## 1=.999999...?

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### Re: 1=.999999...?

Wonderer wrote:
Juggernaut wrote:I don't think math deals with real quantities, excepting the essential quantity to math, of one. And 1/3, and .333... are not numbers exactly, but ratios. When seeing 1/3, what must be know for the number to have significance is one. If it is a pound, a mile, or a bag of sugar, then the 1/3 has some meaning. Apart, as part of a system of abstraction, 1/3 is not an answer, but is a question phrased as an answer.

i'm talking about 1 orange, 2 oranges 1 and a half oranges. thse are real numbers.

i also think that 1 third orange is a real quantity.

our number system simply prevents us from writing it out in digit form.

the amount confusions that result from this simple fact are incredible.

Fractions are not quantities, but ratios in reference to certain quantities. One is the essential concept to all numbers. Number is a conceptual manifold, but all numbers, and all ratios are in relation to one, and that specific ratio of two to one, or three to one, or four to one; only holds true in math. Since reality actually presents us with no ones that are exactly like any others except as conceptual units, which is how we percieve reality anyway, the line between math and reality will always be there. Let me add: If it were possible to concieve of 1/3 as an eactual number, a quantity, then it would not be presented as a problem, one divided by three; but as some solution. In addition, if the world is made up of units, and so concieved, with each idea representing one thing different from all others, then, two oranges does not violence to the concept of orange, but 1/3, or 1/2 of an orange does violence to the concept of orange, since no actual orange can be made from any number of fractions of oranges. I/2 of an orange may depend upon the definition of an orange for its definition, but it is the same with a rotten orange since no rotten orange can define an orange. Does that make sense to you?
Juggernaut
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### Re: 1=.999999...?

Wonderer wrote:
Juggernaut wrote:I don't think math deals with real quantities, excepting the essential quantity to math, of one. And 1/3, and .333... are not numbers exactly, but ratios. When seeing 1/3, what must be know for the number to have significance is one. If it is a pound, a mile, or a bag of sugar, then the 1/3 has some meaning. Apart, as part of a system of abstraction, 1/3 is not an answer, but is a question phrased as an answer.

i'm talking about 1 orange, 2 oranges 1 and a half oranges. thse are real numbers.

i also think that 1 third orange is a real quantity.

our number system simply prevents us from writing it out in digit form.

the amount confusions that result from this simple fact are incredible.

Fractions are not quantities, but ratios in reference to certain quantities. One is the essential concept to all numbers. Number is a conceptual manifold, but all numbers, and all ratios are in relation to one, and that specific ratio of two to one, or three to one, or four to one; only holds true in math. Since reality actually presents us with no ones that are exactly like any others except as conceptual units, which is how we percieve reality anyway, the line between math and reality will always be there. Let me add: If it were possible to concieve of 1/3 as an eactual number, a quantity, then it would not be presented as a problem, one divided by three; but as some solution. In addition, if the world is made up of units, and so concieved, with each idea representing one thing different from all others, then, two oranges does not violence to the concept of orange, but 1/3, or 1/2 of an orange does violence to the concept of orange, since no actual orange can be made from any number of fractions of oranges. I/2 of an orange may depend upon the definition of an orange for its definition, but it is the same with a rotten orange since no rotten orange can define an orange. Does that make sense to you?
Juggernaut
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### Re: 1=.999999...?

Juggernaut wrote:Fractions are not quantities, but ratios in reference to certain quantities. One is the essential concept to all numbers. Number is a conceptual manifold, but all numbers, and all ratios are in relation to one, and that specific ratio of two to one, or three to one, or four to one; only holds true in math. Since reality actually presents us with no ones that are exactly like any others except as conceptual units, which is how we percieve reality anyway, the line between math and reality will always be there. Let me add: If it were possible to concieve of 1/3 as an eactual number, a quantity, then it would not be presented as a problem, one divided by three; but as some solution. In addition, if the world is made up of units, and so concieved, with each idea representing one thing different from all others, then, two oranges does not violence to the concept of orange, but 1/3, or 1/2 of an orange does violence to the concept of orange, since no actual orange can be made from any number of fractions of oranges. I/2 of an orange may depend upon the definition of an orange for its definition, but it is the same with a rotten orange since no rotten orange can define an orange. Does that make sense to you?

I imagine 1 simple as a round black sphere. in order to imagine 1/3, that sphere get's sliced into a 1.3'd piece.

in this manner numbers can be manipulated in any way we currently know of. it's basically a visual fraction.

You said that every number is measured in terms as a ratio from 1. have you every thought of the possibility of measuring every number as a ratio from 4? or 10? or 34576?

