Undergraduate Seminar II
Undergraduate Seminar II MATH 2794
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This 9 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 2794 at University of Connecticut taught by Keith Conrad in Fall. Since its upload, it has received 22 views. For similar materials see /class/205817/math-2794-university-of-connecticut in Mathematics (M) at University of Connecticut.
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Date Created: 09/17/15
ADVICE ON MATHEMATICAL WRITING This handout lists some writing tips when you are dealing with mathematical text 1 NOTATION 1 Do not begin sentences with a symbol Bad x is positive so it has a square root Good Since x is positive it has a square root Bad Let n be an even number n 2m for some m E Z Good Let n be an even number Thus 71 2m for some m E Z Good Let n be an even number so 71 2m for some m E Z Bad One solution is x sin x x is periodic Good One solution is x sin x In this case x is periodic 2 If two mathematical symbols are not part of the same mathematical expression they should never appear next to each other with no words or grammatical marks in between them Bad lfn7 0n2gt0 Good If n 7 0712 gt 0 Good If n 7 0 then 712 gt O 3 When introducing notation make it t the context A lot of the time a choice of notation is just common sense Bad Let m be a prime Good Let p be a prime Bad Let X be a set and pick an element of X say 25 Good Let X be a set and pick an element of X say x Bad Pick two elements of the set X say x and u Good Pick two elements of the set X say x and y Good Pick two elements of the set X say x1 and x2 Good Pick two elements of the set X say x and x 4 Always de ne new notation is it a number a function of what type and be clear about its logical standing Very bad Since 71 is composite n ab Bad Since 71 is composite n ab for some integers a and 1 Good Since 71 is composite n ab for some integers a and b greater than 1 Every integer is a product since 71 71 1 so writing 71 ab alone introduces no constraint whatsoever Bad If a polynomial x satis es n E Z does x have integer coe icients Good If a polynomial x satis es n E Z for every n E Z does x have integer coe icients ADVICE ON MATHEMATICAL WRITING 5 Do not give multiple meanings to the same variable in a proof Bad To show the sum of two even numbers is even suppose a and b are even Then a 2m and b 2m for some integer m We have a b 4m 22m which is even Notice this proof showed the sum of two even numbers is always a multiple of 4 which is nonsense Good To show the sum of two even numbers is even suppose a and b are even Then a 2m and b 271 for some integers m and n We have a b 2m 2n 2m n which is even A a V Avoid overloading meaning into notation Bad Let x gt 0 E Z Good Let x be an integer with z gt 0 Good Let x be a positive integer A I V NEVER use the logical symbols V 3 A V when writing except in a paper on logic Write out what you mean in ordinary language Bad The conditions imply a 0 A b 1 Good The conditions imply a 0 and b 1 Bad lf 3 a root of the polynomial then there is a linear factor Good If there is a root of the polynomial then there is a linear factor Bad If the functions agree at three points they agree V points Good If the functions agree at three points they agree at all points 8 Avoid silly abbreviations or the misuse of standard notations or the use of abbre viations which are used strictly on the blackboard like WLOG st and iff Bad When n is f 271 is an even number Good When n is integral 271 is an even number Good When n is an integer 271 is an even number Bad Let 2 be a C Good Let 2 be a complex number Good Choose 2 E C Bad WLOG we can assume z gt 0 Good Without loss of generality we can assume z gt 0 Bad There is a point z st x gt 0 Good There is a point x such that x gt O 9 If a piece of notation is super uous in your writing don t use it Bad Every differentiable function f is continuous Good Every differentiable function is continuous Good All differentiable functions are continuous Bad A square matrix A is invertible when its determinant is not 0 Good A square matrix A is invertible when det A 7 0 Good A square matrix is invertible when its determinant is not 0 The difference between the use of A in the Bad example and in the rst Good example above is that in the rst Good example something is actually done with ADVICE ON MATHEMATICAL WRITING 3 A we refer to it again in det A In the Bad example the use of A is super uous notation 2 EQUATIONS AND EXPRESSIONS If an equation or expression is important either for its own sake or because you will refer back to it later display the equation on its own line If you need to refer to it later label it as 1 2 and so on on the side Of course if you only need to make