1. 1 · 1 = 1
She made an effort to be clear. She cut a rectangle on the blackboard into neat squares, each of them having the same unit length and we readily agreed that having n of those units on horizontal base line and m on vertical left line results in n · m small squares. Then she announced with a broad smile that because 1 · 1 = 1 we could conclude that the area was n · m · 1, that's to say n · m.
Somehow I didn't like the argument and suggested that one fact has no connection with the other. She made a renewed effort to explain to me how to multiply one by one but I persisted that I was accepting two facts but I believed that they were completely disconnected ones. After several repetitions of our standpoints both of us were fully convinced that the other one was a full idiot. She won the skirmish because she controlled the gradebook.
I recalled the story years later - was it when I attended lectures in measure theory or perhaps general topology? I don't remember. But now I recall it every time I teach Geometria Quantitativa. It seems the best possible setting to consider the topic. Our students have no measure theory or topology at their 8 semester course and I'd rather see them forming their opinion before they go to teach in schools of Santa Catarina state. So I give them an exercise: go measuring all that stuff in a usual way, using the standard arguments: two congruent triangles will have the area of the paralelogram obtained from glueing them together, polygonal figures will be measured breaking them into disjoint triangles and so on and so forth including the usual isometries. Yet a square with unit sides measures 5. Question: will you reach any contradiction in such a system of beliefs?
The practice shows that they heartily dislike the exercise.
2. a 0 = ?
Every now and then it comes to me and I argue: imagine that you've just left notary's office where you declared that your new-born daughter will be called Estela. Your friend meets you in the street and seeing the baby on your shoulder asks her name. Estela you say. How can you prove that it is Estela inquires your friend. And you just raise your shoulders: I don't have to prove anything. I decided to call her like that.
Is it a valid analogy? Could we put anything for a0? Well, could I put any sequence of sounds instead of Estela? The rules may vary from place to place but there will always be some restrictions. And - even more important - I had some good motives that led me to choose Estela among thousands of possibilities.
What are the true motives that lead us to the definition with one on the right? The trouble is that the common false motive is by counting down the number of factors:
The path I usually follow is to some extent anachronic but a mathematician is under no obligation to follow a tortuous path uncovered by historians of science. I do not tell my students that this is exactly how difficult notions were invented. I rather sketch a picture introducing more order than the bygone reality might have contained.
So we start with a platitude that it is easier to add than to multiply. We want to have a magic function that turns multiplication into addition; instead of multiplying two numbers we would add some new ones, call them: souls or spirits of the originals. Or respect tradition and call them logarithms.
I underline: we want it, I italicize: we want it, I repeat the verb want many times, in order to fix it in my students' minds that creating mathematics there are two basic components to start with; the psychological one: I want something, and social one: we agree do adopt some concepts, names, procedures. I repeat again and again:
But wishes and consesus notwithstanding, at first there is not such a function in sight. So we approach the problem from the other direction: we try to construct a function that carries addition to multiplication; if it is invertible, later we will be able to reverse the process.
Thus, I want the function to be strictly increasing. (Well, I might accept a decreasing function. And why avoid function that are not strictly monotone? Just try to plot graph of an invertible function from the unit interval to itself that is neither decreasing nor increasing! It has to be a bit nasty, hasn't it?) There is not much space to improvise, the usual path of enlarging the domain of exponential function is mandatory: start with naturals, pass to integers, rationals and enter the habitual white lies while going to reals. At each step you check that the construct preserves all that had been done so far and doesn't go against your two demands:
I check that postulates 1. and 2. stand firm. And if I admitted k = 1 then the condition 1. and the law of cancellation would yield a 1 = a. I incorporate the condition into the definition and repeat the reasoning with k = 0 . Here the comparison of a 1 = a 0 · a with a = 1 · a convinces me that I should accept a 0 = 1 .
- For any real a > 1 I set
- a 2 = a · a ,
- and inductively for any k > 1
- a k+1 = a k · a .
The tough part of this story is convicing students that I didn't prove anything, that it was my persisting in my wishes that funnelled me into some choices.
3. Is 0 odd?
A straightforward question on the number zero brings an efficient shock treatment to my class. Which of these options is true I ask my students,
(1) zero is oddUsually the vote splits between two latter choices and the class anxiously awaits a verdict establishing the Truth. The answer is I announce, stretching the phrase as much as I can that all three possibilities are correct.
(2) it is even
(3) for zero the question is meaningless
Clearly, for a while they see it as a joke and it may take more than an hour to have them accepting that mathematics need not be about facts and numbers but about paths that do or do not link facts.