1 is only the perceivably simplest ratio.

It doesn't matter that in the real world we cannot divide people in 2, make a perfect sphere, reconstruct oranges or measure a perfect inch.

in our minds we can make the operations perfect.

which is where i get my confidence that 1/3 x 3 = 1

To pervert that equation in anyway, however rational, is a deviation from the simplest form.
Whatever makes you happy.

Wonderer
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### Re: 1=.999999...?

You can say 1/3 X 3 = 1 because it is the reverse of an operation 1 divided by 3, is the same problem, 1/3, which is never solved because it is unsolvable, unless the unit is not a true unit to begin with. If you take 1/3 of a sphere, you change the whole nature of the sphere, having changed its unity and its identity. So, instead, we ask what is 1/3 the volume, or what is 1/3 the weight, which are separate qualities from the sphere that it has as other objects has them.

What we do with impossible problems in math is not different than what we do in reality with impossible problems. We just push them forward. This is a small thing: whether math can be a practical thing of universal utility, or whether even it must conform the the rules of logic and identity as all other concepts. Perhaps I am saying this wrong, Math is not a concept, but a coneptual manifold; yet one is, and an identity. Math is rather, a system of logic based upon qualitative values made quantitative. We have the same problem with all moral realities. Look at justice. It is always going to be unitary, that is, one thing, in concept and identity. But how justice is defined, as a general abstraction is meaningless when applied to specific questions. At a minimum, justice is always going to have two side even if it is one thing. Every instance of it is different, because the situations and the people involved are different. Just so, math says there is such a thing as a one, as a unit; but no one can be shown equal to any other when scrutinized, so equality is a value we judge, just as we judge justice, within a certain context. If we consider the element of time, thing grow even more confused. What justice was yesterday it will not be tomorrow. What a reasonable man might accept as just tomorrow, he may not today. If we multiply one by time, we get an answer that is plus or minus one. Now; it is possible that .999... might fall within the range of plus or minus one, but I do not think it has the range of expression. First it is a variable expression within limits, that is: .9 to ,999...infinity. But one can have meanings from zero, to infinity when multiplied by time. (One can be meaningless, or meaningful) To multiply .999... with time is like multiplying infinity with infinity, or a variable with a variable. Whether or not one can ever get a firm grasp on an answer is the question, and I don't think that could be the object in Math. Math seeks an answer. The answer is not whether .999... equals one; but whether .999... time two, equals two; and whether .999... times three equals three. We can say that 1 is .999... only when we know that as we know one by every ratio of one to every number. And this uncertainty is not the object of math. Though it is the object of justice.

As a practical matter there is no difference between .999... and one. It simply makes problems when it is supposed to make solutions. There is no honor at the end of it. There is nothing to get exercised about. One is just a concept, and math is just a form.
Juggernaut
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### Re: 1=.999999...?

yes. the rules of MATH are constructed to allow for imprecisions like in the OP.

But this is where i remain firm.

If math is suppose to be quantitative logic. (logic of manipulating quantities) then .333[bar] is where math and logic go seperate ways.
Whatever makes you happy.

Wonderer
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### Re: 1=.999999...?

Wonderer wrote:yes. the rules of MATH are constructed to allow for imprecisions like in the OP.

But this is where i remain firm.

If math is suppose to be quantitative logic. (logic of manipulating quantities) then .333[bar] is where math and logic go seperate ways.

Ahh. It's just a little fudge. Not like the big one, that all ones are equal. It makes me want to ask why it works at all, and my guess it that we are only talking about a gross equality to begin with, but also since it is in the form of a relationship, and a process, that people can correct errors as they proceed. Math does not give people the answer, but an answer. At its extreme it is what religion is: certainty without regard for truth.
Juggernaut
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### Re: 1=.999999...?

Wow, this is a really irritating thread.

Let me begin by saying to all the non-mathematicians, there is a *huge* difference between saying, in appropriate confidence in one's own intelligence coupled with humility stemming from lack of direct knowledge, "it seems to me that 0.9b != 1. What do you think?" and between saying, "I don't really know what I'm talking about, but nonetheless, I'm right and centuries of mathematicians are wrong!"

I'm not committing the fallacy of authority when I say that. Math is fundamentally different from the sciences in that, in empirical sciences, nothing is ever truly proven. In physics, there is no such thing as getting something perfectly correct and then moving on. While there is a great deal of moving on, there is always revision to be done to the original theory, or else continually more precise tests to continue to try to shatter the best available explanatory model. In math, there absolutely is such a thing as getting it right and moving on. One does not need to offer challenges to proven material, provided that the proof is solid. If someone were to challenge the proof of Fermat's Last Theorem, it wouldn't be completely unreasonable, as very few people understand the proof. But to seriously challenge the idea that 0.9b = 1 is complete and utter folly.

