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Created the article, per discussion. The discussion is copied here for convenience. Be sure to improve the article and\or discuss changes here. -- Meni Rosenfeld 15:50, 15 January 2006 (UTC)
During my studies, I have encountered the concept of a "formal calculation", in the sense of, roughly, a calculation for which the steps are not completely substantiated, and yet the result can give us insight about the true answer to the problem in question. I want to write an article about that concept, but I haven't found any references to it on the web, so I'm not sure how widely it is used and whether I understand the concept properly. Any ideas? -- Meni Rosenfeld 18:34, 12 January 2006 (UTC)
A formal argument is when you just follow what the syntax seems to suggest your reasoning, without proving the reasoning is sound. Like when you prove that, in a ring, if (1+ab) is invertible, then so is (1+ba) by using power series. Power series don't exist in a ring, but but you can still make formal arguments using them. - lethe talk 21:58, 12 January 2006 (UTC)
Lethe's example is what I would call a heuristic inference. It seems very strange to me to call this "formal": it's good because of informal gut feeling experience, not in virtue of the formal structure of the problem. --- Charles Stewart 22:02, 12 January 2006 (UTC)
Are all in favor of creating a stub, bearing the title "Formal calculation", based on the definition Jitse found, and beating it around until we reach something we can agree upon? -- Meni Rosenfeld 13:40, 13 January 2006 (UTC)
I know that "formal calculation" seems to imply a rigorous one, and actually that did confuse me the first times I encountered the concept. But I got the impression that, while perhaps ambiguous, it is usually used in the sense I described - Much like in the probably more common term formal power series. In this sense, "formal" actually means of form, namely, the form of the objects matter and not their underlying meaning - making the calculation perhaps systematic, but not really rigorous because we are using properties without any justification to why these properties should hold. We could always delete the article later if we can't seem to rich any consensus. -- Meni Rosenfeld 14:59, 13 January 2006 (UTC)
Of course formal power series are ultimately defined in a rigorous way, but the inspiration for this definition comes from a non-rigorous application of properties of convergent power series to arbitary power series. That's where the term "formal" comes from. -- Meni Rosenfeld 15:12, 13 January 2006 (UTC)
I think that this is a good topic for an article, and it may well prove useful for my planned article on Boole's algebraic logic (to be carefully distinguished from Boolean algebra, since Boole's system allows terms that do not have set-valued denotations). They can be seen to be similar to the status of polynomials prior to the discovery of complex numbers: onbe can know the sum and product of the roots of a quadratic and know furthermore that those roots don't exist. If we are to resort to neologism, why not optimistic calculation? --- Charles Stewart (talk) 16:29, 13 January 2006 (UTC)
It appears that the phrase is used in the proposed sense. It also appears to be understood in other ways, and it appears that some folks feel that the proposed sense is not a good sense. For an inclusionist (not necessarily me), Wikipedia should have an article. The article should note the opposition and provide disambiguation. However, a major unresolved question is: What is the primary meaning of "formal calculation"? The answer to that I do not know, but I'm inclined to think it's the "rigorous" sense, not the proposed sense. -- KSmrq T 01:23, 14 January 2006 (UTC)
In a nutshell, I think my original proposition of creating a stub and beating it around is fair. I'll do that now. Be sure to check it out for any flaws\omissions\whatever as I am an inexperienced editor. Formal calculation. -- Meni Rosenfeld 15:20, 15 January 2006 (UTC)
I can't really add more than has been said above, but if there's still some doubt as to how common this is...I can say that I've ran into this not just in lectures or talks but in books. As I can best recollect, you usually say it's just a formal calculation when you are dealing with either infinite series or integrals to "derive" an expression by playing with the original expression and using certain formal rules. These rules are possibly not fully justified, due to matters of convergence, but the resulting expression is sometimes useful. I'm pretty certain this has occurred in plenty of texts. I suggest looking through mathematical physics literature in particular. -- C S (Talk) 16:25, 15 January 2006 (UTC)
Euler-Maclaurin formula#Motivation for the existence -- Meni Rosenfeld 19:18, 15 January 2006 (UTC)
In section 3 of talk:Formal_power_series Oleg and I have a lengthy discussion on the subject. Bo Jacoby 22:20, 15 January 2006 (UTC)
