“A mathematician is a device for turning coffee into theorems.” (P. Erdös)
So while you’re waiting for the coffee to take effect, look through these… [Compiled without attribution from lots of sources. If you know of an original source for any of these, please let us know.]
Quickies
Q. Why did the chicken cross the Möbius strip?
A. To get to the other – er…
Q. What does a mathematician do when he’s constipated?
A. He works it out with a pencil.
Q. What’s the value of a contour integral around Western Europe?
A. Zero, because all the Poles are in Eastern Europe.
An English mathematician was asked by his very religious colleague:
Q. Do you believe in one God?
A. Yes – up to isomorphism.
Q. What’s purple and commutes?
A. An abelian grape.
Q. What’s yellow, and equivalent to the Axiom of Choice?
Alternatively: Q. What’s yellow and pro-choice?
A. Zorn’s Lemon.
Q. Why did the mathematician name his dog “Cauchy”?
A. Because he left a residue at every pole.
Q. Why is it that the more accuracy you demand from an interpolation function, the more expensive it becomes to compute?
A. That’s the Law of Spline Demand.
Q. What’s nonorientable and lives in the sea?
A. Möbius Dick.
Q. What is an ‘ugh’?
A. The dual of a cough.
Q. Why didn’t Newton discover group theory?
A. Because he wasn’t Abel.
Asked how his pet parrot died, the mathematician answered, “Polynomial. Polygon.”
Lumberjacks make good musicians because of their natural logarithms.
Russell to Whitehead: “My Gödel is killing me!”
Did you hear about the geometer who went sunbathing and became a tangent?
My geometry teacher was sometimes acute, and sometimes obtuse, but always right.
Old mathematicians never die; they just lose some of their functions.
Statisticians probably do it.
Algebraists do it in groups.
(Logicians do it) or [not (logicians do it)].
Möbius always did it on the same side.
Remember Americans pronounce z ‘zee’.
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 right!
A little longer
Three men are in a hot-air balloon. Soon, they find themselves lost in a canyon. 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.
Fifteen 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?” “For three reasons. One, he took a long time to answer; two, he was absolutely correct, and three, his answer was absolutely useless.”
A bunch of Polish scientists decided to flee their repressive government by hijacking an airliner and forcing the pilot to fly them to a western country. They drove to the airport, forced their way on board a large passenger jet, and found there was no pilot on board. Terrified, they listened as the airport sirens rang out. Finally, one of the scientists suggested that since he was an experimentalist, he would try to fly the aircraft.
He sat down at the controls and tried to figure them out. The sirens got louder and louder. Armed men surrounded the jet. The would-be pilot’s friends cried out, “Please, please take off now! Hurry!” The experimentalist calmly replied, “Have patience. I’m just a simple Pole in a complex plane.”
Noah’s Ark lands after The Flood and Noah releases all the animals, saying, “Go forth and multiply.” Several months pass and Noah decides to check up on the animals. All are doing fine except a pair of snakes. “What’s the problem?” asks Noah. “Cut down some trees and let us live there,” say the snakes. Noah follows their advice. Several more weeks pass and Noah checks up on the snakes again. He sees lots of little snakes; everybody is happy. Noah says, “So tell me how the trees helped.” “Certainly,” reply the snakes. “We’re adders, and we need logs to multiply.”
Two male mathematicians are in a café. The first one says to the second that the average person knows very little about basic mathematics. The second mathematician disagrees, and claims that most people can cope with a reasonable amount of maths. The first goes off to the toilets, and in his absence his companion calls over the waitress.
He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer one third x cubed. She repeats, “One thir – dex cue”? He repeats, “One third x cubed”. “One thir dex cubed?” Yes, that’s right, he says. So she agrees, and goes off mumbling to herself, “One thir dex cubed…”
The first guy returns and the second proposes a bet to prove his point that most people do know something about basic maths. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks, “What is the integral of x squared?”.
As instructed, the waitress says “One third x cubed,” and while walking away, turns back and adds over her shoulder, “Plus a constant.”
Dubious mathematics
1+1=3, for large values of 1 and small values of 3.
lim ---- lim 3 = 8 8->9 \/ 8 = 3 w->oo
sin x lim ------- = 6 n->oo n
Proof: cancel the n in the numerator and denominator.