1) Wonderer, you ask good questions and justify them with good explanations of your intuition - but you then fail, completely, to be receptive to the information given to you by those more knowledgable. Aporia is *exactly* right in everything he has said, and clearly has some mathematical training. Furthermore, he has explained his position in detail and with admirable patience. Wonderer, you are wrong; 0.9b is equal to 1. Science has been wrong in the past and overlooked some things that in hindsight are obvious; but math doesn't work that way. Once something is truly proven in math, it will no longer be open to debate, EVER, because a valid proof is purely based on logic, and is completely outside any other consideration. Because of this, a broad and enduring mathematical consensus speaks volumes more (and more loudly) than a broad and enduring scientific consensus ever will.

2) Juggernaut, you seem to have interesting philosophical ideas, but at the same time, you seem to have the sort of mind much more suited to philosophical musings than mathematical examination. You seem to approach this thread staunchly from that perspective, and as a result, sadly, you seem to take very little from the accurate and informative posts of Aporia. It may be that your points would be considered valid, or at least interesting, from a Philosophy of Mathematics perspective; but from a pure Mathematics perspective, your objections are wrong. Indeed, more so, they are vacuous, seemingly based solely on the conflation of the symbol with the mathematical value it represents.

3) Math is not the logic of manipulating quantities. Math, more than anything, is simply logic applied to very particular sets of axioms. While it began as quantitative, it is now so abstract that notions of quantity only apply to certain parts of math.

4) INFINITESIMALS. Wonderer, you and others have brought up the idea that 1 - 0.9b = 1/infinity, or an infinitesimal quantity, multiple times. There are rich areas of mathematics devoted to the idea of the infinitesimal, and thankfully due to this, I can tell you how the concept of the infinitesimal relates to your objection.

First, in mathematical systems that possess infinitesimal quantities, they almost invariably possess infinite quantities. In such systems, it is completely accurate to say 1/infinity = infinitesimal, and conversely, 1/infinitesimal = infinity. Furthermore, in such a system, it is completely accurate to say that 0.9b = 1 - infinitesimal, and thus that 0.9b != 1. (Interestingly, in virtually all such cases, 0.9b is not equal to anything.) In this sense, your intuition is completely justified. Unfortunately, the real number system does not possess infinitesimal quantities other than 0.

Let me explain. The real number system consists of all numbers you can make by stringing together a finite collection of digits (from 0 to 9), appending a decimal, and tacking on a potentially infinite collection of digits after the decimal. In other words, every real number can be represented as

a1 a2 a3 ... a(m) (DECIMAL POINT) b1 b2 b3 ... b(n) b(n+1) ...

where the a values don't go forever, but the b values can.

Every real number can be written this way. Period. Anything that cannot be written this way cannot be considered a real number. You have to know the value of every a and b value exactly or else you don't know the real number. And every a and b value has to have a FINITE INDEX associated with it. In other words, you can have a b1000 (the 1000th digit after the decimal) and a b1,000,000, but you cannot have a b(infinity).

So if 0.9b != 1, the question naturally arises, what is 1 - 0.9b? The answer, "an infinitesimal", or 0.0......01, are COMPLETELY unsatisfactory and insufficient. In order to give an answer, you have to specify what all the a and b values are. So if our answer is going to be, in essence, "a bunch of zeroes with a 1 at the end", we have to specify WHERE EXACTLY that "1 at the end" is, or else we aren't talking about a real number. And because you can't have a b(infinity), you can't just say "an infinite number of zeroes". Well, 1 - 0.9b clearly isn't a THOUSAND zeroes and then a 1. It isn't a MILLION zeroes and then a 1. It's smaller than any of those. In fact, no matter how far out you put that 1, (1 - 0.9b) is smaller than that number. Because of the pivotal fact that you can't put that 1 infinitely far out, you can never construct a positive real number that is equal to the difference.

