Talk:Formal calculation

New article

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)

Formal calculation (original discussion)

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)

On the contrary, I think of a "formal calculation" specifically as a calculation in which every step is very clear and verifiable. I'm not sure I know a name for what you're referring to. Meekohi 20:43, 12 January 2006 (UTC)

I think I know roughly what Meni is trying to say. I would have thought you might find it at heuristic or heuristic argument or something similar, but they seem to be run by philosophers. Dmharvey 20:47, 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)

Lethe's reply coincides with my experience. I suspect that it may be hard to find good references, but I remember reading about it recently. Bear with me … -- Jitse Niesen (talk) 22:05, 12 January 2006 (UTC)

Here we are. Stuart S. Antman, Nonlinear Problems of Elasticity, Applied Mathematical Sciences vol. 107, Springer-Verlag, 1995. Page 1 contains the paragraph: "I follow the somewhat ambiguous mathematical usage of the adjective formal, which here means systematic, but without rigorous justification. A common exception to this usage is formal proof, which is not employed in this book because it smacks of redundancy." (his emphasis). -- Jitse Niesen (talk) 22:28, 12 January 2006 (UTC)

I think the term systematic calculation would be far more fitting nomenclature, but that doesn't really carry the connotation of being subtly incorrect that we're looking for. Meekohi 02:02, 13 January 2006 (UTC)

I wouldn't call it incorrect: it is, after all, an excellent heuristic. I'd rather say it was non-well-founded. --- Charles Stewart 02:13, 13 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 don't know, personally I'm fairly opposed. To me the term Formal Calculation distinctly implies that it is rigorously correct. The reference Jitse gave doesn't really give much support in my mind, seeing as he points out this is ambigous usage. If we are going to make an article on it, I think the main article should describe what it means to be rigorous/systematic, and then there should be a short section pointing out that it is possible to be apparently systematic, but still incorrect. Meekohi 14:05, 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)

Formal power series are just sequences over a ring with convolution as multiplication. Since all sums involved are finite, this is a rigerous mathematical topic. Convergent power series is a different topic requiring the ring to be a Banach algebra. In france there is a state wide research association called "Calcul formel", which would probably translate as symbolic calculus or even symbolic algebra. The research and design of computer algebra systems is part of that.--LutzL 15:09, 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 the originally-proposed topic is a 'derivation', universal in (say) theoretical physics. It's not a particularly good topic for an article, though. Charles Matthews 16:20, 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)

I think "formal" in "formal calculation" has the same meaning as in "formal power series". In my experience, it is often used in the following context (for instance, in a talk on Kolmogorov-Arnold-Moser theory which I just attended): We want to prove that a function f_epsilon with a certain property exists for epsilon sufficiently small. We know f_0, so we expand f_epsilon in a power series in epsilon. If this is possible (i.e., if we can find all the coefficients in the power series), we have a "formal solution". To prove that this is actually a solution, we have to show that the power series has a positive radius of convergence.

So, formal is not just optimistic. And I don't think "formal" in this meaning is a neologism either, as Meni, Lethe and I have all heard of "formal" in this meaning. -- Jitse Niesen (talk) 18:08, 13 January 2006 (UTC)

Another example: formal group law. Dmharvey 21:16, 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)

I believe the phrase is commonly used in physics in the sense of "we know this can't possibly be right, but by shoving symbols around on a page, here's what you can come up with". For example, "formally", one has 1+2+3+...=-1/12, which is clearly both "right" and "wrong" in various deep ways. That is, its ambiguous without further clarification about how in the world this could possibly be a valid manipulation; but in physics, further clarification is often too hard to provide. A formal calculation is one step up from handwaving. linas 06:01, 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)

Yeah it seems that there is enough support for the idea now that we should have an article, even though I still don't like the terminology ;) Meekohi 15:28, 15 January 2006 (UTC)

Further discussion

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)

Another example for a formal calculation

Euler-Maclaurin formula#Motivation for the existence --Meni Rosenfeld 19:18, 15 January 2006 (UTC)

Discussion on formal calculation

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)

Stubiness

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 removed the stub notice. Wikipedia:Stub says that stubs are generally below 10 sentences. Well done! -- Jitse Niesen (talk) 08:52, 16 January 2006 (UTC)

false dichotomy between rigor and non-rigor

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)

I, too, am not very comfortable with the dichotomy presented. I've put it merely because it reflects the opinions of some, according to the above discussion. I don't think people use "formal calculation" in a way different from the one emphasized in the article. However, while I agree completely that the basic meaning of "formal" is invariably "of syntax and not of semantics", I believe what this means in practice does vary. A "formal definition" or "formal proof" means one that is completely rigorous, since we disregard our intuitive interpretation and focus on the exact syntax. However, a "formal calculation" is one that is not very rigorous, since we manipulate expressions based on what they look like and not on what they mean.

