Is 1 = 0.999... ? Really?

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Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
12
32%
No, 1 ≠ 0.999...
22
59%
Other
3
8%
 
Total votes : 37

Re: Is 1 = 0.999... ? Really?

Postby phyllo » Tue Feb 09, 2021 4:58 pm

That is all I meant. So if 0.333... represents the 3s being at infinity then you agree that 0.333... = 1/3.
Are you asking the same question that I just answered?

I've already stated multiple times that 1/3=0.333... exactly.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Tue Feb 09, 2021 5:00 pm

Certainly real wrote:To convey what I'm trying to convey more clearly, if I ask the question what is a third of a 1 meter ruler, I would not get the answer, well that's just a third of a one meter ruler. I would either get the answer you cannot have a third of a meter, or it's .333...m or 33.333...cm

But that doesn't mean that you can't divide the meter. It only means that you cannot say it using decimal digits.

Bend that meter into a circle an cut it at exactly 120 degrees and you have your 1/3 meter. Ask someone how long those portions are in centimeters and they cannot give you an exact number.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Tue Feb 09, 2021 5:12 pm

wtf wrote:Great. Then tell me what \(10^\infty\) means "according to the standard mathematical definitions," and please make sure to reference those definitions. Because frankly I'm highly familiar with "standard mathematical definitions," and you have not presented one yet.


I think I already did. \(10^{\infty}\) stands for "10 multiplied by itself infinitely". Every term in that sentence (such as 10, multiplication and infinity) has an established definition. My claim is that all you have to do is use the established rules and deduce the meaning of the expression (in the same way one can deduce the meaning of "2 + 2" without having to look for an answer from mathematicians, books and online encyclopedias.) If you want a longer description, here is one: the statement tells you to calculate the result of \(10 \times 10\) and then take that result and multiply it by \(10\) to get a new result and then take that result and multiply it by \(10\) to get a new result and so on for an infinite number of times. What you get after you're done multiplying all the \(10\)s is what the expression stands for. What exactly is unclear and/or unconventional about that? How can I help you if you are not willing to explain what's unclear and/or unconventional? Merely repeating yourself by saying "It's undefined", "You haven't defined it!" and "Define it!" will get you nowhere. A better approach would be to explain what it means for something to be undefined, and if it proves to be necessary, to present an argument that my expression is undefined.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Tue Feb 09, 2021 5:19 pm

Magnus Anderson wrote:
wtf wrote:Great. Then tell me what \(10^\infty\) means "according to the standard mathematical definitions," and please make sure to reference those definitions. Because frankly I'm highly familiar with "standard mathematical definitions," and you have not presented one yet.


I think I already did. \(10^{\infty}\) stands for "10 multiplied by itself infinitely". Every term in that sentence (such as 10, multiplication and infinity) has an established definition. My claim is that all you have to do is use the established rules and deduce the meaning of the expression (in the same way one can deduce the meaning of "2 + 2" without having to look for an answer from mathematicians, books and online encyclopedias.) If you want a longer description, here is one: the statement tells you to calculate the result of \(10 \times 10\) and then calculate the result of multiplying that result by \(10\) and then to calculate the result of multiplying that result by \(10\) and so on for an infinite number of times. What you get after you're done multiplying all the \(10\)s is the meaning of the expression. What exactly is unclear about that? How can I help you if you are not willing to explain what's unclear? Merely repeating yourself by saying "It's undefined", "You haven't defined it!" and "Define it!" will get you nowhere. A better approach would be to explain what it means from something to be undefined, and if it proves to be necessary, to present an argument that my expression is undefined.

Wikipedia wrote:Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number

- That is ambiguous - not "well-defined". There might be many values that fit into those categories.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Tue Feb 09, 2021 5:32 pm

Perhaps you should start by defining what the word "ambiguous" means.

When we say that what someone is saying is ambiguous, we are merely saying that we cannot determine exactly what that person is talking about ("He might be talking about X, but also, he might be talking about Y".)

Wikipedia seems to be unambiguous about the meaning of the word "infinity". It tells us right at the start that the word means "something that is larger than any real or natural number". What exactly is ambiguous about that?

The fact that the word "infinity" can be used to represent more than one number does not mean that it is ambiguous. It merely means that it is not a specific number but a category of numbers (similar to the word "integer".)

And even if "infinity" is an ambiguous term, what makes you think that logic can't handle ambiguity? Of course, ambiguous terms have limited use but that's a different subject.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Tue Feb 09, 2021 5:55 pm

Magnus Anderson wrote:The fact that the word "infinity" can be used to represent more than one number does not mean that it is ambiguous. It merely means that it is not a specific number but a category of numbers (similar to the word "integer".)

In maths that is exactly what ambiguous means - not specific enough to use.

