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Theorem : All positive integers are equal.
Proof : Sufficient to show that for any two positive integers, A and B,
A = B. Further, it is sufficient to show that for all N > 0, if A
and B (positive integers) satisfy (MAX(A, B) = N) then A = B.
Proceed by induction.
If N = 1, then A and B, being positive integers, must both be 1.
So A = B.
Assume that the theorem is true for some value k. Take A and B
with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence
(A-1) = (B-1). Consequently, A = B.
M__________________________________________________________________________
From: Benjamin.J.Tilly@dartmouth.edu (Benjamin J. Tilly)
Theorem : All numbers are equal to zero.
Proof: Suppose that a=b. Then
a = b
a^2 = ab
a^2 - b^2 = ab - b^2
(a + b)(a - b) = b(a - b)
a + b = b
a = 0
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From: Michael_Ketzlick@h2.maus.de (Michael Ketzlick)
Theorem : 3=4
Proof:
Suppose:
a + b = c
This can also be written as:
4a - 3a + 4b - 3b = 4c - 3c
After reorganising:
4a + 4b - 4c = 3a + 3b - 3c
Take the constants out of the brackets:
4 * (a+b-c) = 3 * (a+b-c)
Remove the same term left and right:
4 = 3
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From: Benjamin.J.Tilly@dartmouth.edu (Benjamin J. Tilly)
Theorem: 1$ = 1c.
Proof:
And another that gives you a sense of money disappearing...
1$ = 100c
= (10c)^2
= (0.1$)^2
= 0.01$
= 1c
Here $ means dollars and c means cents. This one is scary in that I
have seen PhD's in math who were unable to see what was wrong with this
one. Actually I am crossposting this to sci.physics because I think
that the latter makes a very nice introduction to the importance of
keeping track of your dimensions...
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From: clubok@physics11 (Kenneth S. Clubok)
Theorem: 1 = -1 .
Proof:
1 -1
-- = --
-1 1
1 -1
sqrt[ -- ] = sqrt[ -- ]
-1 1
sqrt[1] sqrt[-1]
------- = -------
sqrt[-1] sqrt[1]
1=-1 (by cross-multiplication)
And here's my personal favorite:
Use integration by parts to find the anti-derivative of 1/x. One
can get the amusing result that 0=1. (Until you realize you have to put
in the limits.)
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From: jreimer@aol.com (JReimer)
Theorem: 1 = -1
Proof:
1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = 1^ = -1
Also one can disprove the axiom that things equal to the same thing
are equal to each other.
1 = sqrt(1)
-1 = sqrt(1)
therefore 1 = -1
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From: kdq@marsupial.jpl.nasa.gov (Kevin D. Quitt)
Theorem: 4 = 5
Proof:
16 - 36 = 25 - 45
4^2 - 9*4 = 5^2 - 9*5
4^2 - 9*4 + 81/4 = 5^2 - 9*5 + 81/4
(4 - 9/2)^2 = (5 - 9/2)^2
4 - 9/2 = 5 - 9/2
4 = 5
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baez@guitar.ucr.edu (john baez) writes:
Theorem: 1 + 1 = 2
Proof:
n(2n - 2) = n(2n - 2)
n(2n - 2) - n(2n - 2) = 0
(n - n)(2n - 2) = 0
2n(n - n) - 2(n - n) = 0
2n - 2 = 0
2n = 2
n + n = 2
or setting n = 1
1 + 1 = 2
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From: magidin@uclink.berkeley.edu (Arturo Viso Magidin)
Theorem: In any finite set of women, if one has blue eyes then they
all have blue eyes.
Proof. Induction on the number of elements.
if n= or n=1 it is immediate.
Assume it is true for k
Consider a group with k+1 women, and without loss of generality assume
the first one has blue eyes. I will represent one with blue eyes with
a '*' and one with unknown eye color as @.
You have the set of women:
{*,@,...,@} with k+1 elements. Consider the subset made up of the first
k. This subset is a set of k women, of which one has blue eyes. By
the induction hypothesis, all of them have blue eyes. We have then:
{*,...,*,@}, with k+1 elements. Now consider the subset of the last k
women. This is a set of k women, of which one has blue eyes (the next-to-last
element of the set), hence they all have blue eyes, in particular
the k+1-th woman has blue eyes.
Hence all k+1 women have blue eyes.
By induction, it follows that in any finite set of women, if one has
blue eyes they all have blue eyes. QED
M__________________________________________________________________________
From: Zorro
Theorem:
All positive integers are interesting.
Proof:
Assume the contrary. Then there is a lowest non-interesting positive
integer. But, hey, that's pretty interesting! A contradiction.
QED
I heard this one from G. B. Thomas, but I don't know whether it is due to
him.
