- 1 x 1 = 1
- a
^{0}= ? - Is 0 odd?
- Thales theorem: how to formulate it?
- Thales theorem: why a theorem, first of all?
- The axiom on commuting vectors
- The axiom on the scalar
**1** - Chords
- Many faces of a six element group
- What is “series”?
- Non-trivial trivial permutation
- Remainder for Chinese
- That sixty element group

1.
** 1 · 1 = 1**

At the end of my primary school a lady
tried to make me believe that multiplying lengths of sides of a rectangle
one gets its area. It was her duty to show me that it was just that
way as she was my maths teacher.

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.

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2.
** a ^{0} = ?**

And here comes that zero power *immer wieder*.
How to prove that it is 1?
**a **^{2} = a · a,
a ^{1} = a ,
so that dividing once more by **a** a student is carried to accept that
“**a **^{0} = 1” is the right choice. It is
truly misleading. Equally well a student may reason like this: “we put
** a · a = a **^{2}
because 2 just counts the number of factors! So when I have bare **a**,
there is no multiplication, no factors, so I should take
** a = a **^{0} !”
first **I** want, then **we** agree;

I**want**,
we **agree**.
** a **^{0} story comes to a swift end.
The initial definition is:
** 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 .

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 **a ^{0}**?
Well, could I put

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:

I

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:

- the value at the sum of arguments is the product of values at those
arguments:
- for real numbers
**s , t**there is**f(s + t) = f(s) · f(t)**

- for real numbers
- the greater is an argument, the greater is the respective value:
- if
**s < t**then**f(s) < f(t)**

- if

I check that postulates 1. and 2. stand firm. And if I admitted

- For any real
a > 1I seta,^{2}= a · a- and inductively for any
k > 1a.^{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.

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3.
**Is 0 odd?**

They come to the University with all those certainties … But we want
them to become mathematicians. I know that an apple is an apple
but could it happen that an apple hatches an egg? Dear Freshman,
no need to feel offended, nobody's attacking your personal convictions.
Just confirm it or rebut it.

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.

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4.
**Thales theorem: how to formulate it?**

The geometry class starts. “You won't believe” I tell them “how much you
can achieve with these three old instruments: Thales theorem, Pythagorean
theorem, the theorem on central angle subtending a chord”. Soon I get
a confirmation that they won't, indeed. Out of those three theorems they
usually know the one of Pythagoras - with the understanding that the
well-known sketch of three squares fully justifies the statement. Sometimes
they do well with trigonometric identities without realizing that the health
of trigonometric functions is based on the first of those theorems.
**made** it!” as well as another one: “**We** made 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 “WeThales theorem I have two half-lines that are not parallel to each other. Both of them start in a point that I callthe 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?**

Why should **I** lose my time proving Thales theorem if other guys don't?
It seems that only fundamentalists and suckers do it …

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.”

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6.
**The axiom on commuting vectors.**

A nice exercise with one-line solution asks you to prove that the
axiom on commutativity of vector addition is a consequence of other
axioms of the linear space. All you have to do is to calculate
in two different ways
**(1+1)·(u+v)** for **u, v** in the
space **V**.

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.

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Among my pleasant mathematical reminiscences is a scene from my M.Sc.
exam: the question is the set of axioms of linear space, two of my famous
professors seem to be awaiting something with their bodies thrust forward
and a tension in their faces and I am at ease because I'm sure that they
want to see that one times v is v. In fact, once I write it down, they
smile and relax.

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.

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8.
**Chords**

Should one counter the Beloved Masters? The chances
are that the Beloved Master will never know, so let me try.

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.

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9.
**Many faces of a six element group**

I like *S*_{3}. If I teach group theory, the exercise on
its ten apparitions will wait for
students at the very beginning. But I manage to infiltrate parts of it also
in other classes . It is amusing to hunt for *S*_{3} in
unexpected environments.

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 *S _{3}* 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 *S _{3}*, then you always find
its applications or justifications for its presence.

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I love *Dover Books on Mathematics*. Using Internet I order
one of them and four months later I have it in my hands. I open it - and
immediately I get stuck somewhere between page *vii* and page 3.

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:

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.Roastbeefis 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 variablexwhere you may use addition and multiplication. Fine, now we may pass to the convenient ways of expressing this roastbeef.

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?

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11.
**Non-trivial trivial permutation**

It takes little time to realize that the task “show that the notion of
parity of a permutation is well-defined” has equally good and more tangible
version: “show that trivial permutation cannot be written by an odd
number of transpositions.” It may seem strange that someone would use
*any* time for parity of the function that does exactly nothing;
still, the notion is essential in the process of defining the
*determinant* of a matrix.
**1**. But it cannot survive the complete walk
to the left as there would be its only appearance, contradicting
the assumption that the product gives 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 **e _{n}** 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
**Z _{3}** , you will have to talk about
constructing finite rings - and then all
other rings

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 **A _{n}** of positive natural
numbers up to

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:

- for any
**j**two neighboring transpositions**(1 j)**vanish; - if the sets
**{1,j}**and**{f,g}**are disjoint then the transpositions**(1 j)**and**(f g)**commute; - if
**f**is not**1**then**(f j)(1 j)=(1 f)(f j)**.

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“Why this name of *Chinese Remainder Theorem?*” ask my students.
“It's simple” I tell them. “Our civilization is not inclined to give
them back the Yanghui Triangle but some P.C. guys felt remorses so a
remainder was sent back in return. And everybody knows that Chinese
language is difficult, so we recount the theorem in the most complicated
way that we can invent. In fact, we'd rather tell it in Chinese than in
geometric terms.”
*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?

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 canUnder friendly conditions the intersection of arithmetic progressions is an arithmetic progression.

Good. So it is not Chinese any more.

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Yes, it was inevitable. I also have my proof that **A**_{5}
is simple.

The only possible even permutations of 5 symbols are of types
(12), (123), (12)(34), (12345) - and the maverick e_{5}. 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 **A _{5}** or in

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 **A _{5}**.
Just put two notations in two lines and read the effect as a (conjugating)
permutation.

I.N.Herstein,
*Topics in Algebra,* (1964), p.67

I.D.Macdonald,
*The Theory of Groups,* (1968), pp244-245

N.Jacobson,
*Lectures in Abstract Algebra,* (1964), p.36

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