"Lecture" for Week 7/8

Early Greek homework
Well, I finally finished this grading! You'll notice that I'm learning to give short assignments.

2 (trisectrix): Let's agree to write this one off. I assigned it before I knew that I would post a document containing the equation. That left you (and me) confused about what else you were supposed to do about it. I'm sorry.

3 (Pythagorean triples) and 5 (square root of 5): These were fairly routine for most of you. Those who had trouble will get private comments, and those with especially helpful solutions will be invited to post them.

12 (1:root-3 division): Almost everybody realized that a 30-60-90 triangle gives you two segments in the right ratio. The responses then divided into three categories:

  1. Some people thought that was enough, and stopped.
  2. Some people thought that was not enough, and stopped in frustration.
  3. Three people provided three different methods of dividing a given line segment in the prescribed ratio (rather than attaching a given segment to another one to get a properly divided segment of a different length). De-Vonna's method is the simplest. I'll ask all three to post their solutions.

Essays: The issue of irrationals elicited more interesting responses than the question about an ancient thinker. I'll create a place in the discussion system for you to post your remarks.

Two extra problems of your choice: Too much variety here for comments. For the record, here are the numbers of students choosing each:

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.
More on logarithms
You may be amused by some materials I wrote for calculus students on logarithms, particularly "Logarithmic distribution of random data" and "How to make a slide rule (or logarithmic graph paper)". (In fact, I decided to make this part of this week's assignment.) 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 ea+b = ea eb, 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.
More on cubic equations
[PDF document]