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From: daniel@hagar.ph.utexas.edu (James Daniel)
Aren't multi-valued functions fun? Once you realize what's going on,
though, you can make them into silly proofs pretty much without thinking.
Here's one I just made up:
Object: to prove that i < 0 ( that is, sqrt(-1) < 0 )
Well, ( .5 + sqrt(3/4)*i )^3 = (-1)^3
(most would assert this to be a false statement -- mostly
cuz they'll get the math wrong. It's a true statement.
It's the next statement that is false.)
which means that .5 + sqrt(3/4)*i = -1
So then 1 + sqrt(3)*i = -2
sqrt(3)*i = -1
i = -1/sqrt(3)
Therefore i is a negative number. QED.
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From: julison@cco.caltech.edu (Julian C. Jamison)
Theorem: All numbers are equal.
Proof:
Choose arbitrary a and b, and let t = a + b. Then
a + b = t
(a + b)(a - b) = t(a - b)
a^2 - b^2 = ta - tb
a^2 - ta = b^2 - tb
a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4
(a - t/2)^2 = (b - t/2)^2
a - t/2 = b - t/2
a = b
So all numbers are the same, and math is pointless.
M__________________________________________________________________________
From: pfc@math.ufl.edu (P. Fritz Cronheim)
This one is from Jerry King's _Art of Mathematics_
16/64=1/4 by cancelling the 6's. Here the result is true, but the method
is not. Do the ends justify the means? :)_
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Methods of Mathematical Proof
This is from _A Random Walk in Science_ (by Joel E. Cohen?):
To illustrate the various methods of proof we give an example of a
logical system.
THE PEJORATIVE CALCULUS
Lemma 1. All horses are the same colour.
(Proof by induction)
Proof. It is obvious that one horse is the same colour. Let us assume
the proposition P(k) that k horses are the same colour and use this to
imply that k+1 horses are the same colour. Given the set of k+1 horses,
we remove one horse; then the remaining k horses are the same colour,
by hypothesis. We remove another horse and replace the first; the k
horses, by hypothesis, are again the same colour. We repeat this until
by exhaustion the k+1 sets of k horses have been shown to be the same
colour. It follows that since every horse is the same colour as every
other horse, P(k) entails P(k+1). But since we have shown P(1) to be
true, P is true for all succeeding values of k, that is, all horses are
the same colour.
Theorem 1. Every horse has an infinite number of legs.
(Proof by intimidation.)
Proof. Horses have an even number of legs. Behind they have two legs
and in front they have fore legs. This makes six legs, which is cer-
tainly an odd number of legs for a horse. But the only number that is
both odd and even is infinity. Therefore horses have an infinite num-
ber of legs. Now to show that this is general, suppose that somewhere
there is a horse with a finite number of legs. But that is a horse of
another colour, and by the lemma that does not exist.
Corollary 1. Everything is the same colour.
Proof. The proof of lemma 1 does not depend at all on the nature of the
object under consideration. The predicate of the antecedent of the uni-
versally-quantified conditional 'For all x, if x is a horse, then x is
the same colour,' namely 'is a horse' may be generalized to 'is anything'
without affecting the validity of the proof; hence, 'for all x, if x is
anything, x is the same colour.'
Corollary 2. Everything is white.
Proof. If a sentential formula in x is logically true, then any parti-
cular substitution instance of it is a true sentence. In particular
then: 'for all x, if x is an elephant, then x is the same colour' is
true. Now it is manifestly axiomatic that white elephants exist (for
proof by blatant assertion consult Mark Twain 'The Stolen White Ele-
phant'). Therefore all elephants are white. By corollary 1 everything
is white.
Theorem 2. Alexander the Great did not exist and he had an infinite
number of limbs.
Proof. We prove this theorem in two parts. First we note the obvious
fact that historians always tell the truth (for historians always take
a stand, and therefore they cannot lie). Hence we have the historically
true sentence, 'If Alexander the Great existed, then he rode a black
horse Bucephalus.' But we know by corollary 2 everything is white;
hence Alexander could not have ridden a black horse. Since the conse-
quent of the conditional is false, in order for the whole statement to
be true the antecedent must be false. Hence Alexander the Great did not
exist.
We have also the historically true statement that Alexander was warned
by an oracle that he would meet death if he crossed a certain river. He
had two legs; and 'forewarned is four-armed.' This gives him six limbs,
an even number, which is certainly an odd number of limbs for a man.
Now the only number which is even and odd is infinity; hence Alexander
had an infinite number of limbs. We have thus proved that Alexander the
Great did not exist and that he had an infinite number of limbs.
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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.
M__________________________________________________________________________
From: rmaimon@husc9.Harvard.EDU (Ron Maimon)
Theorem: All dogs have nine legs.
Proof:
would you agree that no dog has five legs?
would you agree that _a_ dog has four legs more then _no_ dog?
4 + 5 = ?
