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06.23.2008

e-Venn Stephen

Tonight’s episode of The Colbert Report was awesome.

Venn-gence

The episode included the following graphic

as a justification for why “America is the best, greatest country God has ever given Man on the face of the Earth,” because all topical humor is that much funnier with Venn diagrams.

Of interest is a relatively nice pair of descriptions of relative complements, including the assertions that

(it’s been coasting for years), and

although I am somewhat interested in knowing what elements exist in the set ( [ Best countries ] U [ Great countries ] ) \ [Earth ], although perhaps that was addressed in 1991 by Star Trek; I’ll have to check my references.

Logic

It’s also worth noting that Colbert asserts that Proof by repitition is an irrefutable fact of logic, which makes a fine contribution to the list of other commonly accepted proof techniques, including

Proof by vigorous handwaving.
Works well in a classroom or seminar setting.

Proof by forward reference.
Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.

Proof by funding.
How could three different government agencies be wrong?

Proof by example.
The author gives only the case n = 2 and suggests that it contains most of the ideas of the general proof.

Proof by omission.
Examples include: “The reader may easily supply the details,” or “The other 253 cases are analogous,” etc.

Proof by deferral.
“We’ll prove this later in the course.”

Proof by picture.
A more convincing form of proof by example. Combines well with proof by omission.

Proof by intimidation.
“Trivial.”

Proof by cumbersome notation.
Best done with access to at least four alphabets and special symbols.

Proof by exhaustion.
An issue or two of a journal devoted to your proof is useful.

Proof by obfuscation.
A long plotless sequence of true and/or meaningless syntactically related statements.

Proof by wishful citation.
The author cites the negation, converse, or generalization of a theorem from the literature to support his claims.

Proof by eminent authority.
“I saw Karp in the elevator and he said it was probably NP-complete.”

Proof by personal communication.
“Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication].”

Proof by reduction to the wrong problem.
“To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem.”

Proof by reference to inaccessible literature.
The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.

Proof by importance.
A large body of useful consequences all follow from the proposition in question.

Proof by accumulated evidence.
Long and diligent search has not revealed a counterexample.

Proof by cosmology.
The negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God.

Proof by mutual reference.
In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown to follow from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.

Proof by metaproof.
A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.

Proof by vehement assertion.
It is useful to have some kind of authority relation to the audience.

Proof by ghost reference.
Nothing even remotely resembling the cited theorem appears in the reference given.

Proof by semantic shift.
Some of the standard but inconvenient definitions are changed for the statement of the result.

Proof by appeal to intuition.
Cloud-shaped drawings frequently help here.

Cookies!

“Me have crazy times in 70s and 80s.  Me like the Robert Downey Jr. of cookies!” –Cookie Monster

This might possibly be my favorite quote ever.

Filed under: Math musings

05.29.2008

Innumeracy revised

Overheard at lunch:

I ordered the 10-piece chicken nuggets but they gave me the 6-piece nuggets instead.  I went up to complain and the guy gave me another 6-piece nugget instead of, like, the 2 pieces I was missing.  I guess he couldn’t count.

Filed under: Math musings, Anecdotes

05.23.2008

Link o’the week

Every number has an interesting property

I have often heard the story about G. H. Hardy’s meeting with S. Rumanujan at the hospital.  According to Hardy:

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Related to this is the following joke (which I have been told by students looking to get a bit of extra credit by showing their mastery of basic proof techniques):

Theorem: Every natural number has an interesting property.

Proof. Assume the contrary. Let A be the least natural number with no interesting property whatsoever. Well, that’s a pretty interesting property about A, so we have a contradiction. Q.E.D.

Logical errors aside, I often find myself looking at natural numbers and trying to find their “interesting” property.

Well, look no more.  Erich Friedman has conveniently listed interesting properties for the first ten thousand natural numbers, starting at 0.

Astute minds will note that ten thousand examples does not a proof make.  Well, fret not, ye of sharp mental acument.  Mike Keith (author of the astounding Cadaeic Cadenza) gives a neat, direct proof that every number has an interesting property by proving that every natural number is the only number to satisfy being expressed as a sum of a certain number of  powers in a certain number of ways.  Enjoy the cleverness!

