Archive for August, 2008

Mathematician of the week: Louis Antoine de Bougainville

August 31, 2008

Louis Antoine de Bougainville was born November 11, 1729, and died August 31, 1811. While his mathematical contributions were modest, he has surprisingly strong name-recognition for an eighteenth-century mathematician…

By 1756, Bougainville had published two volumes on the integral calculus, explicitly presented as a supplement to and extension of L’Hopital’s Analyse des infiniment petits pour l’intelligence des lignes courbes (published in 1696, the first textbook on the differential calculus). Bougainville’s work earned significant praise, including Bougainville’s election to membership in the Royal Society of London. However, this publication also marked the end of Bougainville’s mathematical career.

After joining the French Army in 1754, Bougainville served with some distinction in the French and Indian war. By the early 1760s, Bougainville had joined the French Navy. In 1764, he establishing the first European settlement on the Falkland Islands (Port St. Louis), and during 1766 – 1769, he became the 14th known Western navigator, and first Frenchman, to circumnavigate the globe. During that voyage, his ships came upon the heavy breakers of the Great Barrier Reef, and turned away to the north, toward the Solomon Islands. (Bougainville thus narrowly avoided sailing upon Australia, some three years before James Cook’s expedition which claimed New South Wales for Great Britain.) Bougainville Island (politically part of Paupa New Guinea) was apparently named by Bougainville during this voyage.

The flowering vine bougainvillea is also named for Louis Antoine de Bougainville. A plant native to South America, Bougainville wrote extensively about it for European readers following his circumnavigatory voyage.

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Godzilla’s Dinner Party

August 29, 2008

It’s a little known fact that Godzilla likes to throw dinner parties. Some of these gatherings are formal dinners, with no fewer than four forks, but others are more intimate. Upon occasion Godzilla throws a dinner with only two guests, but then at the last minute a friend drops by and naturally Godzilla invites him to stay for enchiladas with mole sauce (or whatever the evening’s menu). This means that Godzilla has to add a fourth place setting to his triangular table. What to do?

Fortunately, Godzilla’s table is hinged and can turn from an equilateral triangle into a square at a moment’s notice. Allow him to demonstrate. In the mock up below the separate hinged pieces are colored for easy demarcation.

Godzilla prepares by looking at the table.

He slowly starts to separate the pieces:

You can see where the three hinges are (on the outside of the triangle) connecting the four pieces. Godzilla continues to spread them out. There’s a pretty star outline in the center.

He swings those bottom three pieces around…

and up towards to the top

Now it’s starting to look like a square:

And voilà! He’s gotten a square table! The party is saved!

Here’s the final formation in color.

Isn’t that cool?

There’s a slight problem with this design, though, in that there have to be at least four table legs, all close together. Fortunately Greg N. Frederickson designed a triangular table with a large enough piece in the center so that a single pedestal would do, and the six tiny swinging pieces could all be hidden with a linen table cloth. It’s the lead article (“Designing a Table Both Swinging and Stable”) of this month’s College Mathematics Journal. He has some nifty spiff animations here and even more information at the bottom of this page.

For more on these dissections, check out Ivars Peterson’s January 27, 2003 Math Trek column about how chemists were doing exactly the same sort of table spinning as Godzilla (without mentioning him by name, of course) using little plates that would self assemble into different kinds of shapes (although to do the triangle-square hinged switcharoo they had to be connected with thread.)

Time for the dinner party. Pass the enchiladas.

What’s your favorite number?

August 28, 2008

In the middle of class today (History of Math, full of Junior math majors) one of my students asked out of the blue “What’s your favorite number?” I said it was 4, because back when I was in elementary school there was a joke that went around where Person A would ask Person B for their favorite color, favorite animal, and favorite number. After Person B replied, Person A would come back with something like, “You’re going to marry a purple dog with 7 legs. HAHAHAHAHA!!!!!!!” The idea of a blue cat with anything other than 4 legs really bothered me, so my favorite number immediately became 4. The fact that I was going to marry a blue cat apparently didn’t cause any angst at all.

This quickly evolved into me asking my class their favorite numbers. And not only did every single person have one, but they all had clearly thought about this. One person also liked 4, but explained it was the ONLY even number she liked. Other than 4, she really hated even numbers. Just hated them. One person liked 24 because it was her birthday. Two people liked numbers (1 and 30) because they was their jersey numbers. One person had two favorites: 6 and 27. One person liked 144 because (1+4+4)×(1×4×4)=144. [This only works for 0, 1, 135, and 144. Isn’t that cool?] And 19 was the favorite number of not one, but two people. For some reason that struck me as a dark and sophisticated favorite number to have. It’s so….prime.

