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You have hit a fortune from the fortune collection 'math'
containing mathematical fortune entries. Latest update: June 6 2001
This fortune got started in October 1999 and is maintained in
http://www.dynamical-systems.org
The file contains mathematical quotations, jokes, annectotes,
even some mathematical results with short proofs.
To install this file:
1) 'strfile math' produces a file 'math.dat' .
2) Copy both files 'math' and 'math.dat' into '/usr/share/games/fortunes'
or wherever your fortune program keeps the fortunes. You might check
first, if not a newer version of this file or an other file named
'math' already resides there. The 'fortune' program itself is usually in
'/usr/games/fortune'. If you are not system administrator, put the files
'math' and 'math.dat' into a private directory like for example '~/lib'
and replace 'fortune math' by 'fortune ~/lib/math'.
Examples to use this fortune:
fortune math random entries from this file
fortune -m PI= PI with 100 digits
fortune -m E= E with 100 digits
fortune -m Mandelbrot Mandelbrot set (generated with pbmtoascii)
You might want to erase this first entry of the fortune before
installation.
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Every problem in the calculus of variations has a solution, provided
the word solution is suitably understood.
-- David Hilbert
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The real mathematician is an enthusiast per se. Without enthusiasm
no mathematics.
-- Novalis
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There is no royal road to geometry.
Euclid
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One may be a mathematician of the first rank without being able
to compute. It is possible to be a great computer without having
the slightest idea of mathematics
-- Novalis
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Geometry may sometimes appear to take the lead over analysis,
but in fact precedes it only as a servant goes before his master
to clear the path and light him on the way.
-- James Joseph Sylvester
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The essence of mathematics lies in its freedom.
-- Georg Cantor
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Fantasy, energy, self-confidence and self-criticism are the
characteristic endowments of the mathematician.
-- Sophus Lie
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Pure mathematics is the magician's real wand.
-- Novalis
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When a mathematician has no more ideas, he pursues axiomatics.
-- Felix Klein
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The paper "On the nature of turbulence" with F. Takens was
eventually published in a scientific journal.
(Actually, I was an editor of the journal, and I accepted
the paper by myself for publication. This is not a recommended
procedure in general, but I felt that it was justified in this
particular case).
-- D. Ruelle, in Chance and Chaos
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A good topological theorem to mention any time is
the theorem which, in essence, states that however
you try to comb the hair on a hairy ball, you can
never do it smoothly - the so-called 'hairy-ball'
theorem. You can make snide comments about the
grooming of the hosts' dog or cat in the meantime as
you pick hairs off your trouser leg.
-- R. Ainsley in Bluff your way in Maths, 1988
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LARGE NUMBERS: (10^n means that 10 is raised to the n'th power)
10^4 One "myriad". The largest numbers, the Greeks were considering.
10^5 The largest number considered by the Romans.
10^10 The age of our universe in years.
10^22 Distance to our neighbor galaxy Andromeda in meters.
10^23 Number of atoms in two gram Carbon (Avogadro).
10^26 Size of universe in meters.
10^41 Mass of our home galaxy "milky way" in kg.
10^51 Archimedes's estimate of number of sand grains in universe.
10^52 Mass of our universe in kg.
10^80 The number of atoms in our universe.
10^100 One "googol". (Name coined by 9 year old nephew of E. Kasner).
10^153 Number mentioned in a myth about Buddha.
10^155 Size of ninth Fermat number (factored in 1990).
10^(10^6) Size of large prime number (Mersenne number, Nov 1996).
10^(10^7) Years, ape needs to write "hound of Baskerville" (random typing).
10^(10^(33)) Inverse is chance that a can of beer tips by quantum fluctuation.
10^(10^(42)) Inverse is probability that a mouse survives on sun for a week.
10^(10^{51)) Inverse is chance to find yourself on Mars (quantum fluctuations)
10^(10^(100)) One "Gogoolplex", Decimal expansion can not exist in universe.
-- from R.E. Crandall, Scient. Amer., Feb. 1997
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The statement sometimes made, that there exist only analytic functions
in nature, is in my opinion absurd.
-- F. Klein, Lectures on Mathematics, 1893
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The introduction of numbers as coordinates ... is an act of violence...
-- H. Weyl, Philosophy of Mathematics and Natural Science
1949
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Mathematics possesses not only truth but supreme beauty - a beauty cold and austere,
like that of a sculpture
-- Bertrand Russell
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Geometry is magic that works...
-- R. Thom. Stability Structurelle et Morphogenese, 1972
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Ernst Zermelo, who created a system of axioms for set theory,
was a Privatdozent at Goettingen when Herr Geheimrat Felix Klein
held sway over the fabled mathematics department. As Pauli told it,
"Zermelo taught a course on mathematical logic and stunned his
students by posing the following question: All mathematicians in
Goettingen belong to one of two classes. In the first class belong
those mathematicians who do what Felix Klein likes, but what they
dislike. In the second class are those mathematicians who do what
Felix Klein likes, but what they dislike. To what class does Felix
Klein belong?"
Jordan, having listened intently, broke into roaring laughter. Pauli
paused, took a sip of wine and said disapprovingly, "Herr Jordan, you
have laughed too soon". He continued: "None of the awed students
could solve this blasphemous problem. Zermelo then crowed in his
high-pitched voice, 'But, meine Herren, it's very simple. Felix
Klein isn't a mathematician.'" Jordan laughed again. Pauli
drained his wine glass approvingly and concluded with
"Zermelo was not offered a professorship at Goettingen".
-- E.L. Schucking, in 'Jordan, Pauli,Politics,
Brecht and a variable gravitational constant'
Physics Today, Oct. 1999
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In the beginning, everything was void, and J.H.W.H.Conway
began to create numbers. Conway said, "Let there be two
rules which bring forth all numbers large and small. This
shall be the first rule: Every number corresponds to two
sets of previously created numbers, such that no member of
the left set is greater than or equal to any member of the
right set. And the second rule shall be this: One number is
less than or equal to another number if and only if no member
of the first number's left set is greater than or equal to the
second number, and no member of the second number 's right
set is less than or equal to the first number." And Conway
examined these two rules he had made, and behold! they were
very good.