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
=2.2 STATISTICS AND STATISTICANS
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Did you hear the one about the statistician?
Probably....
M__________________________________________________________________________
Statistics means never having to say you're certain.
[With apologies to Erich Segal]
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In earlier times, they had no statistics, and so they had to fall
back on lies. - STEPHEN LEACOCK
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"The group was alarmed to find that if you are a labourer, cleaner or dock
worker, you are twice as likely to die than a member of the professional
classes" [The Sunday Times 31st August 1980]
M__________________________________________________________________________
From: ph2008@mail.bris.ac.uk (CJ. Bradfield)
Statistics is the art of never having to say you're wrong.
Variance is what any two staticticians are at.
(Not that I particularly dislike statisticians... I hate all
mathematicians!!) [sorry mum!]
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97.3% of all statistics are made up.
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it's like the tale of the roadside merchant who was asked to explain how
he could sell rabbit sandwiches so cheap. "Well" he explained, "I have to
put some horse-meat in too. But I mix them 50:50. One horse, one rabbit."
[DARREL HUFF, How to lie with statistics]
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Are statisticians normal?
M__________________________________________________________________________
From: joeshmoe@world.std.com (Jascha Franklin-Hodge) (List of Taglines)
Smoking is a leading cause of statistics.
I could prove God statistically.
43% of all statistics are worthless.
"There are lies, damned lies, and statistics." -Mark Twain
3 out of 4 Americans make up 75% of the population.
Death is 99 per cent fatal to laboratory rats.
M__________________________________________________________________________
Did you know that the great majority of people have more than the average
number of legs? [It's obvious really; amongst the 57 million people in
Britain there are probably 5,000 people who have only got one leg.
Therefore the average number of legs is
(5000 * 1) + (56,995,000 * 2)
---------------------------------- = 1.9999123......
57,000,000
Since most people have 2 legs....... ]
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A statistician is a person who draws a mathematically precise line from an
unwarranted asumption to a foregone conclusion.
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A statistician can have his head in an oven and his feet in ice, and
he will say that on the average he feels fine.
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From: Chris Morton (mortoncp@nextwork.rose-hulman.edu) do it collection
Statisticians do it continuously but discretely.
Statisticians do it when it counts.
Statisticians do it with 95% confidence.
Statisticians do it with large numbers.
Statisticians do it with only a 5% chance of being rejected.
Statisticians do it with two-tail T tests.
Statisticians do it. After all, it's only normal.
Statisticians probably do it.
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From: Mathematics Magazine, December 1990.
Subject: Statisticians
( Excerpted from "Quotes, Damned Quotes" by John Bibby )
If there is a 50-50 chance that something can go wrong, then 9
times out of ten it will. (Paul Harvey News, 1979)
``Give us a copper Guv'' said the beggar to the Treasury
statistician, when he waylaid him in Parliament square. ``I
haven't eaten for three days.'' ``Ah,'' said the statistician, ``and
how does that compare with the same period last year?'' (Russell
Lewis)
``I gather, young man, that you wish to be a Member of
Parliament. The first lesson that you must learn is, when I call
for statistics about the rate of infant mortality, what I want
is proof that fewer babies died when I was Prime Minister than
when anyone else was Prime Minister. That is a political
statistic.'' (Winston Churchill)
``You haven't told me yet,'' said Lady Nuttal, ``what it is your
fiance does for a living.''
``He's a statistician,'' replied Lamia, with an annoying sense of
being on the defensive.
Lady Nuttal was obviously taken aback. It had not occurred to
her that statisticians entered into normal social relationships.
The species, she would have surmised, was perpetuated in some
collateral manner, like mules.
``But Aunt Sara, it's a very interesting profession,'' said Lamia
warmly.
``I don't doubt it,'' said her aunt, who obviously doubted it very
much. ``To express anything important in mere figures is so
plainly impossible that there must be endless scope for
well-paid advice on the how to do it. But don't you think that
life with a statistician would be rather, shall we say,
humdrum?''
Lamia was silent. She felt reluctant to discuss the surprising
depth of emotional possibility which she had discovered below
Edward's numerical veneer.
``It's not the figures themselves,'' she said finally. ``It's what
you do with them that matters.'' (K.A.C. Manderville, The undoing
of Lamia Gurdleneck)
M__________________________________________________________________________
People who do very unusual jobs: the man who counts then number of people
at public gatherings.
You've probably seen his headlines, "Two million flock to see Pope.", "200
arrested as police find ounce of cannabis.", "Britain #3 billion in debt".
You probably wondered who was responsible for producing such well
rounded-up figures. What you didn't know was that it was all the work of
one man, Rounder-Up to the media, John Wheeler. But how is he able to go
on turning out such spot-on statistics? How can he be so accurate all the
time?
