3/3 = 1, correct?

So 3*0.333[bar] = 3*1/3 = 1.

Now multiply 0.333[bar] by 3 the long way. Multiply every digit by 3 (carrying ones where necessary, etc).

You get 0.9999[bar].

So 0.999[bar]=3*0.333[bar]=3*1/3=3/3=1

No. 0.333[bar] does not actually equal 1/3. It is just a representation.

It doesn't true but it would if infinity was real. That is the point of the concept infinity. Infinity means something of which there is nothing larger, which means that .999...=1 because it is infinite. So one cannot exceed it. That is the infinity paradox of maths that is used to make all maths work. It's also very logical too, it's just some people don't get the logic and some people do.

[Wonderer]

I don't mind you guys perusing your own facet of discussion in this thread, but i just ask that you guys try to keep things respectable.

stick to the issues, and watch out for appeals to the person, thanks.

[Wonderer]

Infinity means something of which there is nothing larger, which means that .999...=1 because it is infinite. So one cannot exceed it. That is the infinity paradox of maths that is used to make all maths work. It's also very logical too, it's just some people don't get the logic and some people do.

"Infinity means something of which there is nothing larger, which means that .999...=1 because it is infinite."

I don't get that logic.

to me infinity is something which has no finite quantity. you cannot say that infinity means nothing is larger, because then the largest "thing" in question can have a finite quantity aswell as being considered "infinite".

i don't think we need paradoxes and proofs to make math work, it works regardless of how we perceive infinity.

you say that infinity makes .999[bar] equal one, i say it makes it as close to 1 as possible.

c'est la vie

[Sidhe]

To me infinity is something which has no finite quantity. you cannot say that infinity means nothing is larger, because then the largest "thing" in question can have a finite quantity aswell as being considered "infinite".

This is your problem you just said the same thing as I did, but you don't realise it.

If you add something to infinity it is still all there is.

I don't think there's a problem with maths, I think there's a problem with you and maths and how you understand it. Sincerely.

Everything that is anything else is smaller by definition, because it's a sub set of everything. I know I've said this already but hell I'll give it till 00:00 my time, and if you haven't grasped it I think I'll just have to accept you never will. As I said before some people get concepts like this some people don't, don't make the mistake of thinking what you think is right by definition of you being you. That way lies idiocy. There comes a time particularly in maths and science where a resort to consensus and authority is all there is left, this is that point. You either accept that or abandon hope of ever understanding maths. If the concepts are above you it's analogous to accepting that I can't speak Chinese, and no matter how much I hear it I will not understand it without some means of analogy, like someone pointing at a dog and saying quang or whatever a dog is in Chinese. I've seen this argument a lot and some people really just can't get it. I don't know maybe their brains are just wired differently, maybe they have no maths genes? But it's never been resolved by a series of brilliant mathematicians on the forums I visit going, OMFG Now wait a minute, I've wasted my career, this guy with a highs school level of education is onto something, all maths is wrong and it's back to square 1! I don't think it can because it's just not wrong, it's axiomatically correct and logical. That's why these people use it. There's not some conspiracy to hide the Numberwang code from you.

Sorry that reminds me of this:

[Wonderer]

you'll laugh when you read this, but one of the few times one of my careless typo mistakes causes confusion has come to pass

i actually meant to say "cannot"

if something has a finite value or quantity, it is finite, not infinite.

my sincere apologies

[Three Times Great]

1 does equal 0.9[bar]. its just mathematical fact, no way around it.

numbers are only abstract concepts, and in the abstract, the infinite limit which approaches a certain final finite number, is equal to that number in every way. in fact, technically since every number is only such a limit of infinite sums, ALL numbers are [bar] decimals (or rather, an open set of summed infinitely-receeding fractions) to infinity.

every rational number can be expressed as function curve on a graph, a curve which exponentially meets the number at infinity. yes, saying i have 0.9999999... slices of an apple and add them together, is not the same thing as saying that i have 1 apple. but, numbers are abstract; and you cannot have a material infinity, its a contradiction. so naturally, since infinity only exists in the abstract world of numerical ideas, that is where it is equal to the finite number it meets at infinity.