And the first number was created from the void left set and the
void right set. Conway called this number "zero", and said that
it shall be a sign to separate positive numbers from negative
numbers. Conway proved that zero was less than or equal to
zero, and he saw that it was good. And the evening and the
morning were the day of zero. On the next day, two more numbers
were created, one with zero as its left set and one with zero as its
right set. And Conway called the former number "one", and
the latter he called "minus one". And he proved that minus one
is less than but not equal to zero and zero is less than but
not equal to one. And the evening...
-- D. Knuth, Surreal numbers, 1979
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Mathematics consists essentially of :
a) proving the obvious
b) proving the not so obvious
c) proving the obviously untrue
For example, it took mathematicians until the 1800'ies to
prove that 1+1=2 and not before the late 1970 were they
confident of proving that any map requires no more
than four colors to make it look nice, a fact known by
cartographers for centuries.
There are many not-so-obvious things which can be proved true
too. Like the fact that for any group of 23 people, there is
an even chance two or more of them share birthdays. (With
groups of twins this becomes almost certain. Not quite certain
as you will of course point out: they might all have been born
either side of midnight).
Mathematicians are also fond of proving things which are obviously
false, like all straight lines being curved, and an engaged telephone
being just as likely to be free if you ring again immediately after,
as if you wait twenty minutes.
-- R. Ainsley in Bluff your way in Maths, 1988
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There exists a subset of the real line such that the infimum
of the set is greater then the supremum of the set.
-- Gary L. Wise and Eric B. Hall,
Counter examples in probability and real analysis,
1993, First Example in book
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Transcendental number : A number which is not the root of any
polynomial equation, like pi and e, and which can only be
understood after several hours meditation in the lotus position.
-- R. Ainsley in Bluff your way in Maths, 1988
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There are great advantages to being a mathematician:
a) you do not have to be able to spell
b) you do not have to be able to add up
The illiteracy of mathematicians is taken for granted.
There still persists a myth that mathematics somehow
involves numbers. Many fondly believe that university
students spend their time long dividing by 173 and
learning their 39 times table; in fact, the reverse is
true. Mathematicians are renowned for their inability to
add up or take away, in much the same way as geographers
are always getting lost, and economists are always
borrowing money off you.
-- R. Ainsley in Bluff your way in Maths, 1988
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In this note we would like to offer an elementary 'topological'
proof of the infinitude of the prime numbers. We introduce a
topology into the space of integers S, by using the arithmetic
progressions (from -infinity to +infinity) as a basis. It is not
difficult to verify that this actually yields a topological space.
In fact, under this topology, S may be shown to be normal and hence
metrisable. Each arithmetic progression is closed as well as open,
since its complement is the union of the other arithmetic
progressions (having the same difference). As a result, the union
of any finite number of arithmetic progressions is closed.
Consider now the set A which is the union of A(p), where A(p)
consists of primes greater or equal to p. The only numbers not
belonging to A are -1 and 1, and since the set {-1,1} is clearly
not an open set, A cannot be closed. Hence A is not a finite union
of closed sets, which proves that there is an infinity of primes.
-- H. Fuerstenberg, On the infinitude of primes,
American Mathematical Montly, 62, 1955, p. 353
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The barber in a certain town shaves all the people who don't
shave themselves. Who shaves the barber?
This is meant to be a clever little paradox with no solution
but you can annoy the asker intensely by saying it's easy and
that the barber is a women.
You can then ask the following (a version of Russell's
Paradox, - point this out too): in a library there are some
books for the catalogue section which is a list of all books
which don't list themselves. Shold he or she include this book
in its own list? If so, then it becomes a book which lists
itself, so it shouldn't be in the list of books which don't
and vice versa. This should keep the most determined assailant
at bay while you attack the wine.
-- R. Ainsley in Bluff your way in Maths, 1988
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Hadamard, trying to find a job in a US university,
came to a small university and was received by the chairman
of the department of mathematics. He explained who he was
and gave his curriculum vitae. The chairman said:
'our means are very limited and I can not promise that
we shal take you'. Then Hadamard noticed that among the
portraits on he wall was his own. 'That's me!' he said.
'Well, come again in a week, we shal think about this'.
On his next visit, the answer was negative and his portrait
had been removed.
-- Vladimir Mazya and Tatyana Shaposhnikova, in
Jacques Hadamard, a universal Mathematician,
AMS History of Mathematics Volume 14
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The appropriate object is known as the Cantor set, because it
was discovered by Henry Smith in 1875. (The founder of set
theory, Georg Cantor, used Smith's invention in 1883. Let's
fact it, 'Smith set' isn't very impressive, is it?)
-- Ian Stewart, in Does God Play Dice, 1989 p. 121
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... Thus the erectile organ comes to symbolize the place of
JOUISSANCE, not in itself, or even in the form of an image,
but as a part lacking in the desired image: that is why it is
equivalent to the (-1)^(1/2) of the signification produced above,
of the JOUISSANCE that it restores by the coefficient of its
statement to the function of lack of signifier (-1).
-- Lacan, Ecrits, Paris 1966 (cited in
'Fashionable nonsense' by Alan Sokal and Jean Bricmont)
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Mandelbrot made quite good computer pictures, which seemed to show
a number of isolated "islands" (in the Mandelbrot set M).
Therefore, he conjectured that the set M has many distinct connected
components.
(The editors of the journal thought that his islands were specks of
dirt, and carefully removed them from the pictures).
-- John Milnor, in Dynamics in one complex variable, 1991
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sin, cos, tan, cot, sec, cosec - Formulae derived from the sides of
triangles but which crop up in completely unexpected places. Sins are
extremely common, but rarely do you encounter secs in mathematics.
-- R. Ainsley in Bluff your way in Maths, 1988
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This reminds me of the Hilbert story, which I learned from
my teacher Franz Rellich in Goettingen:
When Hilbert - who was old and retired - was asked at a
party by the newly appointed Nazi-minister of education:
"Herr Geheimrat, how is mathematics in Goettingen, now
that it has been freed of the Jewish influences" he
replied: "Mathematics in Goettingen? That does not EXIST
anymore".