Filed under: Math musings, l.o.t.w.

05.14.2008

Joke time!

Each semester, I offer students a last chance for extra credit by writing their favorite joke or riddle on their crib sheet, with extra credit assigned based purely on how funny I think it is. Here are some of the better ones.

Math contributions

Q: What’s the difference between an extrovert mathematician and an introvert mathematician?
A: The extrovert looks at your shoes when speaking to you.

Q: What kind of clothes does a mermaid wear to math class>
A: An algae bra.

Q: What’s the derivative of a cow?
A: Prime rib.

Q: What’s the shortest math joke?
A: “Let ε < 0."

Q: How do you pick up chicks in a Calculus II class?
A: "If you were f(x), then I’d like to be ∫f(x) dx so I could be the area under your curves.”

Q: What would you hear at a mathematicians dance party?
A: I draw the sine, by Ace of Log-base.

Q: If only DEAD people understand hex, how many people understand hex?
A: 57,005.

Why did Helen Keller struggle with math?
A: She was a woman.
(This is offensive in more ways than I can count.)

Test your personality! How many digits of π do you know?

In topological hell, beer comes in Klein bottles.

There are three types of mathematicians: those who can count and those who can’t.

Alcohol and calculus don’t mix! Never drink and derive!

Two mathematicians walked into a bar, but the third one ducked.

Two polynomials walk into a bar. The bartender, a derivative, asks them “Can I take you order?” The polynomials run out screaming “Help! The bartender threatened to kill me!”

A father is very much concerned about his young son Billy’s bad grades in math. In desparation, he decides to register Billy at a catholic school. After his first term there, Billy brings home his report card. And, lo and behold, he’s getting A’s in math. The father is, of course, pleased, but wants to know: “Why are your math grades suddenly so good?” Billy’s face goes a little white, and he replies, “when I walked into the classroom the first day, and I saw that guy on the wall nailed to a plus sign, I knew one thing: they meant business!”

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 sirens got louder. 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.”

A mathematician went insane and believed that he was the differentiation operator. His friends had him placed in a mental hospital until he got better. All day he would go around frightening the other patients by staring at them and saying “I differentiate you!” One day he met a new patient; and true to form he stared at him and said “I differentiate you!”, but for once, his victim’s expression didn’t change. Surprised, the mathematician marshalled his energies, stared fiercely at the new patient and said loudly “I differentiate you!”, but still the other man had no reaction. Finally, in frustration, the mathematician screamed out “I DIFFERENTIATE YOU!” The new patient calmly looked up and said, “I’m not scared of you. I’m ex.”

When NASA first started sending up astronauts, they quickly discovered that standard ball-point pens would not work in zero gravity. So to combat this problem, NASA engineers spent a decade and several million dollars to develop a pen that would write in zero gravity, upside down, under water, on almost any surface (including glass), and at temperatures ranging from below freezing to 300 degrees. The Russians were faced with the same dilemma as well, but found a different solution: they used a pencil.*

* I’ve always been a fan of this story, however untrue it is.

A computer scientist is selected for a job in Texas because of his amazing pattern-matching abilities. As he’s driving through the Texan desert, he gets incredibly hungry. so he finds a diner along this massive stretch of highway and stops there to eat. “I’ll have a plain cheeseburger,” he says to the waitress. She returns with a massive burger, a foot across, dripping with lettuce and onions and the works. The computer scientist insists there is a mistake, but the waitress replies with a smile: “Everything’s bigger in Texas.” So the guy eats the burger and finds he’s thirsty. “I’ll have a small Coke,” he says to the waitress. She returns with a massive 80-ounce caraffe of carbonated soda. The computer scientist insists there is a mistake, but the waitress replies with a smile: “Everything’s bigger in Texas.” So the guy drinks the drink, only to realize he needs to pee. Since he can’t find the waitress, he looks for clues: he follows some pipes along the ceiling down a hallway and out a door, where he falls into a swim ming pool. In a pattern matching-panic with the words of the waitress ringing in his mind, he cries “Don’t flush! Don’t flush!”

