Archive for November, 2008

5 – 1 = 4, unless it equals 5

November 30, 2008

olympic_pictogram_modern_pentathlonThe Union Internationale de Pentathlon Moderne held their 2008 Congress in Guatemala in November, and the main headline from the Congress was the decision to combine two of the five Modern Pentathlon events into a single competition.

From its introduction at the 1912 Olympics until the end of 2008, the five events of the Modern Pentathlon have included running, fencing, swimming, show jumping, and shooting. (The phrase Modern Pentathlon is meant to distinguish this competition from the Pentathlon, an ancient Olympic event involving competitions in javelin, shot put, wrestling, discus, and running.)

The recent change has involved combining the running and shooting events into a combined event:

Athletes will begin … with a short run to the shooting range, here they take 5 shots before a 1km run. They repeat this 2 further times, in total 15 shots and 3km run. [link]

Unlike the winter biathlon, athletes will not carry their sidearms while running.

The Yahoo News article on this describes it as changing the Pentathlon from five to four events, which raises some obvious doubts as to nomenclature. (Quadrathlon, perhaps?) The UIPM insists, though, that the Pentathlon is not being reduced to four events, since “All five disciplines are still equally represented and competed”.

I anticipate a fantastic Final Jeopardy question: “The combined number of events in the Decathlon, the Heptathlon, and the Modern Pentathlon.”

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Nominations for the 2008 Eddies

November 29, 2008

Here are 360’s nominations for the 2008 Edublog Awards:

Best New Blog: Division by Zero

Best Resource Sharing Blog: Teaching College Math Technology Blog

Best Teacher Blog: JD2718

Each of these three blogs is one we read regularly here at 360.  Check them out, and don’t forget to nominate your favorites by the 30th!

Happy Thanksgiving!

November 27, 2008

thanksgivingWhile you’re baking your turkey and pondering why the temperature continues to rise even after the bird comes out of the oven or thinking about how Game Theory showed that tryptophan affects trust and cooperation, you might also be wondering, “What kind of math did the folk at the first Thanksgiving study?”

The answer is…I don’t know. So I looked around at what textbooks would have been available. I couldn’t find any math texts of the Wampanoags from around 1621, and while I was enticed by a reference to “our earliest native American arithmetic, the Greenwood book of 1729” from David Smith’s History of Mathematics Vol.2, in looking further it seems that the phrase “native American” was used only in contrast to having been originally published in England, Spain, or other non-American country. So as far as the Wampanoags are concerned, I’ll admit to remaining in the dark about what specific kinds of math they would have been learning (formally or informally).

Then I looked into the Pilgrims. As they hailed from England, it seems likely that they’d be using a an English text. And for this they had few choices: there was a 1537 book entitled An Introduction for to : Lerne to Reckon with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym [that last word is Algorithm], which was a translation of a book in Latin by Luca Pacioli. Wikipedia mentions that another textbook was published in 1539, but I couldn’t find reference to the title or author (though I suppose I could add one myself).

A more likely candidate would be Arithmetick, or, The ground of arts by Robert Recorde, first published in 1543 and listed in several places (e.g. this site) as “The first really popular arithmetic in English”. The title is short for The grou[n]d of artes: teachyng the worke and practise of arithmetike, moch necessary for all states of men. After a more easyer [et] exacter sorte, then any lyke hath hytherto ben set forth: with dyuers newe additions. Here’s a woodcut from the 1543 version

robert_recorde-1543

and here’s the title page from the 1658 edition, with its fancy modern spelling:

recorde-title(Heh heh — I just ran that page through Adobe’s Optical Character Recognition, and it pretty much didn’t recognize a thing.)

I couldn’t track down any copies of what exactly was in the book, but this article about medicine in English texts quotes quite a bit from Recorde’s book. For example:

In 1552, Record amplified his arithmetical text with, among other material, a “second part touchyng fractions”. This new section included additional chapters on various rules of proportion, including “The Rule of Alligation”, so-called “for that by it there are divers parcels of sundry pieces, and sundry quantities alligate, bounde, or myxed togyther”.