-- Jurgen Moser, in Dynamical Systems-Past and Present,
Doc. Math. J. DMV I p. 381-402, 1998
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There are two glasses of wine, one white and one red. A teaspoonful of
wine is taken from the red and mixed in with the white. Then a teaspoonful
of this mixture is taken and mixed in with the red. Which is bigger, the
amount of red in the white or the amount of white in the red?
The answer is that the're both the same, because there's the same volume
in each glass, so whatever quantity of red is in the white must be equal
to the quantity of white in the red. However in practice it is impossible
to do this because the white always runs out first at parties and the red
always gets spilt on someone's white trousers.
-- R. Ainsley in Bluff your way in Maths, 1988
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"Suppose you're on a game show and you are given a choice of three
doors. Behind one door is a car and behind the others are goats. You
pick a door-say No. 1 - and the host, who knows what's behind the doors,
opens another door-say, No. 3-which has a goat. (In all games, the host
opens a door to reveal a goat). He then says to you, "Do you want to
pick door No. 2?" (In all games he always offers an option to switch).
Is it to your advantage to switch your choice?"
-- The three doors problem, also known as Monty-Hall Problem
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Pure mathematician - Anyone who prefers set theory to sex.
-- R. Ainsley in Bluff your way in Maths, 1988
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There was a mad scientist ( a mad ...social... scientist ) who kidnaped
three colleagues, an engineer, a physicist, and a mathematician, and locked
each of them in separate cells with plenty of canned food and water but no
can opener. A month later, returning, the mad scientist went to the engineer's
cell and found it long empty. The engineer had constructed a can opener from
pocket trash, used aluminum shavings and dried sugar to make an explosive, and
escaped.
The physicist had worked out the angle necessary to knock the lids off the tin
cans by throwing them against the wall. She was developing a good pitching arm
and a new quantum theory.
The mathematician had stacked the unopened cans into a surprising solution to
the kissing problem; his dessicated corpse was propped calmly against a wall,
and this was inscribed on the floor in blood:
Theorem: If I can't open these cans, I'll die.
Proof: assume the opposite...
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Proof by induction - A very important and powerful mathematical tool,
because it works by assuming something is true and then goes on to
prove that therefore it is true. Not surprisingly, you can prove almost
everything by induction. So long as the proof includes the following
phrases:
a) Assume true for n; then also true for n+1 because.. (followed by
some plausible but messy working out in which n, n+1 appear prominently).
b) But is true for n=0 (a little more messy working out with lots of
zeros sprayed at random through the proof).
c) So is true for all n. Q.E.D.
-- R. Ainsley in Bluff your way in Maths, 1988
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LEMMA: All horses are the same color.
Proof (by induction): Case n=1: In a set with only one horse, it is obvious
that all horses in that set are the same color.
Case n=k: Suppose you have a set of k+1 horses. Pull one of these horses
out of the set, so that you have k horses. Suppose that all of these horses
are the same color. Now put back the horse that you took out, and pull out
a different one. Suppose that all of the k horses now in the set are the
same color. Then the set of k+1 horses are all the same color.
We have k true => k+1 true; therefore all horses are the same color.
THEOREM: All horses have an infinite number of legs.
Proof (by intimidation): Everyone would agree that all horses have an even
number of legs. It is also well-known that horses have fore-legs in front and
two legs in back. But 4 + 2 = 6 legs is certainly an odd number of legs for
a horse to have! Now the only number that is both even and odd is infinity;
therefore all horses have an infinite number of legs.
However, suppose that there is a horse somewhere that does not have an infinite
number of legs. Well, that would be a horse of a different color; and by the
Lemma, it doesn't exist. QED
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Dean, to the physics department. "Why do I always have to give you guys so
much money, for laboratories and expensive equipment and stuff. Why couldn't
you be like the maths department - all they need is money for pencils, paper
and waste-paper baskets. Or even better, like the philosophy department.
All they need are pencils and paper."
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An astronomer, a physicist and a mathematician were holidaying
in Scotland. Glancing from a train window, they observed a black sheep
in the middle of a field.
"How interesting," observed the astronomer, "all Scottish sheep are black!"
To which the physicist responded, "No, no! Some Scottish sheep are black!"
The mathematician gazed heavenward in supplication, and then intoned,
"In Scotland there exists at least one field, containing at least one sheep,
at least one side of which is black."
-- J. Steward in 'Concepts of Modern Mathematics'
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An engineer, a chemist and a mathematician are staying in three adjoining
cabins at an old motel. First the engineer's coffee maker catches fire.
He smells the smoke, wakes up, unplugs the coffee maker, throws it out
the window, and goes back to sleep. Later that night the chemist smells
smoke too. He wakes up and sees that a cigarette butt has set the trash
can on fire. He says to himself, "Hmm. How does one put out a fire?
One can reduce the temperature of the fuel below the flash point, isolate
the burning material from oxygen, or both. This could be accomplished
by applying water." So he picks up the trash can, puts it in the shower
stall, turns on the water, and, when the fire is out, goes back to sleep.
The mathematician, of course, has been watching all this out the window.
So later, when he finds that his pipe ashes have set the bed-sheet on fire,
he is not in the least taken aback. He says: "Aha! A solution exists!"
and goes back to sleep.
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Taking logs - Broadly speaking, any equation which looks difficult
will look much easier when logs are taken on both sides. Taking logs
on one side only is tempting for many equations, but may be noticed.
-- R. Ainsley in Bluff your way in Maths, 1988
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Theorem: A cat has nine tails.
Proof: No cat has eight tails. A cat has one tail more than no cat.
Therefore, a cat has nine tails.
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Prime number - A number with no divisors. Boxes of chocolates always
contain a prime number so that, whatever the number of people present,
somebody has to have that one left over.
-- R. Ainsley in Bluff your way in Maths, 1988
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Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer,
You take one down, and pass it around,
Aleph-null bottles of beer on the wall.
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At the end of a proof you write Q.E.D, which stands not for
Quod Erat Demonstrandum as the books would have you believe, but
for Quite Easily Done.
-- R. Ainsley in Bluff your way in Maths, 1988
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1+1 = 3, for large values of 1
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Group theory - An exceedingly beautiful branch of pure mathematics
used for showing in how many ways blocks of wood can be painted.
