Archive for December, 2011
Algebra, a Symbol-Minded Pursuit
It is reported that when Augustus DeMorgan was asked his age, he responded algebraically:
I was x years old in the year x^{2}.
From personal experience, I can assure you that responding to a simple question with an algebra problem is no way to make friends. But perhaps DeMorgan had better success with this tactic than I.
As it turns out, there is a similar fact regarding my age and year of birth.
In the year x^{2}, my age will x years with the digits of x reversed.
The following anonymous quotation would, I suspect, meet with DeMorgan’s approval:
The human mind has never invented a labor-saving machine equal to algebra.
Almost everyone has an opinion about algebra, and most people have expressed their opinion without anonymity.
- Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.
Fran Lebowitz - One person’s constant is another person’s variable.
Susan Gerhart - [My algebra teacher] kept putting problems up on the board. I just kept following her and erasing the problems. Then she yells at me. I’m like, “Number 1, I like to attack the problem, not the person. That’s the first rule of problem solving. And B, you kinda seem like you’re a trouble maker, because you got to come up with all these fake problems, and it’s really cutting into our pizza time.” And she’s like, “You can’t list things 1 and then B. It’s 1 and 2, or A and B.” And I’m like, “Oh, you don’t like it when I mix numbers and letters together? Like you do in algebra, you hypocrite?”
Mike Vecchione
Prime Time
One of the great joys of my current job is that I get to visit math classes. This is awesome, and I am incredibly grateful to the teachers who invite me to their classrooms. I’ve thought about returning to the classroom myself, but visiting is much better — I get to see magic and interact with kids, but I don’t have to worry about correcting misbehaviors, creating or grading tests, or filling out report cards.
I recently witnessed several great classes at Tincher Prep, a K-8 school in California. The students were the most collectively polite group of kids I’ve ever met, and the faculty was filled to capacity with intelligent, dedicated professionals. Students in first grade measured things with paper cut-outs of their foot, to get an appreciation for why we have standard measures. Kindergarten kids happily sang number songs and then counted by 5’s to figure out that it was the 75th day of the school year. Students in a middle school class were jumping out of their seats with excitement when playing a review game. In every class I visited, students were excited to be learning. What an awesome environment!
In one classroom, students were given the following assignment:
Complete this list of the first 10 prime numbers:
1, 2, ___, ___, ___, ___, ___, ___, ___, ___
John Derbyshire claims that Henri Lebesque was the last mathematician who considered 1 to be a prime number. The primary reason it should not be considered a prime number is that the Fundamental Theorem of Arithmetic — which states that every integer greater than 1 can be represented as the product of a unique set of prime numbers — will not hold. It also causes a problem with Euler’s Totient Function: for prime numbers, φ(n) = n – 1, but this rule is violated if 1 is considered a prime number.
The teacher who posed this problem to students, however, shouldn’t feel bad for including 1 as a prime number. Lots of professionals have trouble figuring out which numbers are prime…
- Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction, every odd integer greater than 2 is prime.
- Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error,
11 is prime, … - Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, …
- Programmer: 3 is prime, 5 is prime, 7 is prime, 7 is prime, 7 is prime, …
- Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 — we’ll do the best we can, …
- Software Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release, …
- Biologist: 3 is prime, 5 is prime, 7 is prime, results have not yet arrived for 9, …
- Lawyer: 3 is prime, 5 is prime, 7 is prime, there is not enough evidence to prove that 9 is not prime, …
- Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime if 2/3 is deducted for taxes, …
- Statistician: Let’s try several randomly chosen numbers: 17 is prime, 23 is prime, 11 is prime, …
- Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student.
- Computational Linguist: 3 is an odd prime, 5 is an odd prime, 7 is an odd prime, 9 is a very odd prime, …
- Psychologist: 3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to
suppress it, … - Casino Card Counters: 3 is prime, 5 is prime, and 7 is prime, but I’ll take 21 over any of them.
Never Good at Math
In The Lexicographer’s Dilemma, author Jack Lynch describes the reaction of strangers when they learn of his vocation:
When I’m introduced at a party as an English professor, people immediately turn apologetic about their grammar and shuffle uncomfortably, fearful of offending me and embarassing themselves. No one feels compelled to confess to engineers that they never got the knack of building bridges, or to doctors that they don’t understand the lymphatic system — but nearly everyone feels a strange obligation to come clean to someone who is supposed to be an expert in “grammar.”