I have expanded the article a bit. I still think it needs additional work, since it doesn't have much content and what It does have is based mainly on my intuition, but does anyone think we should remove its stub status? -- Meni Rosenfeld 07:15, 16 January 2006 (UTC)
I don't like the dichotomy that the intro presents between different contexts of the word formal. I don't view the word as being a contronym. For me, the over-arching meaning of the word "formal" is something like "deals with syntax but not semantics". A formal calculation is one that manipulates the form of symbols, the syntax. It may fail to be rigorous if the calculation is applied to systems which do not follow the rules of syntax used, the semantics. It will be (and can always be made, if it is not) completely rigorous if the system in question is that defined by the syntax, in a sense which is made precise by the notion of adjoint functors, as for example the formal sums of triangles used in simplical homology, which are essentially defined by their syntax, or the enveloping algebra of a Lie algebra. So it's not that people use the word to mean opposite things, it's just that a formal calculation may be not always be applicable where someone may try to apply it. Would you kindly folks like to share your thoughts about my point of view on this matter? - lethe talk + 11:48, 4 February 2006 (UTC)
Formal calculations are without interpretations. Example: 10−17+8 = −7+8 = 1. The intermediate result −7 has no interpretation as a number of pebbles. The detour 10−17+8 = 10+8−17 = 18−17 = 1 eases the minds of those who rely on the limited interpretation of numbers as counting pebbles. Mathematicians object against the 'formal' calculation S=1+2+4+8+...=1+2S implying S=−1 because they rely on a limited interpretation of infinite series. Bo Jacoby 13:36, 22 February 2006 (UTC)
Let me try. There are many examples in mathematics where members of a proper subset of a set has interpretations, but not every member of the set has an interpretation. The first example above is that the subset of nonnegative integers are cardinal numbers of finite sets, while the negative integers are not cardinal numbers of finite sets. The second example above is that the subset of convergent series define real numbers while the divergent series do not define real numbers. A third example is found in the article on multiset where power series with nonnegative integer coefficients define multisets of nonnegative integers, while other power series do not define multisets. Manipulation with elements that do not have interpretations are called formal. As a formal calculation can not be understood in terms of the usual interpretation, it is often rejected as nonsensical. Bo Jacoby 08:54, 23 February 2006 (UTC) PS. See also the fine example in Complex_number#History. Bo Jacoby 09:26, 23 February 2006 (UTC)
If a calculation is formal or not depend on the interpretation. A formal calculation is a logically consistent argument in the set in which it takes place, but not in the subset in which it is interpreted. The example from complex number history is not considered formal any longer, now that complex number interpretation is common knowledge. Nobody considers computations involving negative numbers 'formal' any longer, but once they were. I consider the equation 1+2+4+8+... =−1 to be absolutely true, and I completely disagree with the article saying: "This is of course absurd in the context of reals, where power series originated from". I expect the acceptance of divergent series defining values, to be crucial to elementary particle physics which is presently troubled with perturbation calculations giving divergent results. The argument is simple. Let S=1+2+4+8+...=1+(2+4+8+...)=1+2(1+2+4+...)=1+2S, so S=1+2S, so S=−1 . There is nothing absurd in the result. You expect the sum of positive numbers to be positive ? Well, that is true for finite sums but not for infinite sums. Get accustomed to that, just as you got accustomed to the fact that "the sum of rational numbers is a rational number" is true for finite sums but not for infinite sums. Bo Jacoby 14:12, 23 February 2006 (UTC)
Let take an elementary example. A child understands that 2+2=4 based on counting fingers. The finger method of addition also works for more difficult problems like 6+1 and 3+4. Now he tries 7+4 and realizes that he does not have that many fingers. So he says: "according to the original definition, the sum does not exist to begin with". He considers "7+4=11" to be a formal calculation. The teacher did ask the child to count on his fingers, but now he says: "Forget what I taught you about fingers". The child says: "Why did you teach me in the first place what you now ask me to forget?". The teacher answers: "Because you could not understand the general case until you tried some special cases". So, after some resistance, the intelligent child lets go of the limitations of the original definition and understands that the logic of counting extends the number of fingers. Bo Jacoby 09:10, 24 February 2006 (UTC)