Or, to phrase the entire argument more simply: in the real number system, the only infinitesimal value is 0. Thus, if (1 - 0.9b) is equal to an infinitesimal, then it is equal to 0.

~~~~~~

There are several concepts I've put forth here without attempting to justify why those are true. This is where I retreat to the non-fallacious version of the argument of authority. I went to school for this. I'm a trained mathematician. While we're trying to give you a good intuitive and semi-formal exposure to these concepts, the minutia involved are concepts you can't master overnight, and should read a good book, or take a class, if you want to get down pat. They aren't things you can reasonably expect to get out of a post; nonetheless, they're accessible, so that if you're really interested in debating the topic intelligently, I can refer you to a good book or the appropriate university class so you can get the necessary background.

I will toss this out there, though. You can see that the real numbers don't contain any infinitesimals (other than 0) if you believe these facts:

1) You can divide by any real number other than 0 and get another real number
2) Every real number is either larger than 0, smaller, or equal to 0.
3) If 0 < a < b, 0 < 1/b < 1/a
4) The real numbers don't contain any infinite values. (This isn't to say that infinity doesn't exist - just that infinity doesn't exist as a real number.

If you believe these, then you will believe that if the real numbers contain infinitesimals, they also contain "infinities", which we agree by hypothesis that it doesn't. Here's a more detailed argument:

Let's suppose there were a real number that was an infinitesimal but not 0. Call it m. If m is positive, then 1/m would necessarily be positive. Since m is infinitesimal, any "normal", non-infinitesimal positive real number will be bigger than m, no matter how small it is. (E.g., if m = 0.000....1 [infinitely many zeroes and then a 1], 0.1 is bigger than m. 0.01 is bigger than m. 0.000000000001 is bigger, and so on.) This implies that 1/m is bigger than 10, 100, 1000000000000, and so on, and thus that 1/m must be infinitely big.
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### Re: 1=.999999...?

It is the logic of the thing. What math is is Logic, and the first rule of that logic as I can see it isthat one is one. Now, we are saying something different, that .999... equals one. Which I could see if it were really an operation like 5-4= but that again is an operation of logic. Now, in its full context, .999... can be seen as an operation because 1/3 is an unanswered question, an inigma, and 3X1/3 is also an inigma of sorts, though logic tells us it should be one. Look at it this way, If you were to multiply the transendental pi by three hundred and sixty degrees, you should be able approximate a sphere, but only to approximate it. Since you can only approximate a value for 1/3 you can get no more than an approximation by multiplying that. Consider it this way again. One is not only an identity, but a ratio of one to what ever number you can imagine. But as no one is ever the equal of another in reality, so the ratio of one to one, or one to any number is only appoximate, and now to divide this approximation into three, which is to say by 3, an approximation, and then do this impossible task of finding an equal one third out of the division of three approximations, and multiplying that, by 3; when logic tells us the answer, and our numbers tell us otherwise... .999... is the approximation of one arrived at by dividing and then multiplying approximations of one .

Not one part of the whole entire cosmos is firm. The earth beneath our feet trembles and floats. The fact of life is that we must live with uncertainty, and if uncertainty, then inequality. Math denies this inequality, and does so falsely. Math is a system of logic unto itself, and only when applied to reality measure for measure is it at all useful. So, it should be a tamed dog which does as told. Math is not our master, but we are the masters of it. Don't get tied up in digits. If you owned them all it would not give you one part of terra fima.
Juggernaut
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### Re: 1=.999999...?

twiffy, i have read your entire post.

i think understood all of it. i was already aware of most of it.

At the end when you make a point that 1/m where m is an "infintesimal" is equal to infinity, or "infinitely big".

i would agree. what is the point you were trying to make with that?
Whatever makes you happy.

Wonderer
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### Re: 1=.999999...?

If I may,
Twiffy wrote:1) You can divide by any real number other than 0 and get another real number
2) Every real number is either larger than 0, smaller, or equal to 0.
3) If 0 < a < b, 0 < 1/b < 1/a
4) The real numbers don't contain any infinite values. (This isn't to say that infinity doesn't exist - just that infinity doesn't exist as a real number.

By hypothesis, there is not infinite real number. So any number whose inclusion would entail an infinite number is excluded. Since including an infinitesimal m would entail that its reciprocal, 1/m, be included, and since its reciprocal would be infinite, the infinitesimal cannot be a real number.

Do I have that right, Twiffy?
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Carleas
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### Re: 1=.999999...?

I have already ceded the fact that mathematically, through ways like the sum of an infinite series, we can sum .9b at 1.

I don't however believe that creating .9b is logical in thr first place.

