komplexify!

04.8.2010

Apparently ambiguous times

My spring semester schedule is such that on certain days I have a four-hour break between my classes, which is an ideal time to get research done, although in practice most of that time is wasted on the web.  The other day I glanced up at my wall clock to gauge the time, and was surprised when I saw this:

I first I though the hour hand had fallen off the clock (which had happened to a wristwatch I owned) and was momentarily panicked at the prospect of missing my afternoon class (or rather, being 11 minutes late to it). However, a few seconds later the minute hand moved a smidge more to reveal that it had simply obscured the hour hand entirely, indicating a time of 2:11.

That got me to thinking… What are the other such “apparently ambiguous times,” times at which the hour hand and the minute hand both point in the same direction? Generalizing a bit, what are the “really apparently ambiguous times” at which the hour, minute, and second hands all point in the same direction?

It’s a little fun to figure out, and I won’t deprive you of it until after the jump.

(more…)

Filed under: Math musings

04.1.2010

Pun-ishment

Filed under: Funny pics, Math musings

03.29.2010

A letter to cheating students

To all students  planning on copying their math solutions straight out of the solutions manual:

Please first consider the following story.

Billy needed to compute the general antiderivative of the function 1/x. Stumped, he glanced around the class, and saw that Amy, who always got things right, had written “log x”, so he copied the answer from her.  Of course, Billy was a sharp tack himself, so in order to prevent himself from being caught copying, he rewrote the answer as “timber x!”.

Now, if you happen to be a student planning on copying his or her answers from some outside source, let me be frank with you.

You are as dumb as Billy.

As proof, did you realize that Amy’s answer is wrong?

Did you realize it’s wrong for two different reasons?

No?

You are as dumb as Billy.

Fortunately, this is easy to rectify:

Seriously.  You are not smart enough to get away with cheating.

If  you cannot do a problem on your own, and you are not smart enough to seek help from the professor or a tutor or a peer, then you are not smart enough to get away with cheating.

I’ve tried to explain this to you before, but apparently it hasn’t sunk in.

Let me try again.  I’ll stick to three reasons.

Reason # 1: if you are dumb enough to plan on copying down a problem wholesale, so too are a bunch of your equally dumb classmates.  As a result, a sizable portion of the class will turn in exactly the same solution, down to the freaking formatting.  On a scale of 0 to 100, the chance of that occurring naturally is 0.  Now, whether or not you have any experience with basic probability, your math professor sure as hell does.

Don’t be a Billy!

Reason #2: if you are dumb enough to copy down an entire problem wholesale, you’re dumb enough to have no idea when the thing you’re copying is total garbage.  Earlier in the month I gave an example of the sheer stupidity of which you are capable when you attempt to cheat.  To return to the probabilistic statement, on a scale of 0 to 100, the chance of a sizable portion of the class naturally turning in exactly the same batshit-insanely stupid solution, down to the freaking formatting, is -100.

Don’t be a Billy!

Reason #3: if you are dumb enough to copy down an entire problem wholesale, you’re also dumb enough to be unable to cover your tracks should you decide to “pull a Billy.”  To wit, just a few weeks after the “cot/cosec” debacle alluded to above,  I had students turn the following problem: compute the arclength of the plane surve x = t^3, \, y = t^2 over the interval [0,2].  A quarter of them handed in the following, word for word:

There is absolutely no way to get the second equality from the first one: what did you do — cancel some of the powers of t under the radical, but nothing else? Nor is there any way to get the third equality from the second one: neither integration by parts nor substitution works, and in any case, the expression in parentheses in the fourth line is not the same thing as the expression in parentheses from the third line.  The final equality does follow from the third one, but at this point if you’re clueless enough to have bought the previous two lines, there’s no way your professor is going to believe you can track fractional powers of large integers in your head.