The rule of alligation, we are told, “hath great use in composition of medicines, and also in myxtures of mettalles, and some use it hath in myxtures of wynes, but I wyshe it were lesse used therein than it is now a daies”. Despite Record’s regret about the adulteration of wine, the first problem the Master uses to exemplify his discussion of alligation in fact deals with mixing wines; the second involves a merchant mixing spices; and the remainder involve the mixing of metals. None of the examples involve medicine, although the merchant’s spices are once called “drugges”.

A third part was added in 1582 by John Mellis. Here’s a page (from Newcastle University) showing one of the pictures in the margin (possibly from that year, or possibly a later edition; I couldn’t tell for certain).

Records Arithmetic

Robert Recorde is actually better known for a later book: The Whetstone of Witte, which was published in 1557 and so is another candidate for An Original Thanksgiving Math Text.  It was called because, according to the poem on the front, “Here, if you list your wittes to whette, Moche sharpenesse therby shall you gette,” and appears (using Google books) to have been well-known enough to be referenced by Shakespeare in Act 1 Scene 2 of As You Like It (written around 1600) when Celia says, “…for always the dulness of the fool is the whetstone of the wits. How now, Witte! whither wander you?”

The Whetstone is best known for its introduction of the equal sign =, which Recorde explains below:

whetstone-equal-1

Which reads as

Nowbeit, for easy alteration of equations.  I will pro(?provide?) a few examples, because the extraction of their roots, may the more aptly be wrought.  And to avoid the tedious repetition of these words “is equal to”, I will set as I do often in work use a pair of parallels or Gemowe lines of one length, thus: ======, because no two things can be more equal….

So there you have it: math that some of the first Thanksgiving folk could possibly have studied.

thanksgiving

Perfection

November 26, 2008

perfect-numbersIn the episode “The Boneless Bride in the River” in Season 2 of the TV show Bones, the body of a young woman is found in a river, and it’s discovered that she was likely brought over to the US on a fiancée visa. About 14 ½ minutes into the episode, two of the main characters (Special Agent Seeley Booth and Forensic Anthropologist Dr. Temperance “Bones” Brennan) have this conversation:

Booth: Homeland Security says the fiancée visa was expedited by a lawyer on retainer into a smaller bride agency here in town called “The Perfect Wife”.
Brennan: Oh that sounds archaic.
Booth:
No, you know, in therapy I learned that superlatives like perfect are meaningless.
Brennan:
Not in science. A perfect number is a number whose divisors add up to itself, as in one plus two plus three equals six.
Booth:
Well, in therapy I learned that definitive statements are by their very nature, wrong.
Brennan:
Isn’t the statement “definitive statements are by their very nature wrong”, definitive, and thus wrong?

Speaking of wrong, Brennan was a little bit wrong in her definition: a perfect number is one whose proper divisors add up to itself. But still, neat math in a neat show is always worth a mention. And perfect numbers are pretty neat, because like so much in number theory they’re simple but there are still open problems about them.

A bit of history [where “a bit” apparently means “a lot because I don’t know how to edit today”]: perfect numbers were studied by Pythagoras, which makes the concept at least 2500 years old. Euclid also talked about perfect numbers a few hundred years later in Book IX of The Elements. In particular, Thomas Heath’s translation of Proposition 36 states:

If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

As a translation of this translation, this is saying that if 1+2+22+…+2k is prime, then that sum times the last number in the sum (2k) must be perfect. For example, 1+2=3 is prime so 3·2=6 is perfect. Likewise, 1+2+4=7 is prime, and 7·4=28 is perfect. [As a note, the formula is is sometime written out algebraically with 2k+1-1 used instead of 1+2+22+…+2k; in addition, when it’s written that way it’s sometimes reindexed so that k is used in place of k+1, and the statement becomes “If 2k-1 is prime, then (2k-1)·2k-1 is perfect.”]

A few hundred years after Euclid, Nicomachus wrote some more information about perfect numbers. According to the St. Andrew’s web site, he made
five claims:

(1) The nth perfect number has n digits.
(2) All perfect numbers are even.
(3) All perfect numbers end in 6 and 8 alternately.
(4) Euclid’s algorithm [described above] to generate perfect numbers will give all perfect numbers
(5) There are infinitely many perfect numbers.