Your example fits nicely into the analogy. Define 4+7=ω, meaning many. Any number greater than ten is many. That extension is perfectly plausible and possible and even useful to very young children. The trouble is that arithmetic no longer works, because the nice rule "a+b=a+c implies b=c" no longer applies. The equation 4+7=4+8 is true because ω=ω, but the equation 7=8 is not true. The rules of arithmetic are the rules we want to keep. Any solution to the equation x=1+2x is equal to −1, and the formal expression 1+2+4+8+... is a solution to the equation x=1+2x. The equation 1+2+4+8+...=−1 is no more absurd than the equation 4+7=11, and the equation 1+2+4+8+...=2ω is as damaging to logic as the equation 4+7=ω. Bo Jacoby 14:04, 24 February 2006 (UTC)
I look forward to seeing your theory, showing my example of cancellation incorrect. I accept that choosing any definition is subjective; (this seems to be a fair compromise between calling one option "absolutely true", which I did, and calling the same option "absurd", which you did). I share your taste regarding the projective extension of the real line, (mainly because this option can be extended to the complex plane, where only one infinity is introduced to produce the Riemann sphere). The projective extension even supports the idea of negative numbers being beyond infinity, so you should not object that much against 1+2+4+8+...=−1 . I agree that the breakdown of the cancellation rule in the theory of cardinal numbers didn't stop anyone from discussing that theory, but I expect that Georg Cantor would have liked to preserve the rule if at all possible. Let's list the options. Option 1: 1+2+4+8+...=−1. Option 2: 1+2+4+8+...is undefined. Option 3: 1+2+4+8+...=2ω. Option 1 is a result of a formal calculation. It leads to no logical problems, but to the counterintuitive result that an infinite sum of positive numbers need not be positive. Option 2 is perfectly powerless. Option 3 leads to the breakdown of arithmetics. To me the choice is easy. Consider the following formal calculation. 0=0+0+0+...=(1−1)+(1−1)+(1−1)+...=1−1+1−1+1−1+...=1+(−1+1)+(−1+1)+(−1+1)+...=1+0+0+0+...=1. What can be done to this scary result? First possibility: "Accept for a fact that 0=1". Second possibility: "Ban all divergent series". Third possibility: "Accept that you are not always allowed to insert or delete more than a finite number of pairs of parentheses into an infinite sum". The first possibility leaves us with no arithmetic at all. The second possibility, I believe, was the choice of the mathematician Niels Henrik Abel, and others follow his judgement. The third possibility gives a useful extension of assigning values to series, and no drawbacks except some delightful discussions with fellow mathematicians (See Talk:Formal_power_series). The formal calculation breaks down to this:
Bo Jacoby 14:48, 26 February 2006 (UTC)
Thanks. It looks interesting. Let me see if I understand you right. The sum ω = 1+1+1+... = 1+(1+1+...) satisfies the equation ω=1+ω which has no solution in a nontrivial ring (mathematics). So the ring is the trivial ring where 0=1. All expressions equal 0, no positive numbers exist, and no number is greater than another number. 0+1+2+3... is not really different from 1+2+3+4+... because they are both zero. So this approach did not give you what you are looking for. You were not guided by a formal calculation but rather by a strong intuition. You want badly 1+2+4+8+... to be a positive infinity, but the whole world of arithmetic blows up in the process. Bo Jacoby 18:23, 26 February 2006 (UTC)
I do assure you that the respect is mutual. Bo Jacoby 18:38, 26 February 2006 (UTC). I'll try to translate your system into the powerfull language of formal power series. The series a0+a1+a2+... is identified with the generating function f(x)=a0+a1x+a2x2+... . The symbol x is merely a formal parameter. Shifting to the right corresponds to multiplication with x . Two series have the same 'tail' if the difference between the corresponding generating functions is a polynomial. This is an equivalence relation between functions. So 1+1+1+... corresponds to f(x) = 1+x+x2+... = (1−x)−1. The next example is 1+2x+3x2+...=(1−x)−2. The third example is 1+2x+4x2+...=(1−2x)−1. The fourth example is x+2x2+3x3+...=x(1−x)−2. Finally: 1+(1+1+1+...) corresponds to 1+(1−x)−1=(2−x)(1−x)−1. As these computations do not reproduce your results, this is probably not what you ment. Bo Jacoby 08:51, 27 February 2006 (UTC)
Yes, I do approve your wording. I just wrote the section Polynomial#Using_polynomials_for_extending_the_concept_of_number with examples of formal calculation. We might refer to it from this article. The point is that even if we do not know what a certain algebraic number is, we may still reduce a polynomial of that algebraic number. So if i2=−1, then (1+i)2=1+2i+i2=2i. This is the power of formal calculation, and I think the article should show it as clearly as possible. Bo Jacoby 13:52, 27 February 2006 (UTC)
