Nate Smith

A Question About Infinite Decimals

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This quote is from another thread. It lead me to a question about the nature of infinite decimals.

.999 repeating is "equal" to 1 because they both represent the same value, but you have to understand how an infinitely repeating decimal "represents" a value.

There is no such thing as a number with an infinite number of digits because there is no actual infinity. There is only .9, .99, .999, etc., each with a finite number of digits. The repeating expression, usually written as ".999..." (or with a dot over the last 9) refers to this sequence, written as {.9, .99, .999, ...}.

The limit of that sequence is 1 because the terms become closer and closer to 1 to any arbitrary precision...

This is the only sense in which the sequence represents the value 1 and is what it means to say that .999... "equals" 1: The limit of the sequence is 1.

Let's say I have a length of rope one foot long. I cut 9/10ths of that rope off. Then I cut off 9/10ths of the smaller piece, and then 9/10th of that remaining smaller piece and so on. I would have lengths of rope 0.9' + 0.09' + 0.009' + ...

Of course I could never do that an infinite amount of times, but isn't there a sense in which the rope is "all of the decimals" of the number 0.999..., even while it is whole? In this sense, can't we view the number 1 as existing as 0.999... and not just as the limit of that series?

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This quote is from another thread. It lead me to a question about the nature of infinite decimals.

Let's say I have a length of rope one foot long. I cut 9/10ths of that rope off. Then I cut off 9/10ths of the smaller piece, and then 9/10th of that remaining smaller piece and so on. I would have lengths of rope 0.9' + 0.09' + 0.009' + ...

Of course I could never do that an infinite amount of times, but isn't there a sense in which the rope is "all of the decimals" of the number 0.999..., even while it is whole? In this sense, can't we view the number 1 as existing as 0.999... and not just as the limit of that series?

I don't have a complete answer for you, but you seem to be confusing concretes with abstractions. Numbers are abstractions. When you cut off 9/10 of a rope, you now have 2 ropes. If you cut off 9/10 again, you now have 3 ropes. The fractions represent the relationship of each rope to the length of the original rope. Certainly you can say that the rope is 0.999... when considering the mathematical relationships among parts of the rope because mathematically it is the same as 1. Notice that when you say 9/10 of the rope, you do not use the same process: you end the decimal as 0.90, rather than using 0.9000....

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Let's say I have a length of rope one foot long. I cut 9/10ths of that rope off. Then I cut off 9/10ths of the smaller piece, and then 9/10th of that remaining smaller piece and so on. I would have lengths of rope 0.9' + 0.09' + 0.009' + ...

Of course I could never do that an infinite amount of times, but isn't there a sense in which the rope is "all of the decimals" of the number 0.999..., even while it is whole? In this sense, can't we view the number 1 as existing as 0.999... and not just as the limit of that series?

You can mentally subdivide a rope by the process you describe, and your earlier statement that "There seems to be a respect in which the rope can be divided into an infinite number of lengths (without actually having been divided)" is closer to the truth. That infinite process of mental subdivision qua process does exist, but only as a mental process specified with a finite conceptual mechanism and always resulting in a finite number of 9's, not a completed infinity. (This much is from Aristotle and his meaning of the infinite as "potential".)

You refer to a process consisting of "I cut 9/10ths of that rope off. Then I cut off 9/10ths of the smaller piece, and then 9/10th of that remaining smaller piece and so on" and then say you realize that you could never do the subdivision an infinite number of times to get an infinity of 9's. But then who is going to do it? Where are the infinity of 9's? The rope is not "all of the decimals". The rope is just the rope, sitting there in physical reality with whatever attribute of length it has, but otherwise minding its own non-conceptual business.

All the subdivisions and the sequences of 9's are your cognitive attempt to measure it conceptually, to increasing degrees of precision, counting your own progression of mental subdivisions of a unit -- all of which has nothing to do with the completely non-cognitive rope. That infinite process of subdivision of epistemological units is not a part of the rope and neither is the base unit. You are conceptually comparing the rope's attribute of length with some numbers of your selected units in order to make an objective measurement to some degree of precision. The rope has an actual length, but the mathematics of comparing and counting units is not intrinsic to the rope.

In fact with your rope you don't get very far at all because the flexible strands are moving around and are irregular and frayed at the ends, making a physical measurement of the macroscopic rope beyond a certain relatively low precision impossible. You are referring to a process of conceptual, not physical, subdivision, and you can't use the physical rope as a basis to claim that the infinity of 9's actually exists as part of the rope's measurement of length. Again, the mathematics is not intrinsic.