-- R. Ainsley in Bluff your way in Maths, 1988
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Mathematician: 3 is prime,5 is prime,7 is prime, by induction -
every odd integer higher than 2 is prime.
Physicist: 3 is prime, 5 is prime, 7 is prime,
9 is an experimental error, 11 is prime,...
Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime,
11 is prime,...
Programmer: 3's prime, 5's prime, 7's prime, 7's prime,
7's prime,...
Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 --
we'll do for you the best we can,...
Software seller:3 is prime, 5 is prime, 7 is prime,
9 will be prime in the next release,...
Biologist: 3 is prime, 5 is prime, 7 is prime, 9 --
results have not arrived yet,...
Advertiser: 3 is prime, 5 is prime, 7 is prime, 11 is prime,...
Lawyer: 3 is prime, 5 is prime, 7 is prime, 9 --
there is not enough evidence to prove that it is not prime,...
Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime,
deducing 10% tax and 5% other obligations.
Statistician: Let's try several randomly chosen numbers:
17 is prime, 23 is prime, 11 is prime...
Psychologist: 3 is prime, 5 is prime, 7 is prime,
9 is prime but tries to suppress it,...
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PI=
3.141592653589793238462643383279502884
19716939937510582097494459230781640628
6208998628034825342117067982148086513...
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Euler E=
2.718281828459045235360287471352662497
75724709369995957496696762772407663035
3547594571382178525166427427466391932...
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Mandelbrot set M = { c | Tz=z*z+c has a bounded orbit at z=0 }
"
oMM
MooMMMM
oMMMMMMMM
"MMMMMMM"
oo oo oooMMMMMMMooo oo
""MMMooMMMMMMMMMMMMMMMMMMMo oo o
"MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
oo ooMMMMMMMMMMMMMMMMMMMMMMMMMMMMM""
"MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMo
o ooMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMooo
oo "o o oMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM"
MMMMMMMMMMoo oMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMo
ooMMMMMMMMMMMMM MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM""
MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM""
"oMoMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM"
oMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
""MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMoo
" " "MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
MMMMMMMMMMMMMMMM"MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM"
MMMMMMMMMMM"" MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
MM"""MM""" MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
o" oMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
"M"MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM"
MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
MMMMMMMMMMMMMMMMMMMMMMMMMM" "MM"o
M""" MM"MMMMMMMMMMMMM""M
MMMMMMM
MMMMMMMMM
oMMMMMM""
MMM"
"
"
%
THEOREM: The limit as n goes to infinity of sin x/n is 6.
PROOF: cancel the n in the numerator and denominator.
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A mathematician is a device for turning coffee into theorems.
-- P. Erdos
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Finally I am becoming stupider no more.
-- Epitaph, P. Erdos wrote for himself
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epsilon child
bosses women
slaves men
captured married
liberated divorced
recaptured remarried
trivial beings nonmathematicians
noise music
poison alcohol
preaching giving a lecture
supreme fascist god
died stopped doing mathematics
preach lecture
Joedom UDSSR
Samland USA
on the long wave length communists
on the short wave length fashists
-- from the vocabulary of P. Erdos
'the man who loved only numbers'
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Chebyshev said it, and I say it again
There is always a prime between n and 2n
-- P. Erdos
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Outrage, disgust, the characterization of group theory as
a plague or as a dragon to be slain - this is not an atypical
physist's reaction in the 1930s-50s to the use of group
theory in physics.
-- S. Sternberg
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Anyone who considers arithmetical methods of producing random
digits is, of course, in a state of sin.
-- J. von Neumann
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The mathematician's patterns, like the painter's or the
poet's must be beautiful; the ideas, like the colors or the
words, must fit together in a harmonious way. Beauty is the
first test: there is no permanent place in the world for
ugly mathematics... It may be very hard to define
mathematical beauty, but that is just as true of
beauty of any kind - we may not know quite what we mean by
a beautiful poem, but that does not prevent us from
recognizing one when we read it.
-- G.H. Hardy
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It is a melancholy experience for a professional
mathematician to find himself writing about mathematics.
-- G.H. Hardy
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There is a much quoted story about David Hilbert, who one
day noticed that a certain student had stopped attending
class. When told that the student had decided to drop
mathematics to become a poet, Hilbert replied, "Good-
he did not have enough imagination to become a
mathematician".
-- R. Osserman
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Referee's report: This paper contains much that is new and
much that is true. Unfortunately, that which is true is not
new and that which is new is not true.
-- H. Eves 'Return to Mathematical Circles', 1988.
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Structures are the weapons of the mathematician.
-- N. Bourbaki
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Mathematics is the only instructional material that can be presented
in an entirely undogmatic way.
-- M. Dehn
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Each problem that I solved became a rule which served afterwards to
solve other problems
-- R. Decartes
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For a physicist mathematics is not just a tool by means of which
phenomena can be calculated, it is the main source of concepts and
principles by means of which new theories can be created.
-- F. Dyson
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If the entire Mandelbrot set were placed on an ordinary sheet of paper,
the tiny sections of boundary we examine would not fill the width of
a hydrogen atom. Physicists think about such tiny objects; only
mathematicians have microscopes fine enough to actually observe them.
-- J. Eving
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Sample letter of recommendation:
Dear Search Committee Chair,
I am writing this letter for Mr. Still Student who has applied for a
position in your department. I should start by saying that I cannot
recommend him too highly.
In fact, there is no other student with whom I can adequately compare
him, and I am sure that the amount of mathematics he knows will
surprise you.
His dissertation is the sort of work you don't expect to see these days.
It definitely demonstrates his complete capabilities.
In closing, let me say that you will be fortunate if you can get him
to work for you.
Sincerely,
A. D. Advisor (Prof.)
-- from MAA Focus Newsletter
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To divide a cube into two other cubes, a fourth power or in general
any power whatever into two powers of the same denomination above
the second is impossible, and I have assuredly found an admirable proof
of this, but the margin is too narrow to contain it.