2 * integer = ?
integer / 8 = ?
\(10^{integer} = ?\)

The word "integer" cannot be used - as an integer - because it is mathematically ambiguous - it is a category of values, not a specific value. And "infinity" is equally a category of values - not a specific value.

\(10^{\infty} = ?\)

If you want to use maths and be specific just use \(infA\) instead -
\(10^{infA} = 10^{Natural Number Count} = 10^1x10^2x10^3x...\)
- which is not a natural number itself.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Tue Feb 09, 2021 6:18 pm

What makes you think that I cannot use categories of numbers?

What makes you think that \(b\) in \(a^b\) cannot be a category of numbers?

You think that exponentiation is limited to specific numbers? (Sort of like how wtf thinks it is limited to real numbers?)
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Tue Feb 09, 2021 6:22 pm

The only definition I could find for the ellipsis "..." is in Wikipedia -
The latter formula means the sum of all natural numbers from 1 to 100. However, it is not a formally defined mathematical symbol. Repeated summations or products may similarly be denoted using capital sigma and capital pi notation, respectively:

I have never seen "..." being used to mean anything other than "continue indexed throughout the natural numbers". So without a formal definition in our debate I would have stated -
Ellipsis - "..." =
    "repeated throughout the natural numbers with index starting at either the first digit to the right of the decimal (when a decimal is to its left) or repeated throughout the natural numbers with index starting at the first left hand digit indicated at the left of the symbol."

From there, most of our argumentation would never have occurred - although perhaps you want to debate a "better" definition - but that is a different debate.
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Re: Is 1 = 0.999... ? Really?

Postby Certainly real » Tue Feb 09, 2021 6:28 pm

obsrvr524 wrote:
Certainly real wrote:To convey what I'm trying to convey more clearly, if I ask the question what is a third of a 1 meter ruler, I would not get the answer, well that's just a third of a one meter ruler. I would either get the answer you cannot have a third of a meter, or it's .333...m or 33.333...cm

But that doesn't mean that you can't divide the meter. It only means that you cannot say it using decimal digits.

Bend that meter into a circle an cut it at exactly 120 degrees and you have your 1/3 meter. Ask someone how long those portions are in centimeters and they cannot give you an exact number.


Where .333... is meaningful, it is hypothetically possible to divide the meter into 3 equal parts. But the equality of those parts must be expressed by saying that each part is .333...m long. .333...m long is either meaningful (therefore it has an exact meaning, which means it can serve as an exact answer), or it is not meaningful (which means we cannot divide the meter into three meaningfully equal parts). It would be like saying:

A: I've just divided x into 3 equal parts.
B: How big is each part
A: Unknown in terms of exact size
B: How do you meaningfully know they are exactly equal?

or

A: I've just divided x into 3 equal parts.
B: How big is each part
A: It is exactly 1/3 of the whole of x
B: How big is 1/3 of the whole of x
A: I can't give an exact answer to this
B: Does an exact answer actually exist but you just don't know it, or is it the case that no exact answer exists?
A: No exact answer exists
B: You are saying 1/3 of x is meaningful, yet you are also saying that there is no objective truth regarding what 1/3 of x actually amounts to in terms of size/measure. How can one of those three pieces of x not have a measurable value? If it has a size, it has a measurable value.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Tue Feb 09, 2021 6:46 pm

Certainly real wrote:But the equality of those parts must be expressed by saying that each part is .333...m long. .333...m long is either meaningful (therefore it has an exact meaning, which means it can serve as an exact answer), or it is not meaningful (which means we cannot divide the meter into three meaningfully equal parts). It would be like saying:

I don't think that is true.

It is meaningful to say that 1/3=0.333....

But that does not mean that 0.333... is truly an exact match to 1/3.

What 0.333... represents exactly is a number that is within an infinitesimal of being 1/3 - and that is a useful meaning.
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Re: Is 1 = 0.999... ? Really?

Postby phyllo » Tue Feb 09, 2021 7:22 pm

obsrvr524 wrote:
Certainly real wrote:But the equality of those parts must be expressed by saying that each part is .333...m long. .333...m long is either meaningful (therefore it has an exact meaning, which means it can serve as an exact answer), or it is not meaningful (which means we cannot divide the meter into three meaningfully equal parts). It would be like saying:

I don't think that is true.

It is meaningful to say that 1/3=0.333....

But that does not mean that 0.333... is truly an exact match to 1/3.

What 0.333... represents exactly is a number that is within an infinitesimal of being 1/3 - and that is a useful meaning.
But you have not mathematically demonstrated the existence of this infinitesimal, while others have mathematically demonstrated the lack of an infinitesimal.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Tue Feb 09, 2021 8:49 pm

phyllo wrote:But you have not mathematically demonstrated the existence of this infinitesimal, while others have mathematically demonstrated the lack of an infinitesimal.