The start is a delicate issue as I have to ask what is an even number? and someone will answer: a number that is divisible by 2. Now it is time to ask: is 1/7 divisible by 2? And what about log 3 10? Soon we see that the problem is with integers - or with natural numbers. Once we reach this mark, it is easier to become technical. Let natural numbers start with 1 - as many calculus books still propose - and let even number be a natural multiple of 2; the rest of the set goes to odd numbers. Then zero is an outsider, just as much as 1/7, and the third option stands.
If we include 0 among natural number, would it be forced to become an even number? Not necessarily, logical reasoning may support both parties in the game. Where does concept of evenness come from? From pairing? Perhaps. Just think of kids in a kindergarden. They go for a walk in pairs. Nobody's walking? No pairs. Some kids in pairs and nobody left alone? The number of kids gets the attribute of being even.
It is not rare in colloquial language that some feature becomes a label of the collection. Just recall 5 cent coins that are named nickels or red haired persons who become redheads. So 2, 4 and alike will be called even numbers. And what is not even in this set ends up in the odd group, and we have the option (1).
Then comes the moment that we decide to refine our tools. We might start with all integers right away. And divisibility by 2 will be decided by looking whether the remainder is 0 or 1. So we may as well choose the option (2). Wait a second, is it really as well or is it much better? Is there any reasonable criterion to help us decide which of possible and formally acceptable definitions should win? Doubtlessly, someone in the class will eventually pull another concept out of the limbo: the use. Usefulness. We did not meet to play around, we want to forge tools. And the simpler is their design, the easier will be using them. No, our high school professor did not tell us about it.
Right. But why didn't she? If you went to teach in high school, would you consider it worth to pass these ideas to your students?
They hesitate. That is fine. That moment of hesitation is a good beginning.
4. Thales theorem: how to formulate it?
After a long breath (a gasp, maybe) we start with the Thales Theorem. I enter the excavation process deep inside in collective subconsciousness and after a bit of work a design appears. It is full of names and concepts and all the costumary traps are in there. For example, letters a and b are placed between consecutive point A, B, C on the line but nobody is sure whether b is a segment from A to C or form B to C. Then I give Greek letters to all possible angles, gothic letters to lengths (to distinguish them from respective segments) and the design becomes absolutely unreadable. The final touch is a drawing of two butterflies to ornate the sketch. They don't protest as they got accostumed to all these butterflies in their school textbooks.
When I feel that the despair reached its climax and everybody regrets choosing the wrong course, I divide the class in small groups (there may be four legs in our thinking machines; the fifths would be a tail) and ask them: can you clean up this mess? Your group will feel good if you manage. And you may find it equally pleasing to try to demolish the proposal of your competitors. Be fair but tough. Still, there are some rules: no proper names for objects that appear in the story, no sketches, no specialized terms. A layman listening to you on the phone must stand a chance to understand what you mean.
It never takes less than 2 hours before we get an acceptable formulation. The best that my students worked out in one of the courses goes like that.
Every time the experiment is conducted, the final version is the hybrid that got inputs from several groups. Thus, many students show that they consider themselves well-succeeded authors. There is an unmistakable part We made it! as well as another one: We made it!
Thales theoremI have two half-lines that are not parallel to each other. Both of them start in a point that I call the origin. I cross both of them with two distinct parallel lines that do not pass through the origin. In one of the half-lines I form two segments; they start at the origin and they end at meeting points with the parallels.
I also consider another pair of segments that lie in the parallels and have their final poins in the half-lines. For each of both pairs of segments I form ratio of the greater length to the smaller one.
I claim that both ratios are equal.
Remarks. Sometimes the winning proposal avoids describing half-lines putting V figure in its place. I prefer the former version. I have future introduction of Cartesian coordinates in mind and the common conviction that the origin is something that the newborns are equipped with. The secondary profit of the cited approach is how it makes clear that the origin is choosen, called into existence and later (if need be) named as O or something equally poetical.
5. Thales theorem: why a theorem, first of all?
I do not mean to say that everybody goes around using something of doubtful validity, putting at risk whole theories. I mean that people make sure that Thales theorem is good and useful just assuming it is, so that they can pass immediately to more interesting tasks.
Is it an honest attitude for a working mathematician? Sure it is if everybody does it. Have you ever seen anybody talking of vector space and trying to justify or prove the fact that for a scalar k and vectors u, v one has k(u+v) = ku + kv? Just unthinkable. Let's assume it and pass to serious problems of Linear Algebra.
