"Lecture" for Week 11/12

Renaissance and transition homework

Cardano and Bombelli: No big difficulty here. I asked for two good solutions to be posted, one working from the formula given in the text and one from the algorithm that led to the formula. (See also the following remark.)

Trisection: Here the example given in the text for Bombelli's method was misleading. In general the real part of the cube will not be the same as the real part of the cube root. Bombelli (back in the previous problem) apparently guessed that the real part of the two cube roots would be 3/2 by observing that their sum must be 3 -- remember that he knew this answer to the cubic equation beforehand and merely wanted to rationalize it from the Cardano formula -- and had some kind of confused intuition that the two terms were complex conjugates.

Back to the trisection equation. To analyze it I had to return to the formula for the solutions of the cubic given in the National Bureau of Standards Handbook (ed. by Abramowitz and Stegun), which I'll summarize here for the special case where the quadratic term is zero. (Forced to lapse into TeX here.)

Logarithmic data distribution: As many of you noticed, this doesn't always work precisely as advertised. I think there are two main reasons for that:

  1. not enough data points to get a good statistical distribution; or
  2. the data does not "span several orders of magnitude", or suffers some other selection effect.
A particularly instructive case of the latter is the 25 largest American cities. If we look just at the populations over a million, we see something that decently resembles the expected logarithmic distribution (but a little heavy at the bottom): If we press onward, we encounter the 25th city at 493,559. Naturally, there are quite a few cities bunched above this point. So, when the distribution is arbitrarily cut off there, there is an excess of data points with first digit 5 or 6, and the distribution as a whole doesn't look logarithmic. On the other hand, choosing 25 cities and towns at random from the road atlas would give a better result.

Rule of 72: Everybody chose to do this one; past a certain age, pensions have more resonance than wildlife ecology. Most everybody saw that the relevant formula is

t = ln(2)/ln(1+r),

where 100r = R is the interest rate in percent. Since ln(2) = .69... and ln(1+r) is approximately r for small r, we have t approximately 69/R. So, why 72 instead of 69? For one thing, 72 has lots of divisors, so it makes the back-of-the-envelope calculations come out even. More quantitatively, note the Taylor expansions

ln(1+r) = r - r2/2 + ... and hence

1/ln(1+r) = 1/r [1 + r/2 + ...].

For a typical interest rate like 6%, 1 + r/2 = 1.03. And 69 x 1.03 is not 72, but it's getting there! No number in the numerator will work exactly for all interest rates, but a number slightly greater than 69 will do a decent job for any plausible interest rate.

Calculus exercises

The first 4 were straightforward. I think we and our students are all happy that "subtangent" and "subnormal" have disappeared from the syllabus.

De Sluse: This topic should be X-rated: no high school algebra students allowed to see the calculations! When you look closely, you see that de Sluse is just doing what our textbooks now call implicit differentiation, but in a horribly ambiguous and confused notation.

Galois and Bell
In this week's assignment I recommended the article by Rothman about myths in the biographies of Evariste Galois. Rothman is especially hard on the author Eric Temple Bell. Bell's books on mathematics and mathematicians were immensely popular around the middle of the 20th century; I read several when I was in high school. Today they are out of favor, being regarded as inaccurate and antifeminist (although I think no more sexist than most books about math history from that period and earlier). That's why they are not listed in our suggestions of "general histories" although more than one of them would qualify for that category on the basis of content. It turns out that Bell himself was a strange character (he had some private demons, as they say). There is an article about him by Constance Reid, Am. Math. Monthly 108, 393-402 (2001).