## What are identities in elementary mathematics?

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 What are identities in elementary mathematics? Sergei Akbarov 10/22/10 2:00 PM Dear colleagues,Who can explain me what people in Calculus (or, I do not know, maybe I should say "in elementary mathematics") mean by equality of elementary functions? I mean, everybody knows, of course, that equality of functionsf(x)=g(x)on, say, an interval I means that they coinside in each point x\in I.This is the definition for all, not necessarily elementary functions. But one can expect that elementary functions (I mean, x^a, a^x, sin, cos, etc.) can be defined independently in purely algebraic way, say, like an algebra, generated by a list of identities. Of course, the definition must be such that all the other identities could be deduced from this prearrannged list as corollaries, and this will be equivalent to the definition of this algebra as a subalgebra of all functions on an interval I.People in computer algebra are discussing different ways to teach computer to recognize identities in elementary mathematics, see, e.g the book "A=B"http://www.math.upenn.edu/~wilf/AeqB.htmland the identities they consider are "analytical identities", i.e. to study them we should consider our functional algebra as a subalgebra of the algebra of all functions on I (or of all continuous, smooth, analytical, etc., functions on I). But perhaps there is a way to separate elementary functions from analysis? I mean to axiomatize them in such a way that the question of whether f(x)=g(x) is true or not will be just a question, whether this identity can be deduced in algebraic way from the axioms of the theory?Were there any investigations in this field? Does such approach indeed has a chance to exist, or, maybe there are some negative results?I would greatly appreciate any references, suggestions, etc.Sergei Akbarov Re: What are identities in elementary mathematics? Norbert Marrek 10/23/10 9:00 AM Am 22.10.2010 23:00, schrieb Sergei Akbarov:In Z/2Z the polynomial functions f(X)=X+1 and g(X)=X^2+1have the save values f(9)=g(0)=1 and f(1)=g(1)=0.Would you consider X+1 the same as X^2+1?Aloha        Norbert Re: What are identities in elementary mathematics? tc...@lsa.umich.edu 10/23/10 9:00 AM In article ,Sergei Akbarov wrote:>But perhaps there is a way to separate elementary functions from>analysis? I mean to axiomatize them in such a way that the question of>whether f(x)=g(x) is true or not will be just a question, whether this>identity can be deduced in algebraic way from the axioms of the theory?I think the top two Google Scholar hits on "recognizing zero" shouldpoint you in the right direction.-- Tim Chow       tchow-at-alum-dot-mit-dot-eduThe range of our projectiles---even ... the artillery---however great, willnever exceed four of those miles of which as many thousand separate us fromthe center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences Re: What are identities in elementary mathematics? David Hobby 10/24/10 2:00 PM On Oct 23, 12:00�pm, "tc...@lsa.umich.edu" wrote:> In article ,>> Sergei Akbarov wrote:> >But perhaps there is a way to separate elementary functions from> >analysis? I mean to axiomatize them in such a way that the question of> >whether f(x)=g(x) is true or not will be just a question, whether this> >identity can be deduced in algebraic way from the axioms of the theory?>> I think the top two Google Scholar hits on "recognizing zero" should> point you in the right direction.> --> Tim Chow � � � tchow-at-alum-dot-mit-dot-edu...As an aside, I'd expect axiomatizing the theory of elementaryfunctionsto be tricky.  Compare this to Tarski's High School Algebra Problem,which considered a small subset of all elementary functions and wasstill quite difficult.http://en.wikipedia.org/wiki/Tarski%27s_high_school_algebra_problem---David Hobby Re: What are identities in elementary mathematics? Dan Luecking 10/25/10 5:00 PM On Sat, 23 Oct 2010 17:00:06 +0100 (BST), Norbert Marrek wrote:>Am 22.10.2010 23:00, schrieb Sergei Akbarov:>> Dear colleagues,>>>> Who can explain me what people in Calculus (or, I do not know, maybe I>> should say "in elementary mathematics") mean by equality of elementary>> functions? I mean, everybody knows, of course, that equality of functions>>>> f(x)=g(x)>>>> on, say, an interval I means that they coinside in each point x\in I.>>>> This is the definition for all, not necessarily elementary functions. But>> one can expect that elementary functions (I mean, x^a, a^x, sin, cos, etc.)>> can be defined independently in purely algebraic way, say, like an algebra,>> generated by a list of identities. Of course, the definition must be such>> that all the other identities could be deduced from this prearrannged list>> as corollaries, and this will be equivalent to the definition of this>> algebra as a subalgebra of all functions on an interval I.