Now, the progress of mathematics is so great nowadays that an instructor may introduce all linear space axioms in 7 minutes. (And if you can do something in 7 minutes what used to take an hour before then you have a progress, haven't you?) And I am not sure what percent of new Linear Algebra textbook make a distinction between (a finite-dimensional) vector space and (axiomatically defined general case of a) linear space. Another victim of the progress is the distinction between linear substitution and linear transformation; once they were as different as the value of a function and the function itself. But working in a country that has not yet fully succumbed to fast food and fast mathematics, I ask my linear algebra classes to consider the axiom of k(u+v) = ku + kv (with due universal quantifiers) and we examine the question: is it really Thales theorem? Soon we see that this wording is more general, no exclusion of parallel u and v (it becomes a scissor that may be closed or completely opened) - and that the equality sign makes it two theorems at a time: the Thales theorem and the inverse one.
Then I beg them: keep it always close to your hearts. It is the mother of all homomorphisms. Once it used to be proved. It could be proved, if you really cared.
6. The axiom on commuting vectors.
The nice part of it is not the simplicity of the solution but a depth of the algebraic conclusion that it forces: do you want to have a field that has its non-zero elements acting as isomorphisms of a group? Then make sure that your group is Abelian. But the exercise is a darling of people who care more about axioms than about the worlds they delineate: it is a case that illustrates how you may dispose of a redundant axiom; if you do like axioms you choose them to form a set slim and full of independence.
But how to behave if you want to work with matrices and do not care about the elegance of the introductory set of axioms? Throw away the axiom in question as it may be readily recovered as a theorem or stay with it to have a clear-cut situation from the very beginning of the work? Questions of that type might help the students strike the balance between the common enchantment by the loveliness of the façade and tough demands for efficiency as you carve deeper and deeper that mathematical rock.
7. The axiom on the scalar 1.
I recall the scene when my students try to compose a list of axioms of a theory (say: real numbers or division algebras) and some axioms maliciously go into hiding. I do not blame their math training; I think it is the consequence of living in TV world with few organizing structures and many dispersed images that ask for a crib to furnish a meaningful sequence.
The axiom on multiplication by 1 is the greatest victim. If it seems obvious then it has not been really understood.
Funny: the simpler is the notion, the greater confusion it generates in textbooks. They tend to be correct, concise and elegant, but the meaning has to wait in line to get its share of attention. So I feel it worthwhile to tell my students: Have you got confused with the scalars? Yes, they are numbers; no, they aren't, because you multiply number by a number and not by a vector. Just think of your neighbors, three brothers; they work as bus drivers. When Jim, one of them, puts his blue cap and sats at the wheel he is not Jimmy, he is Mr.Cullen, the driver of the car J18. That is to say, he continues to be your buddy Jim Cullen but now he is at work and his cap shows it. The same happens with his brothers. The moment they go to work, they do become somewhat different, you can tell looking at their caps.
Something similar happens to our numbers. We send them to work with vectors. At first we may think that they stretch vectors, shrink them, turn them the other way round. But when the number 2 gets on its posts to become the double stretcher it is not a simple number any more, its blue cap (that we often forget to buy) makes it a special type of servant; we call it scalar.
Then we look at the axioms and see what they actually say: at start we had a number k. At the goal we have a group endomorphism k that acts on the group V. And entering the ring of endomorphisms numbers appear to have preserved their inner relations: addition and multiplication in the field seems to be mimicked among endomorphisms. That is to say: the passage from the field K to End(V) is a homomorphisms. But there are only two possible homomorphisms from a field into a ring: either it is trivial (every image acts as zero) or it is monomorphism (different numbers have different actions). How can we know which of two cases we are dealing with? Well, just look at the number 1, it has only two choices: either it becomes the trivial homomorphism 0 or it becomes the identity 1. Over.
I've never seen anything so difficult before. Wrong. At your biology class you have put a drop of clean water on a glass, inserted it under the microscope and discovered a complex ecosystem within. We're just starting to use our magnifying glass.
In Geometry Revisited by H.S.M.Coxeter and S.L.Greitzer, the proof of the Steiner-Lehmus theorem (p.14) presents clear ideas in simple language: Of two unequal chords, the shorter, being farther from the center, subtends a smaller angle there and consequently a smaller acute angle at the circumference.
Yes, I understand, you understand, he/she/it understands. But what will happen if I say: I don't understand? The rule of the game is to repeat the argument in different terms. Draw it, redraw it, change approach - and see whether some new version meets your standards of being clear. So I look at the central triangle based on a chord inside the circular section. Then at the fact that the chord is shorter than the arc. Another fact that the circle is convex. All of these are equally clear - but all of them hide exactly the same difficulty.