Lemma: All horses are the same colour.
Proof:
Base case: It is immediate that all horses in a set containing only one horse are the same colour.
Inductive step: Suppose you have a set of k+1 horses. Remove one horse from the set, so that you have k horses. The inductive hypothesis tells us that all these horses are the same colour. Now return the horse you removed and take out a different horse. Again, by the inductive hypothesis, the remaining k horses are all the same colour. Thus all k+1 horses you started with are the same colour.
The result follows by induction. QED.
Theorem: All horses have infinitely many legs.
Proof (i): Everyone would agree that all horses have an even number of legs. It is also well-known that horses have forelegs in front and two legs at the back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a horse to have! So, since we have shown the number of legs on a horse to be both even and odd, there must be infinitely many of them. QED.
Proof (ii): Suppose, for a contradiction, that there exists a horse which does not have infinitely many legs. That would be a horse of another colour; so by the above Lemma, it doesn’t exist. QED.
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. QED.
Theorem: All positive integers are equal.
Proof: It suffices to show that, for any two positive integers a and b, we have a=b; to demonstrate this, we show that, for all positive integers n, if for any two positive integers a and b we have max(a,b)=n then a=b. We proceed by induction.
Base case: If n=1 then a and b, being positive integers, must both equal 1, so a=b.
Inductive step: Assume that the theorem is true for some value k. Take positive integers a and b with max(a,b)=k+1. Then max(a-1,b-1)=k so, by the inductive hypothesis, a-1=b-1; consequently a=b. QED.
The great and the good
John von Neumann supposedly had the habit of simply writing answers to homework assignments on the board (the method of solution being, of course, obvious) when he was asked how to solve problems. Once, one of his students tried to get more helpful information by asking if there was another way to solve the problem. Von Neumann looked blank for a moment, thought, and then answered, “Yes.”
Norbert Wiener was renowned for his absent-mindedness. When he and his family moved from Cambridge to Newton his wife, knowing that he would be of absolutely no help, packed him off to MIT while she directed the move. Since she was certain that he would forget that they had moved and where they had moved to, she wrote down the new address on a piece of paper, and gave it to him. Naturally, in the course of the day, some insight occurred to him. He reached in his pocket, found a piece of paper on which he furiously scribbled some notes, thought it over, decided there was a fallacy in his idea, and threw the piece of paper away.
At the end of the day he went home – to the old address in Cambridge, of course. When he got there he realised that they had moved, that he had no idea where they had moved to, and that the piece of paper with the address was long gone. Fortunately inspiration struck. There was a young girl on the street and he conceived the idea of asking her where he had moved to, saying, “Excuse me, perhaps you know me. I’m Norbert Wiener and we’ve just moved. Would you know where we’ve moved to?” To which the young girl replied, “Yes Daddy, Mommy thought you would forget.”
The great Polish mathematician Waclaw Sierpinski was coincidentally also absent-minded and coincidentally also had to move house. His wife knew of his fallibility as they stood on the street with all their belongings, said to him, “Now, you stand here and watch our ten cases, while I go and get a taxi.” She left him there, eyes glazed and humming absently. Some minutes later she returned, a taxi having been called. Sierpinski challenged her (possibly with a glint in his eye): “I thought you said there were ten cases, but I’ve only counted to nine.” His wife insisted there were ten. “No, count them,” replied Sierpinski, “0, 1, 2, …”
The great logician Bertrand Russell once claimed that he could prove anything if given that 1+1=1. So one day, an undergraduate demanded: “Prove that you’re the Pope.” Russell thought for a while and proclaimed, “I am one. The Pope is one. Therefore, the Pope and I are one.”
Hiawatha Designs an Experiment
Kendall, Maurice (1959)
The American Statistician 13: 23-24
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
Made primaeval 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.
The Zeta Function Song
[Sung to the tune of “Sweet Betsy from Pike”]
Where are the zeros of zeta of s?
G. F. B. Riemann has made a good guess,
They’re all on the critical line, said he,
And their density’s one over 2 pi log t.
