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Q: How many mathematicians does it take to change a lightbulb?
A: $0.999999 \ldots$

Banach-Tarski Paradox

Q: Give me an anagram of Banach-Tarski.
A: Banach-Tarski Banach-Tarski.

Educational Standards

Two captains of industry, Arthur and George, were in a restaurant discussing the state of educational standards, particularly in the field of mathematics. Arthur was convinced they were slipping badly, and that your average college student was completely mathematically illiterate. George, on the other hand, was confident that any student would at least know the basics of calculus.

"I bet you a hundred bucks," said Arthur, "that if you were to ask a random college student a basic question in calculus, he wouldn't understand the question, let alone furnish you with an answer."

"I'll think about that," said George. "Not sure whether to take you up on your bet or not, but I reckon you'd be wrong."

Arthur slipped off to the mens' room at that point, and while he was gone, George called over the waitress Jody. (He knew that was her name because it was written on a badge pinned to her uniform. This appears to be a custom in certain chain diners.)

"I'd like you to help settle a wager between me and my colleague," he said. "When he comes back, I'm going to call you over, and ask you a question, to which you are to answer: one third x cubed."

"Wuntur dex cue?"

"One third x cubed."

"One thurrd ex cuebd."

"That's it, one third x cubed."

"One third ... x cubed."

"That's it, perfect. There's a good tip in it for you."

Arthur returned. George said, "Yes, I think I will take you up on it. A hundred bucks says our waitress can answer such a question. Hey, Jody! What's the indefinite integral of x squared with respect to x?"

"One third x cubed," replied Jody, dutifully.

"You see?" said George, pocketing Arthur's hundred.

As Jody turned away, she called back over her shoulder, "Plus a constant."

George ruefully took Arthur's hundred back out of his pocket and dropped it onto the table.

Pythagoras's Theorem

Once upon a time there were three ladies of the First Peoples of America sitting around the campfire.

On a reindeer skin sat a lady who was the mother of a fine young warrior who weighed $140$ pounds.

On a buffalo skin sat a lady who was the mother of a fine young warrior who weighed $160$ pounds.

The third lady, as well she might, was sitting on the skin of a hippopotamus, as she herself weighed a mighty $300$ pounds.

As you can see:

The squaw on the hippopotamus is equal to the sons of the squaws on the other two hides.

Knot Theory

Student A: "What's your favourite area of mathematics?"
Student B: "Knot theory."
Student A: "Me neither."
-- 1994: Colin C. Adams: The Knot Book: Knot Jokes and Pastimes (attributed to Martin Scharlemann)

Circle Geometry

The roundest knight at King Arthur's round table was Sir Cumference.

He acquired his shape from too much pi.

Sufficiently Large

$1+1 = 3$, for sufficiently large values of $1$.

Why? Because $1.4 + 1.4 = 2.8$.

The result follows after rounding to the nearest integer.

Number Bases

Why do mathematicians get Halloween and Christmas confused?

Because $\mathsf{Dec} \ 25$ equals $\mathsf{Oct} \ 31$.

-- 2009: Ian Stewart: Professor Stewart's Hoard of Mathematical Treasures: Halloween $=$ Christmas (but it's a considerably older joke than that.)


There are $10$ sorts of people in the world: those who understand binary and those who don't.


There are $10$ sorts of people in the world: those who understand ternary, those who don't, and those who thought this was going to be the binary joke.



Only dead people and me understand hexadecimal. So, how many people understand hexadecimal?

deae people.

(And me too. That's deaf people.)


There are $10$ sorts of people in the world: those who understand hexadecimal, and F the rest.

... and just plain innumeracy

There are three sorts of people in the world: those who can count, and those who can't.

... and while we're about it ...

There are $2$ types of people in the world. One type can extrapolate from the available data

Average Number of Hands

Most people in the world have more than the average number of hands.