Now think of what it would mean to try to regard .999... with an actual infinity of 9's as a number. (By "number" here we mean rational number, not real number, so as to not beg the question or invert the proper conceptual hierarchy -- the meaning of the more abstract real numbers is coming up below.) You would have 9 tenths, plus 9 hundredths, plus ... as you described in the beginning except that it would have to go on forever. Each decimal place represents a number of some unit (a subdivision of the base unit), so you would need an infinity of units, which does not exist. ".999..." cannot exist as a completed infinity of decimal places. ".999..." with the dots refers to a finite process of adding decimal places, not a completed infinity of 9's. There is no such thing as "infinite precision" in any measurement because there is no completed infinity of subdivisions of a unit.

As for the conceptual process, you have to distinguish between the sequence and its limit, which are two different mathematical concepts. The limit is a number (in this case 1), which is the limit of the sequence of numbers .9, .99, .999, etc, which means that the numbers in the sequence are closer and closer to the limit 1 the farther you go. The sequence as such -- an open-ended collection of numbers -- is not a number.

But we don't stop there. The real numbers, in contrast to the rational numbers, are based on sequences (following a careful technical process we won't go into here). You can specify .999... as a real number, but that requires first understanding the sequence of numbers .9, .99, .999, etc. and the intended relations between them as the precision increases, which is why it is a higher level abstraction. If you didn't have the concept of that sequence, the "real number" .999... would make no sense to you because you can only hold the concept with a knowledge of the finite mechanism (using the concept of "potential infinite") of the progression in the sequence to which it refers. (The theory of real numbers is much more abstract than your rope example; it makes possible concepts of number in which the degree of precision is left unspecified for a single number in order to allow for measurement with any units at any time, and is required to distinguish the facts giving rise to different kinds of numbers within mathematical theory.)

That is why the real numbers are a higher level abstraction than the rational numbers. As a real number, .999... is a "number", but you have to understand it as a higher level abstraction that requires several lower level concepts to form the more abstract concept of the real number. (I am directly relying on the theory of concepts and abstraction in AR's Introduction to Objectivist Epistemology, which you should be sure to read and understand or this use of the concept "abstraction" won't mean much to you.)

Once you understand that much, then you can say that numbers exist with an "infinite number" of decimal places, but only in the technical sense of the concepts of real numbers based on the sequences. It doesn't mean that .999... exists as a completed whole of infinite 9's. (Meanwhile, of course the number 1 as the limit still exists as a simple number independently of all of this, and 1 is the actual value equivalent to the real number .999...; but not every real number has a rational number as the limit of its sequence, which is why they are required.)

When you do all this properly, you concretize the abstract concepts of real numbers in the form of symbols (just as you do with words for ordinary concepts) so you can deal with them as a mental concrete represented by a symbol. In fact you have to do that -- real numbers are often recognized (including by AR) as representing sequences, but until you have integrated the pieces under a symbol you haven't completed the process of conceptualization. And until you understand what knowledge is subsumed under the concept and how the concept is formed, you don't understand it. The concept then subsumes all the information you used in its formation, but all those details are ordinarily left unspecified and implicit when you use a concept -- the cognitive purpose of concepts is mental unit economy; you forget about those complexities except for when you need them explicitly.

That mental process of conceptualization is what permits you to regard numbers with infinite decimals as numbers and to use them arithmetically and analytically just like integers and rational numbers -- but you use all numbers as abstract objective concepts, not something intrinsic to objects that are being measured. Real numbers necessarily require a higher level abstraction than simple numbers like integers and fractions because of all they conceptually depend on.

This is similar to how you routinely use the concept of furniture as an abstraction subsuming the lower level concepts of table, chair, etc. integrated under the word "furniture" so you can use "furniture" as a mental unit in your thinking. A major difference, however, is that tables, chairs, etc. are concepts of things while the concepts of mathematics are concepts of method, which in turn makes the whole enterprise much more abstract and complex. (If you do it right, and completely, at the scientific level, the technical mathematical theory shows how using sequences as open ended representations of numbers always works out consistently with the laws of arithmetic, thereby validating the more abstract concept and its use in more advanced analytical theory.)

That is how "there a sense in which .999..." is a number and how it is a numeric representation of 1, but it does not mean that "the rope is 'all of the decimals' of the number 0.999..." -- which is a fallacy of intrinsicism -- and it must be regarded as referring to a sequence, even though that fact is left implicit when we use concepts of real numbers.