It's all the same truth - the one that we frequently offer our students proving that when x tends to zero then the quotient of sin(x) by x tends to 1. Inadvertently, it might be that argument that constitutes a vicious circle, as many authors had already stressed (and many other mathematicians ignore it).
I hate to feel cornered by a childish doubt, so I ask Waldir: how would you argue here if I told you that I didn't understand? And if you ask why you should waste your time on it then I admit: the truth is that I actually don't understand. Waldir makes a sour face and says that he needs some time.
We meet two days later and the simplest of defensable arguments is equivalent to saying that for acute angles the sine function is strictly increasing.
We don't like the argument as too sophisticated. But for a time we have to accept its company; other arguments are not much better. Much simpler, I mean.
Most often it is near the entrance doors that such traps are dispersed. Once you get to central parts of a theory you are allowed to use nasty calculations and intertwined reasonings. The pressure to be simple and elementary disappeares. But the beginning, the introduction, has to be simple. And it is difficult to admit that simple things are not as simple as they seem to be.
9. Many faces of a six element group
Indicating a noble genealogy of one's idiosyncrasies makes one feel safer and wiser. So let me tell that when I attended Peter Hilton's conference in Paris (I think it was on nilpotent groups), the host said something to the effect that any mathematician (or was algebraist the term that he used?) should always carry S3 just in case she or he needed to test some silly hypotheses. For example, the one that once the kernel and factor group are nilpotent than the group itself must be nilpotent.
Well, the argument is nice but if it didn't exist it could be invented. The essence is that if you like S3, then you always find its applications or justifications for its presence.
10. What is series?
Last time it happened to me when I reached page 2 of Theory and Applications of Infine Series by Konrad Knopp. I realized that I did not understand what series meant.
The simplest possible criterion of understanding is explain it to someone else. But I cannot tell my students anything about a symbol of the form or formal expression because they have heard me so many times reciting this litany:
Roastbeef is not an expression written with nine Latin letters; it is a notion describing objects with texture, smell and taste; it could be denoted by a nine-letter word in a menu but it would maintain its existence even if all menus disappeared. So, please, do not ever tell me that polynomial is an expression of the form because it will make me laugh. What type of object is polynomial: A mineral? An emotional state? Right, it is a function. How is it constructed? I mean: what type of bricks can you use? Which ways of joining them together are permitted? Oh, I see. So, a polynomial in one real variable is a function composed of real numbers and a real variable x where you may use addition and multiplication. Fine, now we may pass to the convenient ways of expressing this roastbeef.It is not easy for them (and for me) to get rid of the custom of describing mathematical object as expressions. I have to tell them what type of object the series is. If it is a limit of a sequence of partial sums than it is a number but talking about a number that is convergent sounds like a bad joke. If it is a process, a passage to infinity of something, then what is the story of a series being equal to 1? Bad luck. I have to call Waldir.
Waldir, too, is caught by surprise. He murmurs something like you always spoil my Saturdays but I do not care much about it as his geometrical questions often spoil my Sundays. Okay, on Monday we know the way out of trouble but it will not be easy to tell the truth to our students: all their books talk about a series, while we see two different notions of series. They are just like integrals - the ones with discrete domain. So you have a definite series (that might be equal to 1) and you have an indefinite series (that does or does not converge, we shall have to examine it).
Simple? Sure. Obvious? Yeah probably it is but why nobody told me about it when I was working with series all these years?
11. Non-trivial trivial permutation
Yes, I know, the determinant may be introduced in other ways, say: using an axiomatic, multilinear and elegant approach - but you will not do it, if you like your students. And anyway, when calculating volume of a parallelipiped - or solving a system of 3 by 3 linear equations - they encounter formulas involving 3!. They will necessarily notice some regularities about the appearance of the signs.
To prove that parity is well defined is a notoriously tricky business wrote an author in 1968. In a footnote he indicated a book (published at the same period by Cambridge University Press) containing a fallacious demonstration and in the next page he proposed his own proof. The proof, being also incorrect, confirmed veracity of the first statement.
Then you take very famous book by another author and find another proof that is also short and wrong. These two cases have the same slip in common: a conviction that there is a standard way to write the trivial permutation (on n letters), using 0 transpositions. The enchantment with the numerical character of the symbol zero made both of those excellent mathematicians loose from sight that writing the function with 0 transpositions means standard way of not writing it by means of transpositions or denoting it by a symbol en that has no connection with transpositions. But this error is a subtle and instructive bit of mathematics, very different from lots of sheer nonsense written on the topic in quite a few books (I may furnish some titles via e-mail).
Quite revealing is I.Herstein's observation; he uses the polynomial that one usually meets dealing with van der Monde's determinant and precedes the proof with this remark: Frankly, we are not happy with the proof we give of this fact for it introduces a polynomial which seems extraneous to the matter at hand.
