Are you asking the same question that I just answered?That is all I meant. So if 0.333... represents the 3s being at infinity then you agree that 0.333... = 1/3.
I've already stated multiple times that 1/3=0.333... exactly.
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Are you asking the same question that I just answered?That is all I meant. So if 0.333... represents the 3s being at infinity then you agree that 0.333... = 1/3.
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
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.
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
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".)
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:
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.
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:
But you have not mathematically demonstrated the existence of this infinitesimal, while others have mathematically demonstrated the lack of an infinitesimal.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.
phyllo wrote:But you have not mathematically demonstrated the existence of this infinitesimal, while others have mathematically demonstrated the lack of an infinitesimal.
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.
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 \(\frac{1}{3}\).
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.
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.
Magnus Anderson wrote:But then they erroneously equate \(\frac{1}{3}\) with \(0.33\dot3\).
wtf wrote:But that is the DEFINITION of the sum of the infinite series.
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.
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
\({lim_{(x\to c)} f(x)=L}\)
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?
obsrvr524 wrote:In formulas, a limit of a function is usually written as
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.
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
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."
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.
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.
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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