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As for the conceptual process, you have to distinguish between the sequence and its limit, which are two different mathematical concepts. The limit is a number (in this case 1), which is the limit of the sequence of numbers .9, .99, .999, etc, which means that the numbers in the sequence are closer and closer to the limit 1 the farther you go. The sequence as such -- an open-ended collection of numbers -- is not a number.

There is much about your post that I do not fully understand. Clearly there is some foundation that I am missing. I would like to get deeper into the heirarchy of numbers and what you mean by the difference between a sequence and its limit. I also need to understand the terminology better as well.

Let me try and start from the beginning and see how far I can get:

The first numbers that we grasp are the counting numbers (beginning with one). We perceive units and groups of units. Conceptually, these larger groups are counted. Later in our development (not necessarily next though), we conceive of "parts of a whole", in other words, fractions. Our first experience is probably with the concept of "half", one of two equal parts totalling a whole. The notation of course is 1/2. Soon after, we become familiar with thirds, fourths, etc. and eventually parts such as two-thirds or three-fifths and so on.

I take it from what you are saying that things get more complicated when we try and represent these fractions as decimals. For example: Let's say we try and measure a rod (which is of length 1/4 feet). Using our one-foot ruler, we immediately see that it is less than a foot. We proceed to design a better ruler that measures to the nearest 0.1 feet. (Children of course don't design instruments early on; this analogy is supposed to mimic what we learn as long division.) With our ruler, we determine that the rod is between 0.2 and 0.3 feet long. We proceed to design a better ruler that measures to 1/10 of 1/10 of a foot (or 0.01 ft). At this point, we determine that the rod is of length 0.25 feet long.

At this point, have we encountered the concept of a sequence? I think so, just a short finite one though. The sequence is 0.2, 0.25.

Now let's imagine that we use the same process to measure a rod of length 1/3 feet. On first measure, it is 0.3, then 0.33, then 0.333, etc. We have probably all divided 1 by 3 using long division and seen how the process continues indefinitely. If at any time we stop (which of course we have to), we don't have a decimal representation of 1/3. The sequences is 0.3, 0.03, 0.003, ... and only as a limit do we have 1/3. Am I understanding you correctly so far?

Is the example of 1/3 a a higher-level abstraction than 1/4? If so, how exactly.

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Let me rephrase that last part. Abstractions like one-third and one-fourth are on the same level, but are you saying that the decimal representation 0.333... is a higher-level abstraction than 0.25? If so, how?

(By the way, is one-third a concept? Don't concepts need to be held by a single word, not a hyphenated word?)

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Let's say we try and measure a rod (which is of length 1/4 feet). Using our one-foot ruler, we immediately see that it is less than a foot. We proceed to design a better ruler that measures to the nearest 0.1 feet. (Children of course don't design instruments early on; this analogy is supposed to mimic what we learn as long division.) With our ruler, we determine that the rod is between 0.2 and 0.3 feet long. We proceed to design a better ruler that measures to 1/10 of 1/10 of a foot (or 0.01 ft). At this point, we determine that the rod is of length 0.25 feet long.

At this point, have we encountered the concept of a sequence? I think so, just a short finite one though. The sequence is 0.2, 0.25.

That is the beginning of the process, but strictly speaking a mathematical sequence of numbers is an infinite set of numbers in a particular order as indexed by the positive integers. The issue of limits doesn't arise without infinite sequences. In the case of {.2,.25} you have specified a process in which you stop at the last finite step. For infinite sequences there is no "last step" so there is an essential difference.

Also, as an aside to keep the meaning of a sequence clear, the elements of a sequence do not in general repeat the previous elements in their notation like they do in a decimal expansion. With {.3,.33,.33,...}, as a new '3' is appended in each new element all the previous 3's are repeated in their same decimal positions in all the subsequent elements. In contrast, and more representative, with a sequence like {1/n} the terms are 1, 1/2, 1/3, etc. (converging to 0) with no such repetition. The concept of a sequence is broader than the examples of infinite decimal representations.

