The following jokes are mainly drawn from Dundes et al.'s *Foolproof: a sampling of mathematical folk humor*. Because jokes are always funnier once you've had them explained to you, each joke has an optional explanation.

What's purple and commutes?

An abelian grape.

In an *abelian group*, it doesn't matter what order you multiply elements together - they *commute*.

Grapes are sometimes purple.

What's hot, chunky, and acts on a polygon?

Dihedral soup.

The *dihedral groups* represent the symmetries of a regular polygon.

Soup is often chunky, and best served hot.

What's sour, yellow, and equivalent to the Axiom of Choice?

Zorn's Lemon.

The *Axiom of Choice* says that you can always pick a representative element from a non-empty set, no matter how it's described. You can create useful models of set theory with or without this axiom.

*Zorn's Lemma* states that a partially ordered set containing upper bounds for every totally ordered subset necessarily contains at least one maximal element. This is true if and only if the axiom of choice is true.

Lemons are sour and yellow.

What's the value of a contour integral around Western Europe?

Zero, because all the Poles are in Eastern Europe.

A *contour integral* integrates a function over a curve. Its value can be calculated just by looking at the direction of the contour around any *poles*.

Residents of Poland are called *Poles*.

What do you get if you cross an elephant and a banana?

\( \left\lvert \text{elephant} \right\rvert \times \left\lvert \text{banana} \right\rvert \times \sin(\theta) \)

The *cross product* finds a vector at right-angles to two given vectors. The cross product of vectors \(a\) and \(b\) is \(|a| \times |b| \sin(\theta)\), where \(\theta\) is the angle between the vectors.

What do you get if you cross a mosquito with a mountain climber?

You can't cross a vector with a scalar.

The cross product only works on two vectors, not scalars.

A mosquito is often a disease *vector*, while a mountain climber *scales* mountains.

What's a polar bear?

A rectangular bear after a coordinate transform.

Polar coordinates describe points using an angle and a distance from the origin. You can convert from rectangular to polar coordinates using a *transformation*.

What goes "Pieces of seven! Pieces of seven!"

A parroty error.

A *parity error* is when you're off by one.

It's a cliché that parrots say "Pieces of eight! Pieces of eight!"

Why can't you grow wheat in \( \mathbb{Z}/6\mathbb{Z} \)?

It's not a field.

A *field* is a set with defined addition, subtraction and multiplication operations. \( \mathbb{Z}/6\mathbb{Z} \) is the group of integers *modulo* 6. It's not a field because there's no way of defining division on it.

Wheat is grown in fields.

What's big, grey, and proves the uncountability of the reals?

Cantor's diagonal elephant.

*Cantor's diagonal argument* shows that the real numbers are not *countable* - they can't be arranged into a list like the counting numbers can.

Elephants are big and grey.

What's grey and huge and has integer coefficients?

An elephantine equation.

A *diophantine equation* is a polynomial equation where the coefficient of each term is an integer.

Elephants are huge and grey.

What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?

An antique tractorization domain.

The *fundamental theorem of arithmetic* states that every number greater than 1 has a unique factorization into prime numbers. A *unique factorisation domain* is a *ring* which also has this property.

Farmers use tractors.

Why are group theorists especially prone to sleeping sickness?

This disease comes from the Tietze fly.

*Tietze transforms* are operations which change the presentation of a group, without changing its structure.

Tsetse flies spread sleeping sickness.

What is black and white and fills space?

A piano curve.

The *Peano curve* is a kind of space-filling curve - it visits every point in 2D space, despite being a 1D curve.

Piano keyboards are black and white.

How many geometers does it take to screw in a lightbulb?

None. You can't do it with a straightedge and compass.

Euclidean geometry studies the constructions you can make with just a straightedge (ruler) and compass. For example, given a circle, you can't construct a square with the same area using only a straightedge and compass.

How many analysts does it take to screw in a lightbulb?

Three. One to prove existence, one to prove uniqueness, and one to derive a nonconstructive algorithm to do it.

Many proofs in functional analysis don't show explicitly how to construct an object with a certain property. Instead, they could prove just that the object *exists*, or that there's only one such object. Sometimes a proof will give an *algorithm* to produce an example, but it can't be carried out in finite time.

How many mathematicians does it take to screw in a lightbulb?