Now I’m feeling all jealous of everyone else’s favorite numbers and thinking I should let go of the whole elementary school thing. I’ve noticed that I’ve used 87 three times in the last two days as a synonym for “a lot” [as in, “You can reach me by email because I check 87 times a day”] so maybe 87 is my subconscious favorite number.

What’s yours? Why?

(The Onion had this question in a survey back in 1996. I was living in Madison at the time and saved it, but can’t find it now — it’s somewhere in the Archaeological Dig that is my desk. But you can see it the results online here.)

Mathematician of the week: Johann Lambert

August 27, 2008

Johann Lambert was born on August 26, 1728. (His father, Lukas Lambert, was a tailor, and was not involved in the discovery of a primality test for Mersenne numbers.)

By the time he was a teenager, Lambert found himself working full-time for his father, while pursuing his academic studies at night. Over the ensuing decade, Lambert held a number of jobs while teaching himself mathematics and science. He first attracted attention on the academic stage with an article on heat, published in 1755. A mere three years later Lambert published a book on the passage of light through various media.

Lambert struggled throughout his life to find academic employment, but in spite of that he was quite prolific, publishing over 150 articles and books by the time of his death at the age of 49.

He had two particularly significant mathematical contributions:

  • His Theorie der Parallellinien (1766) was a study of the consequences of the negation of the parallel postulate, including the correlation between the sum of the angles of a triangle and its area in a non-euclidean geometry.
  • In 1768, Lambert gave the first rigorous proof of the irrationality of π. Specifically, he showed that if x is a nonzero rational number, then both e^x and \tan x must be irrational, using the continued fraction representations of those two functions.

Google Calculator has a little trouble

August 26, 2008

Seriously, we don’t mean to write only about math mistakes and promise to have something non-mistakey tomorrow and even Thursday. But for today, it turns out that Google Calculator is having a spot of trouble. If you type in a big calculation like 800000000000010-800000000000007 you get the expected answer of 3, but if you become all sneaky and change that to 800000000000010-800000000000008 then you get 0. The article “Google’s calculator muffs some math problems” on cnet news yesterday has other examples to play around with. Google released the statement:

We are aware that the calculator tool in Google Web search is not working properly for certain calculations, and we are looking into this problem further. We apologize for any problems that this causes our users.

which suggests that this might turn out to be more than floating point error. Maybe.

As a bonus mistake, the article above refers to Google wanting to raise $2,718,281,828 in its IP0 several years ago [those digits look familiar to anyone?] but calls it $2,718,281,828 billion instead of $2.7 billion. As one commenter points out, $2,718,281,828 billion is a lot of money.

Regretting the Error

August 25, 2008

We often post Math Mistakes on this site: times when a little checking could have gone a long way. The same thing applies to journalism, and on the site Regret the Error Craig Silverman posts “media corrections, retractions, apologies, clarifications and trends regarding accuracy and honesty in the press”. They’re often amusing, like the recent correction:

A picture purporting to show Apple’s corporate headquarters in Cupertino (Google pipped – Apple the new king of Silicon Valley as market value overtakes hi-tech rival, page 3, August 15) in fact showed Symantec’s headquarters nearby

Not surprisingly, there are a lot of mistakes that have to do with numbers and math. A LOT of mistakes. These are a little different than the math mistakes we usually post, because the problem is typically in the reporting rather than the original story, but they still include such stories as:

The Star Ledger, August 20, 2008
Due to an editing error, the For Collectors column in Saturday’s Abode section reported incorrectly that the 2009 Double Eagle gold coin would sell for $20. While the coin will have a $20 denomination, it will contain an ounce of 24-karat gold and will sell for approximately $900, depending on the value of gold at… (Story)

New Scientist, August 4, 2008
We said that Australian companies “forecast spending $800 between 2002 and 2013 on geothermal exploration” (19 July, p 24). That should have been $800 million. (Story)

The New York Times, July 15, 2008
An article last Monday about the United States Olympic swimming trials, including the accomplishments of Dara Torres at age 41, misstated the age of a Canadian swimmer from the 1972 Games. Brenda Holmes was 14, not 44, when she competed for Canada. (Story)

For a list of many of the mistakes (often titled “Fuzzy Numbers”) see here.