"We can't" admits Wheeler blithely. "Frankly, after the first million we
stop counting, and round it up to the next million. I don't know if you've
ever counted a papal flock, but, not only do they look a bit the same,
they also don't keep still, what with all the bowing and crossing
themselves."
"The only way you could do it accurately is by taking an aerial photograph
of the crowd and handing it to the computer to work out. But then you'd
get a headline saying "1,678,163 [sic] flock to see Pope, not including
35,467 who couldn't see him", and, believe me, nobody wants that sort of
headline."
The art of big figures, avers Wheeler, lies in psychology, not statistics.
The public like a figure it can admire. It likes millionaires, and
million-sellers, and centuries at cricket, so Wheeler's international
agency gives them the figures it wants, which involves not only rounding
up but rounding down.
"In the old days people used to deal with crowds on the Isle of Wight
principle -- you know, they'd say that every day the population of the
world increased by the number of people who could stand upright on the
Isle of Wight, or the rain-forests were being decreased by an area the
size of Rutland. This meant nothing. Most people had never been to the
Isle of Wight for a start, and even if they had, they only had a vision of
lots of Chinese standing in the grounds of the Cowes Yacht Club. And the
Rutland comparison was so useless that they were driven to abolish Rutland
to get rid of it.
"No, what people want is a few good millions. A hundred million, if
possible. One of our inventions was street value, for instance. In the old
days they used to say that police had discovered drugs in a quantity large
enough to get all of Rutland stoned for a fortnight. *We* started saying
that the drugs had a street value of #10 million. Absolutely meaningless,
but people understand it better."
Sometimes they do get the figures spot on. "250,000 flock to see Royal
two", was one of his recent headlines, and although the 250,000 was a
rounded-up figure, the two was quite correct. in his palatial office he
sits surrounded by relics of past headlines - a million-year-old fossil, a
#500,000 Manet, a photograph of the Sultan of Brunei's #10,000,000 house -
but pride of place goes to a pair of shoes framed on the wall.
"Why the shoes? Because they cost me #39.99. They serve as a reminder of
mankind's other great urge, to have stupid odd figures. Strange, isn't it?
They want mass demos of exactly half a million, but they also want their
gramophone records to go round at thirty-three-and-a-third, forty-five and
seventy-eight rpm. We have stayed in business by remembering that below a
certain level people want oddity. They don't a rocket costing #299 million
and 99p, and they don't want a radio costing exactly #50."
How does he explain the times when the figures clash - when, for example,
the organisers of a demo claim 250,000 but the police put it nearer
100,000?
"We provide both sets of figures; the figures the organisers want, and the
figures the police want. The public believe both. If we gave the true
figure, about 167,890, nobody would believe it because it doesn't sound
believable."
John Wheeler's name has never become well-known, as he is a shy figure,
but his firm has an annual turnover of #3 million and his eye for the
right figure has made him a rich man. His greatest pleasure, however,
comes from the people he meets in the counting game.
"Exactly two billion, to be precise."
MILES KINGTON writing in The Observer, 3 November 1986
M__________________________________________________________________________
From: goble@infonaut.com (Clark Goble)
You know how dumb the average guy is? Well, by definition, half
of them are even dumber than that. -- J.R. "Bob" Dobbs
*M_________________________________________________________________________
From: Kirk Lindberg (kalindberg@mmm.com)
Q: What is the definition of a statistician?
A: Someone who doesn't have the personality to be an accountant.
*M_________________________________________________________________________
Did you hear about the Statistician that couldn't get laid?
He decided a simulation was good enough.
*M_________________________________________________________________________
From: rogers@sasuga.Hi.COM (Andrew Rogers)
"She was only the statistician's daughter, but she knew all the standard
deviations."
*M_________________________________________________________________________
From: en4bmhd@bs47c.staffs.ac.uk (Hendrik De Vloed)
All probabilities are 50% ... either something happens, or it doesn't!
From: brc2@Lehigh.EDU
Correction...
My doctor told me I only have a 50% chance of making it- but he said there's
only a 15% of even that.
*M_________________________________________________________________________
From: ahilditc@awadi.com.au & ts@uwasa.fi (Timo Salmi) &
Juhani Heino
A:I'll bet that 99% of people who read the question don't!
T:That's a mean thing to say.
J:Yes, it was. I guess that person is too regressed.
As a matter of fact, I'm 75.4 % sure about that.
T:Incidentally, did you know that using non-linear regression in
research is currently out of line.
*M________________________________________________________________________
From: jlevine@rd.hydro.on.ca (Jody Levine)
80% of all statistics quoted to prove a point are made up on the spot.