Nicomachus’s word was law as far as perfect numbers were concerned, and his claims, while unproven, were believed for decades centuries a really really long time. Of course, there were only four perfect numbers known at that time (6, 28, 496, and 8128), so they really didn’t have much to go on. In reality, the 5th and 6th perfect numbers (33,550,336 and 8,589,869,056 respectively) disprove claims (1) and the “alternately” portion of (3), but it took a while for someone to discover those.

(Which leads to a little tangent: who did discover those numbers? Ismail ibn Ibrahim ibn Fallus (1194-1239) made a list of ten numbers he thought were perfect. Three of them weren’t, but the other seven were. Sadly, a lot of other mathematicians had no idea about this list: mathematicians in Western Europe had to wait another 350 years for those numbers to enter their collective psyche, during which time a few other mathematicians found the 5th and 6th perfect numbers and were equally ignored.)

But even after it was known that Nicomachus’s claims weren’t themselves perfect (ba DUM!), mathematicians continued to study the numbers. In the 1600s Pierre Fermat tried to find patterns, and ended up discovering his Little Theorem as a consequence. Marin Mersenne also spent some time on it, and in fact his exploration of when 2k-1 is prime, as a part of that theorem of Euclid’s mentioned above, led to the notion of Mersenne primes (primes of the form 2k-1 where k itself is prime).

In the 1700s Leonhard Euler entered the fray. He couldn’t prove that Euclid’s formula generated all perfect numbers, but he did show that it generated all even perfect numbers. And a bunch of other mathematicians spent a lot of time trying to show that numbers were or were not perfect (which was related to showing that specific numbers of the form 2k-1 were or were not prime), a challenging task in the pre-computer days.

Not that we’re doing much better now. As of this moment, we still only know of 46 perfect numbers, and they’re pretty big. We do know a few cool things about perfect numbers in general:

  • Even perfect numbers end in 6 or 8.
  • Even perfect numbers are triangular numbers (e.g. 6=1+2+3 and 28=1+2+3+4+5+6+7) where the ending digit is one less than a power of 2.
  • The reciprocals of the divisors of perfect numbers all add up to 2:
    \frac{1}{1} + \frac{1}{2}+\frac{1}{3} + \frac{1}{6}=2
    \frac{1}{1} + \frac{1}{2}+\frac{1}{4} + \frac{1}{7}+\frac{1}{14} + \frac{1}{28}=2

But there’s a lot we still don’t know:

  • We don’t know if all perfect numbers are even.
  • We don’t know if there are a finite or infinite number of perfect numbers.

In other words, perfection continues to eludes us.

Nicolaus Copernicus: lost and found

November 25, 2008

copernicus_krakowHe was born 535 years ago as Mikolaj Kopernik or Nicolaus Koppernigk, and he died 70 years later. In between, he proposed that the sun and not the earth is at the center of the universe, which was a bit of a shock at the time.

When he died in 1543, he was buried in Frombork Cathedral in Northern Poland but his exact grave was never marked. Then four years ago the Bishop (Jacek Jezierski) requested help from archaeologist Dr. Jerzy Gassowski in finding the grave. It took a few years, but a grave was indeed found in an appropriate spot. But was this Copernicus? The body was the right age, but that’s hardly conclusive. What is fairly conclusive is DNA evidence, except that there weren’t exactly databases set up at the time. What they needed was something like a piece of his hair.

And that, it appears, is exactly what they had. Some of Copernicus’s books (that he himself owned, not that he wrote) are still around and on display. In one of those books were four hairs. Dr. Marie Allen tested the DNA, and it turned out that two of those hairs belong to the body under the cathedral.

Proof? Maybe not. But as CSI meets De revolutionibus, it’s pretty cool.

For more information, see the Post-Gazette or CNN.

A Paltry Geometric Dilemma, Part 2

November 24, 2008

Via Internet Time-Wasters III at Making Light, here’s the World’s Hardest Easy Geometry Problem:

From the site:

Using only elementary geometry, determine angle x. Provide a step-by-step proof.