This passage struck me, because I often get a similar reaction when people learn that I’m a math professional. I cannot begin to count the number of times I’ve heard a taxi driver, a real estate agent, a waiter, a waitress, a flight attendant, and even a local newscaster utter the following line:
Math folks often get upset by this statement. I often hear them lament, “Illiterate people would never tell you that they can’t read, but no one has a problem telling you that they’re not good at math.”
That’s a bad analogy. When someone claims to be bad at math, what they usually mean is that they never figured out how to factor a trinomial in Algebra I or that they were tripped up by two-column proofs in Geometry. While I don’t have data to prove it, I suspect that they are able to count, and I would further guess that they have little trouble with the four basic operations. So while you may hear someone admit, “I was never very good at math,” you will likely never hear, “I can’t count.”
By comparison, when someone admits, “I can’t read,” they are admitting that they never acquired the most basic skill associated with letters and words. Reading is the linguistic equivalent of counting. Were it the case that someone was unable to count, she would likely be as embarrassed about it as an illiterate person would be about her inability to read.
Ever the cynic, when someone tells me, “I was never very good at math,” my first thought is usually, “Your teachers were not very good at teaching math.” Every student has the ability to shine mathematically, but it usually takes a teacher who is willing to pull back the curtain and show them what we mathy folks already know — that the beauty of mathematics lies far beyond rote computation, in a realm where exploration, failure and epiphany provide an infinity of pleasure.
Stepping down from my soapbox, here is a passage about the beauty of mathematics from Dirk Gently’s Holistic Detective Agency by Douglas Adams:
The things by which our emotions can be moved — the shape of a flower or a Grecian urn, the way a baby grows, the way the wind brushes across your face, the way clouds move, their shapes, the way light dances on the water, or daffodils flutter in the breeze, the way in which the person you love moves their head, the way their hair follows that movement, the curve described by the dying fall of the last chord of a piece of music — all these things can be described by the complex flow of numbers.
That’s not a reduction of it, that’s the beauty of it.
Ask Newton.
Ask Einstein.
Ask the poet (Keats) who said that what the imagination seizes as beauty must be truth.
He might also have said that what the hand seizes as a ball must be truth, but he didn’t, because he was a poet and preferred loafing about under trees with a bottle of laudanum and a notebook to playing cricket, but it would have been equally true.
The Book of Tebow
Editor’s Note: The following post is more about (American) football than math, but it does contain some humor (or, perhaps more correctly, it contains material similar to the other material that poses as humor on this blog). Just be forewarned. Read at your own peril.
I fell in love with Denver on a family trip in 1982. My favorite colors as a kid were blue and orange. So I was already a fan of the Denver Broncos when they acquired my favorite college player, John Elway, on May 2, 1983. (Ironically, my wife and I acquired our twin sons on May 2, also, albeit more than two decades later.) On Sunday afternoons growing up, I’d watch my hometown Pittsburgh Steelers at one o’clock, and I’d hope that NBC would show Elway and the Broncos during the late game.
So this whole Tebow thing? Yeah, I feel a little like I’m jumping on a bandwagon. Then again, I’ve been a fan of the Broncos for 28 years, so cut me a little slack.
Plus, it’s just so damned compelling. Any quarterback can win football games, but it takes a rare talent to repeatedly perform miracles. You better believe that I have already set our DVR to record tomorrow’s Broncos-Patriots game.
I also love the hype and the humor. The nickname “God’s Quarterback” seems to have stuck, and this great joke has surfaced:
And on the seventh day, God rested so he could watch his son play quarterback for the Denver Broncos.
A few days ago, a headline in The Christian Post caught my eye:
Tim Tebow ‘God’s QB,’ But Does God Care About Football?
