Archive for March, 2008

Hyperinflation

March 31, 2008

Zimbabwe has been in the news these past few days (having just held a presidential election).  In reading the latest reports on the vote counting, nearly every article gave as background material two statistics describing the economic hardships in Zimbabwe:  the unemployment rate is between 80% and 90%, and the annual rate of inflation is 100,000%.

Both of those facts are stunning, but I felt I could understand the first one — I can conceive of life in a society with an 80% unemployment rate.   But the inflation rate is so extreme, I found it difficult to continue reading, and felt a need to put that information into a context I could grasp.

To convert the interest rate to a daily rate, we imagine a daily inflation  rate of r, which in turn causes consumer prices to rise on average by a factor of $1+r$ per day.  Each subsequent day involves a further multiplication by the factor $1+r$.  Over the course of a year, we have prices increasing by a factor of 1000, and hence $(1+r)^{365} = 1000$, and $r = \sqrt[365]{1000} - 1 \approx 0.019$.

Thus the daily inflation rate is just under 2%.  A product costing $100 on a given day, would cost$101.91 the second day, $103.86 the third,$105.84 the fourth, $107.86 the fifth, and so on. In late March 2008, President Mugabe ordered food prices to be lowered to their levels of February 12, the day that pay raises took effect for public employees. By how much would prices have risen in a 45 day period, if the annual inflation rate is 100,000%? After 45 days, a$100 item would cost   $\ 100\cdot(1000)^{45/365}\approx \ 234.35$.  When Mugabe ordered food prices rolled back to their levels from one and a half months earlier, he was cutting prices by nearly 60%.

The Times Online (London) has a recent article detailing the brutal reality of these economic hardships, the consequences when prices for some goods and services can triple in just three days, the challenges of purchasing soap, cooking oil, and other necessities.

7 things about me you’d not likely guess

March 30, 2008

I’ve been memed! Sol on Wild About Math! tagged me last week to list 7 things about myself that you be unlikely to guess. Hmmmmm, what can I share that you might not know…. Click to find out.

March 29, 2008

Today is the Second Annual University of Rochester Mathematical Olympiad, so in honor of the students at all the schools (including ours!) that are participating, here’s the math video that my students play before their exams.

Good luck everyone! And if you want to try your hand at some other contests, check out last year’s test or the Math contest problems link over on Wild About Math!

A Question of Style

March 28, 2008

There’s an interesting discussion going on over at Ars Mathematica about proof styles, beginning with the question, “What makes a well-written proof?” I don’t have a complete answer, but some of the adjectives that come to mind are: understandable, lucid, concise (which can sometimes conflict with “understandable”), motivated, clever (I suspect there might be disagreement on that one). “Correct”, I suppose, ought to be #1, but let’s take that as a given.

A good example of what I think is a well-written proof is Eisenstein’s proof* of the Law of Quadratic Reciprocity.

Head on over and let Walt know what you find well-written.

* G. Eisenstein, Geometrischer Beweis des Fundamentaltheorems für die quadratischen Reste, J. Reine Angew. Math. 28 (1844), 246-248; Math. Werke I, 164-166; Engl. Transl. Quart. J. Math. 1 (1857), 186-191, or A. Cayley: Coll. Math. Papers III, 39-43**

** Found at Proofs of the Quadratic Reciprocity Law

March 27, 2008

This weekend marks the second weekend of the NCAA basketball tournament (aka March Madness). For many, the NCAA tournament is the last chance to watch some of the best college basketball around. For others, it’s a time to fill out brackets, decide which 10-7 upset to pick, flip a coin on the 8-9 games, and cross your fingers.

If you’ve ever participated in an NCAA office pool, you’re familiar with at least one way of scoring correct bracket picks. For example, each correct pick is worth 1, 2, 4, 8, 16, and 32 points, in the 1st, 2nd, 3rd, 4th, 5th, and 6th rounds, respectively. This makes each round worth the same number of points, and thus gives equal weight to each round. But is that the best way to do things?

When I was still in graduate school, several of us (looking for another way to kill time so as to avoid doing real work) decided to examine different schemes for scoring brackets. Not that we were participating in an office pool – just in case we ever wanted to join one. Or start one. Or run one for several years.

With the above scheme (1, 2, 4, 8, 16, 32), there are 192 points available in the entire tournament, and each round is worth 1/6 of those points, giving each round equal weight. This places a great deal of weight on the final game, and for whatever reason, we found that unacceptable. Of course, being able to determine the overall winner out of 64 (now 65) teams ought to be significant, but we didn’t want an incorrect pick to essentially eliminate someone from the contest. So we started analyzing other natural-looking sequences of length 6, computing the relative weights of each round for each sequence:

Points per game (Rds. 1-6) Relative round weights (%)
1, 2, 4, 8, 16, 32 16.7, 16.7, 16.7, 16.7, 16.7, 16.7
1, 2, 3, 4, 5, 6 26.7, 26.7, 20, 13.3, 8.3, 5
1, 2, 3, 5, 8, 13 23.4, 23.4, 17.5, 14.6, 11.7, 9.5
1, 2, 3, 5, 10, 15 22.4, 22.4, 16.8, 14, 14, 10.4
2, 3, 5, 7, 11, 13 29.8, 22.3, 18.6, 13, 10.2, 6

Certainly there are other “natural” sequences that could be analyzed. We ended up choosing liking 1, 2, 3, 5, 10, 15 (although I no longer remember why). Perhaps it was a happy medium between schemes which weighted the final game too low (5 or 6%) and schemes that weighted it too high (16.7%). Regardless, we congratulated each other on an excellent nerding.

Later that spring, I discovered that Microsoft has an Excel template for scoring NCAA brackets, and I went berserk playing with it… but that’s a story for another day.

Shall we play a game?