-- P. de Fermat
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Mathematics is not only real, but it is the only reality. That is that
entire universe is made of matter, obviously. And matter is made of
particles. It's made of electrons and neutrons and protons. So the entire
universe is made out of particles. Now what are the particles made
out of? They're not made out of anything. The only thing you can say
about the reality of an electron is to cite its mathematical properties.
So there's a sense in which matter has completely dissolved and what is
left is just a mathematical structure.
-- M. Gardner
%
God does arithmetic.
-- K.F. Gauss
%
Don't just read it; fight it! Ask your own questions, look for your own
examples, discover your own proofs. Is the hypothesis necessary? Is
the converse true? What happens in the classical special case? What about
the degenerate cases? Where does the proof use the hypothesis?
-- P.R. Halmos
%
God not only plays dice. He also sometimes throws the dice where they
cannot be seen.
-- S.W. Hawking
%
'Wir muessen wissen. Wir werden wissen.'
(We have to know. We will know.)
-- D. Hilbert (engraved in tombstone)
%
Physics is much too hard for physicists.
-- D. Hilbert
%
Hofstadter's Law: It always takes longer than you expect, even when
you take into account Hofstadter's Law.
-- D.R. Hofstadter, Goedel-Escher-Bach
%
The science of mathematics presents the most brilliant example of how
pure reason may successfully enlarge its domain without the aid of
experience.
-- E. Kant
%
A topologist is one who doesn't know the difference between a doughnut
and a coffee cup.
-- J. Kelley
%
Say what you know, do what you must, come what may.
-- S. Kovalevsky
%
God made the integers, all else is the work of man.
-- L. Kronecker
%
There is no branch of mathematics, however abstract, which may not some
day be applied to phenomena of the real world.
-- N. Lobatchevsky
%
Medicine makes people ill, mathematics make them sad and theology
makes them sinful.
-- M. Luther
%
The mathematician who pursues his studies without clear views of this
matter, must often have the uncomfortable feeling that his paper and
pencil surpass him in intelligence.
-- E. Mach
%
I tell them that if they will occupy themselves with the study of
mathematics they will find in it the best remedy against the lusts of
the flesh.
-- T. Mann
%
Today, it is not only that our kings do not know mathematics,
but our philosophers do not know mathematics and - to go a step
further - our mathematicians do not know mathematics.
-- J.R. Oppenheimer
%
Mathematics consists of proving the most obvious thing in the least
obvious way.
-- G. Polya
%
However successful the theory of a four dimensional world may be,
it is difficult to ignore a voice inside us which whispers:
"At the back of your mind, you know a fourth dimension is all
nonsense". I fancy that voice must have had a busy time in the
past history of physics. What nonsense to say that this solid
table on which I am writing is a collection of electrons
moving with prodigious speed in empty spaces, which relative to
electronic dimensions are as wide as the spaces between the
planets in the solar system! What nonsense to say that the thin
air is trying to cursh my body with a load of 14 lbs. to the
square inch! What nonsense that the star cluster which I see
through the telescope, obviously there NOW, is a glimpse into a
past age 50'000 years ago! Let us not be beguiled by this voice.
It is discredited...
-- Sir Arthur Eddington
%
The first million decimal places of pi are comprised of:
99959 0's
99758 1's
100026 2's
100229 3's
100230 4's
100359 5's
99548 6's
99800 7's
99985 8's
100106 9's
--David Blatner, the joy of pi
%
Math historians often state that the Egyptians thought pi = 256/81.
In fact, there is no direct evidence that the Egyptians conceived
of a constant number pi, much less tried to calculate it. Rather,
they were simply interested in finding the relationship between the
circle and the square, probably to accomplish the task of precisely
measuring land and buildings.
--David Blatner, the joy of pi
%
2000 BC Babilonians use pi=25/8, Egyptians use pi=256/81
1100 BC Chinese use pi=3
200 AC Ptolemy uses pi=377/120
450 Tsu Ch'ung-chih uses pi=255/113
530 Aryabhata uses pi=62832/20000
650 Brahmagupta uses pi=sqrt(10)
1593 Romanus finds pi to 15 decimal places
1596 Van Ceulen calculates pi to 32 places
1699 Sharp calculates pi to 72 places
1719 Tantet de Lagny calculates pi to 127 places
1794 Vega calculates pi to 140 decimal places
1855 Richter calculates pi to 500 decimal places
1873 Shanks finds 527 decimal places
1947 Ferguson calculates 808 places
1949 ENIAC computer finds 2037 places
1955 NORC computer computes 3089 places
1959 IBM 704 computer finds 16167 places
1961 Shanks-Wrench (IBM7090) find 100200 places
1966 IBM 7030 computes 250000 places
1967 CDC6600 computes 500000 places
1973 Guilloud-Bouyer (CDC7600) find 1 Mio places
1983 Tamura-Kanada (HITACM-280H) compute 16 Mio places
1988 Kanada (HITAC M-280H) computes 16 Mio digits
1989 Chudnovsky finds 1000 Mio digits
1995 Kanada computes pi to 6000 Mio digits
1996 Chudnovsky computes pi to 8000 Mio digits
1997 Kanada determines pi to 51000 Mio digits
--David Blatner, the joy of pi
%
The following is a transcript of an interchange between defence
attorney Robert Blasier and FBI Special Agent Roger Martz on
July 26, 1995, in the courtroom of the O.J. Simpson trial:
Q: Can you calculate the area of a circle
with a five-millimeter diameter?
A: I mean I could. I don't...math I don't ...
I don't know right now what it is.
Q: Well, what is the formula for the area of a circle?
A: Pi R Squared
Q: What is pi?
A: Boy, you ar really testing me. 2.12... 2.17...
Judge Ito: How about 3.1214?
Q: Isn't pi kind of essential to being a scientist
knowing what it is?
A: I haven't used pi since I guess I was in high school.
Q: Let's try 3.12.
A: Is that what it is? There is an easier way to do...
Q: Let's try 3.14. And what is the radius?
A: It would be half the diameter: 2.5
Q: 2.5 squared, right?
A: Right.
Q: Your honor, may we borrow a calculator?
[pause]
Q: Can you use a calculator?
A: Yes, I think.
Q: Tell me what pi times 2.5 squared is.
