math quotes

The statement sometimes made, that there exist only analytic functions
in nature, is in my opinion absurd.
-- F. Klein, Lectures on Mathematics, 1893

pi/4 = 1-1/3+1/5-1/7+1/9 ...
-- Wilhelm von Leibniz

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

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

The essence of mathematics lies in its freedom.
-- Georg Cantor

A mathematician is a device for turning coffee into theorems.
-- P. Erdos

Chebyshev said it, and I say it again
There is always a prime between n and 2n
-- P. Erdos

The real mathematician is an enthusiast per se. Without enthusiasm
no mathematics.
-- Novalis

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

Pure mathematician - Anyone who prefers set theory to sex.
-- R. Ainsley in Bluff your way in Maths, 1988

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

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

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

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

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

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

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

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!

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

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

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

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 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

Pure mathematics is the magician's real wand.
-- Novalis

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

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

II III V VII XI XIII XVII XIX XXIII XXIX ...

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'

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

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