I hope this clarifies some of the confusion. However, I'd be more than happy if you rewrite the intro in a way you're more comfortable with. -- Meni Rosenfeld (talk) 14:05, 20 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)

I'm not sure I understand... Can you clarify what you meant? -- Meni Rosenfeld (talk) 14:00, 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)

Okay, I don't believe that saying "a formal calculation is rejected" is entirely correct. Of course no one considers a formal calculation to be a logically consistent argument, but it is still a useful tool, either for simplifying calculations, or as an inspiration for the creation of new "interpretations". For example, 1+2+4+8+... has no meaning when discussing reals, but it has a very precise meaning, and is equal to -1, when discussing 2-adics instead. -- Meni Rosenfeld (talk) 11:38, 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)

Now I understand (more or less) what you are trying to say. And I think we differ mainly on how we call things. When I said "this is absurd", I did not mean anything like "we must haunt down everyone who says this like the dogs that they are" (humorously exaggerating, of course), but rather "this does not agree with the usual definition of infinite sums in real analysis". Expanding the definition to include such sums is fine, but it is important to remember that this definition is not a result of the original definition, which does not say anything about such sums. It is an acceptable expansion driven by the desire to make certain algebraic properties hold. Your "proof" that S=-1 may demonstrate why it is desirable to define it in a specific way, but it is by no means a proof that it holds, since according to the original definition, the sum S does not exist to begin with, so you can't perform operations on it, let alone assume that you can play with it algebraically like you would do with ordinary numbers (which is just as bad as assuming the sum of positive numbers must be positive).

Having clarified this point, do you think we have a conceptual disagreement, or just that the phrase you mentioned needs a rewrite? -- Meni Rosenfeld (talk) 14:38, 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)

I understand the analogy, but I think the difference is that with the addition, there is apparently only one plausible extension, but with infinite sums there isn't. The definition S=-1 builds on some properties of the original definition, while completely ignoring others - And this, I believe, can be done in more than one way. Such a definition is based on purley algebraic concepts, disregarding the analytical implications. Here is an alternative interpretation: Since we have

$\sum _{k=0}^{n}q^{k}={\frac {q^{n+1}-1}{q-1}}$

We can decide to let

$\sum _{k=0}^{\infty }2^{k}={\frac {2^{\omega }-1}{2-1}}=2^{\omega }-1$

And build a suitable framework around this concept, which gives a specific meaning to expressions like $2^{\omega }$, and this can be done in such a way that $2^{\omega }$ is greater than any real number, and obviously positive. I believe this is a part of what they do in non-standard analysis (but I'm no expert on that topic). So my point is that what you dicuss is a plausible extension, just not the only plausible extension. The extension will depend on which properties we want to keep, and which we are willing to discard. That is why when we discuss more elementary concepts, we can't just assume that S must be equal -1 and so forth. -- Meni Rosenfeld (talk) 09:37, 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+...=2^ω is as damaging to logic as the equation 4+7=ω. Bo Jacoby 14:04, 24 February 2006 (UTC)

I respectfully disagree. First, your specific example of cancellation isn't correct - I can present to you the theory I've come up with to deal with this issue (I honestly don't know if it was published before, although it probably was), but my guess is that you won't like it anyway - Although I do like it. And addition definitely does cancel, although admittedly other rules are discarded. This leads us to the point that which rules we want to keep is subjective. In ordinal and cardinal numbers we don't have the rules of arithmetic (cancellation included), but that didn't stop anyone from dicussing them. If you'll allow me to use your own words, "You expect all the rules of arithmetic to hold? Well, that is true for finite sums but not for infinite sums. Get accustomed to that.". I respect your preference to a specific set of rules over another, which you undoubtedly share with many other mathematicians and perhaps even the majority, but it is nowhere near absolute. To give another example: When attempting to extend the real number line, among the ways to do it are affinely, introducing a positive and negative infinity, and projectively, introducing an unsigned infinty. Both give up arithmetic rules, but they are still dicussed. The affine extension keeps the order relation, which makes it more popular; The projective extension discards order, but it allows division by zero and related consequences, making it a personal favorite of mine. Neither is more "correct" than the other; Which one is more elegant is subjective. -- Meni Rosenfeld (talk) 08:20, 26 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+...=2^ω. 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:

0 = 0+0+0+... = (1−1)+(1−1)+(1−1)+...