Of course, if you look to the answer in the back of the book, it reads

If you haven’t found the flaw yet, let me just put in in terms you’ll understand:

It’s one thing to cheat.  It’s another thing to cheat badly.  It pisses off your professors to no end, and if they’re like me, they’ll make you sign a letter of Academic Dishonesty that gets submitted with the Dean of Students and appears in big ol’ letters on your transcripts as a punishment for EPIC STUPIDITY.

Don’t be a Billy!

Sincerely,

– Every math professor you’ve ever had.

Filed under: Math musings

03.17.2010

Eponymy

It has always amused me that,despite mathematicians’ twin loves of hero worship and precision, we’re just as bad about correctly naming things as anybody else.  For example,

  • Arabic numbers were invented in India.  By Hindus.
  • The Leibniz formula \displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots was discovered before Leibniz by his contemporary James Gregory, and 300 years before either of them by Madhava of Sangamagram.
  • Euler’s number e was discovered by Jacob Bernoulli, while Euler’s formula e^{i\theta} = \cos(\theta) + i \, \sin(\theta) was discovered by Cotes.
  • l’Hospital’s Rule was discovered by Johann Bernoulli.
  • Pell’s equation was solved by Lord Brouncher
  • The Gaussian distribution was introduced by de Moivre.
  • Cramer’s Rule was discovered by Maclaurin…
  • But Maclaurin series were discovered by Taylor.
  • Burnside’s Lemma was proved by Cauchy and Frobenius…
  • But Frobenius’ Theorem by proven by  Deahna.
  • Stoke’s Theorem was discovered by Lord Kelvin.
  • The Mandlebrot set was discovered by Fatou and Juila.

Interestingly enough, these are all examples of something called Stigler’s Law, which states that “No scientific discovery is named after its original discoverer.”  It was put forth in 1980 by statistician Stephen Stigler… and is based on work in the 1940s by sociologist Robert Merton.

Filed under: Math musings

03.9.2010

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. Although it’s a little late this year (these are from Fall 2009!), here are some of the better ones.

Mathy contributions

Q: What did the zero say to the eight?
A: Nice belt!

Q: Why was ten afraid of seven?
A: 7 8 9!

Q: Why was 3 afraid of \pi?
A: He was being irrational.

Q: What did the circle say to the tangent line?
A: Stop touching me!

Q: Why do you rarely find mathematicians at the beach?
A: Because they can get a tan from sine and cosine. They don’t need the sun.

Q: Why are math books always sad?
A: They have so many problems.

Q: What do you get when you add three apples to two apples?
A: A liberal arts college math problem.

Q: Why was the student’s failed exam wet?
A: It was below C level.

Q: If the natural log is \displaystyle \int_1^x \frac{dt}{t}, what’s the unnatural log?
A: Duraflame.

Q: Did you here the joke about the empty set?
A: Don’t worry. It doesn’t have a point.

Q: What’s the worst part about math jokes?
A: If you get them, you probably don’t have friends.

“I heard the government wants to put a tax on the mathematically ignorant.”
“Funny… I thought that’s what the lottery was.”

Suppose we know that if you don’t study, then you’ll fail. It then follows that

\begin{array}{r@{\,=\,}l}  \mbox{no study} & \mbox{fail} \\+ \mbox{study} & \mbox{no fail} \\ \hline \mbox{study} + \mbox{no study} & \mbox{fail} + \mbox{no fail} \\ \mbox{study}(1 + \mbox{no}) & \mbox{fail}(1 + \mbox{no}) \\ \therefore \mbox{study} & \mbox{fail} \end{array}

That is, you’re gonna fail either way. Might as well play video games.

Happy face arithmetic:

Math puns:

  • What kind of undergarments does a mermaid wear?  An algebra!
  • What did the acorn say when it grew up?  Geometry!
  • What do you call a teapot boiling on Mt. Everest?  Hypotenuse!
  • What do you get when you divide the circumference of a pumpkin by its diameter?  Pi!

What is 2 x 2?