\(\displaystyle \) \(\) \(\displaystyle \frac {\sin x} {\mathrm n}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \mathrm {si} \ x \frac{\mathrm n} {\mathrm n}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 6\) $\quad$ $\quad$

The proof that $\mathrm u \left({x \, \mathrm u! \, s}\right)^{-1} = 9$ is left as an exercise for the reader.

Even more Six

\(\displaystyle \dfrac{9 \left({8 - x}\right)} {\left({9 - 8}\right) x} + \dfrac{8 - x} {9 - 8} + \dfrac{11 - x} {x - 1}\) \(=\) \(\displaystyle x\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle 6\) $\quad$ $\quad$

What happens when you rotate the above through $180^\circ$?

\(\displaystyle 9\) \(=\) \(\displaystyle x\) \(\displaystyle \impliedby\) $\quad$ $\quad$
\(\displaystyle x\) \(=\) \(\displaystyle \frac{1 - x} {x - 11} + \frac{8 - 6} {x - 8} + \frac{x \left({8 - 6}\right)} {\left({x - 8}\right) 6}\) $\quad$ $\quad$

All Odd Numbers Are Prime

Proof by inductive argument:

$1$, that's prime (well not technically, but that's just mathematical double-talk).
$3$, that's prime.
$5$, that's prime.
$7$, that's prime.
$9$, that's prime (although when I measured it, it looked like it might not be - experimental error, ignore that one)
$11$, that's prime.
$13$, that's prime.

We can extrapolate from there.

Black Friday customer proof:

$1$, that's prime, it's obvious it is, no argument there. I said, no argument there.
$3$, that's prime.
$5$, that's prime.
$7$, that's prime.
$9$, that's prime YES IT IS -- DON'T ARGUE! YOU ARE STUPID!
$11$, that's prime.
$13$, that's prime.

Any more slackwit stupid people out there want to argue wit' me?


Hear about the constipated mathematician?

He worked out logs with a pencil.

Axiom of Intuition

The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
-- Jerry Bona

Log Cabin

\(\displaystyle \int \frac 1 {\text{cabin} } \rd \, \text{cabin}\) \(=\) \(\displaystyle \ln \text{cabin} + C\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \text{houseboat}\) $\quad$ $\quad$
-- Can apparently be found in Gravity's Rainbow by Thomas Pynchon


Q: How many Bourbakists does it take to change a lightbulb?
A: Changing a lightbulb is a special case of a more general theorem concerning the maintenance and repair of an electrical system. To establish upper and lower bounds for the number of personnel required, we must determine whether the sufficient conditions of Lemma $2.1$ (Availability of personnel) and those of Corollary $2.3.55$ (Motivation of personnel) apply. If these conditions are met, we derive the result by an application of the theorems in Section $3.11.23$. The resulting upper bound is, clearly, a result in an abstract measure space, in the weak-$*$ topology.


Q: How many lightbulbs does it take to change a lightbulb?
A: One, if it knows its own Gödel number.
-- 2005: P. Renteln and A. DundesFoolproof: A Sampling of Mathematical Folk Humor (Notices of the AMS Vol. 52no. 1

Principia Mathematica

Lyrics: Colin Fine
Music: Burt Bacharach

What do you get if you take a set
Add an associative operation
Give it an identity, make everything invertible?
A-a-ah, a-a-a-a-a-a-ah, it's a group
A-a-a-a-ah, it's a group.
What do you get if you take a group
Add an associative operation
Make it distributive over the first one?
A-a-ah, a-a-a-a-a-a-ah, it's a ring
A-a-a-a-ah, it's a ring.
Don't tell me it's too hard for you
Cos I can prove it so I know it's true
Out of a handful of simple axioms
I can build mathematics to satisfaction
What do you get if you take a ring
Make the multiply commutative
Give a reciprocal for each non-zero element?
A-a-ah, a-a-a-a-a-a-ah, it's a field
A-a-a-a-ah, it's a field.
(And not a skew field either!)
What do you get if you take a field
And you take a group with a scalar multiply
Make it associative with the field multiply
A-a-ah, a-a-a-a-a-a-ah, it's a vector space
A-a-a-a-ah, it's a vector space.
You can go on like this all day
Building structures in this kind of way
You end up feeling pretty cocksure
When you get into categories with meta-structure!
What do you get with some proofs and rules
Some axioms to give you a formal system?
You try to prove that it's consistent,
A-a-ah, a-a-a-a-a-a-ah ...??
Godel's Theorem!