A: 19
Q: Do you want to write down 19? Square millimeters, right?
The area. What is one tenth of that?
A: 1.9
Q: You miscalculated by a factor of two, the size, the
minimum size of a swatch you needed to detect EDTA
didn't you?
A: I don't know that I did or not. I calculated a little
differently. I didn't use this.
Q: Well, does the area change by the different method of
calculation?
A: Well, this is all estimations based on my eyeball. I
didn't use any scientific math to determine it.
--David Blatner, the joy of pi
%
To those who do not know Mathematics it is difficult to get across a
real feeling as to the beauty, the deepest beauty of nature. ...
If you want to learn about nature, to appreciate nature, it is
necessary to understand the language that she speaks in.
-- Richard Feynman in
"The Character of Physical Law"
%
All science requires Mathematics. The knowledge of mathematical things is
almost innate in us... This is the easiest of sciences, a fact which is
obvious in that no one?s brain rejects it; for laymen and people who are
utterly illiterate know how to count and reckon.
-- Roger Bacon
%
Pure mathematics consists entirely of such asseverations as that, if
such and such a proposition is true of anything, then such and such
another proposition is true of that thing... It's essential not to discuss
whether the proposition is really true, and not to mention what the
anything is of which it is supposed to be true... If our hypothesis is
about anything and not about some one or more particular things,
then our deductions constitute mathematics. Thus mathematics may
be defined as the subject in which we never know what we are talking
about, nor whether what we are saying is true.
-- Bertrand Russell
%
The more ambitious plan may have more chances of success
-- G. Polya, How To Solve It
%
THEOREM: Every natural number can be completely and unambiguously
identified in fourteen words or less.
PROOF:
1. Suppose there is some natural number which cannot be unambiguously
described in fourteen words or less.
2. Then there must be a smallest such number. Let's call it n.
3. But now n is "the smallest natural number that cannot be unambiguously
described in fourteen words or less".
4. This is a complete and unambiguous description of n in fourteen words,
contradicting the fact that n was supposed not to have such a description!
5. Since the assumption (step 1) of the existence of a natural number that
cannot be unambiguously described in fourteen words or less
led to a contradiction, it must be an incorrect assumption.
6.Therefore, all natural numbers can be unambiguously described in fourteen
words or less!
%
THEOREM: 1=2
PROOF:
1. Let a=b.
2. Then a^2 = ab,
3. a^2 + a^2 = a^2 + ab,
4. 2 a^2 = a^2 + ab,
5. 2 a^2 - 2 ab = a^2 + ab - 2 ab,
6. and 2 a^2 - 2 ab = a^2 - ab.
7. Writing this as 2 (a^2 - a b) = 1 (a^2 - a b),
8. and cancelling the (a^2 - ab) from both sides gives 1=2.
%
II III V VII XI XIII XVII XIX XXIII XXIX ...
%
"Can you do addition?" the White Queen asked.
"What's one and one and one and one and one and one
and one and one and one and one?"
"I don't know," said Alice, "I lost count.".
-- Lewis Carrol alias Charles Lutwidge Dodgson,
Alice's Adventures in Wonderland
%
"She can't do Subtraction", said the White Queen. "Can you do
Division? Divide a loaf by a knife -- what's the
answer to that?"
"I suppose --" Alice was beginning, but the
Red Queen answerd for her. "Bread and butter, of course ..."
-- Lewis Carrol alias Charles Lutwidge Dodgson,
Alice's Adventures in Wonderland
%
Theorem: the square root x of 2 is irrational.
Proof: x=n/m with gcd(n,m)=1 implies 2=n^2/m^2 which is
2 m^2=n^2 so that n must be even and n^2 a multiple of 4.
Therefore m is even. This contradicts gcd(n,m)=1.
%
It is still an unending source of surprise for me to see
how a few scribbles on a blackboard or on a sheet of paper
could change the course of human affairs.
-- Stanislaw Ulam.
%
Of all escapes from reality, mathematics is the most
successful ever. It is a fantasy that becomes all the
more addictive because it works back to improve the
same reality we are trying to evade. All other escapes-
sex, drugs, hobbies, whatever - are ephemeral by comparison.
The mathematician's feeling of triumph, as he forces
the world to obey the laws his imagimation has created,
feeds on its own success. The world is premanently
changed by the workings of his mind, and the certainty
that his creations will endure renews his confidence as no
other pursuit.
-- Gian-Carlo Rota
%
A good mathematical joke is better, and better mathematics
than a dozen mediocre papers.
-- John Edensor Littlewood
%
pi/4 = 1-1/3+1/5-1/7+1/9 ...
-- Wilhelm von Leibniz
%
THEOREM (A. Katok) There exists a measurable set E of area one in
the unit square (0,1) x [0,1] together with a family of disjoint
smooth real analytic curves G(y) which fill out this square, so that
each curve G(y) intersects E in at most one single point.
PROOF. Define for p in (0,1) the piecewise linear map T on [0,1]
by T(x)=x/p for x in A=[0,p) and f(x)=(x-p)/(1-p) for x in B=[p,1).
It is easy to see that T is measure-preserving. Denote by T^n(x) the
n'th iterate of the map, that is T^n(x)=T(T^(n-1)(x)). For fixed p, code
x by an infinite sequence b(n)=0 if x(n)=T^n(x) in A and b(n)=1 else.
In terms of this coding, T corresponds to the shift map. By the strong
law of large numbers, for Lebesgue almost every x in [0,1], the
frequency of 1's in the associated symbol space is defined and equal
to (1-p). Let E be the subset of (p,x) in (0,1) x [0,1] such that
the frequency of 1's is equal to 1-p. It is a measurable set. Because
the intersection of E with each line {p} x [0,1] has full Lebesgue
measure, Fubini's theorem implies that E has Lebesgue area 1.
For x in [0,1] define y(p,x) = sum b(n) 2^n, where b(n) is the coding
of x. The sets G(y) = { (p,x) | y(p,x)=y } are disjoint and each G(y)
is a smooth real analytic curve.
