Specs for "Project3". First Draft, 6 November 2000.
Please follow any revisions on our website. Title:
"Problems and Examples from Medieval Mathematics."
This project covers the period between Archimedes and Newton. It consists of
a connected document (e.g. 3 hole folder)of revised, rewritten, corrected,
elaborated homework problems listed below. There are two alternatives:
Alternative 1. If you feel you cannot do an adequate job on all of the problems,
and your midterm score was very low, you may substitute a
corrected solution to the midterm.
Alternativ2.It may consist of your journal, provided it covers a substantial
part of the course (roughly, your entries go past the middle of the notebook).
Figurative Arithmetic
(a) (Al-Khowarixmi) Exercise 1 on pg 83 of Edwards.
(b) Use Theon's algorithm to find the square-root of these six numbers
1.234, 12.34, 123.4 and 4.321, 43.21, 432.1. Explain the number of decimals
places you chose to calculate. Are there really six different problems here?
Anhyphereisis a.k.a. King Midas' Tiling Problem
Recall that Euclid's algorithm consists in "taking the smaller from
the larger" until this is no longer possible.
(a) Tile some handy "real rectangles". What did you learn from that?
(b) Consider the rectangle whose short side has length 3 units, and whose
long side has length 1 more than the hypotenuse of a 3 by 1 right triangle.
Are its sides commensurable, is its side:side ratio rational? why/whynot.
Weierstrass's Definition of Real Numbers
Here is the most useful version of the definition. Note how well it fits
our concept of a decimal.
Given a double sequence of rational numbers with these four properties:
a0 <= a1 <= a2 <= ...
b0 >= b1 >= b2 >= ...
every ai <= every bj
lim(n->oo) bn - an = 0
(note how we usually rewrite these in a more compact form)
then there exists a unique real number x between the ai and the bj.
Compare this to the Greek notion of "compression" (see Edwards)
(a) Why is X = 1.2112111211112... a real number?
(b) Express 1/3 in binary (binimals as opposed to decimals). Explain how
your answer is an algorithm for constructing 1/3 by "halving and doubling".
Bishop's Exercises on Area Preserving transformations.
(a) Reword Ex.1 so that the same algorithm (differentiate termwise,
multiply by x, evaluate) applies 3 times).
(b) Ex. 2 and Ex. 3. Be sure to distinguish between a simple shear, a
Cavalieri shear, a linear transformation, and their area preserving properties.
Bishops's Exercises
(a) Ex. 4. and Ex. 5. Why does a monozoid have area? How does a
monotonote non-decreasing (call it "rising") function define a monozoid?
(b) A 1:1, onto transformation from a plane to another that preserves lines
(i.e. the {U=T(X)| X lie on a line L} is itself a line), is called _affine_.
Show how an affine transformation always has the form:
u = a x + b y + p
v = c x + d y + q
or, in vector notation, T(X) = Ax + By + P .
So, now you have two ways of solving Ex.7. Either you work directly from
the definition of "affine" (tricky argument) or you use this additional
information on what affine transformations "look like".