when we multiply .3b by 3, if we could do it all "properly" wouldn't it come out to be 1?

the only reason for .3b and .9b to even exist is because of the impracticality of the infinite versus our finite existence.

What is .9b/1 ?

You spoke of infinitesimals always requiring infinites, well the infinite 9's seem to do the trick.

Maybe i'm just stupid, but this infinitly long string of number business just seems overly complicated for something so simple as 1/3 x 3 =1
Whatever makes you happy.

Wonderer
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### Re: 1=.999999...?

Carleas wrote:If I may,
Twiffy wrote:1) You can divide by any real number other than 0 and get another real number
2) Every real number is either larger than 0, smaller, or equal to 0.
3) If 0 < a < b, 0 < 1/b < 1/a
4) The real numbers don't contain any infinite values. (This isn't to say that infinity doesn't exist - just that infinity doesn't exist as a real number.

By hypothesis, there is not infinite real number. So any number whose inclusion would entail an infinite number is excluded. Since including an infinitesimal m would entail that its reciprocal, 1/m, be included, and since its reciprocal would be infinite, the infinitesimal cannot be a real number.

Do I have that right, Twiffy?

do "infinite digits" count as a disqualifier?
Whatever makes you happy.

Wonderer
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### Re: 1=.999999...?

Wonderer wrote:I have already ceded the fact that mathematically, through ways like the sum of an infinite series, we can sum .9b at 1.

I don't however believe that creating .9b is logical in thr first place.

when we multiply .3b by 3, if we could do it all "properly" wouldn't it come out to be 1?

the only reason for .3b and .9b to even exist is because of the impracticality of the infinite versus our finite existence.

What is .9b/1 ?

You spoke of infinitesimals always requiring infinites, well the infinite 9's seem to do the trick.

Maybe i'm just stupid, but this infinitly long string of number business just seems overly complicated for something so simple as 1/3 x 3 =1

You said a mouffull there:infinite v finite existence.

How far can math get from what we can verify before we all write it off as hogwash? I mean, if it were God we were speaking of as an infinite, many math heads would say we cannot know it; but they think because numbers say it instead of a priest, that we can know it. I'd say, have the knowledge; and then abstract it, and don't bother abstracting further than you can verify. In the real world .9 is one, and .99 is one, and .999 is one. Look at the actual variation of units, and since none can be more than one, all other units in comparison are so much less than one, and yet, one. But this we can know with out a means of extreme abstraction. It does not require a mathematical proof since it has a social proof. There is no proof for one as .999... -that does not do damage to the most basic concept of math. One is one. It is already an abstraction of reality not useful because it is exact in any sense, but because it stays put, and holds its value. Ruin one, and the whole of math falls like a house of digits.
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### Re: 1=.999999...?

Juggernaut,

Here is a reply to your post of Jul 2. Forgive the long time it took me. I felt like we were talking past each other and couldn't figure out a way to connect our intellectual universes. But I want to try again.

In math only one, as a number has meaning, and there is only one 'number' necessary in concept.

What is the intellectual content of this statement? What do you mean by one "having meaning"? How do you define that?

Meaning, as I understand it, is built up from a web of defining associations and distinctions. For instance, one has meaning as a concept because (for example) I can tell you when you have put one quarter in front of me, and when you have not. Because I can use the concept to analyze and structure my world, the concept has meaning. But the same is true of two, three, sqrt(2), pi, and 0.9[bar]. It's just that the analysis and structuring I can perform with them is a bit more complex.

I agree with you that in some sense, we construct all of our numbers from one. Or at least that's one approach you can take -- traditionally you use a set-theoretic construction and start with the empty set (which is "one" object). But I don't understand how you can define "meaning" in such a way that only the number one has meaning, unless you have something radically unusual in mind when you say "meaning".

Since all other numbers get their meaning based upon one, they are signs, and one is the concept.

Again, explain this statement. How does numbers "getting their meaning based upon one" make numbers "signs", while only "one" is a true concept? What do you mean by sign and concept that you make statements like this? They have no concrete meaning to me whatsoever, and seem like just so much hot air.

In one of these dictionaries around here Idea was in part defined as: one thing. If this is so then the conceptual duck is a single duck having the qualities of all ducks.

Again, what does this mean concretely? Suppose I were to take this statement as written. Then I could go to a pond, point out a duck, and tell my friend "since this duck is a duck, it has the qualities of all ducks, hence it is a 'conceptual' duck!"

How can my friend understand this? Is he to just stand and nod solemnly in agreement? What is the intellectual meat of this sentence?

