M629 Lecture 7

"Lecture" for Week 7

The transition, or precalculus, period

Some procedural matters

We have reached a point in the course where Stillwell starts to give more material than we can cover in one semester. I have omitted some chapters from the reading assignments, except for the biographies at the end. (We can't skip Pascal, since he figures in number theory and probability as well as projective geometry.) The chapters left out are those I consider somewhat specialized or off the main track, but that is a personal judgment. Given my physics background, I consider mechanics elementary but projective geometry and elliptic curves to be off the main track. Some of you may disagree.
There is quite a lot of reading about the calculus era (next week), so I moved some of "transition" into last week and some calculus into this one. Logarithms were in last week's reading, but the problems are coming up now.
Although it overlaps slightly with the regular reading, something I really do want to you read is this: J. V. Grabiner, "The Changing Concept of Change: The Derivative from Fermat to Weierstrass", Mathematics Magazine 56 (1983) 195-206. It stretches from the transition period to the 19th century, but I've put it in the reading assignment for the calculus week. It's never too early to start.

Logarithms

In Fall '03 the textbook exercises on the late renaissance period did not inspire me at all, so I substituted some materials I had written for calculus students about logarithms. The M. 629 students liked them, so I've put them in the homework again. For you young folks who don't know what a slide rule is: Suppose you took two ordinary rulers and put the "0" end of one against the point 3.12 on the other, then read off on the second ruler the number opposite the point 4.29 on the first. You would get the sum 3.12 + 4.29, a rather clumsy way of adding two numbers. But now suppose that the labels on the rulers are the exponentials of the distances of the points from the ends, or, equivalently, each number is plotted at a position equal to its logarithm. Then, because of the law of exponents e^a+b = e^a e^b, the number you read off is the product of the two numbers you located on the rulers. This is actually a practical (cost-effective) way of multiplying numbers to 3-place accuracy -- or was, until Hewlett-Packard devastated the slide rule market around 1970.

Euler's totient function

On page 24 of "The Transition ..." you hit Euler's generalization of Fermat's "little" theorem, which is stated in terms of something called "phi". According to the first number theory book I found in my office [M. R. Schroeder, "Number Theory in Science and Communication, with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity", p. 113], phi(m) is the number of positive integers r smaller than m and relatively prime to m (i.e., r and m have no common factors, or (r,m) = 1). Thus, for example, if m is prime, everything smaller is relatively prime to it, so phi(m) = m-1. If you look closely, you will see this definition at work in the proof on p. 24.