I have a Speed Bump cartoon on the door to my office suggesting, in fact, that He does:
I look forward to all that will follow. The legend of Tim Tebow continues to grow, and no doubt organized religion will begin to take advantage of the publicity. I suspect a rewrite to the Good Book before too long…
The Gospel According to Tebow 1 God created Tebow in his own image, in the image of God created He him. And God said unto him, Go forth, and run and score, and replenish hope in the city of Denver: and have dominion over the dolphins of the sea; and over the fowl of the air, over cardinals and eagles and ravens; and over cowboys and redskins and titans; and over every living thing that moveth upon the turf. 2 And lo, He made a great arena, called as the Stadium of the Authority of Sports, which was ten-thousand cubits from the one rim to the other. It was round all about, and a line of thirty-one thousand, four-hundred fifteen and nine-hundred twenty-six thousandths cubits (approximately) did compass it round about. And the incorrect approximation of pi previously appearing in scriptures was thus smote, and it was good. 3 He placed the stadium above the water five-thousand two-hundred eighty feet, providing a wonderful number with which to demonstrate the law of divisibility by eleven. 4 And He bade him, play your best, and do not be discouraged in half the first, or by thine rating of eighty-three-point-four, or by trailing your opponents at the end of quarter third; play well when the end is nigh, and best your enemy after regulation time has expired. 5 Lastly God said unto him: kneel before me, with but one knee upon the earth and a clenched fist upon thine brow, and let photographers take pictures; and all the peoples of the earth shalt imitate thee and post their pictures at www.tebowing.com, thus begetting an international phenomenon. 6 And Tebow did as commanded, and it was good. |
Math Rebus Puzzles
What word or phrase is represented by the following picture?
Puzzles like this are sometimes called Pictogram Puzzles, Word Picture Puzzles, or Hieroglyphs.
Kevin Stone at BrainBashers calls them Brain Bats.
Lots of folks refer to them as Rebus Puzzles. And while I don’t think that’s exactly right (what I think of as a rebus can be found here), that’s the word I’m going to use, too, because it’s the shortest.
Whatever. Enjoy the MJ4MF Rebus Quiz. Note that each picture is somehow mathematical, even if the answer isn’t.
Download the MJ4MF Rebus Quiz (PDF)
If you’re a classroom teacher, the dread day-before-break is fast approaching. The MJ4MF Rebus Quiz is a great activity for students who have too much energy to sit still and too little focus to learn anything. (Permission is granted for the MJ4MF Rebus Quiz to be used with students for non-commercial educational purposes.)
The answer key and a copy of the exam can be found at http://mathjokes4mathyfolks.com/rebus.html.
144 Gross Jokes
Recently, I gave a presentation that contained 288 jokes. But most members of the audience were turned off, claiming it was two gross.
But one guy really liked it. “Your jokes are funny,” he said, “though I don’t think my wife would like your humor. How many off-color jokes do you know?”
“I have a collection of 144 gross jokes,” I told him.
“Wow!” he said. “How did you find the time to collect 20,736 jokes?”
Sorry. Just seems like 12/12 is a good day to be making such jokes.
The word dozen comes from the Old English word doziene, which comes from the Old French word dozaine, which is a derivative of the Latin word duodecim (duo = two, decim = ten).
Warning! Off-color joke approaching!
A man calls his friend and asks, “What has a two-inch penis and hangs down?”
“I dunno,” says his friend.
“A bat,” says the man. “Now, what has a twelve-inch penis and hangs up?”
“I dunno,” says the friend.
Dial tone…
The following is a list of my favorite things that come in groups of 12.
- Signs of the Chinese Zodiac — what’s not to love with dragons, roosters, and pigs?
- Angry Men — sure, it’s a little sexist with an all-male cast, but three of those males were Jack Klugman, Ed Begley and Henry Fonda, and it’s ranked #6 in the IMDB Top 250.
- Donuts — mmm, donuts…
- Eggs — can’t really have a list of dozens that doesn’t include eggs, right?
- Inches in a Foot — how many inches in a nose?
- Labours of Hercules — though I can’t decide which was the best, cleaning shit out of stables or stealing a belt from a woman.
- Players on a Canadian Football Team — in the U.S., it’s 11 players on a 100-yard field; in Canada, it’s 12 players and a 110-yard field; the next country to don a football league must have 13 players on a 120-yard field, to follow the little known but never broken n + 1 players on a 10n-yard field edict.
- Ounces in a Troy Pound — because, really, who needs Avoirdupois?
- Function Keys on a PC Keyboard — F7 is the most-used function key on my laptop, since Shift-F7 lets me synonym search in Word.
- Roses — red if you’re nice, black if you’re naughty.
- Face Cards in a Deck — jacks, queens, and kings.