That is not true.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Tue Feb 09, 2021 9:27 pm

obsrvr524 wrote:Would you agree that if the string of 3s in 0.333... actually got to infinity then 0.333... = 1/3 ?


phyllo wrote:I already covered this. There is no time or process involved.

There is no "got to infinity" or "getting to infinity". It's at infinity. It's always equal.


obsrvr524 wrote:That is all I meant. So if 0.333... represents the 3s being at infinity then you agree that 0.333... = 1/3.


No matter how many 3s are implied by the ellipsis, the expression is never equal to \(\frac{1}{3}\). In other words, even with a \(3\) located at the position with an index of \(-infA\) (or any other negative infinity), the number would still be less than \(\frac{1}{3}\).

The larger the number of \(3\)s, the closer the number is to \(\frac{1}{3}\). But it can never be exactly \(\frac{1}{3}\).
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Re: Is 1 = 0.999... ? Really?

Postby phyllo » Tue Feb 09, 2021 9:42 pm

That is basically contradicted by all the mathematical results shown in this thread.

Sure, you can say that but at the same time you are saying that the sum of a converging geometric series is wrong. (For example)

Which result do you prefer to retain?

I prefer not to throw away the work on geometric series.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Tue Feb 09, 2021 9:56 pm

Magnus Anderson wrote:No matter how many 3s are implied by the ellipsis, the expression is never equal to \(\frac{1}{3}\). In other words, even with a \(3\) located at the position with an index of \(-infA\) (or any other negative infinity), the number would still be less than \(\frac{1}{3}\).

Do you think that you can Resolution Debate that proposal?
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Feb 10, 2021 2:26 am

No problem. But let's not start a new debate until we review and summarize our first debate. Also, I'd want to make certain things clear in advance. (And I won't be available until the weekend or so.)

But I'm surprised you disagree. What's your position on the issue?
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Feb 10, 2021 3:06 am

phyllo wrote:That is basically contradicted by all the mathematical results shown in this thread.

Sure, you can say that but at the same time you are saying that the sum of a converging geometric series is wrong. (For example)

Which result do you prefer to retain?

I prefer not to throw away the work on geometric series.


I think you're presenting a false dichotomy. You either accept that \(0.33\dot3 < \frac{1}{3}\) and throw away the work on geometric series or you reject that \(0.33\dot3 < \frac{1}{3}\) and keep the work on geometric series. In reality, you can accept that the two numbers aren't equal AND keep the existing work by making an adjustment to it.

What they have shown is that the limit of \(0.3 + 0.03 + 0.003 + \cdots\) is \(\frac{1}{3}\). I don't dispute that. But then they erroneously conclude that \(0.33\dot3 = \frac{1}{3}\). It's like someone correctly showing that the ceiling of \(1 + 5 \div 10\) is \(2\) but then erroneously concluding that \(1 + 5 \div 10 = 1\).
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Re: Is 1 = 0.999... ? Really?

Postby wtf » Wed Feb 10, 2021 3:08 am

Magnus Anderson wrote:What they have shown is that the limit of \(0.3 + 0.03 + 0.003 + \cdots\) is \(\frac{1}{3}\). I don't dispute that.


But that is the DEFINITION of the sum of the infinite series.


Magnus Anderson wrote:But then they erroneously equate \(\frac{1}{3}\) with \(0.33\dot3\).


It's not an erroneous equating. It's a definition.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Wed Feb 10, 2021 3:19 am

wtf wrote:But that is the DEFINITION of the sum of the infinite series.


What's the point of repeating this point? I've already responded to it. Why not address my response instead?

I am fully aware that's how they define it. The thing is that that definition conflicts with existing definitions.

If we define symbols "2" and "+" the way we normally define them, then the meaning of "2 + 2" can be deduced to be the same as the meaning of "4". You are thus not free to declare that "2 + 2" means "10" and not "4".

"0.3 + 0.03 + 0.003 + ..." has its own meaning and someone (whoever that is) declaring that it means something else is introducing a contradiction in their system of thought.

That expression DOES NOT stand for a limit.

(In the same exact way that "1 + 2 / 4" does not stand for the ceiling of "1 + 2 / 4".)
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Re: Is 1 = 0.999... ? Really?

Postby wtf » Wed Feb 10, 2021 4:31 am

Magnus Anderson wrote:
wtf wrote:But that is the DEFINITION of the sum of the infinite series.


What's the point of repeating this point? I've already responded to it. Why not address my response instead?

I am fully aware that's how they define it. The thing is that that definition conflicts with existing definitions.

If we define symbols "2" and "+" the way we normally define them, then the meaning of "2 + 2" can be deduced to be the same as the meaning of "4". You are thus not free to declare that "2 + 2" means "10" and not "4".