Carnival of Mathematics #39

August 22, 2008

The Carnival is here again! Celebrate the nearing end of August by visiting the 39th Carnival of Mathematics over at It’s the Thought that Counts, a blog by A and Z about “politics, society, science, morality, religion, and whatever else comes to mind”. (One post of theirs from earlier this month that I particularly enjoyed: Gallons per mile, which examined whether miles per gallon is the most appropriate way to measure a car’s efficiency.)

Returning to the carnival, this edition features a neat problem about the number 39 (If 39 people are sitting around a circular table and no one is in the correct place, prove that there is some rotation with at least two people in the correct place. Only they said it with more storytelling flair.), and a great collection of other posts. Enjoy!

The Summer Newsletter is Here!

August 21, 2008

The Summer Newsletter of the Nazareth College Math Department is up! Okay, so it’s appearing at the end of summer, but that’s the beauty of having no actual deadline: if you’ve decided to spend your summer having a cutie pie baby boy join your family (like Batman) or traveling and seeing how long you can go without mowing the lawn (like TwoPi and myself), August is still technically summer so you’re good.

We normally name each issue after a different mathematician, but this issue is named The Godziller after our good friend Godzilla, who shows up here from time to time. And because we’re nothing if not lazy efficient, several of the articles are “gently used” blog posts from the past couple months. But there’s lots of local news about our students, and the last page has a neat Sudoku Puzzle and some math problems (of varying difficulty) to work on which, if you solve them, will lead to fame and fortune. Or at least a hearty Congratulations and link in the next newsletter.

An old math mistake: crab boat buyout

August 20, 2008

I was just looking some stuff up on Google (always eager to find math mistakes in the news), and I ran across this story from June 2004 of a math mistake that caused all sorts of trouble in the crab boat industry. Apparently there was a program in which federal fishing authorities would buy back crabbing boats [to compensate for their being too many boats for the amount of crab available], with the buyouts related to the fishing history of the individual boat. The problem occurred with boats that had multiple owners: if a boat had three owners, for example, then the catch was counted three times (once for each owner) and the boat’s total was incorrectly listed as being three times as large as it actually was. The article continued:

This led to another mistake. The NOAA officials then used the flawed history totals to calculate how much crab would be divided among the remaining boats — and how the fleet would repay the buyback loan.

According to a later article the mistake was apparently rectified and 25 boats bought out (compared to the original 28 that were planned).

Speaking of math mistakes, on God Plays Dice last week there was a quote from The New York Times which referred to 300,000 million Chinese playing basketball. That’s a lot of people.

Babel Fish, Snickers, Godzilla, and Garfield

August 19, 2008

Yesterday’s post was about how Pole Vault conversion between Metric and Imperial is not symmetric: 16′ 1″ converts to 4.90 meters, but 4.90 meters only converts to 16′ 3/4″.

It turns out that language translation programs aren’t symmetric either. For example, Babel Fish translates the phrase Gozilla sure eats a lot of Snickers bars! as Godzilla mange sure beaucoup de barres de Snickers! in French, but then if you translate the French phrase back into English you get Godzilla eats sour much bars of Snickers! Sour (surely?) Snickers — bleh. Godzilla can have ’em.

(Not surprisingly, translation via Babel Fish is also not transitive: if you translate Godzilla mange sure beaucoup de barres de Snickers! into Dutch you get Godzilla eet zurig vele staven van Snickers! but if you translate Gozilla sure eats a lot of Snickers bars! into Dutch directly you get Zekere Gozilla eet heel wat bars van Giechels!.)

The lack of symmetry in translation programs resulted in a great set of cartoons in Garfield Lost in Translation on Blogoscoped, in which Garfield cartoons were translated into Chinese and then back into English. You can see the cartoons here, but here’s one example:

Author Philipp Lenssen comments here that he had to use Babel Fish for the initial translation into Chinese because:

Google’s translation were – unfortunately for the purpose of this – far too good to be funny most of the times, even when trying multi-language chains (e.g. English to Japanese to German to English).

There’s more of this on The Language Log and also on The Lansey Brother’s Blog (using the Gettysburg Address).

Speaking of Garfield, if you haven’t seen Garfield Minus Garfield it’s worth a look. Each day a new Garfield cartoon is posted with all the animals and their monologues removed. The results look like this:

Ah, Garfield. Providing amusement on so many levels.

How to lose inches without even trying.

August 18, 2008

I was just looking over this morning’s paper, and reading the story “Russian champion disses Jenn” about how pole valuter Yelena Isinbayeva was pretty sure that she was going to win the gold [which she did later today], and that Jenn Stuczynski was unlikely to surpass her. The third paragraph in the story read:

Asked if she was annoyed by media suggestions that Stuczynski was a challenger after her U.S. record vault of 16 feet, 3/4 inch (4.90 meters) earlier this season, Isinbayeva was utterly dismissive.