1−1+1−1+1−1+... = 1/2

1+(−1+1)+(−1+1)+(−1+1)+...=1+0+0+0+... = 1.

Bo Jacoby 14:48, 26 February 2006 (UTC)

About "absurd", I have rephrased it to be more neutral. I agree with the ideas you present, but still differ in taste. About my ideas, we first consider arbitary sequences of reals (seen as functions from N to R, where N here includes 0): We say that two sequences have the same tail (I'm still working on the terminology) if they are equal for almost any n (that is, for all values of n beyond an arbitary point). This is obviously an equivalence relation, so we shall name every equivalence class a "new number", and also call it the tail of every one of its elements. Now we denote by f(ω) the tail of the sequence a_n = f(n). For example the tail of 1,1,1,... is 1; The tail of 0,1,2,3,4,... is ω; The tail of 0,1,4,9,16,... is ω²; And the tail of 1,2,4,8,... is 2^ω. It is important to note that shifting a sequence changes its tail, for example the tail of 1,2,3,4,5,... is ω+1. Now we can define an infinite sum to be the tail of the sequence of its partial sums. For example:

$\sum _{n=0}^{\infty }1=\omega$

$\sum _{n=0}^{\infty }n={\frac {\omega ^{2}+\omega }{2}}$

$\sum _{n=0}^{\infty }2^{n}=2^{\omega }-1$

We can also define arithmetic operations on numbers as element-wise operations on their representative sequences (which can be easily shown to be well-defined). In particular, if a+b=a+c, it means that if a_n, b_n and c_n are representatives of a, b and c, then for almost any n, a_n + b_n = a_n + c_n and therefore b_n = c_n, which thus holds for almost any n, meaning that b=c.

Now here's the trick (the part which you won't like): Since shifting a sequence changes its tail, the same applies for series, too. So for example, 0+1+2+3+4+5+... is very different from 1+2+3+4+5+6+...; The latter is always one element ahead of the former, and is thus greater than it by ω. So this means that an infinite sum can't be intuitively interpreted as simple addition, and we must use an exact notation like Σ if we wish to avoid inaccuracies.

An additional point is that ω, let alone 2^ω, are greater than any real number x, since n>x for almost any n. Among other things, this means that the sum we discussed is positive.

I guess you don't like this structure, but in my opinion it is very rich (allowing, among other things, defining different magnitudes of infinities and infintesimals), and keeps the idea that the sum of positive quantities is positive (which to me, is just as important as some arithmetic rules). -- Meni Rosenfeld (talk) 16:07, 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 told you you wouldn't like it. The point is that you can't write 1+1+1+1+... = 1+(1+1+1+1+...) since infinite sums aren't just addition. The former is "an infinite sum of 1's", the latter is 1 plus an infinite sum of 1's. Basically this means that the latter has one more 1 than the former. But like I said, you really do have to use a more precise notation to deal with it consistently. And no, the whole world of arithmetic does not blow up in the process, you just interpret things a bit differently (who said interpreting infinite sums would be easy?). To me it looks very logical, and I respect the fact that you don't feel the same - But I expect this respect to be mutual, which it doesn't currently seem to be. -- Meni Rosenfeld (talk) 18:33, 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)

I haven't worked with formal power series enough to be sure if the disagreement between them and my system is that terrible. Currently it seems to me to be merely another negative point for it - Which still does little to subtract from my belief in its elegance. Perhaps my opinion will change in the future, but the fact remains that this is still a possible interpretation (even if arguably not a very useful one). Perhaps I will also come up with (or learn of) a theory that still addresses my main concerns while having less drawbacks. The point is that I still believe it demonstrates why a statement like "1+2+4+8+...=-1 is absolutely true" is unwise, while I agree that saying it is absurd is equally unwise. And I do hope you approve of the new wording I gave it. -- Meni Rosenfeld (talk) 11:39, 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)