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Created the article, per discussion. The discussion is copied here for convenience. Be sure to improve the article and\or discuss changes here. -- Meni Rosenfeld 15:50, 15 January 2006 (UTC)
During my studies, I have encountered the concept of a "formal calculation", in the sense of, roughly, a calculation for which the steps are not completely substantiated, and yet the result can give us insight about the true answer to the problem in question. I want to write an article about that concept, but I haven't found any references to it on the web, so I'm not sure how widely it is used and whether I understand the concept properly. Any ideas? -- Meni Rosenfeld 18:34, 12 January 2006 (UTC)
A formal argument is when you just follow what the syntax seems to suggest your reasoning, without proving the reasoning is sound. Like when you prove that, in a ring, if (1+ab) is invertible, then so is (1+ba) by using power series. Power series don't exist in a ring, but but you can still make formal arguments using them. - lethe talk 21:58, 12 January 2006 (UTC)
Lethe's example is what I would call a heuristic inference. It seems very strange to me to call this "formal": it's good because of informal gut feeling experience, not in virtue of the formal structure of the problem. --- Charles Stewart 22:02, 12 January 2006 (UTC)
Are all in favor of creating a stub, bearing the title "Formal calculation", based on the definition Jitse found, and beating it around until we reach something we can agree upon? -- Meni Rosenfeld 13:40, 13 January 2006 (UTC)
I know that "formal calculation" seems to imply a rigorous one, and actually that did confuse me the first times I encountered the concept. But I got the impression that, while perhaps ambiguous, it is usually used in the sense I described - Much like in the probably more common term formal power series. In this sense, "formal" actually means of form, namely, the form of the objects matter and not their underlying meaning - making the calculation perhaps systematic, but not really rigorous because we are using properties without any justification to why these properties should hold. We could always delete the article later if we can't seem to rich any consensus. -- Meni Rosenfeld 14:59, 13 January 2006 (UTC)
Of course formal power series are ultimately defined in a rigorous way, but the inspiration for this definition comes from a non-rigorous application of properties of convergent power series to arbitary power series. That's where the term "formal" comes from. -- Meni Rosenfeld 15:12, 13 January 2006 (UTC)
I think that this is a good topic for an article, and it may well prove useful for my planned article on Boole's algebraic logic (to be carefully distinguished from Boolean algebra, since Boole's system allows terms that do not have set-valued denotations). They can be seen to be similar to the status of polynomials prior to the discovery of complex numbers: onbe can know the sum and product of the roots of a quadratic and know furthermore that those roots don't exist. If we are to resort to neologism, why not optimistic calculation? --- Charles Stewart (talk) 16:29, 13 January 2006 (UTC)
It appears that the phrase is used in the proposed sense. It also appears to be understood in other ways, and it appears that some folks feel that the proposed sense is not a good sense. For an inclusionist (not necessarily me), Wikipedia should have an article. The article should note the opposition and provide disambiguation. However, a major unresolved question is: What is the primary meaning of "formal calculation"? The answer to that I do not know, but I'm inclined to think it's the "rigorous" sense, not the proposed sense. -- KSmrq T 01:23, 14 January 2006 (UTC)
In a nutshell, I think my original proposition of creating a stub and beating it around is fair. I'll do that now. Be sure to check it out for any flaws\omissions\whatever as I am an inexperienced editor. Formal calculation. -- Meni Rosenfeld 15:20, 15 January 2006 (UTC)
I can't really add more than has been said above, but if there's still some doubt as to how common this is...I can say that I've ran into this not just in lectures or talks but in books. As I can best recollect, you usually say it's just a formal calculation when you are dealing with either infinite series or integrals to "derive" an expression by playing with the original expression and using certain formal rules. These rules are possibly not fully justified, due to matters of convergence, but the resulting expression is sometimes useful. I'm pretty certain this has occurred in plenty of texts. I suggest looking through mathematical physics literature in particular. -- C S (Talk) 16:25, 15 January 2006 (UTC)
Euler-Maclaurin formula#Motivation for the existence -- Meni Rosenfeld 19:18, 15 January 2006 (UTC)
In section 3 of talk:Formal_power_series Oleg and I have a lengthy discussion on the subject. Bo Jacoby 22:20, 15 January 2006 (UTC)