Now, there are many phenomena that seem to have meaning even though as infinites they cannot be concieved of. We all sort of give a subjective value to existence, or the cosmos, or to God; but we cannot measure the meaning against the reality...With finite reality we stand on firmer ground, and I think it is for that reason, in part, that people sometimes thought the concept preceeded the reality. It is because every concept represents a measure of perfection that reality does not.

Again, does this have a concrete meaning? Do the nouns and verbs refer to anything specific at all?

Every conceptual one is equal, and no real one is equal.

Okay, at this point I'll leave off asking you to get specific and try to guess your meaning. Although the request still stands.

If only 'conceptual' ones are equal, then why not interpret math as referring to only conceptual ones?

In saying 3-2 approximates 1, it is because only in math does one equal one.

So are you saying 3-2 = 1 but only "in math", or are you saying 3-2 only approximates 1 even in math? And what would it mean to work with 3-2 and 1 "outside of math"? They are mathematical concepts, there is no way to speak about them outside math. For example, what does it mean for numbers to approximate each other outside of math?

It is not something that can be demonstrated.

Demonstrated where, to what standards, how, why? What do you mean by demonstrate?

Rather you need the insight to realize how futile is math at giving an exact picture of reality.

I never said math can give an exact picture of reality. Math is an exact analysis of a system of abstract ideas which approximate or resemble certain parts of reality.

If you must think of math in a real-world context, don't think of it as giving approximate answers to real-world questions. Instead, it is giving exact answers to abstract questions, questions which approximate or resemble certain real-world questions in useful ways. For example:

Math question:
Q: What is 3-2?
A: 1.

Approximate real-world question:
Q: If I have three oranges and John steals two oranges from me, how many do I have left?
A: One orange.

If you derive the answer to the real-world question by doing math rather than looking in your current orange stock, you risk missing out on key facts. For example, maybe John felt some remorse and brought the oranges back, so you really still have three oranges. Or (as was discussed in the 1+1=24 thread) one could mess up the conversion of the real-world data to/from math, and get a nonsense answer that way. These errors come about not because math is inexact, but because the conceptual resemblance between math and the real world is incomplete or incorrectly conceived by the math user.

To summarize, the process of applying math to the real world can be modeled as a flow-chart:

real-world question --(approximately model as)--> math question --(solve exactly)--> math answer --(approximately model as)--> real-world answer.

This is why it makes no sense to say that 3-2 "approximately" equals 1. The approximation steps occur outside the domain of math, purely in the domain of modeling. There is no such thing as 1 or 3-2 "in the real world". They are purely mathematical objects. 1 orange is a real object, but 1 and 1 orange are not the same. 1 is an abstract mathematical object, 1 orange is a real physical sensible orange. The moment you add a quantitative modifier to a real-world noun, you have engaged in the process of approximate mathematical modeling and have thus left the strict domain of exact, pure mathematics.

aporia
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### Re: 1=.999999...?

Now, we are saying something different, that .999... equals one. Which I could see if it were really an operation like 5-4= but that again is an operation of logic.

What is an "operation of logic"? What makes 5-4 = 1 an "operation of logic" and 0.9[bar] = 1 not?

Keep in mind -- I can't emphasize this enough -- the proof that 0.9[bar] = 1 has nothing to do with the silly grade school proof where you write

1 = 1/3 * 3 = 0.3[bar] * 3 = 0.9[bar].

By the standards of real mathematics that is no proof at all, because in grade school neither 0.3[bar] nor 0.9[bar] are defined using rigorous logic. The real proof uses logic of the same sort that would be used to prove that 5-4=1, or that 1=1, or to construct the number 1 in the first place. 0.9[bar] = 1 is no less a logical statement than 5-4 = 1, it's just logic you're not familiar with.

You, Juggernaut, seem to be under the impression that the 1/3 thing is somehow essential to 0.9[bar] = 1, and are trying to critique it from that standpoint -- witness all your talk about ratios of 1 to 3 etc. But 1/3 * 3 is completely irrelevant, and middle school teachers have unfortunately done us all a disservice by presenting that "proof" as though it were the real reason 0.9[bar] = 1.

The middle part of your post is about 1/3 * 3, so I will let it alone...

The fact of life is that we must live with uncertainty, and if uncertainty, then inequality. Math denies this inequality, and does so falsely.

Pure math, strictly construed, says nothing about reality, as I discussed previously. It cannot confirm or deny uncertainty in the world, it knows nothing of the world's uncertainty. It is purely a logical system which bears some resemblance to many real world systems, and can be used to understand them. It is the approximation steps which introduce uncertainty in applying math to the real world. The pure math itself is exact.

Math is a system of logic unto itself, and only when applied to reality measure for measure is it at all useful. So, it should be a tamed dog which does as told. Math is not our master, but we are the masters of it.

