tingilinde

tools for recreational math

Mathematica 7 Home Edition is the real thing (minus 64 bit capability) as far as I can tell with what amounts to a hobbyist only license.   The commercial license is well over $2000, but for $295 you can play with math in the privacy of your own home as long as it is only for hobbyist purposes.

Really neat stuff.

means and harmonic means

Today I was reading a "research note" by an automotive analyst who was talking about what would be required to increase the fleet average mileage for Ford to meet the new CAFE standards.  Reading through it something seemed fishy and sure enough - he had used a simple mean for his back of the envelope calculation.

Consider a very small car company that makes 5 cars a year.  They get 19, 21, 22, 24 and 100 mpg respectively.  The mean is 37.2 mpg.  The company comes very close to the new standards with 80 percent of their fleet doing poorly and one spectacular car rounding things out.

The problem is miles per gallon is a flakey measure.  If you are interested in how much fuel you buy and really want to compare cars, a better approach is gallons per mile (or 100 miles if you want bigger numbers -- some europeans use liters per 100 km).  We are interested in the inverse - a rate of sorts.

If you use something as wacky as mpg, the way to attack the problem is to calculate the harmonic mean.

Using the same five cars we now get 25.33 mpg - not close to being good enough.  

The CAFE standard uses a weighted harmonic mean which is just: , where the weights w_i are associated with the data x_i.  The unweighted example is just the special case where each weight equals 1.

_____

It may be that you have never heard of a harmonic mean, but if you think about this class of problem a bit you either work in gallons per mile or derive a harmonic mean on your own. The point is that people have a lot of data available and do things with it.  The trick is knowing what to do.  This guy should have known better - he must have been unfamiliar with the subject mater.  

A friend uses the phrase "naked without data"  (a nice way to frame things), but that presupposes you know how to handle the data. You may think you are clothed, but you are always naked if you are innumerate.  (apology to Hans Christian Anderson)

the gift of symmetry

Some people pay to have stars named as gifts - unfortunately, besides feeling good about themselves, there is no recognition of this as such things are not for sale. But Marcus du Sautoy offers to name a symmetrical object in hyperspace for you. Donations go to help people in Guatemala.

(via the Guardian's Science Weekly podcast)

His new book on symmetry is available now. Not only is Marcus a well-known mathematician, but he is a good popularizer of math and an engaging speaker.

ready for the 15th?

Euler will be 300 on April 15.

One of the more remarkable mathematicians and the guy who first noted the most beautiful equation ever: e^iπ + 1 = 0.

What is your list of the best mathematicians ever? If I had to pick four, they would be: Archimedes, Newton, Euler and Gauss.

multiplication on a gerbert's abacus

and numerous other obscurities. Along the way you get a sense of the invention of numbers across civilizations and centuries.

If you know anyone who loves numbers a fantastic last minute gift would be Georges Ifrah's The Universal History of Numbers. Get a copy for yourself if numbers are a personal passion.

math from the subcontinent

The audio for today's lunchtime walk was this week's program from the BBC's In Our Time.

Indian Maths ...

Recommended listening (which is true of most of these programs)

find it on your friendly podcast aggregator

name that equation

an amusing use of cpu time...

[10n/(10n - 1)]k

and more fun

on generating interesting sequences with simple fractions. (via Science News)

an excerpt

Calculate 100/89. You get the decimal expansion 1.1235955056 . . .

Look closely, and you'll see that this fraction generates the first five Fibonacci numbers (1, 1, 2, 3, and 5) before blurring into other digits. Recall that, starting with 1 and 1, each successive Fibonacci number is the sum of the two previous Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.

Calculate 10000/9899. This time, you get 1.0102030508132134559046368 . . .

This fraction generates the first 10 Fibonacci numbers (using two digits per number). Going further, the fraction 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number).

Note that, in successive fractions, two 0s are appended to the numerator and a 9 to the beginning and end of the denominator.

Will the next fraction, 100000000/99989999, generate the first 20 Fibonacci numbers? Does the pattern continue forever? The answer appears to be yes.

math

math as done by students with creative, if wrong, approaches

poincaré in the new yorker

Manifold Destiny
Who really solved the Poincaré conjecture?

A nice piece Sylvia Nasar and David Gruber in the current August 28, 2006 issue of The New Yorker.

recommended (unfortunately only in the print edition, but that works well in the bathtub)

Of course it is old news, but Grigory Perelman rejected the Fields Medal today.