In prime mover's defence, he remembers meeting Colin Fine at one or two SF conventions in the 1980's. He recently found this poem, a copy of which he got hold of in approximately 1987, lurking near the bottom of a pile of old magazines.

Mathematical Double-talk

[Then he] proved that an automorphic resonance field has a semi-infinite number of irresolute prime ideals.
-- Colin Fine again, as it appears in Pyramids by Terry Pratchett. The above line was the result of a specific request for an example of plausible-sounding mathematical double-talk.

Medical Conditions

Ring epimorphism? I had one of them once. The doctor had to give me suppositories.
-- Mrs. Prime.mover.

Imaginary numbers

After eating too much food, the mathematician announced: "$\sqrt {\left({-1 / 64}\right)}$."

-- Kit Yates, via Twitter

A constant argument

$i$: "Be rational."
$\pi$: "Get real!"


Three logicians walk into a pub.

The barmaid asks: "Are you all having beer?"

The first logician replies: "I don't know."

The second logician replies: "I don't know."

The third logician replies: "Yes."

-- Peter Rowlett, via Kit Yates, via Twitter

More beer

$\aleph_0$ mathematicians walk into a bar. The first one orders a pint of beer. The second one orders half a pint of beer. The third orders a quarter of a pint of beer. The fourth one orders an eighth of a pint of beer. After hearing the seventh order, the barman pours two pints, and says, "You guys should know your limits."

Surreal variant

$\aleph_0$ mathematicians walk into a bar. The first orders one beer. The second one orders two beers. The third one orders three beers. The bartender stops them, says: "You guys are idiots," and pours out $-\dfrac 1 {12}$ of a beer.


What goes: $3.1415 \text{baa}$?

Shepherd's pi.

What is the volume of a pizza whose radius is $z$ and whose thickness is $a$?

From Volume of Cylinder: $\text{pi} z z a$.

More $\pi$

I've memorised all the digits of pi. Now I just have to remember what order they come in.
-- (Traditional)

Proof by Contradiction

A mad scientist captures a mathematician and locks him in a room full of cans of food - but no can opener.

Checking on the cell several weeks later, the mathematician is lying dead, but he has written one last message on the dust on the floor:


If I cannot open these cans of food, I will die.


Suppose not.


The Mathematicians' Party

Once upon a time, all the mathematicians who had ever lived attended a great big party, in order to let their hair down and enjoy themselves for once.

Plenty of physicists attended, and quite a few chemists and biologists came too.

Our roving reporter was at the door, and these are some of the things he observed ...

Click on the link in the title of this section to go to the party yourself.

Optimists versus Pessimists

Consider the real interval:

$\mathbb I := \left[{a \,.\,.\, b}\right)$

An optimist regards $\mathbb I$ as half-open.

A pessimist regards $\mathbb I$ as half-closed.

Author Prank

I have started covering a book (Introduction to Boolean Algebras) whose authors state:

"The verification that $A$ (a Ring of Idempotents, LF) becomes a Boolean ring in this way is an amusing exercise in ring axiomatics.", p.5

It's indeed most enjoyable to watch people write out $\left({x \oplus y}\right) \circ \left({x \oplus y}\right) = x \oplus y$...

Möbius Strip

I had a fight with a Möbius strip (or Möbius band for those of a prurient mentality).

I'm ashamed to say I lost. It was completely one-sided.

