- Homework 4.3 is graded
- #18: induction works nicely
- You've got to prove things in general;
you can only rarely prove things by example.
- Due today, 11/30: a one-page, typed proposal for your paper/project. It should give me a good idea of what your focus is.
- For Wednesday: hand in the remaining problems from chapter 4 (4.4 and 4.5) and the first problem set (5.3) from chapter 5. This is the last problem set I'll collect. Remaining problems will make up part of your final exam.
- For Monday, 12/7: please provide a one-page, typed revised proposal for
your paper/project. It should let me know that you're making
good progress!
- Lessons from 6.1/6.2
- Commercial from Chris; next (in order): Jason, Adam, Debra
- Lessons from 6.3
- One last word from Alcuin of York:
Spencer's solution of the men and sisters river crossing! - Section 6.2: The Liber_Abaci and Liber Quadratorum (the Book of Squares)
- "The greatest mathematician of the Middle Ages was Leonardo of Pisa, better known as Fibonacci.... [T]he mathematical renaissance of the West dates from him." (c. 1170-1250)
- The Leaning_Tower_of_Pisa and he are just about twins.
- Read the opening sentences of his Liber Abaci (p. 280):
- These are the nine figures of the Indians:
- 9 8 7 6 5 4 3 2 1
What a start! Arabic numerals introduced, via his "Book of Counting" (his father ran a customshouse). The arabic numbers were first introduced in the Codex Vigilanus around 976 -- p. 282). By the beginning of printed books, c. 1450, our arabic numerals had taken the form in which we know them today.
- "The Liber Abaci embodied virtually all the arithmetical knowledge of Fibonacci's time...." (p. 280)
- Even so, it was not translated into English until 2002, on its 800th anniversary!
- Let's do one problem from the Liber Abaci: #3, p. 287 (we'll come back for more in a bit!).
- The Liber_Quadratorum (the Book of Squares)
- Devoted to Diophantine problems of second degree, including the tournament problems from the court of Emperor Frederick II (proposed by John_of_Palermo
- He also established the Brahmagupta-Fibonacci_identity (see exercise #8, p. 288), which says that the set of sums of two squares is closed under multiplication.
- Notice Fibonacci's notation :
- (arabic in origin) for writing fractions (see p. 285)
- right to left....
- Remember that Fibonacci borrowed from the Abu Kamil commentaries on al-Khowarizmi's work.
- Jordanus_de_Nemore (c. 1225).
- Another high point of the 13th century mathematical scene in Europe.
- May have been at the University of Toulouse (founded 1229).
- More on his murky past....
- Known almost exclusively through his works:
- De Triangulus -- medieval geometry at its best ("rigorous -- and frequently new -- proofs of Euclidean Theorems" -- p. 286)
- De Numeris Datis -- on advanced algebra
- Notation, again: Jordanus used letters to represent numerical quantities throughout, advancing the evolution of algebraic notation
- Forgotten for awhile, but resurrected by Francois Vieta 350 years later!
- Numerical illustrations used Roman numerals
- There followed a period without any particular luminous successors (until the opening salvos of the Renaissance).
- "Fibonacci is remembered today mainly because a nineteenth century French number theorist, Edouard Lucas, attached his name to a sequence that appeared in a trivial problem in the Liber Abaci." (p. 289)
- The Problem: bunnies (p. 289). Here is the solution in the Liber Abaci
- A model for recursive definition:
with
and
.
- We've already talked about them a little (ratio of successive is golden, see p. 294)
- Let's look at the famous Fibonacci puzzle, p. 293, which is based on the identity
- Fibonacci
Nim (we'll adjust the result of #1, p. 295)
- The Fibonacci numbers provide a unique decomposition of the natural numbers, analogous to the primes.
- Every natural number is either Fibonacci, or can be written as the sum of non-consecutive Fibonacci numbers in a unique way (up to order).