You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc.

Small hints are available on the site, but they’re very small. Think of this as something to keep you busy while you’re digesting all that turkey.

(The first Paltry Geometric Dilemma.)

Carnival of Mathematics #44

November 21, 2008

clown at the carnivalIt’s two months before the 44th president will be in the White House, but the 44th Carnival of Mathematics is already here!  It’s got everything from controversy and fraud to games and a request to include some good mathematics in the list of 1,000,000 good things!

This carnival is being hosted by Maxwell’s Demon, the blog of Edmund Harriss, who does both mathematics and art.    On his blog he has a neat post on rep-tiles here (which leads to fractal dragons, on which is based the crocheting project Here Be Dragons at Woolly Thoughts), and on his homepage he shows some of his artwork here and here, including the Octagonal Gasket (like Sierpinski’s Triangle, but with octagons)

octagonal_gasket

and these wooden fractal puzzle pieces that can be put together in different ways

wooden_aab

(Both are licensed under Creative Commons)

Enjoy!

A complicated proposal

November 20, 2008

wedding_cake_2002Matt Snyder and Tanya Hein are getting married this Saturday in Madison. No, I don’t know them, but I read about Matt’s proposal in this afternoon’s Wisconsin State Journal. They’d been dating a few years and were both ready to get engaged, but Matt decided to make it special. And nothing says special like a mathematical treasure hunt. The first key (to one of several locked drawers in a treasure chest) led to a coded message, which ended up instructing Tanya to drive to another location. There she found another key, which fit into another drawer, which had another puzzle in it. Each puzzle, when solved, instructed her to go to a different location where there was another key to another drawer that held — you guessed it — another puzzle.

Each location had a reason — the J.T. Whitney’s parking lot was the site of their first kiss. Some of the encrypted messages, involving complex letter codes, linear algebra and trigonometry, were highly difficult to solve.

Presumably there was no partial credit for getting close to a solution. But neither was there a time limit — this was no weekend jaunt. It took two months before Tanya finally got to the proposal. Her response was to give Matt an envelope with a puzzle inside to decipher.

I can only imagine what their seating chart will be.

Speaking of proposals, are there really a thousand yellow daisies in this scene (about a minute in)? It says there are, and estimation is not my own personal strength in mathematics, but it still seems to me that there’s actually quite a few more than that. Anyone know?

KenKen

November 19, 2008

kenkenproblemBarbie might think math is hard, but perhaps she’d be willing to try this game.  I just heard about it yesterday around the water cooler copier (Thanks Lynn!).  It’s a little bit like Killer Sudoku, but with operations other than just addition.

The initial grid is a square, perhaps 4×4 or 6×6.  Like Sudoku, the digits 1-4 or 1-6 are put in cells so that each row contains exactly one of the digits and each column contains exactly one of the digits.

In addition, there are groups of cells (like the cages in Killer Sudoku).  In the corner is a number and an operation.  If the digits in a group are put together with that operation, they form the given number.  For example, if there are two cells with “8x” in the corner, the digits multiply to 8, and so must be 1&8 or 2&4 in some order.  Likewise, “2/” (using / for division) could be 1&2, 2&4, 3&6, or 4&8 where the pairs are in either order.  While you can’t repeat digits in any row or column, you can repeat digits in a group of cells.

Here’s one to try, courtesy of SGBailey.

kenkenproblem

(You can find the answers in the place that always gives you answers.)

If you search for KenKen there seem to be several sites where you can get your daily fix (including the New York Times), but most require registration.  And apparently in some of the harder versions the operation is left off so that you know the answer, but not whether it is a sum, product, quotient, or different.  I can totally imagine a variation that uses multiple operations in a single group of cells, but as far as I know that hasn’t been invented yet.

Math Careers: Computer Animation (Games and Movies)

November 17, 2008

pixar_-_front_gatesVideo gaming.  Creating cool movies.  If you want to do computer animation, you need math.

