"Lecture" for Week 5

State of the Class Address

Well, the book is getting much better now! I was disappointed in the first three chapters, but we are now to the point where Katz's treatment is clearly better than Allen's.

On the other hand, your complaints that the instructions are not clear for many of the exercises are well founded. I would like to make a list of exercises for which the instructions could be improved. Also, some way of annotating or commenting upon your papers that does not take up an unreasonable amount of the grader's time (or mine) needs to be found. For some reason that problem was not so visible when I taught the course before. I do recall that then I put generic comments, or even complete solutions, into the "lectures" for later weeks, and also that I asked students to post solutions to individual exercises on [the predecessor of] Vista when those solutions were particularly illuminating.

Please nominate your favorite exercises for either of these treatments (sharpened statements for the students of the future, or posted solutions for students of the present). I will start the process below with the Chinese remainder problems (5.19 and 5.20).

Islamic and Medieval mathematics

I really have nothing interesting to add here. I did make an attempt to find out why Leonardo of Pisa is called "Fibonacci". Both Katz and Allen say that the name was not applied to him until the 19th century; Katz obviously dislikes the name and brings it up as seldom as possible. But Carl Boyer's history-of-math book says that it just means "son of Bonaccio" -- Leonardo being himself one of those rich merchants' sons that he and his successors made a living teaching, according to Katz p. 214.

Betraying my physics background (and my years as a calculus teacher), I find the most interesting figure of this period to be Oresme, who apparently was the first to draw graphs of functions representing quantities other than positions, and had some understanding of the relations between position, velocity, and acceleration. I wish authors would tell us how his name should be pronounced -- "ORAME" like a modern French name, or what?

Qin's method for solving congruences

Thanks to Angela Walker for forcing the issue here in a relatively timely manner. The clue to the subtle point is Katz's final comment, "Qin always adjusted matters so that the final coefficient was positive." Unfortunately, Katz didn't adjust the exercises that way. Sometimes when you apply Qin's counting-box algorithm, you get a 1 in the bottom right spot instead of the top right. In that case, first of all, the relevant remainder is the number in the bottom left (instead of the top left). And, second, that number must be interpreted as negative. (Watch what happens to the coefficients of 65 throughout the example in Katz's paragraph (ending in the sentence I quoted) compared to Figure 5.11.) This negative value for x_i is a valid solution of the congruence [P_i x_i = 1 mod m_i], but it is natural to add m_i to it to get a solution in the standard range. Alternatively, you can use x_i as is in the formula for N, but be prepared to add M to N to get a positive value.

In Exercise 19, we first get x_1 = 0. The trouble comes in finding x_2, where the counting board
1 3
0 4
reduces by the algorithm to
1 3
1 1 <-- LOWER LEFT IMPLIES x_2 = - 1.
and we must stop there, not go on to
4 0
1 1
Thus we can take x_2 = m_2 -1 = 3, and we get N = 0 + 1x3x3 = 9. Alternatively, keep x_2 = -1 and compute N = 0 + 1x3x(-1) = -3, and adding 12 we get 9.