Now let's imagine that we use the same process to measure a rod of length 1/3 feet. On first measure, it is 0.3, then 0.33, then 0.333, etc. We have probably all divided 1 by 3 using long division and seen how the process continues indefinitely. If at any time we stop (which of course we have to), we don't have a decimal representation of 1/3. The sequences is 0.3, 0.03, 0.003, ... and only as a limit do we have 1/3. Am I understanding you correctly so far?
[out of order]I would like to get deeper into ... what you mean by the difference between a sequence and its limit. I also need to understand the terminology better as well.
Actually the sequence would be written as {.3, .33, .333,...}. If you use {.3, .03, 003,...} you have to add the previous terms at each step to get the "sequence of partial sums" {.3, .3 + .03, .3 + .03 + .003,...} for the infinite series whose limit is 1/3. (The limit of {.3, .03, 003,...} taken literally would be 0, which is not what you mean.) But that is a technical distinction between a "sequence" and a "series" that doesn't affect the main issue here.

For {.3, .33, .333, ...} the "sequence" is the collection of the expanding decimal numbers, which is not the same concept as the single number 1/3 that is the limit. It's not just that no one of the decimals is 1/3, but that the sequence of numbers is a collection and not a single number. In that sequence, none of the terms are equal to the limit, but even with a sequence like {1, 1, 1, 1,...} with all 1's forever and a limit of 1, the sequence itself is not the same as its limit: The sequence is the collection of elements {1,1,1,...} in which each term happens to have the same value as the limit, but the limit is the single number 1 as opposed to a collection of numbers. The limit is the result of a process represented by the sequence.

[out of order]I take it from what you are saying that things get more complicated when we try and represent these fractions as decimals....
Is the example of 1/3 a higher-level abstraction than 1/4? If so, how exactly.

Let me rephrase that last part. Abstractions like one-third and one-fourth are on the same level, but are you saying that the decimal representation 0.333... is a higher-level abstraction than 0.25? If so, how?

Yes. .25 is the same as 1/4 because .25 is another representation of the subdivision using a combination of different units -- 1/10 and 1/100 -- in accordance with the decimal system. The decimal system of compound units is more conceptually complex than a simple unit fraction, but both representations directly specify a specific number. (There is still also a distinction between the set {.2,.25} and the single number .25.)

In greater contrast, for .333... you need to know the decimal system plus the additional concept of an infinite sequence. Then you have to apply that concept to understand how an infinite decimal representation can represent a number and what you mean by that, in this case the limit. The sequence {.3,.33,.333,...) represents a mathematical method for getting closer and closer to 1/3 using successive subdivisions of tenths. Without the idea of (the rational number) 1/3 as the limit of a sequence, .333... would make no sense because you can't have a completed infinity of decimal places or a completed infinity of subdivisions of the unit. So the complication is not so much in going from simple fractions of a unit to decimal fractions, but rather in what you do with infinite decimal fractions.

While there is no such thing as "infinite precision" using infinite subdivisions, you do have the exact number 1/3 and the concept of 1/3 as a limit which you use in specifying how the sequence represents a value. Those conceptual dependencies in the hierarchy of knowledge are why .333... is a higher level abstraction than .25 (or .33).

.25 could be generated by the successive improvement in accuracy represented by the two step finite "sequence" {.2, .25} in which the last term is the one you want, but you don't need the concepts of infinite sequences or the limit for that. For .333... there is no "last term".

To regard .333... as representing a decimal number it is understood that when you use it in decimal form (as opposed to just using the limit 1/3) you terminate it at some finite number of digits when you use it in an actual calculation or specification of a measurement. The concept .333... means that the number is equal to 1/3 within some degree of precision that is left unspecified in the concept. The number must have some precision, but what degree of precision is omitted when you form the more abstract concept.

All that depends on two additional concepts that you don't have with .25: You have to know that there is an infinite sequence that in fact converges to 1/3 (and that it keeps getting closer at each step because it doesn't wander before coming back), and you have to have the concept of "implicit precision" in order to form the abstraction of an infinite decimal number. That is how you can say the infinite decimal expansion .333... is a number: in any calculation depending on a specific quantity to a specific precision it is one of the numbers in the sequence that is equal to 1/3 within the required precision. The higher level abstraction refers to the collection of decimals in the sequence analogous to the concept man referring to all men. You can use the higher level abstraction of an infinite decimal as a number only because it implies a specific number through the limit and the fact that the use of the concept this way is consistent with the laws of arithmetic.

(That's how it works for any infinite decimal with a repeated pattern, which always has a rational number as its limit, but there are additional complexities in the higher level abstraction of irrational numbers like sqrt(2), which have no rational limit.)