  • A junior mathematician: 4.
  • A tenured mathematician: I don’t know what the answer is, but I can prove it exists.
  • Physicist (after consulting technical references): Between 3.98 and 4.02.
  • An engineer (after consulting a slide rule): 3.99.
  • A logician: I think you need to define 2 x 2 more precisely.
  • A philosopher: What you do mean by 2 x 2?
  • A sociologist: I don’t know, but it was nice talking about it.
  • A behavioral ecologist: A polygamous mating system.
  • A college student: 4.  (The when asked by astonished colleagues how he knew, replies “I memorized it.”)

I thought it was a great idea to name my child after \pi… until the first time he misbehaved, and I had to call him by his full name.

Students nowadays are clueless about mathematics. Why, just the other day a student came into office hours asking if General Calculus was an ancient Roman war hero.

Mathematical pick-up lines:

  • If you were cos2x, then I’s be sin2x so that you and I could be 1.
  • I wish I was your second derivative so I could fill up your concavity.
  • It’s not the magnitude of the vector, it’s how you apply the force.
  • (Hacker’s pick-up line) Solve 2 u x = 106 x2 y for u.

The only arithmetic a man needs in life: add the girl, subtract the clothes, divide the legs, and pray to God you don’t multiply.

A physicist, a mathematician and a computer scientist were discussing the relative merits of having a wife or a girlfriend.  “For sure a girlfriend is better,” says the physicist. “You still have the freedom to experiment.”  “No, no, it’s better to have a wife,” says the mathematician, “because the sense of security you get.”  “No, no, you’re both wrong,” replies the computer scientist. “It’s best to have both so that when the wife thinks you’re with the mistress and the mistress thinks you’re with your wife, you can be with your computer without anyone disturbing you.”

A professor of mathematics sent a fax to his wife: “Dear Wife, You must realize that you are 54 years old, and I have certain needs which you are no longer able to satisfy. I am otherwise happy with you as a wife, and I sincerely hope you will not be hurt or offended to learn that by the time you receive this letter, I will be at the Grand Hotel with my 18-year-old teaching assistant. I’ll be home before midnight. Your Husband.”  When he arrived at the hotel, there was a faxed letter waiting for him that read as follows: “Dear Husband, You, too, are 54 years old, and by the time you receive this letter, I will be at the Breakwater Hotel with the 18-year-old pool boy. Since you are a mathematician, you will appreciate that 18 goes into 54 more times than 54 goes into 18. Therefore, don’t wait up.  Your Wife.”

No wonder the mathematician’s marriage is falling apart: he’s into scientific computing… and she’s incalculable!

A mathematician is a device for turning coffee into theorems.
A computer scientist is a device for turning coffee into code.
An engineer is a device for turning coffee into urine.

An engineer thinks his equations are an approximation of reality.
A physicist thinks that reality is an approximation of his equations.
A mathematician doesn’t care.

Billy needed to integrate the function 1/(1+x). Stumped, he glanced around the class, and saw that Amy, who always got things right, had written “log(1+x)”, so he copied the answer from her. Of course, Billy was a sharp tack himself, so in order to prevent himself from being caught copying, he rewrote the answer as “timber(1+x)”.

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.”

“An engineer, a physicist, and a mathematician are staying at a hotel. That night, the engineer awakes to smell smoke. He goes out into the hallway and sees a fire, so he fills the trash can from his room with water and douses the fire (and most of the hallway too) before going back to sleep. Later, the physicists awakes to smell smoke. He goes out into the hallway and sees a fire. After a few mental calculations involving the flame velocity, water pressure, ballistic trajectory, and so forth, he fills the trash can from his room with a minimal amount of water and effectively douses the fire before going back to sleep. Later, the mathematician awakes to smell smoke. He foes out into the hallway and sees a fire. He also sees the trash can in his room and the sink in his bathroom and concludes “A solution exists” before going back to sleep.