[Proof: set p(0)=p,p(1)=1-p. From x(n)=b(n) p(0)+x(n+1) p(b(n)) follows
x=x(p,y)=p(0)(b(1)+p(b(1))(b(2)+p(b(2))(b(3)+...) ...) ...)
=p(0)(b(1)+b(2) p(b(1)) +b(3) p(b(1)) p(b(2))
+b(4) p(b(1)) p(b(2)) p(b(3)) +...)
Set p(0)=p=(1+t)/2, p(1)=1-p=(1-t)/2. If |t| x(p,y). Since a given symbol
sequence b(n) can have at most one limiting frequency
lim (b(1)+ ... + b(n))/n = 1-p, it follows that each G(y) can intersect
E in at most a single point (p,x(p,y)).
-- John Milnor, Mathem. Intelligencer, Vol 19, 1997
%
It has been said that the First World War was the chemists' war because
mustard gas and chlorine were empolyed for the first time, and that the
Second World War was the physicists war, because the atom bomb was
detonated. Similarly, it has been argued that the Third World War would
be the mathematicians' war, because mathematics will have control over
the next great weapon of war - information.
-- Simon Singh, in 'The code book'
%
Never speak more clearly than you think.
-- Jeremy Bernstein
%
What, in effect are the conditions for the construction of formal
thought? The child must not only apply operations to objects - in
other words, mentally execute possible actions on them - he must
also 'reflect' those operations in the absence of the objects which
are replaced by pure propositions. Thus 'reflection' is thought
raised to the second power. Concrete thinking is the representation
of a possible action, and formal thinking is the representation of
a representation of possible action... It is not surprising,
therefore, that the system of concrete operations must be completed
during the last years of childhood before it can be 'reflected' by
formal operations. In terms of their function, formal operations do
not differ from concrete operations except that they are applied to
hypotheses or propositions whose logic is an abstract translation of
the system of 'inference' that governs concrete operations.
-- Jean Piaget
%
An integer 2^n-1 is called a Mersenne number. If it is prime,
it is called a Mersenne prime. In that case, n must be prime.
Known examples are n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107,
127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941,
11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091,
756839, 859433, 1257787, 1398269, 2976221, 3021377. It is not
known whether there are infinitely many Mersenne primes.
%
A positive integer n is called a perfect number if it is equal to
the sum of all of its positive divisors, excluding n itself.
Examples are 6=1+2+3, 28=1+2+4+7+14. An integer k is an even perfect
number if and only if it has the form 2^(n-1)(2^n-1) and 2^n-1 is
prime. In that case 2^n-1 is called a Mersenne prime and n must be
prime. It is unknown whether there exists an odd perfect number.
%
WILSON'S THEOREM: p prime if and only if (p-1)!==-1 (mod p)
PROOF. 1,2, ..., p-1 are roots of x^(p-1)==0 (mod p). A congruence
has not more roots then its degree, hence
x^(p-1) -1 == (x-1)(x-2) ... (x-(p-1)) mod p. For x=0, this gives
-1 == (-1)^(p-1) (p-1)! == (p-1)! which is also true for p=2.
-- from P. Ribenboim,
'The new book of prime number records'
%
There is keen competition to produce the largest pair of twin primes.
On October 9, 1995, Dubner discovered the largest known pair of twin
primes p,p+2 where p=570918348*10^5120 - 1. It took only one day
with 2 crunchers. The expected time would be 150 times longer! What
luck!
-- from P. Ribenboim,
'The new book of prime number records'
%
How to catch a lion:
- THE HILBERT METHOD. Place a locked cage in the desert.
Set up the following axiomatic system.
(i) The set of lions is non-empty
(ii) If there is a lion in the desert, then there is a lion in the cage.
Theorem. There is a lion in the cage
- THE PEANO METHOD. There is a space-filling curve passing through every
point of the desert. Such a curve may be traversed in as short a time as
we please. Armed with a spear, traverse the curve faster than the
lion can move his own length.
- THE TOPOLOGICAL METHOD. The lion has a least the connectivity of a torus.
Transport the desert into 4-space. It can now be deformed in such a way as
to knot the lion. He is now helples.
- THE SURGERGY METHOD. The lion is an orientable 3-manifold with boundary
and so may be rendered contractible by surgery.
- THE UNIVERSAL COVERING METHOD. Cover the lion by his simply-connected
covering space. Since this has no holes, he is trapped.
- THE GAME THEORY METHOD. The lion is a big game, hence certainly a game.
There exists an optimal strategy. Follow it.
- THE SCHROEDINGER METHOD. At any instant there is a non-zero probability
that the lion is in the cage. Wait.
- THE ERASTOSHENIAN METHOD. Enumerate all objects in the desert: examine
them one by one; discard all those that are not lions. A refinement
will capture only prime lions.
- THE PROJECTIVE GEOMETRY METHOD. The desert is a plane. Project
this to a line, then project the line to a point inside the cage. The
lion goes to the same point.
- THE INVERSION METHOD. Take a cylindrical cage. First case: the lion
is in the cage: Trivial. Second case: the lion is outside the cage.
Go inside the cage. Invert at the boundary of the cage. The lion is
caught. Caution: Don't stand in the middle of the cage during the
inversion!
%
Euler's formula: A connected plane graph with n vertices, e edges and f faces
satisfies n - e + f = 2
Proof. Let T be the edge set of a spanning tree for G. It is a subset of the
set E of edges. A spanning tree is a minimal subgraph that connects all the
vertices of G. It contains so no cycle. The dual graph G* of G has a vertex
in the interior of each face. Two vertices of G* are connected by an edge if
the correponding faces have a common boundary edge. G* can have double edges
even if the original graph was simple. Consider the collection T* of edges
E* in G* that correspond to edges in the complement of T in E. The edges of
T* connect all the faces because T does not have a cycle. Also T* does not
contain a cycle, since otherwise, it would seperate some vertices of G
contradicting that T was a spanning subgraph and edges of T and T* don't
intersect. Thus T* is a spanning tree for G*. Clearly e(T)+e(T*)=2.
For every tree, the number of vertices is one larger than the number of
vertices. Applied to the tree T, this yields n = e(T)+1, while for the tree
T* it yields f=e(T*)+1. Adding both equations gives n+f=(e(T)+1)+(e(T*)+1)=e+2.