One day a farmer called up an engineer, a physicist, and a mathematician and asked them to fence of the largest possible area with the least amount of fence. The engineer made the fence in a circle and proclaimed that he had the most efficient design. The physicist pointed out that fencing off half of the Earth was certainly a more efficient way to do it. The mathematician just laughed at them. He built a tiny fence around himself and said “I declare myself to be on the outside.”

Other good ones

Q: What did the volcano say to his sweetheart?
A: I lava you.

Q: When does Michael Jackson go to bed?
A: When the big hand touches the little hand.

Two fish are in a tank. One turns to the other and says, “Hey, you know how to drive this thing?”

Two muffins are in an oven. One muffin says “We gotta get outta here or we’re gonna die!” The other muffin looks over and say “OH MY GOD! A talking muffin!”

CA businessmen from South Dakota goes on a business trip to Florida. His wife was on a business trip and was planning to meet him there the next day. When he reached his hotel, he decided to send his wife a quick e-mail. Unable to find the scrap of paper on which he had written her e-mail address, he did his best to type it in from memory. Unfortunately, he missed one letter and his note was directed instead to an elderly preacher’s wife, whose husband had passed away only the day before. When the grieving widow checked her e-mail, she took one look at the monitor, let out a piercing scream, and fell to the floor in a dead faint. Hearing the scream, her family rushed into the room and saw this note on the screen: Dearest Wife, Just got checked in. Everything prepared for your arrival tomorrow. P.S. Sure is hot down here!

A man meets a woman at a bar and they go to her place. They’re undressing and he drops his trousers. She points to his messed up knees and asks what happened. He says “When I was young I contracted kneesles.” She says “you mean measles.” He says “No, I actually got kneesles.” She shrugs and continues undressing. When he removes his socks she looks at his sorry toes and asks about them. He says “shortly after the kneesles, I contracted toelio.” She says “You mean polio?” He says “No, I got toelio.” She shrugs it off, until he drops his shorts. She looks again and says “Don’t tell me — smallcox.”

Two blondes are out hunting in the woods when one of them collapses. She doesn’t seem ro be breathing and her eyes are glazed. The other blonde whips out her cell phone and calls 9-1-1. She cries “My friend is dead! What can I do?” The operator replies, “Calm down… I can help! First, let’s make sure she’s dead.” There is a silence, followed by sound of a rifle blast. Back on the phone, the blonde says “Okay, now what?”

One snowy day, a blonde was sitting in her car. Her dad taught her that if she was stuck in a blizzard to wait for a plw to come by and then follow it. So she waited and, sure enough, a plow came rolling by, and she started to follow it. After about a half-hour of trailing it, the plow pulled over and the driver stepped out of the cab and walked over to the blonde’s car. She rolled down the window and the plow driver said “It’s okay that you’re following me, but just to let you know, I’m gonna be plowing this parking lot all day.”

And the winner is…

Q: Did you hear about the two criminals who stole a calendar?
A: They each got sixth months.

Further editions…

You can find some of the previous iterations of Joke Time here and here and here and here.

Filed under: Math musings, Humor

05.6.2008

Questions from a liberal arts real analysis final

Question 1: What is the shortest mathematical joke?
Let epsilon be less than zero.

Question 2: What did the the mathematician say when he let epsilon go to zero?
“Whoops! There goes the neighborhood!”

Question 3: What did the mathematician say about his dog Epsilon to the vet?
“Let Epsilon be fixed.”

Question 4: What do you call the irrational fear of convergent series?
Zenophobia.

Question 5: Why was Cauchy convicted for being a vagrant?
He had no means of showing support.

Question 6: Why does the series 1 + x/1! + x2/2! + x3/3! + … approximate ex so well?
It was Taylor-made for it.

Question 7: What is this?

  1. EA + EA + EA + EA + …,
  2. EA + EA + EA + EA + …,
  3. EA + EA + EA + EA + …,
  4. EA + EA + EA + EA + …,

Fourier series.

Bonus question for engineers: What is the fundamental theorem of analysis?
Every function has a Taylor series that converges to the function and stops after the linear term.

Highlight under the questions for the solutions.

Filed under: Math musings, Humor
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