## "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.