One of our recent graduates ended up working in just this field.  Actually, he wasn’t even a math major:  he was a theater major who picked up a math minor in his last two years.   He’s now doing graphics programming, and it requires a ton of math:  he mentioned that the most important course turned out to be Linear Algebra.   In all fairness, since he’s officially a software engineer I have to assume he had some computer science as well.  But we don’t even offer a computer science degree, and he started work in this right out of college, so I don’ t know how much formal computer science training he had.

The article Math in the Movies from 2007 gives similar information.  There’s a 90-second video there, which I can’t seem to reproduce here, but in part of it the announcer Cindy Demus(?) says:

Trigonometry helps rotate and move characters, while algebra creates the special effects that make images shine and sparkle.  Calculus helps light up a scene and new math techniques turn images like this [flat and blocky] into this [smooth and more realistic].

Then Tony DeRose, a computer scientist from Pixar Animation Studios, added

I remember as a mathematics student thinking, “Well, where am I ever going to use simultaneous equations?” And I find myself using them every day, all the time now.”

So with some sort of computer background and good math skills, that job could be yours as well.

Generating Pythagorean Triples

November 15, 2008

500px-illustration_to_euclids_proof_of_the_pythagorean_theorem2svgA recent post at jd2718 noted that for all Pythagorean triples (a, b, c), that is to say, for all positive integer solutions to a^2 + b^2 = c^2, it turns out that 60 divides the product abc.

In the course of exploring proofs of that result, I suggested a proof using the representation (2uv, u^2-v^2, u^2+v^2), which generates all primitive Pythagorean triples (triples which have no common factors).  Proving that 60 divides their product proves the general result, as every Pythagorean triple has the form (ka, kb, kc), where k is a positive integer and (a, b, c) is primitive.

In the ensuing discussion, a question arose as to how we know that all possible Pythagorean triples can be found in this way.   Since it is easier for me to post LaTeX code to this blog rather than in the comments on the jd2718 blog, I’ll present the proof here.

Suppose that a^2 + b^2 = c^2, with positive integers a, b, c having no common factors.  This implies that exactly one of a and  b is even (since if both are even, then so is c, leading to a contradiction, while if both a and b are odd, then both a^2 and b^2 are congruent to 1 mod 4, while c^2 would be congruent to 0 mod 4, not 2 mod 4).

So without loss of generality, assume that a is even.  Thus a^2 = (c-b)(c+b) is also even, and since (c-b) and (c+b) differ by an even number, both of those factors must be even.

Furthermore, exactly one of (c-b) and (c+b) is 2x(an odd number), since if both factors are divisible by 4, it would follow that both b and c would be even (since b and c are \frac{ (c+b)\pm (c-b)}{2}.)   Likewise, any odd common factor of (c-b) and (c+b) would also then be a common factor of both b and c, from which it follows that

  • (c+b) and (c-b) have no odd common factors
  • one of these expressions is of the form 2 u^2 (for some odd u)
  • and hence the other expression is of the form 2 v^2, where v^2 = \frac{a^2}{4u^2}, where u and v are positive integers

It now follows that c = u^2 + v^2, and b = u^2 - v^2.  We can find a using a^2=(c+b)(c-b) = (2u^2)(2v^2), and thus a = 2uv.

Conclusion:  every primitive Pythagorean triple has the form (2uv, u^2-v^2, u^2+v^2), for positive integers u and v.

Justifying an addictive game: it’s GEOMETRY!

November 13, 2008

small-pearlOne of my students recently sent me an email with a link to the game Shinju. (Thanks Chris Z!) He included the note, “I thought you would enjoy this game I came across. It deals with Maxi geometry!”

The game is played on an 8×8 board that is partially covered with shells, one of which has a golden pearl inside. You click on a shell, and it either reveals the hidden pearl (You Win!) or it gives a number indicating how many steps away the pearl is, where each step can be horizontal, vertical, or diagonal. It’s fun to play the game, and I found it surprisingly addicting. The analysis of the game, including how to win, is beyond the jump.

Math Confusion in the News: percent

November 12, 2008

dollarLast week Governor Arnold Schwarzenegger proposed a temporary (3-year) sales tax increase in California to help close the budget deficit. Some newspapers, however, are mixing up the amount of the increase in an effort to get the news out.