Amusing names

Is there an Emma Lehmer Lemma?


Let $f = \displaystyle \sum_{i \mathop = 0}^n a_i x^i$ be a polynomial.

If the polynomial coefficient $a_n$ of $f$ is $1$, then $f$ is monic

If the polynomial coefficient $a_{n-1}$ of $f$ is $0$, then $f$ is depressed.

So, a polynomial of the form:

$f = x^n + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0$

can be called monic depressive?


A woman sends her logician husband to the shops. "Get me a loaf of bread," she said, "and if they have eggs, get me a dozen."

The husband returns from the shop with twelve loaves of bread.

Yes or no?

A woman has a baby, and the midwife immediately hands it to her logician husband.

"Well?" says the woman, "is it a boy or a girl?"

"Yes," he replies.

The Evils of Drink

Booze and calculus don't mix.

Don't drink and derive.


A visiting Professor of Linguistics was delivering a lecture.

"In the grammars of many languages throughout the world, a double negative expresses a positive. On the other hand there are some languages, such as Russian and the English of Chaucer, in which a double negative remains a negative. However, there is not a language in the world in which a double positive can express a negative."

A voice from the back of the room piped up: "Yeah, right."

-- Taken from the Facebook page of George Takei on 9th March 2014, but has been around for considerably longer than that.

Rhetorical Questions

Q: What do you get when you cross a joke with a rhetorical question?
A: Quite.

Physicist Mathematician and Engineer Jokes

The Red Rubber Ball

A physicist, mathematician and engineer were asked to determine the volume of a red rubber ball.

The physicist filled a beaker with water, immersed the ball, caught the runoff and measured its volume.

The mathematician set up and solved an appropriate triple integral, then measured the diameter of the ball and plugged in the number.

The engineer looked around on his bookshelf, then asked, "Has anyone got a red rubber ball volume table? I've only got the ones for blue and purple."

The Burning Hotel

The engineer is awakened by a smell and gets up to check it. He finds a fire in the hallway, sees a nearby fire extinguisher and after extinguishing it, goes back to bed.

Later that night, the physicist gets up, again because of the smell of fire. He quickly gets up and sees the fire in the hallway. After calculating air pressure, flame temperature and humidity as well as distance to the fire and projected trajectory, he extinguishes the fire with the least amount of fluid.

Variant 1

Finally, the mathematician awakes, only again to find a fire in the hallway. He instantly sees the extinguisher and thinks, "A solution exists!", and heads back into his room.

Variant 2

The punchline has been left as an exercise for the reader.

Variant 3

The mathematician awakens, and finds another fire in the hallway. He looks out the door, then goes back to bed. The house ends up burning down, but the physicist and engineer manages to save the mathematician. When asked why he didn't put out the fire, he says: "I saw the fire, I saw the extinguisher, the solution was trivial."

Variant 4

Then the mathematician awakens, and finds that the embers of the fire are still burning. After giving much thought to the problem, he gets up and lights it up to an actual fire. Then he goes back to sleep, satisfied that the problem has been reduced to a previously solved one.

Variant 5

Then the mathematician awakens, and finds another fire in the hallway. He quickly tears pages out of his notebook, lighting them on fire one by one. He then runs down the hall sliding sheets of burning paper under the other guests' doors.

After the building burns to the ground the fire marshal asks the mathematicians how the fire spread so fast.

He responds: "I thought distributing the problem would lead to finding a solution faster."


$\dfrac {12 + 144 + 20 + 3 \sqrt 4} 7 + \left({5 \times 11}\right) = 9^2 + 0$
A dozen, a gross, and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
Is nine squared and not a bit more.
-- Leigh Mercer

My cat, mathematically-trained,
Says "Your topology's too coarse-grained,
Quantum mechanics
Sends you into blind panics
Because you're not well-enough brained."