This was followed shortly by a quote from Isinbayeva:

“They said, ‘Wooooo’ when she jumped 4.90 (16 feet, 1 inch), but I jumped this height four years ago. It is nothing special.”

Personally I think that vaulting over 16 feet is pretty special indeed: I believe these are the only two women who have ever done it. But what caught my eye was that 4.9 meters was stated as the equivalent to 16 feet, 3/4 inch in the first case, but was translated to 16 feet, 1 inch in the second.

So I checked. It turns out that 16 feet, 3/4 inch is 489.585 cm, which does round to 4.90 meters. Furthermore, 4.90 meters is 16 feet, 0.91 inches, which rounds to 16 feet, 1 inch. So my initial thought was that everyone was just rounding.

Then I checked the USA Track and Field conversion site which had the same hedging, but in the opposite direction — everything is rounded down instead of up. It says 16′ 1″ should be converted to 4.90 meters, but 4.90 meters should be converted to only 16′ 3/4″ . And what should 16′ 3/4″ be converted to? To 4.89 meters. Which converts to 16′ 1/2″. Which converts to 4.88 meters. Which converts all the way down to an even 16′. And of course 16′ converts to 4.87m, which converts to 15′ 11 3/4″, which — hold on to your hats here — also converts to 4.87m. Finally, a fixed point!

And by transitivity of conversion, we have that 16 feet, 1 inch is equivalent to 15 feet, 11 3/4 inch.

Photo (cropped) of Yelena Isinbayeva by Eckhard Pecher, published under Creative Commons Attribution 2.5.

Cut your monthly payments in half!

August 17, 2008

There’s an ad that’s been playing on the radio lately, advertising to cut your monthly credit card payments in half by working with some sort of loan shark lending company. Is this really such a good deal?

It depends, of course, on how much you need the extra money right now, but I decided to play around with the numbers a little bit just to get a sense of how much those lower monthly payments actually cost. By way of example, I assumed the amount of debt was an even $7,000, because that’s somewhere around what different sources say is the mean credit card debt per household. I also initially assumed that the interest rate was 10% (compounded monthly) which is high for mortgages but low for credit cards. Finally, I assumed that the interest rate would stay the same, and that the lower monthly payments would be offset by being in debt for a longer amount of time.

Let’s suppose that you owe that $7,000 and are paying 10% annual interest, and you’ve decided that you want to be debt-free in 5 years. You can accomplish this by paying $148.73 per month, for a grand total of $8,923.80. That’s your initial debt of $7,000, plus $1,923.80 in interest.

What if you want to cut that monthly payment in half? You’ll need to extend your payments not to 10 years, but to between 15 and 16 years because of the magic of compound interest. Keeping the $7,000 and the 10% interest rate, equal payments spread over 15 years would require a monthly payment of $75.22. And the benefit for only having to pay (just over) half as much each month? You’d pay a grand total of $13539.60. That’s right, the amount paid in interest — $6,539,60 — is more than triple what you would have paid over 5 years!

If your interest rate is higher, like 15%, the picture is even bleaker. Paying off $7,000 over 5 years at 15% annual interest results in a monthly payment of $166.53, for a total payback of $9,991.80. But in this case, cutting that monthly payments in half is impossible. The least you could pay is $87.50 per month, and that would just cover the interest so your actual debt would never drop down. Not really good money management.

I do believe that sometimes cutting the payments down for a while is the best scenario for an individual, even worth the added cost. But when I hear ads like this I’m reminded of a friend who’s job for a short time was to try and “help” people by making offers just like this. She hated it, and she quietly cheered when a customer would check more into the numbers and realize that it wasn’t such a good deal in the long run. She quit that job as soon as she could.

I want to make a Chocolate Chip so Big…..

August 14, 2008

Last night at dinner, out of the blue, out 8-year old announced, “I want to make a chocolate chip so big that if it was 6 ft tall, a person would be a millimeter.” We all pondered the Giant Chocolate Chip, and then he asked if it would cover the United States. I said I didn’t think so, and he asked if it would cover any state. “Maybe Rhode Island,” was my reply.