-- from M.Aigner, G. Ziegler "Proofs from THE BOOK"
%
Theorem: e = sum(k) 1/k! is irrational.
Proof. e=a/b with integers a,b would imply N = n! (e - sum(kb because n! e and n!/k! were both integers. However,
0n) n!/k!=1/(n+1) + 1/(n+1)(n+2) + ...<1/(n+1) + 1/(n+1)^2 + ...=1/n
(second sum is a geometric series) for every n is not possible.
-- from M.Aigner, G. Ziegler "Proofs from THE BOOK"
%
After a few years at MIT, the Mathematician Norbert Wiener moved to a larger house.
His wife, knowing his nature, figured that he would forget his new address and
be unable to find his way home after work. So she wrote the address
of the new home on a piece of paper that she made him put in his shirt pocket.
At lunchtime that day, the professor had an inspiring idea. He pulled the
paper out of his pocket and used it to scribble down some calculations. Finding
a flaw, he threw the paper away in disgust. At the end of the day he realized
he had thrown away his address, he now had no idea where he lived.
Putting his mind to work, he came up with a plan. He would go to his old
house and await rescue. His wife would surely realize that he
was lost and go to his old house to pick him up. Unfortunately, when he
arrived at his old house, there was no sign of his wife, only a small girl
standing in front of the house. "Excuse me, little girl" he said "but do you
happen to know where the people who used to live here moved to?" "It's okay,
Daddy," said the little girl, "Mommy sent me to get you".
Moral 1. Don't be surprised if the professor doesn't know your name by the end
of the semester.
Moral 2. Be glad your parents aren't mathematicians. if your parents are
mathematicians, introduce yourself and get them to help you through the
course.
- From the introduction of "How to ace calculus" by
C. Adams, A. Thompson and J. Hass
%
David Hilbert was one of the great European mathematicians at the turn of the
century. One of his students purchased an early automobile and died in one of
the first car accidents. Hilbert was asked to speak at the funeral. "Young Klaus"
he said, "was one of my finest students. He had an unusual gift for doing
mathematics. He was insterested in a great variety of problems, such as..."
There was a short pause, follwed by "Consider the set of differentiable functions
on the unit interval and take their closure in the ..."
Moral 1. Sit near the door.
Moral 2. Some mathematicians can be a little out of touch with reality. If your professor
falls in this category, look at the bright side. You will have lots of
funny stories by the end of the semester.
- From the introduction of "How to ace calculus" by
C. Adams, A. Thompson and J. Hass
%
In a forest a fox bumps into a little rabbit, and says,
"Hi, junior, what are you up to?"
"I'm writing a dissertation on how rabbits eat foxes," said the rabbit.
"Come now, friend rabbit, you know that's impossible!"
"Well, follow me and I'll show you."
They both go into the rabbit's dwelling and after a while the rabbit
emerges with a satisfied expression on his face.
Along comes a wolf. "Hello, what are we doing these days?"
"I'm writing the second chapter of my thesis, on how rabbits devour wolves."
"Are you crazy? Where is your academic honesty?"
"Come with me and I'll show you." ......
As before, the rabbit comes out with a satisfied look on his face and
this time he has a diploma in his paw. The camera pans back and into the
rabbit's cave and, as everybody should have guessed by now, we see an
enormous mean-looking lion sitting next to the bloody and furry remains
of the wolf and the fox. The moral of this story is:
It's not the contents of your thesis that are important --
it's your PhD advisor that counts.
- Unknown Usenet Source
%
It is true that a mathematician who is not also something of a
poet will never be a perfect mathematician.
- K. Weierstrass,
Quoted in D MacHale,
Comic Sections (Dublin 1993)
%
THEOREM: All triangles are equilateral.
PROOF:
1) Given an arbitrary triangle ABC. Construct the middle orthogonal
on AB in D and cut it with the line dividing the angle at C. Call the
intersection E. Form the normal from E to AC in F and from E to BC
in G. Draw the lines AE und BE.
C *
/ \
/ \
*F *G
/ E* \
/ | \
/ | \
/ |D \
A*---------*------------*B
2. The angles ECF and ECG are gleich. The angles EFC and EGC are both
right angles. Because the triangles ECF and ECG have also EC common,
they myust be congruent. Therefore CF=CG and EF=EG.
3. The sides DA and DB are equal. The angle EDA and EDB are both
right angles. Because the triangles EDA and EDB have also ED in common,
they have to be congruent and EA=EB.
4. The angle EGB and EFA are both right angle. Also, EF=EG and EA=EB.
Therefore both triangles EGB and EFA are congruent. Therefore FA=GB.
5. Since CF=CG and FA=GB, addition of the sides gives also CA=CB.
6. Having proved that two arbitrary sides are equal, all are equal.
%
I married a widow, who had an adult stepdaughter. My father, a widow
and who often visited us, fell in love with my stepdaughter and married her.
So, my father became my son-in-law and my stepdaughter became my stepmother.
But my wife became the mother-in-law of her father-in-law. My stepmother,
stepdaughter of my wife had a son and I therefore a brother, because
he is the son of my father and my stepmother. But since he was in the same
time the son of our stepdaughter, my wife became his grandmother and I became the
grandfather of my stepbrother. My wife gave me also a son. My stepmother,
the stepsister of my son, is in the same time his grandmother, because he is
the son of her stepson and my father is the brother-in-law of my child, because his
sister is his wife. My wife, who is the mother of my stepmother, is therefore
my grandmother. My son, who is the child of my grandmother, is the grandchild
of my father. But I'm the husband of my wife and in the same time the grandson of
my wife. This means: I'm my own grandfather.
%
I never could make out what those damned dots meant.
-- Lord Randolph Churchill (1849-1895)
Brittish conservative politician, referring
to decimal points.
%
The mathematician has reached the highest rung on the ladder of human
thought.
-- Havelock Ellis
%
Let no one ignorant of mathematics enter here.
-- Plato, Inscription written over the
entrance to the academy
%
I knew a mathematician, who said 'I do not know as much as God. But I
know as much as God knew at my age'.
-- Milton Shulman, Candian writer