a reference to a displayed equation or expression immediately before or after it appears you could avoid a label and say by the above equation77 etc A H V Bad As a special case of the binomial theorem x y4 m4 4mg 1 6m2y2 lmys 31 suppose several lines of text are here By the equation 8 lines up we see Good As a special case of the binomial theorem 21 x y4 m4 4mg 1 6m2y2 lmys 31 suppose several lines of text are here By equation 21 we see If a single computation involves several steps especially more than two present the steps in stacked form A to V Bad x13m12x1m22m1m1m33x23m1 Bad 22 13 22 12z 1 m13 x2 2m 1z 1 m13 x33z23m1 Good z13 z12z1 z22m1z1 x33z23m1 Equations do not stand by themselves They appear as part of a sentence and should be punctuated accordinglyl If an equation ends a sentence place a period at the end of the line If an equation appears in the middle of a sentence use a comma after the equation if one would naturally pause there Sometimes no punctuation is needed after the equation The following three examples illustrate each possibility A DJ V Good We call 0 a critical point of 1 when 1 is differentiable and NM 0 Good When 1 is differentiable and mo satis es fmo 07 we call 0 a critical point A H V A to V A DJ V A g V ADVICE ON MATHEMATICAL WRITING Good When 1 is differentiable any mo where H900 0 is called a critical point That the equation was displayed separately in each case simply serves to highlight its importance to the reader It could have been included within the main text and punctuation rules of course apply in the same way The words critical poin 7 were set in italics to emphasize that this particular term is being de ned Some books put de ned terms in bold in the de nitions 3 PARENTHESES AND COMMAs Avoid pointless parentheses in mathematical expressions Bad m A A y m2 A yz The parentheses on the right have no purpose Good m A A y 2 A yz Bad If 7 is a factor of the product a1a2 01 then Good If 7 is a factor of the product a1a2 01 then Bad The length is a factor of p A 1 Good The length is a factor of p A 1 Good a A b2 A a A 32 b2 A 02 A 201 A 20w Good a A b2 A a A 32 b2 A 02 A 201 A 20w This example is good only if the writer wants the reader to View 2 A 02 as a single part of the right side Use parentheses to avoid confusing the meaning between a subtraction sign and a negative sign in a mathematical expression Very bad a A b A c Am A be If you look at the right side you can see the writer meant for the left side to be the product of a A b and Ac but the left side instead looks like a plus 1 minus 0 Bad a A 1 Ac AacA be Good a A bAc Am A be Commas are natural places to pause brie y but not as fully as a period If you read something in your head you should be able to notice badly placed commas either because no pause should occur or because a period should be there instead of a comma Bad The condition we want is a 21 Good The condition we want is a 21 Bad The set is in nite we pick a large nite subset of it Good The set is in nite We pick a large nite subset of it While If then 7 is a common phrase it is bad English to write Let then 7 with a comma as the separator Very Bad Let n be an even number then 71 2m for some m E Z Good Let n be an even number Then 71 2m for some m E Z Good Let n be an even number so 71 2m for some m E Z ADVICE ON MATHEMATICAL WRITING 5 4 USE HELPFUL WORDS 1 Tell the reader where you are going Good We will prove this by induction on 71 Good We will prove this by induction on the dimension Good We argue by contradiction Good Now we consider the converse direction Good But x is actually continuous To see why consider Good The inequality a g b is strict a lt 1 Indeed if there was equality then 2 Use key words to show the reader how you are reasoning These include since because on the other hand observe note At the same time vary your choice of words to avoid monotonous writing This may require you to completely rewrite a paragraph Bad We proved for any a that if a2 is even then a is even Now suppose a8 is even Since a8 a42 we obtain that a4 is even Then a2 is even Then a is even Good We proved for any a that if a2 is even then a is even Now suppose a8 is even Then by successively applying the result we proved to a a2 and a we see that a is even A DJ V Watch your spelling If you aren t sure of the difference between necessary and neccessary or discriminate and discriminant look it up Canadian students may use their own avour of spelling but non native English speakers should be careful not to let the grammatical rules of their native language affect their writing in English where those rules are different Use it s only to mean it is The word its like his and her refers to possession Bad It s clear that x has a real root since it