The dissatisfaction could be linked to the fact that once you have polynomials, there is a ring of coefficients in the background; the simplest choice would be Z. But it is infinite, a very strong cannon against a function that does nothing on n elements. You need only three values, 0, +1, -1. If you decide to use Z3 , you will have to talk about constructing finite rings - and then all other rings Zm would be asking: why not me?
So you can imagine how swell I felt when, in early eighties I found a neat way out of the trouble. I taught it to my students and asked with false modesty: why no book brings you something as simple as that? But one day my students told me: our book does! It is basically your proof.
It was the third edition of A First Course in Abstract Algebra by John B.Fraleigh, Addison-Wesley, 1982. It states that the proof comes from the article Even and Odd Permutations by William I.Miller, published in 1971 in the Journal of the Mathematics Associations of Two-Year Colleges, (v.5, p.32). The book is not the first choice for popular courses. Too bad teaching the topic might have been easier since 20 or 30 years.
Here is my rendering of the argument. Take the set An of positive natural numbers up to n and proceed with inductive proof on n. Dealing with transpositions you have to start the induction with n=2. There is only one possible transposition (1 2) and the product (1 2)m is not trivial for odd m. Next, assume that identity map ek is actually always expressed by a product of an even number of transpositions if k<n and then you prove the same statement for en. You take any product of transpositions that results in en and try to modify it so that the symbol 1 does not appear in the modified form. Thus, the modified form is a product of transpositions that operate on less than n symbols. And if your modifications leave the number of factors intact or diminish it by an even number than your claim is proven.
The modifications come when you look for the first transposition on the right that contains 1 and take it for a walk to the left. The traffic rules are:
12. Remainder for Chinese
Does it have to be presented in such an obscure way? No. Forget the crazy formulas for simultaneous solutions of n congruences. Let me start at the very beginning. The Peano axioms.
To begin with, we want to have the consecutive numbers to be able to count. It is a pity but they already have the absurd name of natural numbers. Fine, we will not fight about the name - we would lose. So, what are they? One, two, three and so on?
Experience taught my students that translating and so on to etcetera (written as ellipsis) makes it look more scientific but the message that we do transmit is: you know what I mean. And when we say you know what I mean it is clear sign that each of us may mean something different. So the question is: how to get rid of etcetera? We try to forsee the undesirable meanings of etcetera that should be excluded: moving along a path that divides into two branches; or going along two separate paths; or going round in a circle; or entering a circle while following a 9-alike trajectory. So, step by step we are brought to inventing Peano's axioms as a natural defensive measure. And now we may plot on a blackboard that sequence of dots. It is regular. It is simple. It is reassuring.
But there comes here the question of a scale. If I moved quite away I would not tell a difference between the sequence of natural numbers and the sequence «1007, 2007, 3007 . . . ». So we see that an arithmetic progression is a close cousin of natural numbers - certainly there remains regularity and simplicity.
Now let us take simultaneously two distinct arithmetic progressions. What a helpless mess When we saw them separately there were spacings of r and of s units. And now? Can we determine what type of spaces will appear? Is space 0 guaranteed? No? Oh, I see, the progression 1 plus multiples of 4 will never meet progression 3 plus multiples of 4. Some experimenting with different data brings us to invent anew the Chinese Remainder Theorem:
Two technical points remain. The first is about "friendly conditions" - express them in terms of greatest common divisors. And later, how to find the starting point and the difference? Well, this can be solved using Bézout's identity. The essential point is: you can see it! And when I say see I mean see. Do you visualize in on circles of lengths r,s, lcm(r,s)? Could you solve a system of congruences without any algebra?Under friendly conditions the intersection of arithmetic progressions is an arithmetic progression.
Good. So it is not Chinese any more.
13. That sixty element group
The only possible even permutations of 5 symbols are of types (12), (123), (12)(34), (12345) - and the maverick e5. Talking of type I mean that another samples are obtained inserting another set of symbols.
That exchange of symbols is known in group theory as the conjugation. A justification for introducing one more proof is that one gets inexpensive training in the art of conjugating. A part of the training consists in checking whether one conjugates in A5 or in S5, the group of all permutations on 5 symbols.
Initially, leave the last type (of 5-cycles) aside. Chose any other type and check that any two permutations of that type are conjugated in A5. Just put two notations in two lines and read the effect as a (conjugating) permutation.
Topics in Algebra, (1964), p.67
The Theory of Groups, (1968), pp244-245
N.Jacobson, Lectures in Abstract Algebra, (1964), p.36