"0.3 + 0.03 + 0.003 + ..." has its own meaning and someone (whoever that is) declaring that it means something else is introducing a contradiction in their system of thought.

That expression DOES NOT stand for a limit.


It does stand for a limit, that's the ONLY thing it can stand for.

Once you agree that the limit of .3, .33, .333, ... is 1/3, we're done.

I truly don't understand your point. Addition is a binary operation. The only way to define addition of infinitely many summands is to define it as the limit of the sequence of partial sums. Having done that, we're done. You agree to all this ... then you say no. This I truly don't get.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Wed Feb 10, 2021 5:16 am

wtf wrote:Once you agree that the limit of .3, .33, .333, ... is 1/3, we're done.

I truly don't understand your point. Addition is a binary operation. The only way to define addition of infinitely many summands is to define it as the limit of the sequence of partial sums. Having done that, we're done. You agree to all this ... then you say no. This I truly don't get.
Wikipedia wrote:Limit
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.[1] Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

In formulas, a limit of a function is usually written as

\({lim_{(x\to c)} f(x)=L}\)
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Wed Feb 10, 2021 5:20 am

Magnus Anderson wrote:No problem. But let's not start a new debate until we review and summarize our first debate. Also, I'd want to make certain things clear in advance. (And I won't be available until the weekend or so.)

But I'm surprised you disagree. What's your position on the issue?

Who said I disagree?

So who would you want to debate?
I can moderate.
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Re: Is 1 = 0.999... ? Really?

Postby wtf » Wed Feb 10, 2021 5:28 am

obsrvr524 wrote:In formulas, a limit of a function is usually written as



Look up the definition of the sum of an infinite series. It's defined as the limit of the sequence of partial sums.

Secondly, you posted a handwavy Wiki definition of limit, not the technical definition. What is the point of flaunting your lack of mathematical understanding? If you don't know what a limit is, copypasting Wiki won't help.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Wed Feb 10, 2021 5:58 am

wtf wrote:
obsrvr524 wrote:In formulas, a limit of a function is usually written as



Look up the definition of the sum of an infinite series. It's defined as the limit of the sequence of partial sums.

A series sum is a function as long as there are variables for the limits (which there are).

Wikipedia wrote:Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

Series (mathematics)
The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is,

\(\sum_{i = 1}^{∞} a_i = \lim_{(n → ∞)} \sum_{i = 1}^{n} a_i \)

When this limit exists, one says that the series is convergent or summable, or that the sequence \({ (a_{1},a_{2},a_{3},\ldots )}\) is summable. In this case, the limit is called the sum of the series

They say that the limit is "called" the sum. That admits that it is actually a limit and that it is merely "called the sum".

  • So they provide a definition for "limit"
  • They describe the infinite series of partial sums as "approaching" the limit.
  • And they admit that limit is merely "called" "the sum" for convergent series.

Add to that -
obsrvr524 wrote:The only definition I could find for the ellipsis "..." is in Wikipedia -
Wikipedia wrote:The latter formula means the sum of all natural numbers from 1 to 100. However, it is not a formally defined mathematical symbol. Repeated summations or products may similarly be denoted using capital sigma and capital pi notation, respectively:

I have never seen "..." being used to mean anything other than "continue indexed throughout the natural numbers". So without a formal definition in our debate I would have stated -
Ellipsis - "..." =
    "repeated throughout the natural numbers with index starting at either the first digit to the right of the decimal (when a decimal is to its left) or repeated throughout the natural numbers with index starting at the first left hand digit indicated at the left of the symbol."

- and I think we have every definition issue raised throughout this entire thread.
And it seems that James has agreed with Wikipedia's definitions throughout.

wtf wrote:Secondly, you posted a handwavy Wiki definition of limit, not the technical definition. What is the point of flaunting your lack of mathematical understanding? If you don't know what a limit is, copypasting Wiki won't help.

If you don't like Wikipedia, show us a better source.
Or better yet - go talk to them and correct them. :D
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Re: Is 1 = 0.999... ? Really?

Postby Certainly real » Wed Feb 10, 2021 8:41 am

obsrvr524 wrote:It is meaningful to say that 1/3=0.333....


But that does not mean that 0.333... is truly an exact match to 1/3.


Then your '=' must be taken to mean near to but not an exact match to, or, nearly equally to but not truly equal to.

What 0.333... represents exactly is a number that is within an infinitesimal of being 1/3 - and that is a useful meaning.


I'm looking at this from a non-mathematical point of view. I don't think you can have multiple infinitesimals. To me, infinitesimal is that which separates all possible measures and things from each other. To me, only one existing thing can take the measure of infinitesimal and infinite. I call It Existence/God.
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