*M________________________________________________________________________
From: Sunita Saini
A stats major was completely hung over the day of his final
exam. It was a True/False test, so he decided to flip a coin for the
answers. The stats professor watched the student the entire two hours as
he was flipping the coin...writing the answer...flipping the
coin...writing the answer. At the end of the two hours, everyone else
had left the final except for the one student. The professor walks up to
his desk and interrupts the student, saying:
"Listen, I have seen that you did not study for this statistics test, you
didn't even open the exam. If you are just flipping a coin for your
answer, what is taking you so long?
The student replies bitterly (as he is still flipping the coin):
" Shhh! I am checking my answers!"
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
=2.3 MATHEMATICIANS
From: Hugh Robinson
Okay, here's mine. I am told that it's true, but...
A certain well-known pure mathematician had a wife who, while
intelligent, was not into mathematics. However, by continued
practice, she learnt to distinguish between the conversations
of algebraists and analysts. So when he had guests to dinner
who were talking about mathematics, if they were analysts, she
would introduce at a suitable pause in the conversation:
"But what happens at the boundary?"
Whereas, if they were algebraists, she would say:
"But do the roots lie in the field?"
By this means she was always able to impress his visitors by
her knowledge of mathematics.
(No, don't write and ask for the punchline. That's all.)
M__________________________________________________________________________
Three men are in a hot-air balloon. Soon, they find themselves lost
in a canyon somewhere. One of the three men says, "I've got an idea.
We can call for help in this canyon and the echo will carry our voices
far."
So he leans over the basket and yells out, "Helllloooooo! Where are
we?" (They hear the echo several times.)
15 minutes later, they hear this echoing voice: "Helllloooooo! You're
lost!!"
One of the men says, "That must have been a mathematician."
Puzzled, one of the other men asks, "Why do you say that?"
The reply: "For three reasons. (1) he took a long time to answer, (2)
he was absolutely correct, and (3) his answer was absolutely useless."
M__________________________________________________________________________
A small, 14-seat plane is circling for a landing in Atlanta. It's
totally fogged in, zero visibility, and suddenly there's a small
electrical fire in the cockpit which disables all of the instruments
and the radio. The pilot continues circling, totally lost, when
suddenly he finds himself flying next to a tall office building.
He rolls down the window (this particular airplane happens to have
roll-down windows) and yells to a person inside the building, "Where
are we?"
The person responds "In an airplane!"
The pilot then banks sharply to the right, circles twice, and makes a
perfect landing at Atlanta International.
As the passengers emerge, shaken but unhurt, one of them says to the
pilot, "I'm certainly glad you were able to land safely, but I don't
understand how the response you got was any use."
"Simple," responded the pilot. "I got an answer that was completely
accurate and totally irrelevant to my problem, so I knew it had to be
the IBM building."
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Mathematicians are like Frenchmen: whatever you say to them they
translate into their own language and forthwith it is something
entirely different. (Johann Wolfgang von Goethe)
M__________________________________________________________________________
Old mathematicians never die; they just lose some of their functions.
From: Tim.Nelson@Canada.ATTGIS.COM (list of Old * Never Die, they just)
OLD MATH TEACHERS never die, they just reduce to lowest terms
OLD MATHEMATICIANS never die, they just disintegrate
OLD MATHEMATICIANS never die, they just go off on a tangent
OLD NUMERICAL ANALYSTS never die, they just get disarrayed
OLD TRIGONOMETRY TEACHERS never die, they just lose their identities
M__________________________________________________________________________
From: banghar4@studentb.msu.edu (Rick Banghart)
Two math professors are in a restaurant. One argues that the average person
does not know any math beyond high school. The other argues that the average
person knows some more advanced math. Just then, the first one gets up to use
the rest room. The second professor calls over his waitress and says, "When
you bring our food, I'm going to ask you a mathematical question. I want you to
answer, 'One third x cubed.' Can you do that?"
The waitress says, "I don't know if I can remember that. One thurr... um..."
"One third x cubed," says the prof.
"One thir dex cue?," asks the waitress.
"One"
"One"
"Third"
"Third"
"X"
"X"
"Cubed"
"Cubed"
"One third X cubed"
"One third X cubed"
The waitress leaves, and the other professor comes back. They resume their
conversation until a few minutes later when the waitress brings their food.
The professor says to the waitress, "Say, do you mind if I ask you something?"
"Not at all"
"Can you tell me what the integral of x squared dx is?"
The waitress pauses, then says, "One third x cubed."
As she walks away, she stops, turns, and adds, "Plus a constant!"
M__________________________________________________________________________
Some famous mathematician was to give a keynote speech at a
conference. Asked for an advance summary, he said he would present a
proof of Fermat's Last Theorem -- but they should keep it under their
hats. When he arrived, though, he spoke on a much more prosaic
topic. Afterwards the conference organizers asked why he said he'd
talk about the theorem and then didn't. He replied this was his
standard practice, just in case he was killed on the way to the
conference.
M__________________________________________________________________________
How many mathematicians does it take to screw in a lightbulb?