I have expanded the article a bit. I still think it needs additional work, since it doesn't have much content and what It does have is based mainly on my intuition, but does anyone think we should remove its stub status? -- Meni Rosenfeld 07:15, 16 January 2006 (UTC)
I don't like the dichotomy that the intro presents between different contexts of the word formal. I don't view the word as being a contronym. For me, the over-arching meaning of the word "formal" is something like "deals with syntax but not semantics". A formal calculation is one that manipulates the form of symbols, the syntax. It may fail to be rigorous if the calculation is applied to systems which do not follow the rules of syntax used, the semantics. It will be (and can always be made, if it is not) completely rigorous if the system in question is that defined by the syntax, in a sense which is made precise by the notion of adjoint functors, as for example the formal sums of triangles used in simplical homology, which are essentially defined by their syntax, or the enveloping algebra of a Lie algebra. So it's not that people use the word to mean opposite things, it's just that a formal calculation may be not always be applicable where someone may try to apply it. Would you kindly folks like to share your thoughts about my point of view on this matter? - lethe talk + 11:48, 4 February 2006 (UTC)
Formal calculations are without interpretations. Example: 10−17+8 = −7+8 = 1. The intermediate result −7 has no interpretation as a number of pebbles. The detour 10−17+8 = 10+8−17 = 18−17 = 1 eases the minds of those who rely on the limited interpretation of numbers as counting pebbles. Mathematicians object against the 'formal' calculation S=1+2+4+8+...=1+2S implying S=−1 because they rely on a limited interpretation of infinite series. Bo Jacoby 13:36, 22 February 2006 (UTC)
Let me try. There are many examples in mathematics where members of a proper subset of a set has interpretations, but not every member of the set has an interpretation. The first example above is that the subset of nonnegative integers are cardinal numbers of finite sets, while the negative integers are not cardinal numbers of finite sets. The second example above is that the subset of convergent series define real numbers while the divergent series do not define real numbers. A third example is found in the article on multiset where power series with nonnegative integer coefficients define multisets of nonnegative integers, while other power series do not define multisets. Manipulation with elements that do not have interpretations are called formal. As a formal calculation can not be understood in terms of the usual interpretation, it is often rejected as nonsensical. Bo Jacoby 08:54, 23 February 2006 (UTC) PS. See also the fine example in Complex_number#History. Bo Jacoby 09:26, 23 February 2006 (UTC)
If a calculation is formal or not depend on the interpretation. A formal calculation is a logically consistent argument in the set in which it takes place, but not in the subset in which it is interpreted. The example from complex number history is not considered formal any longer, now that complex number interpretation is common knowledge. Nobody considers computations involving negative numbers 'formal' any longer, but once they were. I consider the equation 1+2+4+8+... =−1 to be absolutely true, and I completely disagree with the article saying: "This is of course absurd in the context of reals, where power series originated from". I expect the acceptance of divergent series defining values, to be crucial to elementary particle physics which is presently troubled with perturbation calculations giving divergent results. The argument is simple. Let S=1+2+4+8+...=1+(2+4+8+...)=1+2(1+2+4+...)=1+2S, so S=1+2S, so S=−1 . There is nothing absurd in the result. You expect the sum of positive numbers to be positive ? Well, that is true for finite sums but not for infinite sums. Get accustomed to that, just as you got accustomed to the fact that "the sum of rational numbers is a rational number" is true for finite sums but not for infinite sums. Bo Jacoby 14:12, 23 February 2006 (UTC)
Let take an elementary example. A child understands that 2+2=4 based on counting fingers. The finger method of addition also works for more difficult problems like 6+1 and 3+4. Now he tries 7+4 and realizes that he does not have that many fingers. So he says: "according to the original definition, the sum does not exist to begin with". He considers "7+4=11" to be a formal calculation. The teacher did ask the child to count on his fingers, but now he says: "Forget what I taught you about fingers". The child says: "Why did you teach me in the first place what you now ask me to forget?". The teacher answers: "Because you could not understand the general case until you tried some special cases". So, after some resistance, the intelligent child lets go of the limitations of the original definition and understands that the logic of counting extends the number of fingers. Bo Jacoby 09:10, 24 February 2006 (UTC)