Three thousand, four hundred and sixty
Five million, six hundred and fifty
Three thousand, six hun-
Dred and seventy one
Point four seven five six one three
-- Unknown attribution

Physics Jokes


"We don't serve faster-than-light particles in here," says the barman.

A tachyon goes into a bar.

Quantum Mechanics

Heisenberg and Schrödinger in a car speeding down the freeway. Predictably, they are stopped by a traffic policeman.

"Do you know how fast you were going?" asked the cop.

"No, but I know exactly where I was," replied Heisenberg.

"You were actually travelling at 85 miles per hour," admonished the cop, sternly.

"Oh great," replied Heisenberg, angrily. "Now I'm lost!"

Deciding to give the car an inspection, the cop opens the trunk.

"Did you know you've got a dead cat in here?" he asks.

"Well, I do now!" replied Schrödinger.

Computer Science Jokes

Natural Numbers

Q: "Why do computer scientists have nine fingers?"
A: "Zero, one, two, three, four; five, six, seven, eight, nine."

Computer Prayer

Our Program which art in Memory,
"Hello World!" be Thy Name.
Thy Operating System come, Thy Commands be done,
at the Printer as it is on the Screen.
Give us this day our Data Dump,
and forgive us our I/O Errors
as we backup those whose Files are faulty.
Lead us not into frustration, and deliver us from email
for Thine is the Algorithm, the Application, and the Solution,
looping forever and ever.

Computer Encoding

Link to a ROT26 encoder:

Computer Programming

There are only two really difficult things in computer programming: cache invalidation, naming things, and off-by-one errors.

Law of Small Numbers

Everything is fast for small values of $n$.

Diophantus Updated

Amanda is $21$ years older than her son John.

In $6$ years from now, Amanda will be $5$ times as old as John.

Where is John's father?

Let $M$ be the age in years of Amanda now.

Let $C$ be the age in years of John now.


\(\displaystyle M\) \(=\) \(\displaystyle C + 21\) $\quad$ $\quad$
\(\displaystyle M + 6\) \(=\) \(\displaystyle 5 \left({C + 6}\right)\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle C + 21 + 6\) \(=\) \(\displaystyle 5 \left({C + 6}\right)\) $\quad$ substituting for $M$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle C + 27\) \(=\) \(\displaystyle 5 C + 30\) $\quad$ simplifying $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle -3\) \(=\) \(\displaystyle 4 C\) $\quad$ subtracting $C + 30$ from both sides $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle C\) \(=\) \(\displaystyle -\frac 3 4\) $\quad$ $\quad$

So the child is $-\dfrac 3 4$ years old, that is, $-9$ months.

That is, the child, will be born in $9$ months time.

So, right now, John's father is inside Amanda.


Q: What goes: two bloody thirds?
A: A vulgar fraction.


An engineer, physicist and mathematician visit a farm.

The farmer gives them a challenge to enclose his sheep with the smallest amount of fence possible.

The engineer starts and makes a circle, declaring: "A circle has the greatest area compared to the circumference".

Next goes the physicist. He extends the fence infinitely far before reducing the fence, forcing the sheep next to one another before declaring "This circle is as small as it can get."

Lastly goes the mathematician, who has pondered upon it all. He builds a small fence around himself before declaring: "I define myself as on the outside."

Word Problems

Pat's first day in school:

Teacher: Pat, if I give you two apples, and then I give you another apple, how many apples will you have?
Pat: I'll have four apples, Miz.
Teacher: No, it's three.
Pat: No, Miz, I'll have four.
Teacher: NO Pat, two plus one equals three -- what on earth makes you think it's four?
Pat: I already have an apple in my lunch box.


Q. What's a rectangular bear?
A. A polar bear after a coordinate transform.


Gödel's First Incompleteness Theorem: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and.

Cauchy Condescension Test

It obviously converges, but I can't be bothered to explain why.


I'm going to this year's Fibonacci convention.
It'll be as big as the last two put together.