This statement of Riemann’s has been like a trigger,
And many good men, with vim and with vigour,
Have attempted to find, with math’matical rigour,
What happens to zeta as mod t gets bigger.
The names of Landau and Bohr and Cramèr,
And Hardy and Littlewood and Titchmarsh are there,
In spite of their efforts and skill and finesse,
In locating the zeros no-one’s had success.
In 1914 G. H. Hardy did find,
An infinite number that lay on the line,
His theorem, however, won’t rule out the case,
That there might be a zero at some other place.
Let P be the function pi minus li,
The order of P is not known for x high,
If square root of x times log x we could show,
Then Riemann’s conjecture would surely be so.
Related to this is another enigma,
Concerning the Lindelhof function mu(sigma)
Which measures the growth in the critical strip,
And on the number of zeros it gives us a grip.
But nobody knows how this function behaves,
Convexity tells us it can have no waves,
Lindelhof said that the shape of its graph,
Is constant when sigma is more than one half.
Oh, where are the zeros of zeta of s?
We must know exactly, we cannot just guess,
In order to strengthen the prime number theorem,
The path of integration must not get too near ’em.
Impure Mathematics: the cautionary tale of Polly Nomial
Once upon a time (1/t) pretty little Polly Nomial was strolling across a field of vectors when she came to the boundary of a singularly large matrix. Now Polly was convergent, and her mother had made it an absolute condition that she must never enter such an array without her brackets on. Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the basis that it was insufficient and made her way in amongst the complex elements.
Rows and columns closed in on her from all sides. Tangents approached her surface. She became tensor and tensor. Quite suddenly two branches of a hyperbola touched her at a single point. She oscillated violently, lost all sense of directrix, and went completely divergent. She tripped over a square root that was protruding from the erf and plunged headlong down a steep gradient. When she rounded off once more, she found herself inverted, apparently alone, in a non-Euclidean space.
She was being watched, however. That smooth operator, Curly Pi, was lurking inner product. As his eyes devoured her curvilinear coordinates, a singular expression crossed his face. He wondered, “Was she still convergent?” He decided to integrate properly at once.
Hearing a common fraction behind her, Polly rotated and saw Curly Pi approaching with his power series extrapolated. She could see at once by his degenerate conic and dissipative that he was bent on no good.
“Arcsinh,” she gasped.
“Ho, ho,” he said, “What a symmetric little asymptote you have. I can see your angles have lots of secs.”
“Oh sir,” she protested, “Keep away from me. I haven’t got my brackets on.”
“Calm yourself, my dear,” said our suave operator, “your fears are purely imaginary.”
“I, I,” she thought, “perhaps he’s not normal but homologous.”
“What order are you?” the brute demanded.
“Seventeen,” replied Polly.
Curly leered. “I suppose you’ve never been operated on.”
“Of course not,” Polly replied quite properly, “I’m absolutely convergent.”
“Come, come,” said Curly, “let’s go to a decimal place I know and I’ll take you to the limit.”
“Never,” gasped Polly.
“Abscissa,” he swore, using the vilest oath he knew. His patience was gone. Coshing her over the coefficient with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places, and began smoothing out her points of inflection. Poor Polly. The algorithmic method was now her only hope. She felt his digits tending to her asymptotic limit. Her convergence would soon be gone forever.
There was no mercy, for Curly was a Heaviside operator. Curly’s radius squared itself; Polly’s loci quivered. He integrated by parts. He integrated by partial fractions. After he cofactored, he performed Runge-Kutta on her. The complex beast even went all the way around and did a contour integration. What an indignity – to be multiply connected on her first integration. Curly went on operating until he completely satisfied her hypothesis, then he exponentiated and became completely orthogonal.
When Polly got home that night, her mother noticed that she was no longer piecewise continuous, but had been truncated in several places. But it was too late to differentiate now. As the months went by, Polly’s denominator increased monotonically. Finally she went to l’Hôpital and generated a small but pathological function which left surds all over the place and drove Polly to deviation.
The moral of our sad story is this: If you want to keep your expressions convergent, never allow them a single degree of freedom.
The only dirty joke in maths
Q. What’s the square root of 69?
A. Eight something.