(By the way, is one-third a concept? Don't concepts need to be held by a single word, not a hyphenated word?)
Every number, including fractions, is a concept. The hyphenated word "one-third", as well as the symbol "1/3" consisting of two numbers joined by the slash, combines symbols into a single symbol whose parts indicate that the concept depends on both 1 and 3 in a special way -- The hyphen in "one-third" or the slash in "1/3" join the two numbers as a single symbol. The notation indicates that the fraction is a higher level abstraction (fractional numbers are a subdivision of "number") depending on both integers.

This follows AR's principle that a "qualified instance" of a concept using two words designating a "compound concept" used as if it were a single concept can be converted into a single concept by hyphenating the words into one symbol.

"One third" as two words means you have only one subdivision, which subdivision consists of the base unit broken into thirds. "One third" could also informally be used as synonymous with "one-third" as the single number if you don't follow the rule carefully -- "one third" and "one-third" have the same value -- but strictly speaking they should be combined with the hyphen when representing a single concept.

It is understood in mathematics that the symbol "1/3" can also emphasize the operation of division with two distinct numbers, and when it appears as a term in an equation you have the option of regarding it either as the division (like a "compound concept") or as a single number, in accordance with the laws of arithmetic which guarantee that the two uses represent the same value.

The decimal representation of numbers combining a string of digits also provides a single symbol for each number. The decimal point and the string of individual digits also represent individual quantities for the subdivisions in tenths -- analogous to the role of "1" and "3" in "1/3" -- but that does not mean the number isn't a concept. The concatenation of symbols into a single symbol has the same role as a word in ordinary language, concretizing the integration for a concept under a single mental unit.

It is significant that mathematics provides a systematic way to name numbers so you can keep track of them without having to remember a completely different word for each of them that you might ever encounter, which would be impossible -- even worse than the Chinese language with its tens of thousands of subtly different symbols. This naming system itself provides for economy of thought both in systematically symbolizing the number and concisely specifying the precise quantity in terms of multiple sub-units under the decimal system. But you don't have to use that system of symbols all the time either; if they become too unwieldy to refer to for some specific application because the shear number of digits for a large or high precision number with many decimal places subverts the purpose of a symbol as a single concrete that you can hold in your mind, then you can temporarily use other symbols for what you need, knowing that in principle you still have a way to relate such numbers to the other numbers and the relevant sub-units. You have a systematic method for generating the symbol for any number, and even though you only directly encounter relatively few of them in a life time, you have a way to deal with anything you do encounter under the open-ended concept of number and its system of names.

And more than that, in calculations you don't even have to stop and think what each number means, or explicitly focus on long strings of digits used for each number, because mathematics provides methods for dealing with them in which the meanings of the concepts themselves are left implicit. You can even do it entirely mechanically on a computer as long as you implement the method within its proper context. It's an amazing cognitive system.

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Now let's imagine that we use the same process to measure a rod of length 1/3 feet. On first measure, it is 0.3, then 0.33, then 0.333, etc. We have probably all divided 1 by 3 using long division and seen how the process continues indefinitely. If at any time we stop (which of course we have to), we don't have a decimal representation of 1/3. The sequences is 0.3, 0.03, 0.003, ... and only as a limit do we have 1/3.
Actually the sequence would be written as {.3, .33, .333,...}. If you use {.3, .03, 003,...} you have to add the previous terms at each step to get the "sequence of partial sums" {.3, .3 + .03, .3 + .03 + .003,...} for the infinite series whose limit is 1/3. (The limit of {.3, .03, 003,...} taken literally would be 0, which is not what you mean.) But that is a technical distinction between a "sequence" and a "series" that doesn't affect the main issue here.

My mistake. I meant to represent the sequence as {.3, .33, .333,...}, not {.3, .03, 003,...}.

That is the beginning of the process, but strictly speaking a mathematical sequence of numbers is an infinite set of numbers in a particular order as indexed by the positive integers.

As an aside, the definitions I've seen don't require sequences to be infinite. There are finite sequences as well as infinite ones.

The issue of limits doesn't arise without infinite sequences.

Agreed.

I think I've got a much better understanding of what you're saying. I'll keep thinking about this.

Would you agree with these two assertions?

1) The concepts of limits and mathematical infinities arise (at least in this situation) as a means to reconcile two previously formed concepts. Once we have the concepts of fractions (parts-of-a-whole) and decimals (decimal representations of fractions), we quickly see that some decimal representations will go on forever (like 1/3). At this point concepts like mathematical infinity and limits arise to reconcile these previous two concepts. If we were using a base-3 number system instead of base-10, one-third would simply be represented as 0.1 and we wouldn't need limits or sequences (but we would need them for other fractions though).