Sherlock Holmes and Dr. Watson went on a camping trip. After a good meal and a bottle of wine they laid down for the night and went to sleep. Some hours later, Holmes awoke and nudged his faithful friend. “Watson, look up at the sky and tell me what you see.”  Watson replied, “I see millions and millions of stars.”  “And,” Holmes asked, “what does that tell you?”  Said Watson, “Astronomically, it tells me that there are millions of galaxies and potentially billions of planets. Astrologically, I observe that Saturn is in Leo. Horologically, I deduce that the time is approximately a quarter past three. Theologically, I can see that God is all powerful and that we are but small and insignificant and finally, meteorologically, I suspect that that we will have a beautiful day for hiking tomorrow. What does it tell you, Holmes?”  Holmes was silent for a minute, then spoke.  “It tells me, dear Watson, that some bastard has stolen our tent.”

A mathematician is flying a 6-hour nonstop flight from California to Florida. Shortly after take-off, the pilot announces that one of the engines had to be turned off due to mechanical failure. “But don’t worry,” he adds, “we’re safe, and we’ve got three engines running perfectly. The only noticeable effect will be that our flight time will be 7 hours instead of 6.” A half-hour later, the pilot announces that a second engine had to be turned off due to mechanical failure. “But don’t worry,” he adds, “we’re safe, and we’ve still got two engines running perfectly. The only noticeable effect will be that our total flight time will be 9 hours instead of 6.” Another half-hour later, the pilot announces that a third engine had to be turned off due to mechanical failure. “But don’t worry,” he adds, “even with one engine we’re still perfectly safe. However, we’re now looking at a total flight time will of 13 hours instead of 6.” “Great,” grumbles the mathematician. “At this rate, when the next engine goes it’s going to take 19 hours to get there.”

A mathematics major is walking across campus when his classmate rides up to him on a new bicycle. “Where did you get the bike from?” he asks. “It’s a Thank you present from that freshman girl I’ve been tutoring,” the math major explains. “Yesterday she called me and told that she had passed her math final and wanted to drop by to thank me in person. She arrived at my place on her bicycle. When I had let her in, she took all her clothes off, smiled, and said to take anything I wanted!” His friend stares at him for a moment, and then replies, “Good choice. I doubt the clothes would’ve have fit.”

Black holes are where God divided by zero.

Math problems? Call \displaystyle 1-800-\big[(10x)(13i)^2 \big] - \frac{\sin(x)}{2.362x} \bigg|_{x=\sqrt{e}}.

The number you have dialed is imaginary. Please rotate your telephone 90 degrees and try again.

Typical student response to seeing the limit definition of the derivative for the first time:

I also got a lot of charts this time round:

Other good ones

Q: Where did the one-legged man work?
A: IHOP.

A cop was on his horse waiting to cross the street when a little girl rode up beside him on her shiny new bike. “Nice bike you got there,” said the cop, “did you get it for Christmas?” “Yes sir,” said the little girl. The cop looked the bike over, and then handed the girl a $5 ticket for a safety violation. “Next time,” he said, “tell Santa to put a reflector light on the back of it.” The girl looked at the ticket, and then at the cop. “Nice horse you got there,” said the girl, “did you get it for Christmas?” “Sure,” said the cop, humoring her. The girl looked the horse over. “Next time,” she said, “tell Santa that the dick goes underneath the horse, not on top.”

“Bob, how’d you get that black eye?” “Well, my wife came home yesterday after shopping for cars. She told me she wanted something that can go from 0 to 160 in 2 seconds. So I got her a bathroom scale

“Honey, I bought a new toilet brush!” “I know, dear, I know. I still prefer toilet paper, though.”

Three Chinese brothers, Bu, Chu, and Fu, came to live in the United States. They decided to change their names to acclimate to the nation. Bu changed his name to Buck, and Chu changed his name to Chuck, and Fu was deported back to China.

And the winner is…

You know you’ve been a physics major too long when someone asks you “What’s new?” and you reply, “That’s c divided by \lambda.”

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