>>>> People in computer algebra are discussing different ways to teach computer>> to recognize identities in elementary mathematics, see, e.g the book "A=B">>>> http://www.math.upenn.edu/~wilf/AeqB.html>>>> and the identities they consider are "analytical identities", i.e. to study>> them we should consider our functional algebra as a subalgebra of the>> algebra of all functions on I (or of all continuous, smooth, analytical,>> etc., functions on I). But perhaps there is a way to separate elementary>> functions from analysis? I mean to axiomatize them in such a way that the>> question of whether f(x)=g(x) is true or not will be just a question,>> whether this identity can be deduced in algebraic way from the axioms of the>> theory?>>>> Were there any investigations in this field? Does such approach indeed has a>> chance to exist, or, maybe there are some negative results?>>>> I would greatly appreciate any references, suggestions, etc.>>>> Sergei Akbarov>>>>>>In Z/2Z the polynomial functions f(X)=X+1 and g(X)=X^2+1>have the save values f(9)=g(0)=1 and f(1)=g(1)=0.>Would you consider X+1 the same as X^2+1?I certainly would, because surely x^2 = x is one of the identities one would include as an axiom for elementaryfunctions over Z/2Z.Similarly, I would consider e^x e^y the same as e^{x+y} over the reals (thought not over the ring of n x n matrices).DanTo reply by email, change LookInSig to luecking Re: What are identities in elementary mathematics? mjc 10/28/10 1:00 AM >> I certainly would, because surely x^2 = x is one of the> identities one would include as an axiom for elementary> functions over Z/2Z.>> Similarly, I would consider e^x e^y the same as e^{x+y}> over the reals (thought not over the ring of n x n> matrices).>> DanI would consider both of these to be theorems, not axioms (the firstfrom 1+1=0, the second from the definition of e^x and properties ofexponentiation and limits). Re: What are identities in elementary mathematics? Ilya Zakharevich 10/28/10 12:00 PM On 2010-10-28, mjc wrote:>>>> I certainly would, because surely x^2 = x is one of the>> identities one would include as an axiom for elementary>> functions over Z/2Z.>>>> Similarly, I would consider e^x e^y the same as e^{x+y}>> over the reals (thought not over the ring of n x n>> matrices).> I would consider both of these to be theorems, not axioms (the first> from 1+1=0,I doubt it.  In F_4, 1+1=0, but x^2 is not equal to x.> the second from the definition of e^x and properties of> exponentiation and limits).I do not know what is a "limit".  And I wonder which "property ofexponentiona" would imply "e^x e^y is the same as e^{x+y}"...Ilya  [P.S.  Is it my imagination, or had the quality of moderation         slipped a bit during the last year?] Re: What are identities in elementary mathematics? Dan Luecking 10/29/10 4:30 AM On Thu, 28 Oct 2010 09:00:03 +0100 (BST), mjc wrote:>>>> I certainly would, because surely x^2 = x is one of the>> identities one would include as an axiom for elementary>> functions over Z/2Z.>>>> Similarly, I would consider e^x e^y the same as e^{x+y}>> over the reals (thought not over the ring of n x n>> matrices).>>>> Dan>>I would consider both of these to be theorems, not axioms (the first>from 1+1=0, the second from the definition of e^x and properties of>exponentiation and limits).I was referring to the original poster's discussion: axioms for a system used to determine equality of elementary functions in a purely algebraic way. This system needs a list of rules or identities for transforming one function into another. These would be the _axioms_ of such a system, and they could certainly be required to depend on what the variables represent.For example, the algebra of elementary functions (over C) is defined to be the smallest set of formulas that contains all constants, z and e^z, and is closed under the usual arithmetic operations plus also functional composition and inversion. The usual field axioms would have to be part of the set of rules (or axioms), and some of the laws of exponents (since not all follow algebraicly from the fieldaxioms).Over C, sin z can be defined in terms of e^z, and varioustrig identities follow from identities for e^z. But over R, one would have to add at least sin x to the starting list. And some more identities for things like sin(x+y).Finally, if our elementary functions are going to be operating over Z/2Z, we need identities or axioms appropriate for that system. Surely we would want our system to include x^2 = x as this is both useful and, in the form (x-1)x = 0, concisely expresses the single defining property of Z/2Z among fields: every element is either 0 or 1. If we did not include it we would have to include something else that expresses the difference between Z/2Z and R or C.My main point was that the set of identities (or axioms) differs depending on what the variables in our functions represent. So x^2 is the same as x in Z/2Z (and not in R)because some axioms make it so. In my previous post, I did not say that the e^{x+y} = e^x e^y was an axiom, but that it is an identity valid for real variables and not for functions of matrix variables. (Still, it doesn't follow from the field axioms unless x and y are rational, so _some_ axiom would still have to be added, and the rules of the game were that it has to be algebraic.) DanTo reply by email, change LookInSig to luecking