- Keys on a Phone Keypad — yet only eight have letters associated with them… weird.
Holiday Gifts to Avoid for the Math Geek on Your List
There are lots of lists with suggestions for what to buy a math geek. Don’t believe me? Just do a Google search for “gifts for math geeks.”
As a public service, I’m providing a list of gifts that no one, under any circumstances, should purchase for a math geek. If you’re not a mathy person, take note. Every item on this list will only bring disappointment to the mathy people in your life. If you are a mathy person, print this list, and tuck it into your mom’s purse or leave it on your sweetie’s pillow.
I’ve railed against this one before, and for good reason. Absolutely the worst math gift EVER. The expression for nine is 3(π – .14). Apparently the designer of this clock face isn’t aware that π has a non-terminating decimal representation. And the expression for seven is 52 – x^{2} + x = 10, which has two solutions, 7 and -6. Let’s hope folks don’t start making dinner reservations for “negative six o’clock.” Sheesh.
Acme’s Klein Bottle Wine Bottle
You can take my word that this is a bad gift, or you can listen to the designer. The manufacturer describes it by saying, “As impractical as it is elegant.” Fact. The description is a litany of flaws: “Wine is trying to go down while the air is trying to go up the spout. Result is slow filling. Pouring wine out is equally frustrating.” And, “Not only are these difficult to fill and empty, but cleaning them is a real challenge.” And the piece de la resistance, “They’re easy to tip over, especially when empty.” On the flip side, their web site includes this gem, too: “Now with a LIFETIME GUARANTEE — you will live your entire life, or your money back.”
A better yet equally geeky option is the Klein Bottle Opener, which is practical if you’d like to use a non-orientable manifold to get the liquid out of a boundaryless compact two-manifold homeomorphic to the sphere. Unlike the Acme Klein Bottle, this tool works very well, and you look super cool using it. (Be careful; there are cheaper versions of this item that are only decorative. They won’t open bottles.)
Okay, sure, this hard-copy volume has “a lot of doodles, notes, and puzzles in the margins,” but, really, you’re paying for a bunch of comics that are available for free at xkcd. What’s more, the math geek on your list has already read all of these comics! (If not, then she isn’t a true math geek. Please call 1-877-NOT-GEEK, and we’ll revoke her license immediately.)
Want a book your math geek will really love? Might I recommend Math Jokes 4 Mathy Folks? (Okay, that was even too shameless for me.)
Need a synonym for geeky? How about dorky, nerdy, dweeby, techy, or studious? Want an antonym for geeky? Try stylish.
Math geeks don’t want silver cuff links. More importantly, they don’t need them. Honestly, what would they wear them with? Plaid flannel shirts don’t have cuff link holes. And for $195, do you really want to buy something that will collect dust in his dresser drawer?
Best of ×, Worst of ×
The following is a very old joke:
“My life is all arithmetic,” the young businesswoman explained. “I try to add to my income, subtract from my weight, divide my time, and avoid multiplying.”
A similar joke, modified for current times, has been floating around Twitter.
Obama is great at math. He divides the country, subtracts jobs, adds debt and multiplies misery.
There are lots of math jokes that involve multiplication.
What tool is used most often in math class?
Multipliers.Where do math teachers eat dinner?
At the times table.Minister: Noah said, “Go forth and multiply!”
Congregant: What did he want the first, second and third to to — add, subtract, and divide?Teacher: Why are you doing your multiplication problems on the floor?
Student: Because you told me not to use tables!
And finally a joke for the upcoming holiday…
If you multiply Santa Claus by i, does that make him real?
Results for My Favorite Game
Thanks to everyone who participated in the online version of my favorite game. Thanks, especially, to those people who helped to share it via Twitter, Facebook, and other blogs.
Though I am posting the results today, I will continue to leave the form online. Though I had only planned to let the contest last one week, entries continue to roll in, and I see no reason to forbid people from playing. From time to time, I’ll update this page… such updates will occur at the intersection of two events: when enough entries warrant an update, and when the muse hits me.
Without further adieu, here are the results.
With 1,042 entries divided into groups of 100, there were 10 complete games played. The charts below show the results for each game. (Sorry if they’re a little hard to read. Click on the images to view them full-size in a separate window. There are two images below — games 1-5 are shown in the top image, and games 6-10 are shown in the bottom image.)