There is no such thing as a perfectly tamed dog. All of our pets have their own uncompromisable nature at their core. To deny that core nature is to ruin the pet. To know where the taming ends and ruining begins requires extensive experience and wisdom.

Removing 0.9[bar] = 1 from math because it is counterintuitive is like removing a dog's hair and legs so that it will stop scratching. It ruins the nature and structure of both to accommodate a limited conception of what they must be. But you have never seen that mathematical structure so you can't know what is being ruined. It's like you've never seen a dog hunt, so you can't know how immeasurably ruined the dog is by the removal of its hair and legs.

Don't ruin the dog. Understand the dog for what it is, and if you don't like it get a different pet.

aporia
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### Re: 1=.999999...?

Tooo much for me right now with my hog half torn apart in the garage and my hands all covered with grease. But trust me, and be patient. I will say 1 has relationship already well established between itself and all other numbers. So, if 1 is thought equal to .999... then .999... must show the same ratio between itself and the other numbers as one has, and I think that is an impossible task.
Next; meaning is value. We think of one as a value. In fact, one is more meaning since to arrive at an equal value for one, one must disregard a lot of inequaliites between ones. So, we say a cow is a cow is a cow, just as one is a one is a one. The value is conserved because we are counting wholes, individuals, and units; and not counting equalities. Only in math is one equal to one. In real life we concieve of things, and all conceptions, cow to cow, cat to cat, dog to dog; are identical, which is only a sort of equality. And certainly it can be more complex. All lines are lines, and the differences between them do not change their conserved quality, their identity. All governments are governments even if they do not govern, but rule by terror. So, if I say the only number in math with meaning is one, it is because all numbers as a system of signs have their meaning in ratio to one. For the meaning of all numbers to not be nebulous and muddy, the value, as a abstraction of reality, of one has to remain one. So long as that meaning is conserved, the system of signs stands upon that base. I'll try to answer in more detail. Later.
Juggernaut
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### Re: 1=.999999...?

This topic is very interesting. Though using the Cauchy idea, the reality is that it is simply wrong. The sequences never truly converge even though they appear to at some point near infinity in the convergence. It's all relative to the place of the supposed convergence and the values immediately preceding that convergence.

Rocket

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### Re: 1=.999999...?

Rocket wrote:This topic is very interesting. Though using the Cauchy idea, the reality is that it is simply wrong. The sequences never truly converge even though they appear to at some point near infinity in the convergence. It's all relative to the place of the supposed convergence and the values immediately preceding that convergence.

There are not more than one one, but there are more than one of me, and when you're looking at me I'm going to hit you, and when you are looking at the other me, I'm going to hit you again, and I and I am going to keep it up until you are pulling your eyes out...

Perhaps, if it did converge, it would go beyond at some point, say at infinity squared...
Juggernaut
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### Re: 1=.999999...?

This is a blast from the past.

I still hold that in reality the numbers do not converge to infinity. I accept .333 bar simply as an imperfect decimal which does not in reality reflect the value of 1/3.

In theory it does, but we cannot crunch theory into a calculator.

Thus we have various imperfections...
Whatever makes you happy.

Wonderer
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### Re: 1=.999999...?

Wonderer wrote:This is a blast from the past.

I still hold that in reality the numbers do not converge to infinity. I accept .333 bar simply as an imperfect decimal which does not in reality reflect the value of 1/3.

In theory it does, but we cannot crunch theory into a calculator.

Thus we have various imperfections...

Well; as the Muslims say: Only Allah is perfect, and that is why I love them... Perfection for human beings or human forms or human reason is impossible... Good enough is good enough, and close enough is close enough....
Juggernaut
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### Re: 1=.999999...?

Juggernaut wrote:
Wonderer wrote:This is a blast from the past.

I still hold that in reality the numbers do not converge to infinity. I accept .333 bar simply as an imperfect decimal which does not in reality reflect the value of 1/3.

In theory it does, but we cannot crunch theory into a calculator.

Thus we have various imperfections...

Well; as the Muslims say: Only Allah is perfect, and that is why I love them... Perfection for human beings or human forms or human reason is impossible... Good enough is good enough, and close enough is close enough....

There is a train of thought which argues anything perfect will not create something that is not perfect.

Rocket

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### Re: 1=.999999...?

Rocket wrote:
Juggernaut wrote:
Wonderer wrote:This is a blast from the past.

I still hold that in reality the numbers do not converge to infinity. I accept .333 bar simply as an imperfect decimal which does not in reality reflect the value of 1/3.

In theory it does, but we cannot crunch theory into a calculator.