A horse walks into a bar and orders a beer. The bartender asks him, "Would you like that in a glass?" The horse replies, "I think not." And POOF! he disappears.
Philosophy students will no doubt find this joke hilarious, because René Descartes' most famous saying was I think therefore I am. But to tell you that before the rest of the joke would be putting Descartes before the horse.


Hardy-Weinberg Equation:

$p^2 + 2 p q + q^2 = 1$

Easy-Weinberg Equation:

$p = 1$

Wives and Mistresses

An engineer, a physicist and a mathematician were talking in the pub.

"It's a disaster," said the engineer. "My wife's just found out about my mistress. And after all that calculation -- to the last decimal place -- of times and places and schedules, and she goes and reads my notebook!"

"I completely understand," replied the physicist. "My mistress just found out about my wife -- she came to my lab and found the apparatus for the industrial diamonds I manufactured for our anniversary the other month -- like a fool I'd left the die for the engraving of our names entwined in a heart, and now the cat is out of the bag."

"Dunno what you're complaining about," said the mathematician complacently. "My wife and my mistress both know all about each other."

"But doesn't that cause all sorts of problems?" exclaimed the engineer.

"Surely that's a foolish situation!" opined the physicist.

"Not at all," said the mathematician insouciantly. "When I'm not with my wife, she thinks I'm with my mistress. And when I'm not with my mistress, she thinks I'm with my wife. But in fact I'm actually in the library getting some mathematics done."

Contour Integrals

What's the value of:

$\displaystyle \oint_{\mathcal C} \operatorname{Europe}\left({z}\right) \rd z$

where $\mathcal C$ is a closed contour circling Western Europe?

By the Residue Theorem, the integral is zero. The Poles are in Eastern Europe.

(Comment: They're not, actually, freedom of movement within the European Community has allowed for them to move at will throughout the continent and there are indeed many Poles in Western Europe. But they can be considered removable.)

That famous chicken

Why did the chicken cross the Möbius strip?

To get to the other ... -- no, wait ...

Separated at birth?

KarlWilhelmFeuerbach.jpg EvaristeGalois.jpg
Karl Wilhelm Feuerbach Évariste Galois

On meticulousness

Why should I have to "dot my "i"s and cross my "t"s?

I know that $\mathbf i \cdot \mathbf i = 1$ and $\mathbf t \times \mathbf t = \mathbf 0$.

-- Fargle

Scottish Sheep are Black

A trainload of scientists have just crossed the border from England into Scotland. (Substitute a frontier of your choice.)

Looking out of the window, an anthropologist sees a black sheep.

Anthropologist: "Ooh look! Sheep in Scotland are black."
Logician: "Well, at least some sheep in Scotland are black."
Chemist: "On this occasion, we have observed that the experiment resulted in a sheep which is black. Until we have repeated the experiment in strictly controlled conditions, we cannot be certain that it will always result in a black sheep."
Statistician: "Oh come on! The entirety of the sample space consists of sheep which are black. It's perfectly appropriate from that to deduce that, to a considerable degree of accuracy, that all sheep in Scotland are black."
Physicst: "No, you're talking rubbish. At this stage, all we can tell with any certainty is that Scotland contains at least one sheep which is black."
Mathematician: "Now let's be strictly accurate here. What we do know is that in Scotland there exists at least one sheep, and this sheep is black on at least one side."

Height and Length

Several engineers are trying to erect a flagpole, but are having a very difficult job of it.

Their directions were to cut the pole to a certain height. However, once the pole is standing straight up, they are having a hard time cutting the pole down to the correct height, by the awkwardness of using a saw on a ladder.

A mathematician walks by and asks what they're doing. They tell her. She helpfully (and a bit condescendingly) explains that it would be much easier to cut the pole to the right size before setting it upright.

As she walks away, the engineers look at each other, annoyed.

"Figures that a mathematician wouldn't know the difference between height and length."