Or maybe not. Curious about just how big that chocolate chip would have to be, I bought some “research supplies” and measured them. It turns out that a chocolate chip is 8 millimeters tall (with a tiny bit of variation, but not enough for me to measure effectively). According to the bastion of truth, the average height of an adult male in the US is 175.8 centimeters (just over 5′9″) and the average height of an adult female in the US is 162.0 centimeters (almost 5′4″). Therefore I declare the average height of an adult in the United States to be 168.9 cm. This gives us:

\frac{\mbox{Giant Chocolate Chip}}{\mbox{person (168.9 cm)}} = \frac{\mbox{6 feet}}{\mbox{1 mm}}

That’s a lot of different units there. Let’s convert to centimeters. The millimeter is just 0.1 cm, and we know that 6 feet is 6·12·2.54 cm since there are 12 inches in a foot and 2.54 centimeters in an inch. Speaking of which, did you know that there are EXACTLY 2.54 centimeters in an inch? That even though inches are way older than centimeters, that inches have been redefined to be exactly 2.54 centimeters? Apparently this happened in 1958, but I only found out about it last month.

So anyway, we know that 6 feet is 182.88 cm, which gives us:

\frac{\mbox{Giant Chocolate Chip}}{\mbox{person (168.9 cm)}} = \frac{\mbox{182.88 cm}}{\mbox{0.1 cm}}

This means that the height of the Giant Chocolate Chip is (168.9)·(1828.8)=308884.3 centimeters. That’s about 3.1 kilometers, or about 1.9 miles. That’s a BIG chocolate chip.

But not big enough to cover Rhode Island. The diameter of a regular chocolate chip is 1 centimeter, which is really cool if you are looking for a mental way to envision centimeters. That means that the diameter of the giant chocolate chip is about 2.4 miles, giving an area of 3.76 square miles. And Rhode Island is actually 1,545 square miles, so you’d need 410 giant chocolate chips to match the area.

But this got me thinking — there’s a lot of chocolate in that humongous chocolate chip. What if we melted it down and spread it out over Rhode Island? How deep would it be? Thinking of a chocolate chip as a cone (which ignores the little swirl on the top), the volume of chocolate is (1/3)·(area of the base)·(height), which for our Godzilla chip is about 2.4 cubic miles. And if it was really hot and the giant chocolate chip melted, it would cover Rhode Island to a depth of 99 inches.

Yummmmmmm!

Mathematician of the Week: Alonzo Church

August 12, 2008

Alonzo Church was born on June 14, 1903, and died August 11, 1995. Essentially his entire early academic career took place at Princeton University, having completed his AB (1924) and his PhD (1927, under Oswald Veblen) there, and then serving as a professor of mathematics from 1929 until 1967. (After retiring from Princeton in 1967, he taught at UCLA as a professor of mathematics and philosophy until 1990.)

Church’s most significant mathematical contribution was the creation (with Stephen Kleene) of the λ-calculus, a formal system in the language of functions.

Church is probably best remembered for Church’s Thesis, the claim that every effectively computable function is in fact a function that is definable in his λ-calculus. Kurt Gödel balked at this claim, and introduced the primitive recursive functions as a more natural alternative to model the notion of effective computability. Stephen Kleene, a student of Church, showed that in fact the functions definable in the λ-calculus exactly correspond to Gödel’s primitive recursive functions. By the late 1930s, another notion of computability had been put forward by Alan Turing, and it too had been shown to be equivalent to λ-definability.

The sets of λ-definable functions, primitive recursive functions, and functions implementable as Turing Machines, are identical sets of functions. This agreement of three diverse approaches to formalizing the vague notion of “effectively computable” is viewed as strong evidence that all three approaches have in fact captured that concept. At its most general, Church’s Thesis is the claim that effective computability is equivalent to these three formalizations. Given that “effectively computable” is unlikely to ever be formally defined, Church’s Thesis remains an unproven (and unprovable) claim.

Alas, Church’s Thesis first appeared in 1936, and was not a part of Church’s (doctoral) thesis of 1927 (Alternatives to Zermelo’s Assumption, an attempt to create a logic in which the axiom of choice is false).

Math Fails

August 11, 2008

Two photos from The Fail Blog:

fail owned pwned pictures
see more pwn and owned pictures

salefail.jpg
see more pwn and owned pictures

The Fail Blog is a fantastic place to visit, though not all the photos are safe for work [and certainly many of the comments are safe for neither work nor home.]

Edited 8/28 to add: Apparently that second photo is actually a promotional poster for The Simpsons — the KWIK-E-MART in the corner should have tipped me off.  But the fact that one can buy a twin pack for more than twice the price of one (as mentioned in the comments below) meant that I never question the reality of this supposed promo!  And the first picture is real, I believe.