s degree is odd Bad lts clear that x has a real root since its degree is odd Good It s clear that x has a real root since its degree is odd Good Since x has odd degree clearly it has a real root Write like this if you can t remember the difference between its and it s Bad lts surely true that starting your nal draft on the last day will leave its mark in your work Good It s surely true that starting your nal draft on the last day will leave its mark in your work 5 TYPES OF MATHEMATICAL RESULTS In mathematics results are labelled as either a theorem lemma or corollary What s the difference 0 A theorem is a main result 0 A lemma is a result whose primary purpose is to be used in the proof of a theorem but which on its own is not considered signi cant or as interesting 0 A corollary is a result that follows from a theorem It could be a special case of the theorem or a particularly important consequence of it 6 ADVICE ON MATHEMATICAL WRITING So theorems stand on their own a lemma always comes before a theorem and corollaries always come after a theorem The order in which these appear then is always Lemma Theorem Corollary There is no reason a theorem must have a lemma before it or a corollary after it But if you have a string of lemmas which don t lead to a theorem for instance then it will look strange to anyone experienced with mathematical writing Here are two examples First we give a lemma and a theorem whose proof depends on the lemma Lemma 51 In the integers if d is a factor ofa and b then at is a factor of am by for any integers z and y Proof Since at is a factor of both a and b we can write a dm and b dn for some integers m and n Then for any x and y we have am by dmm dny dmm ny which shows at is a factor of am by D Theorem 52 Ifa and b are integers and amo byo 1 for some integers m0 and yo then a and b have no common factor greater than 1 Proof This will be a proof by contradiction Suppose there is a common factor at gt 1 of a and b Applying Lemma 51 to the particular combination azo bye 1 is a factor of am byo so at is a factor of 1 But there are no factors of 1 which are greater than 1 so we have a contradiction Therefore a and b have no common factor greater than 1 D Therem 52 uses Lemma 51 but the statement of Lemma 51 was deemed by the author writing it to be worth isolating on its own So it becomes a lemma rather than appear completely inside the proof of Theorem 52 Perhaps the author anticipates other uses of Lemma 51 and so wants to state it separately Next we give a theorem in linear algebra and a corollary which follows from the theorem Theorem 53 For any two square matrices A and B detAB det Adet B The proof of this is hard and is not included here Corollary 54 An invertible matrix has a nonzero determinant Proof If A is invertible say of size n x n then AB In for some matrix B Taking the determinant of both sides Theorem 53 tells us det Adet B detIn 1 so det A 7 0 D Why do we need lemmas at all Could we call everything a theorem Yes but the point of the three different names lemma theorem corollary is to indicate to the reader how the writer views the comparative standing of the different results Although lemmas are principally intended to be used for the proof of a more important result sometimes a lemma turns out to be a very signi cant result and is even named after someone but is still referred to as a lemma for historical reasons Examples include Hensel s lemma in number theory Nakayama s lemma in commutative algebra the RiemanniLebesgue lemma in harmonic analysis and the Schwarz lemma in complex ADVICE ON MATHEMATICAL WRITING 7 analysis These named lemmas are so standard in the literature that for instance to refer to Hensel s theorem or Nakayama s theorem would generate very strange looks In addition to the mathematical results in a paper terminology is used and may need to be de ned for the reader Remember that a de nition is not a theorem or anything like that It s just a description of a new word Here are two de nitions De nition 55 A geodesic is a curve that locally minimizes lengths between points De nition 56 When all the elements in a partially ordered set are comparable to each other we call it a totally ordered set Here is a bad de nition De nition 57 For differentiable functions fz and gm the derivative of their sum 1 995 is He 9W Why isn t this a de nition Because the derivative is de ned in a separate way using limits and you need to prove that f z g m is really the value of the derivative of fz In other words this de nition is in fact a theorem 6 FONTS There are four points to make about italic and non italics font in mathematical writing 0 Single letters that stand for something are set in italics a b a bi z and y The quadratic formula is