One, who gives it to six Californians, thereby reducing it to an
earlier riddle.
-- from a button I bought at Nancy Lebowitz's table at Boskone
Q: How many topologists does it take to change a light bulb?
A: It really doesn't matter, since they'd rather knot.
From:BRIAN6@VAXC.MDX.AC.UK (who has a lightbulb collection)
Q: How many mathematicians does it take to screw in a lightbulb?
A: None. It's left to the reader as an exercise.
A: Just one, once you've managed to present the problem in terms he/she
is familiar with.
In earlier work, Wiener [1] has shown that one mathematician
can change a light bulb.
If k mathematicians can change a light bulb, and if one more simply
watches them do it, then k+1 mathematicians will have changed the
light bulb.
Therefore, by induction, for all n in the positive integers,
n mathematicians can change a light bulb.
Bibliography:
[1] Weiner, Matthew P., <11485@ucbvax>, "Re: YALBJ", 1986
Q: How many statisticians does it take to change a lightbulb ?
A: This should be determined using a nonparametric procedure, since
statisticians are NOT NORMAL.
A: Walt Pirie to hold the bulb and one psychologist, one economist,
one sociologist and one anthroplogist to pull away the ladder.
A: One -- plus or minus three (small sample size).
(Notes: Someone has been asking this as a bonus question on statistics exam
papers for quite a while. Judging from some of his own students' exam answers,
it depends on whether the lightbulb is negatively or positively screwed.)
Q: How many light bulbs does it take to change a light bulb?
A: One, if it knows its own Goedel number.
(Could somebody please explain this one to me ! I think it's something to do
with the maths/logic theories of Kurt Goedel, about it being impossible to
prove things.)
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"A mathematician is a device for turning coffee into theorems"
-- P. Erdos
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Moebius always does it on the same side.
Statisticians probably do it
Algebraists do it in groups.
(Logicians do it) or [not (logicians do it)].
From: Chris Morton (mortoncp@nextwork.rose-hulman.edu) do it collection
Logicians do it consistently and completely.
Mathematicians do it associatively.
Mathematicians do it commutatively.
Mathematicians do it constantly.
Mathematicians do it continuously.
Mathematicians do it discretely.
Mathematicians do it exponentially.
Mathematicians do it forever if they can do one and can do one more.
Mathematicians do it functionally.
Mathematicians do it homologically.
Mathematicians do it in fields.
Mathematicians do it in groups.
Mathematicians do it in imaginary planes.
Mathematicians do it in numbers.
Mathematicians do it in theory.
Mathematicians do it on smooth contours.
Mathematicians do it over and under the curves.
Mathematicians do it parallel and perpendicular.
Mathematicians do it partially.
Mathematicians do it rationally.
Mathematicians do it reflexively.
Mathematicians do it symmetrically.
Mathematicians do it to prove themselves.
Mathematicians do it to their limits.
Mathematicians do it totally.
Mathematicians do it transcendentally.
Mathematicians do it transitively.
Mathematicians do it variably.
Mathematicians do it with Nobel's wife.
Mathematicians do it with a Minkowski sausage.
Mathematicians do it with imaginary parts.
Mathematicians do it with linear pairs.
Mathematicians do it with odd functions.
Mathematicians do it with prime roots.
Mathematicians do it with relations.
Mathematicians do it with rings.
Mathematicians do it with their real parts.
Mathematicians do it without limit.
Mathematicians do over an open unmeasurable interval.
Mathematicians have to prove they did it.
Set theorists do it with cardinals.
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A mathematician is a person who says that, when 3 people are supposed
to be in a room but 5 came out, 2 have to go in so the room gets
empty...
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My geometry teacher was sometimes acute, and sometimes
obtuse, but always, he was right.
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From: lyon@netcom.com (Lyman Lyon)
Physics professor is walking across campus, runs into Math Professor.
Physics professor has been doing an experiment, and has worked out an
emphirical equation that seems to explain his data, and asks the Math
professor to look at it.
A week later, they meet again, and the Math professor says the equation
is invalid. By then, the Physics professor has used his equation to
predict the results of further experiments, and he is getting excellent
results, so he askes the Math professor to look again.
Another week goes by, and they meet once more. The Math professor tells
the Physics professor the equation does work, "But only in the trivial
case where the numbers are real and positive."
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From: gw@molly.informatik.Uni-Koeln.DE (Georg Wambach)
What is the difference between an applied mathematician and a pure
mathematician?
Suppose a mathematician parks his car, locks it with his key and walks
away. After walking about 50 yards the mathematician realizes that he has
dropped his key somewhere along the way. What does he do? If he is an
applied mathematician he walks back to the car along the path he has
previously traveled looking for his key. If he is a pure mathematician he
walks to the other end of the parking lot where there is better light
and looks for his key there.