Thus we have various imperfections...

Well; as the Muslims say: Only Allah is perfect, and that is why I love them... Perfection for human beings or human forms or human reason is impossible... Good enough is good enough, and close enough is close enough....

There is a train of thought which argues anything perfect will not create something that is not perfect.

Great...Show me something perfect... We conceive of our forms as perfect...The conceptual circle is a perfect circle... No circle in reality is perfect... One, is a form, and a perfect form... As such, one, as a form equals one, as a form...Show me the one in reality that is equal to any other one...The problem is not only with the form: One, upon which all numbers are based; but with the form: Equality, which is a form we help relate all forms through....

Like many questions in philosophy, this one tells more about the people answering it than any sort of objective truth.. If we can allow our selves to say one third equals .3333... then of course .9999.... must equal one... As a practical matter the fact cannot be demonstrated, and yet, logically it must be true...What would the point be of representing a monad with a series infinitely more complex??? Is it not the job of logic and math to symplify???Yes, and to be more exact...Using math as a form has inevitably pushed people toward a more exacting reality...In reality, no One is One, but as a result of math, and our consequential technology, one is now, in a technological sense, much closer to one.... I would argue that the most finely tuned devices we have are our thermonuclear weapons, and the intercontinental balistic missiles meant to carry them; and all considered, both deal in gross equalities... Look... Is there a critical mass in each bomb??? I would bet, more than... Is there enough explosives to create the correct assembly speed??? I would bet, more than...Perhaps ninety percent of the time, when I drive down the road I do not have enough cylinders in my motor, but perhaps, twice as many; but that gives me enough motor for nearly every conceivable situation... Does so much over kill make sense??? Sure, because if I had enough engine for eigthy percent of my driving it would be one hundred percent useless...When I get there, I have to get a hundred percent there, and so my transportation has to be able to get there, and back... I hope this does not bore you... The question is never what it seems...
Juggernaut
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### Re: 1=.999999...?

It's correct. Infinity is nonsense.

10(.999[bar]) would be = 9.99[bar]0, which is nonsense (nothing can follow after the "[bar]", i.e., beyond infinity).

The difference between 1 and .999[bar] would be = 0.000[bar]1, which is nonsense (see above). So the difference between 1 and .999[bar] is = 0.000[bar], which is 0.
"Someone may object that the successful revolt against the universal and homogeneous state could have no other effect than that the identical historical process which has led from the primitive horde to the final state will be repeated. But would such a repetition of the process--a new lease of life for man's humanity--not be preferable to the indefinite continuation of the inhuman end? Do we not enjoy every spring although we know the cycle of the seasons, although we know that winter will come again?" (Leo Strauss, "Restatement on Xenophon's Hiero".)

Sauwelios
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### Re: 1=.999999...?

1=1.0

3/3=1.0

1/3=.333333...

(3)x(1/3)=(3)x(.333333...)

1=.999999... <> 1.0

The fractal conversion to decimal placeholders breaks whole numbers into particles. (whole = sum of parts)

Simple. Finished. Done. Kaput.

Say nothing more about it.
Philosophers trade & deal in water, you know? ~ The Universe. Everybody drinks water. Sophists choke on water.

If I have learned of anything in life thus far, then I have learned of the absolute existence of Good & Evil ~ Good & Evil exist absolutely. Some men are born to do Good. Some men are born to do Evil. These two forces reassemble & represent the Primal Movement of the Human Spirit, Human Drama, and Human Trauma. Those who pay homage to God are the Good. Those who resist this tribute are Evil. And only the godless are powerless to choose their Fate & Destiny at all.

So come now and let us crucify this Evil! Let us crucify Jesus Christ of Nazareth once again! Let us watch Rome burn to the ground!

Form and Void
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### Re: 1=.999999...?

Sauwelios wrote:It's correct. Infinity is nonsense.

10(.999[bar]) would be = 9.99[bar]0, which is nonsense (nothing can follow after the "[bar]", i.e., beyond infinity).

The difference between 1 and .999[bar] would be = 0.000[bar]1, which is nonsense (see above). So the difference between 1 and .999[bar] is = 0.000[bar], which is 0.

As Aristotle said; there is a ratio between 1 and 2, 1 and 3, and etc... But there is not ratio between 1 and 0... And part of the drawback in considering one as .999... is that it cannot be shown as a ratio to all the other numbers...Even if it is logically correct it is practically pointless... Can we ever really say that 1/.999... equals one??? Occam's razor cuts both ways...So if 1/3 is .333....; then .333...X 1 is 1...The only possible way to arrive at .999... is by the reverse of an operation...Why then, and what would be the point of representing one as so many superfluities??? The object of math is to make clear what in nature seems complex, and is not to make complex by math what in reality is clear...
Juggernaut
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