Error #1: “Governor Proposes 1.5 Percent Sales Tax Hike” from MyFox Los Angeles

The proposed increase isn’t actually 1.5 percent (which wouldn’t be all that much). It’s 1.5 percentage points, which makes it about a 20% increase from the current 7.25% state sales tax. I suspect that most people understand what the headline intends, however, because using “percent” instead of “percentage point” is fairly common. (Kudos to the LA Times for being precise in their story!)

Error #2: “Schwarzenegger proposes 1.5-cent sales tax increase to close budget gap” from the San Jose Mercury News

This headline is just wrong. A 1.5¢ tax? And it’s not just in the title, but in the body of the story. Several other newspapers made the same mistake, either running the Mercury News story without correction or writing their own story about the 1.5-cent increase (I’m looking at you, Sacramento Bee). Indeed, these 1.5-cent increase stories were common enough that I actually double checked that it wasn’t some new terminology for “percentage points”.

Incidentally, the word “per cent” is only 440 years old, and “per centage” only 222 years old. Tidbits from the OED!

Some WWI tidbits for Veterans Day

November 11, 2008

uncle_sam_pointing_fingerI was just reading the article “Dr. Veblen Takes a Uniform:  Mathematics in the First World War” by David Alan Grier (from the American Mathematical Monthly, Dec 2001).  The full article is about the Captain Oswald Veblen and the math folk who worked with him at a military facility in Aberdeen, Maryland, but there are plenty of additional tidbits  For example:

Over 150 mathematicians served in the First World War. Many took conventional military roles but half found ways to employ their mathematics. They worked as surveyors, assisted cartographers, and taught trigonometry to officer candidates.  (p. 923)

A lot of the work involved making ballistic tables, which was not such a simple task:

These mathematicians did part of their computing at the actual test ranges, where they served as observers, data collectors, and range officers. On the water range, a large range that ended in the Chesapeake Bay, the mathematicians were stationed on towers along the shore. After they observed the splash of a shell hitting the water, they would compute the range of the shot and telephone their result to the firing station. At the firing station, a second mathematician would adjust the range for changes in temperature, humidity, and wind. Once the series of shots was completed, a mathematician at a central office would compute ballistics coefficients and create range tables.  (p. 927)

Which sounds about as exciting as alphabetizing names for the phone book.  By hand.  In the mud.  Nonetheless, it had to be done and so it was.  Another set of computations, not as cold as the above,  involved solving differential equations numerically to computer trajectories.  The main staff started doing this, but then they got bored and made the enlisted men take over.  In Washington DC, however, there weren’t enough men to do they job and so the Diff Equ was handed off to women:

By early summer, [Forest] Moulton hired eight women. All had graduated from prominent universities during the prior two years. All had been mathematics majors. They came from University of Chicago, Brown University, Cornell University, Northwestern University, Columbia University and the George Washington University. For these women, the war was an opportunity to play a role, perhaps only briefly, on the public stage.  (p. 929)

Rosie the Riveter, meet Connie the Computer!  Actually, the only women mentioned by name is Elizabeth Webb Wilson, who turned down nine other job offers before taking on this job.  And apparently in their spare time, at least some of the women involved also fought for suffrage.

All in all, it was a good read.  Happy Veterans Day!

Math Careers: Astronaut

November 10, 2008

shanekimbroughAdditional training required.

Witness Lieutenant Colonel Robert Shane Kimbrough (who goes by Shane). He’s heading up into the sky Friday on the Space Shuttle Endeavor with Colonel Eric Boe and flight engineer Sandra Magnus. Lt. Col Kimbrough didn’t get his undergraduate degree in math (it was in Aerospace Engineering), but he got his MS in Operations Research from Georgia Tech and taught math at West Point. He’s given talks to school kids about how important math and science were in his being able to become an astronaut. And most recently he recognized his high school Calculus teacher (Sandy Sturgeon) in an official NASA letter because she was “instrumental in helping him get where he is today” and made calculus both real and fun.

For more information, see his official NASA biography or this recent article on ajc.com.

This is the premier post in a new series inspired by the common question, “What can I do with a degree in Math?”