not 7b i xbz 7 4ac 2a but rather 7b i V b2 7 4m 2a 39 Doesn t the second version look better You bet it does Single function letters are also in italics like fz and em not fx or eX yuckl The Fibonacci numbers are written as F not as Fn or worse FD Numbers in mathematical expressions are never in italics the polynomial is 2 7 3x 1 not 2 7 3x 1 Basically italic numbers look awful and should be avoided in all circumstances Traditional functions whose label uses several letters are written in non italic font sin 0 cos oz and log t not sin 0 cos 04 or log t Lemmas theorems and corollaries are typeset wholly in italics De nitions and examples are not in italics except that when you introduce a technical term for the rst time it may be a good idea to typeset it in italics so it stands out See the previous section for illustrations of this You should open up a math book and notice this traditional way of typesetting if you were not explicitly aware of it before 7 PROOFREAD Before submitting your work read it over again If you don t think something is well written rewrite it 8 ADVICE ON MATHEMATICAL WRITING 8 EXERCISES The following sentences are all based on actual student writing Can you nd the errors in them Some have more than one 1 So we have that z y 2 Let a E A then for some I E B we have a lt b 3 Consider the matrix M it can be rewritten in the following way 4 We have x E A this implies it is positive 5 From the hint we know that ab c multiplying by 1 1 gives 1 a lc 6 When p is an prime number it is 2 or it is odd 7 Assume that p is not prime let p my with 1 lt x g y 8 Clearing denominators allows you to obtain a2 bc 9 fz is continuous over the inteval 0 1 10 We also now that a is even 11 Let x E S then we have x gt O 12 There is some m not in S else we would have x gt 0 13 Now assume that H C G then we have that every element of H is in G 14 From the last equation we get X 8 15 Let x E H we need to show z gt O 16 Let 1 2 be a totally ordered sequence the following argument shows it has an upper bound 17 When fz is a polynomial then it is integrable We know A B is invertible 19 It s not a trivial result since 0 and 1 the two obvious choices can fail to have the property 20 To show that z y First suppose z gt y 21 The smallest prime factor of n is at least 100 22 We need only two show that x g 0 23 Assume that a b we will get a contradiction 24 Pick two consecutive Fibonacci numbers Fn and Fn 1 25 Let p be a odd prime number 26 The Pythagorean theorem says that for some right triangle with sides a b and c a 61 cc 27 Since p is prime p 71 gt O 28 Then we can apply the concept awry away 29 We need to nd a and b to solve our equation 30 Let n 0 then we can suppose m gt n 31 For i gt 0 a O which is a contradiction 32 Consider m we will show that it is positive 33 The theorem is true for any x a positive number 34 When m gt O Vn we can write zn y 35 If n equals to an even number than it can be written as 2m for some integer m 36 This number is also prime so it is in the set 37 There are an in nite possible ways that the digits might be wrong 38 However I can t stop the paper yet there are some drawbacks ADVICE ON MATHEMATICAL WRITING 9 39 Very little is known about Pythagoras because none of his writings have survived and that it is unknown which work credited to him was actually his work An increasing sequence converges if and only its bounded above Much of work credited to Euclid is probably due to his students This method is not very affective Although we may think there are examples beyond those in our list7 it turns out that there isn t The subtraction of two odd numbers is even The Pythagorean theory has many proofs Not only has integrals been used to compute areas7 but for other applications too All off these functions are differentiable Lambert proved that pi is lrrational Lindemann proved pi as transcendental Now7 that we have seen how to derive the formula Hopefully it is less mysterious The fundamental theorem of calculus was invented by Newton and Leibniz AAAA qgtqgtqgtqgt wwHO VVVV Lets inscribe a triangle in the circle The equation is illustrated on the following picture First we establish some notaiton to make the concept percise A consequance of the theorem is that the size of each nite eld is a prime power We can see that sine the number is positive its a square The diagram below can help when we are lacking of explaining the algebra Another de nition I need to include is an isometry which is a function that doesn t change distances like a rotation 59 Euler s proof was originally found in Stark s book in 1970 60 There arent any simple proofs known of this theorem cummcncncncncncnugtugtugtugtugtugt AAAAAAAAAAAAAAA
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