March 27, 2008

No, not Global Thermonuclear War* — more along the lines of Chutes and Ladders. The Pittsburgh Post-Gazette reported today on a study by Geetha B. Ramani and Robert S. Siegler in which children who played a numerical board game for four 15-20 minute periods over two weeks showed improvement in certain math skills even two months later. Click for more information about the study.

Musical Pi, Part 3

March 26, 2008

Finally, the last four tracks in the suite of π-based music, composed by Jon Turner. (See also part 1 and part 2.)

8. Quest 4 Pi 2
MM=288 175mm 2:26
The first part is the same as 1; after the central cadence on C, the harmony no longer changes, and this forms a coda. The guitar continues to shred the rhythm over the final C7sus harmony.
9. Circle of the Great Spirit 2
MM=72 94mm 4/4 5:13
19 digits of π, theme of 1, in 4/4 with variations:
Introduction: 457/0.
Theme: 31848/9 0/5 949/1 3B/6 9186/2 64/1 -4/0 5/0 7/0.
Variation 1: each duration is divided into two half-length durations.
(Variation 2: is track 1 above, CGS1, in triple meter, 3/4.)
Variation 3: durations are divided into 4, creating a rhythmic crescendo typical of classical variations.
Theme: closing anthem.
Coda: 457/0 eight times.

Bonus track(s):

10. Arc Tango X
MM=170 784mm 4/4 18:06
Same as 2, but continues way beyond 160 to 768.
Long jam already, in flux, could go way beyond…
11. Quest 4 Pi complete
MM=288 341mm 4:44
Finished on FZ birthday 07!
1 and 2 continuous.

Happy New Year (Again!)

March 25, 2008

So, did anyone head out to any New Year’s Eve parties last night? Not surprising, I suppose, since the New Year has been happening on January 1 in many parts of the world for quite some time. Exactly how long depends on what country you’re in.

The early Romans considered March 1 the first day of the year, which is why September, October, November, and December mean 7th, 8th, 9th, and 10th month respectively. In 153 B.C.E, however, the New Year was set at January 1. Even though that’s the same date that many people use today, its adoption (like that of the Gregorian Calendar) wasn’t completely straightforward. Click to discover all the different dates for the New Year, and the confusion resulting from neighboring countries starting the year at different times.

Fun with Statistics

March 24, 2008

Each day I get a brief email update from Science in the News, and one of the summaries today was of an article from the Washington Post about how parents of more than one child spend more quality time with first-born children than with later-born siblings. I glanced at the article and found a list of data that, depending on how you look at it, could be used to justify any one of a number of conclusions. Click to discover how the more children you have, the more/less time you spend with them [depending on how you look at it].

Easter and The Gregorian Calendar

March 23, 2008

Happy Easter! Since Easter played a key role in moving away from the Julian calendar, it seems fitting to talk about the adoption of the Gregorian calendar today.

As mentioned earlier, the Roman calendar had been having all sorts of problems until the time of Julius Caesar, who right before his death got everything back on track by adding a leap day (a second February 24th) every four years. This worked really well except for one small problem: the solar year isn’t exactly 365.25 days long, it’s about 11 minutes/year less than that. But 11 minutes is hard to notice, so everything seemed hunky dory for a very long time. Click to read more about the adoption of the Gregorian calendar and the fun that came along with different countries adopting it at different times!

The 29th Carnival is here!

March 22, 2008

The 29th Carnival of Mathematics is over at Quomodocumque, and it’s full of lots of good posts (including one of our own!). There were several blogs I didn’t recognize, so I’m looking forward to checking them out. Indeed, getting distracted with that very activity is the reason it’s taken me an hour so far to write this two-paragraph post.

Quomodocumque is a blog for sharing interesting things encountered by mathematician and author Jordan S. Ellenberg, a professor at the University of Wisconsin, Madison (the alma mater of two of the bloggers here!)

Can anyone tell me where the word “quomodocumque” comes from? I’m certain it’s Latin, but whether it’s a phrase or the name of something, or a made-up work I’m not sure.

Calculus in the tannery

March 21, 2008

How do you calculate the surface area of an irregular figure? If it’s a cowskin, you can buy an Electronic Area Measuring Machine to find it for you! The LOTO/SOFT model, for example, features continuous measure modeling, only 20 millimeters between photoelectric elements, and output in m², dm², tenths of dm², quarters of a square foot, tenths of square foot, or square inches. Not the right model for you? Don’t worry, there are several more to choose from.

I’m not completely sure how these machines work, but I’m guessing from the pictures and descriptions that each photoelectric element measures the length of skin that passes under it. Then, using the fact that the sensors are 20mm apart (so Δx=20mm) the machine must use the Trapezoid Rule, Simpson’s Rule, or some other form of numeric integration to estimate the surface area. Hooray — calculus in action!

Thanks to Mary Louise for telling me about this last week. She saw it on the “Tannery” episode of Dirty Jobs.

Sweater Math

March 20, 2008

Today (March 20) is what would have been the 80th birthday of Mr. Rogers. Fred Rogers was no stranger to math: if there are any young children around, you can find seven mathy Mr. Rogers activities here (the math mostly being of the kind found in recipes, and appropriate for the younger crowd).

In honor of Mr. Rogers and his penchant for sweaters, March 20 has been declared Sweater Day. So if you like to wear sweaters, this is the day to do it (as explained by Mr. McFeely in the video below): Click to see the video and read about some sites that use math for calculating how to knit the correct size of sweaters.

Musical Pi, Part 2

March 19, 2008

Following up (at long last) on Musical Pi, Part 1, we present the remaining nine pieces (plus bonus tracks!) in the suite of music based on π, composed by Jon Turner, professor of musical composition at Nazareth. (see also part 1 and part 3) Click for the next 7 pieces.