Your example fits nicely into the analogy. Define 4+7=ω, meaning many. Any number greater than ten is many. That extension is perfectly plausible and possible and even useful to very young children. The trouble is that arithmetic no longer works, because the nice rule "a+b=a+c implies b=c" no longer applies. The equation 4+7=4+8 is true because ω=ω, but the equation 7=8 is not true. The rules of arithmetic are the rules we want to keep. Any solution to the equation x=1+2x is equal to −1, and the formal expression 1+2+4+8+... is a solution to the equation x=1+2x. The equation 1+2+4+8+...=−1 is no more absurd than the equation 4+7=11, and the equation 1+2+4+8+...=2ω is as damaging to logic as the equation 4+7=ω. Bo Jacoby 14:04, 24 February 2006 (UTC)
I look forward to seeing your theory, showing my example of cancellation incorrect. I accept that choosing any definition is subjective; (this seems to be a fair compromise between calling one option "absolutely true", which I did, and calling the same option "absurd", which you did). I share your taste regarding the projective extension of the real line, (mainly because this option can be extended to the complex plane, where only one infinity is introduced to produce the Riemann sphere). The projective extension even supports the idea of negative numbers being beyond infinity, so you should not object that much against 1+2+4+8+...=−1 . I agree that the breakdown of the cancellation rule in the theory of cardinal numbers didn't stop anyone from discussing that theory, but I expect that Georg Cantor would have liked to preserve the rule if at all possible. Let's list the options. Option 1: 1+2+4+8+...=−1. Option 2: 1+2+4+8+...is undefined. Option 3: 1+2+4+8+...=2ω. Option 1 is a result of a formal calculation. It leads to no logical problems, but to the counterintuitive result that an infinite sum of positive numbers need not be positive. Option 2 is perfectly powerless. Option 3 leads to the breakdown of arithmetics. To me the choice is easy. Consider the following formal calculation. 0=0+0+0+...=(1−1)+(1−1)+(1−1)+...=1−1+1−1+1−1+...=1+(−1+1)+(−1+1)+(−1+1)+...=1+0+0+0+...=1. What can be done to this scary result? First possibility: "Accept for a fact that 0=1". Second possibility: "Ban all divergent series". Third possibility: "Accept that you are not always allowed to insert or delete more than a finite number of pairs of parentheses into an infinite sum". The first possibility leaves us with no arithmetic at all. The second possibility, I believe, was the choice of the mathematician Niels Henrik Abel, and others follow his judgement. The third possibility gives a useful extension of assigning values to series, and no drawbacks except some delightful discussions with fellow mathematicians (See Talk:Formal_power_series). The formal calculation breaks down to this:
Bo Jacoby 14:48, 26 February 2006 (UTC)
Thanks. It looks interesting. Let me see if I understand you right. The sum ω = 1+1+1+... = 1+(1+1+...) satisfies the equation ω=1+ω which has no solution in a nontrivial ring (mathematics). So the ring is the trivial ring where 0=1. All expressions equal 0, no positive numbers exist, and no number is greater than another number. 0+1+2+3... is not really different from 1+2+3+4+... because they are both zero. So this approach did not give you what you are looking for. You were not guided by a formal calculation but rather by a strong intuition. You want badly 1+2+4+8+... to be a positive infinity, but the whole world of arithmetic blows up in the process. Bo Jacoby 18:23, 26 February 2006 (UTC)
I do assure you that the respect is mutual. Bo Jacoby 18:38, 26 February 2006 (UTC). I'll try to translate your system into the powerfull language of formal power series. The series a0+a1+a2+... is identified with the generating function f(x)=a0+a1x+a2x2+... . The symbol x is merely a formal parameter. Shifting to the right corresponds to multiplication with x . Two series have the same 'tail' if the difference between the corresponding generating functions is a polynomial. This is an equivalence relation between functions. So 1+1+1+... corresponds to f(x) = 1+x+x2+... = (1−x)−1. The next example is 1+2x+3x2+...=(1−x)−2. The third example is 1+2x+4x2+...=(1−2x)−1. The fourth example is x+2x2+3x3+...=x(1−x)−2. Finally: 1+(1+1+1+...) corresponds to 1+(1−x)−1=(2−x)(1−x)−1. As these computations do not reproduce your results, this is probably not what you ment. Bo Jacoby 08:51, 27 February 2006 (UTC)
Yes, I do approve your wording. I just wrote the section Polynomial#Using_polynomials_for_extending_the_concept_of_number with examples of formal calculation. We might refer to it from this article. The point is that even if we do not know what a certain algebraic number is, we may still reduce a polynomial of that algebraic number. So if i2=−1, then (1+i)2=1+2i+i2=2i. This is the power of formal calculation, and I think the article should show it as clearly as possible. Bo Jacoby 13:52, 27 February 2006 (UTC)