\(0.99999\ldots\)

It only takes one mathematician: \(0.99999\ldots = 1\).

Let \(x = 0.999\ldots\). Then \begin{align} 10x &= 9.999\ldots \\ 9x &= 9.999\ldots - 0.999\ldots \\ &= 9 \\ \therefore x &= 1 \end{align}

How many lightbulbs does it take to change a lightbulb?

One, if it knows its own Gödel number.

Kurt Gödel used *Gödel* numbers to encode propositions about numbers as numbers, and hence show that any theory of mathematics is either incomplete or inconsistent. The joke plays on the idea of self-reference.

Why did the chicken cross the road?

Erdős: It was forced to do so by the chicken-hole principle.

Many of Paul Erdős's proofs used the *pigeonhole principle* to show that all other possibilities for a proposition have been exhausted.

Why did the chicken cross the road?

Fermat: It did not fit on the margin on this side.

Pierre de Fermat famously wrote in the margin of one of his books, next to the statement of his Last Theorem, “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

Why did the chicken cross the Möbius strip?

To get the same side.

A *Möbius strip* is a kind of non-orientable surface, which only has one “side”.

What's an anagram of Banach Tarski?

Banach Tarski Banach Tarksi

The *Banach-Tarski paradox* uses the Axiom of Choice to rearrange the points on a sphere into two spheres of the same size as the original.

Why don't jokes work in base 8?

Because 7 10 11.

Base 8 is like the decimal system, but each digit is worth a power of 8 instead of a power of 10. The old joke "Why was 6 afraid of 7? Because 7 8 9" doesn't work in base 8, when 8 and 9 are written as 10 and 11 respectively.

Aleph-null bottles of beer on the wall,

Aleph-null bottles of beer,

You take one down, and pass it around,

Aleph-null bottles of beer on the wall.

\(\aleph_0\) (pronounced "aleph-null") is the *cardinality* of the natural numbers, i.e. "how many natural numbers there are". There are infinitely many natural numbers, so taking one away doesn't change the cardinality.

An engineer, a physicist, and a mathematician are staying in a hotel. The engineer wakes up and smells smoke. He goes out into the hallway and sees a fire, so he fills a trash can from his room with water and douses the fire. He goes back to bed. Later, the physicist wakes up and smells the smoke. He opens his door and sees a fire in the hallway. He walks down the hall to a fire hose and after calculating the flame velocity, distance, water pressure, trajectory, and so forth, extinguishes the fire with the minimum amount of water and energy needed. Later the mathematician wakes up and smells smoke. He goes to the hall, sees the fire and then the fire hose. He thinks for a moment and then exclaims, “Ah, a solution exists!” and then goes back to bed.

Mathematicians are more concerned with finding a solution than actually applying it.

A mathematician and an engineer are on a desert island. They find two palm trees with one coconut each. The engineer shinnies up one tree, gets the coconut, and eats it. The mathematician shinnies up the other tree, gets the coconut, climbs the other tree and puts it there. “Now we’ve reduced it to a problem we know how to solve.”

Rather than giving an argument from first principles every time, if a mathematician can restate a problem in terms of problems they already know how to solve, their work is done.

A mathematician, a physicist, and an engineer were traveling through Scotland when they saw a black sheep through the window of the train. “Aha,” says the engineer, “I see that Scottish sheep are black.” “Hmm,” says the physicist, “You mean that some Scottish sheep are black.” “No,” says the mathematician, “All we know is that there is at least one sheep in Scotland, and that at least one side of that one sheep is black!”

Mathematicians are trained not to make logical leaps without justification.

A mathematics professor was lecturing to a class of students. As he wrote something on the board, he said to the class “Of course, this is immediately obvious.” Upon seeing the blank stares of the students, he turned back to contemplate what he had just written. He began to pace back and forth, deep in thought. After about 10 minutes, just as the silence was beginning to become uncomfortable, he brightened, turned to the class and said, “Yes, it IS obvious.”

What is "obvious" is a subjective judgement, and you can easily make the mistake of thinking something is obvious by missing a crucial detail.

Three statisticians went duck hunting. A duck flew out and the first statistician took a shot, the shot went a foot too hight. The second statistician took his shot and the shot went a foot too low. The third statistician said, "We got it!"

The mean of the two shots is exactly on target. The third statistician has forgotten that the mean of a sample is not always present in the sample.