I told this joke to my brother (he is a "pure"). He answers:
"But we have not dropped our keys!" Hence, I suggest a slight
modification:
Suppose a _tax_payer_ parks his car, locks it with his key and walks
away. After walking about 50 yards the tax payer realizes that he
has dropped his key somewhere along the way. He asked a mathematician
to help him. What does the mathematician do? (...)
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
=2.4 POETRY
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From: chrisman@ucdmath.ucdavis.edu (Mark Chrisman)
"Aleph-0 bottles of beer on the wall,
Aleph-0 bottles of beer;
Take one down, pass it around,
Aleph-0 bottles of beer on the wall!
Aleph-0 bottles of beer on the wall..."
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One and one make two,
But if one and one should marry,
Isn't it queer-
Within a year
There's two and one to carry.
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Geometry keeps you in shape.
Decimals make a point.
Einstein was ahead of his time.
Lobachevski was out of line.
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"IF" (School Maths version)
===========================
If you can solve a literal equation
And rationalise denominator surds,
Do grouping factors (with a transformation)
And state the factor theorem in words;
If you can plot the graph of any function
And do a long division (with gaps),
Or square binomials without compunction
Or work cube roos with logs without mishaps.
If you possess a sound and clear-cut notion
Of interest sums with P and I unknown;
If you can find the speed of trains in motion,
Given some lengths and "passing-times" alone;
If you can play with R (both big and little)
And feel at home with l (or h) and Pi,
And learn by cancellation how to whittle
Your fractions down till they delight the eye.
If you can recognise the segment angles
Both at the centre and circumference;
If you can spot equivalent triangles
And Friend Pythagoras (his power's immmense);
If you can see that equiangularity
And congruence are two things and not one,
You may pick up a mark or two in charity
And, what is more, you may squeeze through, my son.
[Times Educational Supplement 19th July 1947]
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This poem was written by Jon Saxton (an author of math textbooks).
((12 + 144 + 20 + (3 * 4^(1/2))) / 7) + (5 * 11) = 9^2 + 0
Or for those who have trouble with the poem:
A Dozen, a Gross and a Score,
plus three times the square root of four,
divided by seven,
plus five times eleven,
equals nine squared and not a bit more.
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'Tis a favorite project of mine
A new value of pi to assign.
I would fix it at 3
For it's simpler, you see,
Than 3 point 1 4 1 5 9.
("The Lure of the Limerick" by W.S. Baring-Gould, p.5. Attributed to
Harvey L. Carter).
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If inside a circle a line
Hits the center and goes spine to spine
And the line's length is "d"
the circumference will be
d times 3.14159
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If (1+x) (real close to 1)
Is raised to the power of 1
Over x, you will find
Here's the value defined:
2.718281...
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Here's a limerick I picked up off the net a few years back - looks better
on paper.
3_
\/3
/
| 2 3 X pi 3_
| z dz X cos(--------) = ln (\/e )
| 9
/
1
Which, of course, translates to:
Integral z-squared dz
from 1 to the cube root of 3
times the cosine
of three pi over 9
equals log of the cube root of 'e'.
And it's correct, too.
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Not a joke, but a humorous ditty I heard from some guys in an
engineering fraternity (to the best of my recollection):
I'll do it phonetically:
ee to the ex dee ex,
ee to the why dee why,
sine x, cosine x,
natural log of y,
derivative on the left
derivative on the right
integrate, integrate,
fight! fight! fight!
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Other cheers:
E to the x dx dy
radical transcendental pi
secant cosine tangent sine
3.14159
2.71828
come on folks let's integrate!!
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E to the i dx dy
E to y dy
cosine secant log of pi
disintegrate em RPI !!!
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square root, tangent
hyperbolic sine,
3.14159
e to the x, dy, dx,
sliderule, slipstick, TECH TECH TECH!
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e to the u, du/dx
e to the x dx
cosine, secant, tangent, sine,
3.14159
integral, radical, u dv,
slipstick, slide rule, MIT!
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E to the X
D-Y, D-X
E to the X
D-X.
Cosine, Secant, Tangent, Sine
3.14159
E-I, Radical, Pi
Fight'em, Fight'em, WPI!
Go Worcester Polytechnic Institute!!!!!!
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Hiawatha Designs an Experiment
Hiawatha, mighty hunter,
He could shoot ten arrows upward,
Shoot them with such strength and swiftness
That the last had left the bow-string
Ere the first to earth descended.
This was commonly regarded
As a feat of skill and cunning.
Several sarcastic spirits
Pointed out to him, however,
That it might be much more useful
If he sometimes hit the target.
"Why not shoot a little straighter
And employ a smaller sample?"