The graphs above do not reflect all 100 entries for each game. In each game, many numbers greater than 35 were chosen. However, 84.1% of all selected entries were 35 or less.
The winning numbers, respectively, were 3, 3, 20, 4, 7, 2, 16, 4, 4, 2.
If you’re interested in the raw data, download this Excel spreadsheet.
Congratulations to Loïc Grobol of Chécy, France! Loïc was the winner of the 4th game, and his name was randomly selected as the overall winner of the signed copy of Math Jokes 4 Mathy Folks and the specially-designed, one-of-a-kind random number generator. (There is beautiful symmetry that Loïc chose the number 4, that he was the winner of the 4th game, and that the prize has a 4 in the title.)
As shown in the graphs above:
- On average, the number 1 was chosen by 10.4% of entrants. (Edward Early of St. Edward’s College said, “I use that [game] as a bonus question on a test in every class I teach. I believe 1 has never been the winning number, even in a class with only 6 students.” I have played this game over 100 times with various size groups. In my experience, the number 1 has only won twice… by the same woman, who — in a show of incredible bravado — chose the number in consecutive rounds of the game.)
- The number 2 won twice, but it was chosen rather infrequently — less than 1/3 as often as 1. My suspicion is that people figure if you’re gonna go big, go REALLY big… why choose 2 when you can choose 1?
- Besides 1, the number most often chosen was 17, which was selected 5.5% of the time. This seems to corroborate numerous studies that found 17 to be the most commonly selected random number.
- The number 151 was the greatest number chosen more than once.
The chart below shows the frequency of the top nine guesses.
Among the most interesting entries were 666, 1012, 1337, 53,479, and 3,010,994.
The most amusing entry was 10, with the accompanying note, “I’ll choose the base later.”
And you may be wondering… if all 1,000 entries were considered as just one game, what number would have won? That distinction would have gone to 32.
By running this contest, I learned about two interesting uses of this game in classrooms.
Edward Early said that he uses the following version as a bonus question.
Write a positive integer in the blank: _______
How this will be graded: The least positive integer that is submitted by exactly one person will be worth 5 points. The next-smallest will be worth 4 points, and the next-smallest after that will be worth 3 points. All other positive integers submitted by exactly one person will be worth 2 points. Positive integers submitted by more than one person will be worth 1 point. Anything other than a positive integer will receive no credit. Do not ask me to explain this question.
Not surprisingly, with these modifications comes a change in strategy — Edward said that some students choose a large random number, just to ensure they receive 2 points.
Matt Skoss of Possum Educational Services and the Northern Territory Dept of Education and Training said that he’s used this game for years with his kids at school.
Pick the lowest prime number, composite number, surd, cube number or triangular number, etc., depending upon what I’d like the kids to think about.
What an excellent use of a simple game!
Werner, A Man Loved by Airlines
Know what this is?
BEHIHIHIHIHIHIRG
(Answer at bottom of post.)
I am not certain that the preceding or following jokes are funny. I am certain, however, that today is Werner Heisenberg‘s birthday. If you’re not familiar with his work, you might want to read about the Heisenberg Uncertainty Principle before continuing. Not that it’ll make the following jokes any funnier — in fact, if you require an explanation of the content prior to reading these jokes, well, that will almost surely guarantee that you will not find them funny — but perhaps you’ll feel a little smarter. (A more technical description of the principle can be found here.)
Why was Heisenberg’s wife unsatisfied?
When he had the time, he didn’t have the energy; and when he had the position, he didn’t have the momentum.
Heisenberg was out for a drive when a traffic cop stopped him. The cop says, “Do you know how fast you were going?”
Heisenberg replies, “No, but I know where I am.”
Werner Heisenberg, Kurt Gödel and Noam Chomsky walk into a bar.
Heisenberg looks around the bar and says, “Because there are three of us and because this is a bar, it must be a joke. But the question remains, is it funny or not?”
Gödel thinks for a moment and says, “Well, because we’re inside the joke, we can’t tell whether it’s funny or not. We’d have to be outside looking in.”
Chomsky looks at both of them and says, “Of course, it’s funny. You’re just telling it wrong.”
Hmm… maybe I should have shared some information about Gödel and Chomsky, too.
Riddle Answer: Heisenberg (HI’s in BERG)