[Sidhe]

1 does equal 0.9[bar]. its just mathematical fact, no way around it.

numbers are only abstract concepts, and in the abstract, the infinite limit which approaches a certain final finite number, is equal to that number in every way. in fact, technically since every number is only such a limit of infinite sums, ALL numbers are [bar] decimals (or rather, an open set of summed infinitely-receeding fractions) to infinity.

every rational number can be expressed as function curve on a graph, a curve which exponentially meets the number at infinity. yes, saying i have 0.9999999... slices of an apple and add them together, is not the same thing as saying that i have 1 apple. but, numbers are abstract; and you cannot have a material infinity, its a contradiction. so naturally, since infinity only exists in the abstract world of numerical ideas, that is where it is equal to the finite number it meets at infinity.

Interestingly although 1 is the prime numberwang, .999... is not numberwang, however .999... lim x---->{infinity-1} is of course numberwang. Not a lot of people know that.

I think it's all been said infinities don't exist maths axioms do and they are correct by definition that's what an axiom is. You can't argue that the concept of 1 is illogical any more than you can that .999...=1. It's pointless.

[Juggernaut]

1 does equal 0.9[bar]. its just mathematical fact, no way around it.

numbers are only abstract concepts, and in the abstract, the infinite limit which approaches a certain final finite number, is equal to that number in every way. in fact, technically since every number is only such a limit of infinite sums, ALL numbers are [bar] decimals (or rather, an open set of summed infinitely-receeding fractions) to infinity.

every rational number can be expressed as function curve on a graph, a curve which exponentially meets the number at infinity. yes, saying i have 0.9999999... slices of an apple and add them together, is not the same thing as saying that i have 1 apple. but, numbers are abstract; and you cannot have a material infinity, its a contradiction. so naturally, since infinity only exists in the abstract world of numerical ideas, that is where it is equal to the finite number it meets at infinity.

Interestingly although 1 is the prime numberwang, .999... is not numberwang, however .999... lim x---->{infinity-1} is of course numberwang. Not a lot of people know that.

I think it's all been said infinities don't exist maths axioms do and they are correct by definition that's what an axiom is. You can't argue that the concept of 1 is illogical any more than you can that .999...=1. It's pointless.

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...

[gib]

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.

[Three Times Great]

1 does equal 0.9[bar]. its just mathematical fact, no way around it.

numbers are only abstract concepts, and in the abstract, the infinite limit which approaches a certain final finite number, is equal to that number in every way. in fact, technically since every number is only such a limit of infinite sums, ALL numbers are [bar] decimals (or rather, an open set of summed infinitely-receeding fractions) to infinity.

every rational number can be expressed as function curve on a graph, a curve which exponentially meets the number at infinity. yes, saying i have 0.9999999... slices of an apple and add them together, is not the same thing as saying that i have 1 apple. but, numbers are abstract; and you cannot have a material infinity, its a contradiction. so naturally, since infinity only exists in the abstract world of numerical ideas, that is where it is equal to the finite number it meets at infinity.

Interestingly although 1 is the prime numberwang, .999... is not numberwang, however .999... lim x---->{infinity-1} is of course numberwang. Not a lot of people know that.

I think it's all been said infinities don't exist maths axioms do and they are correct by definition that's what an axiom is. You can't argue that the concept of 1 is illogical any more than you can that .999...=1. It's pointless.

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...

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.

[Juggernaut]

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....

[aporia]

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?

[Juggernaut]

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...

[aporia]

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.

[Juggernaut]

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...

[aporia]

Here's how it works:

1. 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.

2. 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].

[Wonderer]

Here's how it works:

1. 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.

i would say 1/10^(n+1)

2. 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].

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.

[aporia]

i would say 1/10^(n+1)

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.

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

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.

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.

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.

[Juggernaut]

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...

[Juggernaut]

Here's how it works:

1. 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.

2. 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].

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...

[Juggernaut]

Here's how it works:

1. 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.

2. 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].

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...

[aporia]

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.