Hiawatha, who at college
Majored in applied statistics,
Consequently felt entitled
To instruct his fellow man
In any subject whatsoever,
Waxed exceedingly indignant,
Talked about the law of errors,
Talked about truncated normals,
Talked of loss of information,
Talked about his lack of bias,
Pointed out that (in the long run)
Independent observations,
Even though they missed the target,
Had an average point of impact
Very near the spot he aimed at,
With the possible exception
of a set of measure zero.
"This," they said, "was rather doubtful;
Anyway it didn't matter.
What resulted in the long run:
Either he must hit the target
Much more often than at present,
Or himself would have to pay for
All the arrows he had wasted."
Hiawatha, in a temper,
Quoted parts of R. A. Fisher,
Quoted Yates and quoted Finney,
Quoted reams of Oscar Kempthorne,
Quoted Anderson and Bancroft
(practically in extenso)
Trying to impress upon them
That what actually mattered
Was to estimate the error.
Several of them admitted:
"Such a thing might have its uses;
Still," they said, "he would do better
If he shot a little straighter."
Hiawatha, to convince them,
Organized a shooting contest.
Laid out in the proper manner
Of designs experimental
Recommended in the textbooks,
Mainly used for tasting tea
(but sometimes used in other cases)
Used factorial arrangements
And the theory of Galois,
Got a nicely balanced layout
And successfully confounded
Second order interactions.
All the other tribal marksmen,
Ignorant benighted creatures
Of experimental setups,
Used their time of preparation
Putting in a lot of practice
Merely shooting at the target.
Thus it happened in the contest
That their scores were most impressive
With one solitary exception.
This, I hate to have to say it,
Was the score of Hiawatha,
Who as usual shot his arrows,
Shot them with great strength and swiftness,
Managing to be unbiased,
Not however with a salvo
Managing to hit the target.
"There!" they said to Hiawatha,
"That is what we all expected."
Hiawatha, nothing daunted,
Called for pen and called for paper.
But analysis of variance
Finally produced the figures
Showing beyond all peradventure,
Everybody else was biased.
And the variance components
Did not differ from each other's,
Or from Hiawatha's.
(This last point it might be mentioned,
Would have been much more convincing
If he hadn't been compelled to
Estimate his own components
From experimental plots on
Which the values all were missing.)
Still they couldn't understand it,
So they couldn't raise objections.
(Which is what so often happens
with analysis of variance.)
All the same his fellow tribesmen,
Ignorant benighted heathens,
Took away his bow and arrows,
Said that though my Hiawatha
Was a brilliant statistician,
He was useless as a bowman.
As for variance components
Several of the more outspoken
Make primeval observations
Hurtful of the finer feelings
Even of the statistician.
In a corner of the forest
Sits alone my Hiawatha
Permanently cogitating
On the normal law of errors.
Wondering in idle moments
If perhaps increased precision
Might perhaps be sometimes better
Even at the cost of bias,
If one could thereby now and then
Register upon a target.
W. E. Mientka, "Professor Leo Moser -- Reflections of a Visit"
American Mathematical Monthly, Vol. 79, Number 6 (June-July, 1972)
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A mathematician named Klein
Thought the Mobius Band was divine.
Said he, "If you glue
The edges of two
You get a weird bottle like mine."
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A challenge for many long ages
Had baffled the savants and sages.
Yet at last came the light:
Seems old Fermat was right--
To the margin add 200 pages.
-- Paul Chernoff
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_There Once Was a Breathy Baboon_ by Sir Arthur Eddington
There once was a breathy baboon
Who always breathed down a bassoon,
For he said, "It appears
That in billions of years
I shall certainly hit on a tune."
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
=2.5 QUOTES
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From: ph2008@mail.bris.ac.uk (CJ. Bradfield)philosophy:
A few of my favourite quotes about mathematics:
"A mathematician is a blind man in a dark room looking for a black cat
which isn't there" - Charles Darwin
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"A person who can, within a year, solve x^2 - 92y^2 = 1 is a mathematician."
-- Brahmagupta
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Anyone who cannot cope with mathematics is not fully human. At best he
is a tolerable subhuman who has learned to wear shoes, bathe and not
make messes in the house. -- Lazarus Long, "Time Enough for Love"
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Sex is the mathematics urge sublimated. -- M. C. Reed.
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"The good Christian should beware of mathematicians and all those who
make empty prophecies. The danger already exists that mathematicians
have made a covenant with the devil to darken the spirit and confine
man in the bonds of Hell." -- St. Augustine
P.S. Augustine did really say that, but in his time there was no difference
between mathematicans and astrologists. Astrologists told the future,
which was diabolic.
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As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to reality.
-- Albert Einstein
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Mathematics contains much that will neither hurt one if one does not know
it nor help one if one does know it. - J.B. Mencken
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
--
Joachim Verhagen Email:J.C.D.Verhagen@fys.ruu.nl
Department of molecular biofysics, University of Utrecht
Utrecht, The Netherlands.