2) When someone argues that 0.999... isn't equal to 1, essentially he is making a stolen-concept fallacy. That person doesn't understand the origins of infinite decimals and how they can only have value and meaning in their limits.

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In order to clarify some of your language, I want to lay out the development of numbers and what I think is the origin and solution to some of these questions. The development that I am laying out is in no way supposed to be comprehensive. I've included only what I think is essential to this problem.

We begin by grasping the counting numbers (1, 2, 3 ...). The counting numbers continue with our “placeholder” system for powers-of-ten. (For example, 5,092 represents 5 one-thousands, 0 one-hundreds, 9 tens and 2 ones.)

I’m not sure if the concept of "power" is explicit or implicit at this point. I think it is only implicit. When I first learned numbers, I remember understanding how ten ones is ten, ten tens is one-hundred, ten one-hundreds is one-thousand, etc. without being aware of exponents. It is a later integration of exponents to realize that our number system is a place holder system for powers of 10 (ex: 6217 = 6(10^3) + 2(10^2) + 1(10^1) + 7(10^0)).

We later make the abstraction parts-of-a-whole, or fractions. The first would probably be 1/2 and then other subdivisions like 1/3, 1/4, etc. These eventually lead to parts such as 2/5 & 4/7.

A later development is the decimal system. Prior to this, we have the counting numbers, and our system using digits 0-9 to indicate how many of each of the "powers of 10" are in a given quantity. The decimal system is a realization that we can also represent quantities by using divisions of ten (i.e. tenths, hundredths, etc.) in addition to using multiples of ten (tens, hundreds, etc.). In other words, the decimal system represents an integration of our previous base-10 system with the concepts of fractions.

Prior to the decimal system, we might have said that we have three-and-one-fourth cups of water, represented as 3 1/4. But with the decimal system, we can indicate this as three and two-tenths and five-hundredths, or 3.25. As we all know, this development makes calculations much more efficient.

The decimal system integrated with our system for the counting numbers lets us represent all quantities as multiples of ten and divisions of ten. For example, 472.53 = 4(100) + 7(10) + 2(1) + 5(1/10) + 3(1/100).

I think it is at this step that we first run into the “problem” of infinite decimals. Some parts-of-wholes can easily be changed from fractional form to decimal form (such as 1/2 = 0.5 or 1/4 = 0.25). But others run into the problem of “going on forever” (such as 1/3 = 0.333… or 25/99 = 0.2525…).

If my understanding is correct, the decimal form of representing fractions is the origin of the infinite decimals. And consequently these infinite decimals only have meaning as the limit of that sequence, as that fraction. When we refer to the quantity 0.555…, we refer to 5/9, because that is its origin; that is the only sense in which it can refer to a quantity.

Please correct me if I am mistaken on this point. I have questions to follow if we’re in agreement so far.

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I know the question has been adequately answered, but I find this derivation gives some further insight:

Let x=9.99999...

So 10x=99.99999...

(In both cases, the decimals continue infinitely).

10x-x=99.99999...-9.99999...

So 9x=90 (as the decimal components cancel, leaving us with 99-9=90)

Or x=10

This reasoning can be applied to any repeated decimal sequence:

x=0.252525...

100x=25.252525...

100x-x=25.252525...-0.252525 (the decimal components will cancel)

99x=25

x=25/99

For x=1/7, the decimal sequence is "142857" repeated, so it would be necessary to subtract x from 1000000x in order to cancel the decimal components. This reasoning circumvents the nature of limits and simply takes the infinite repetition as a given. However it is interesting nonetheless from a mathematical standpoint.

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Of course I could never do that an infinite amount of times, but isn't there a sense in which the rope is "all of the decimals" of the number 0.999..., even while it is whole? In this sense, can't we view the number 1 as existing as 0.999... and not just as the limit of that series?

We can "get away" with so writing 1.0, but those three little dots following the 9's are demanding convergence and limit be defined. To do the thing with rigor one must work with sequences whose n-th terms are well defined functions of n and prove that the sequence converges.

In the case of a power series 1 + x + x*x + x*x*x ..... and x < 1 we have an easy time of it. One can easily prove the limit is 1/(1-x).

We can show the finite series 1 + x + x^2 + ... + x^N = (1 - x^(N+1))/1-x by simple algebra. If x is less than 1 then x^(N+1) goes to zero (again with a simple) proof and the result follows. No diddling with infinities except to show that higher powers of x get as small as you like when x has a magnitude less than 1.

Bob Kolker

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