Mathematics 153
Notes from April 10, 2020, lecture session

Groupme:
There is an account for this class: https://groupme.com/join_group/59059982/zEUxn9fJ
 Relevant directory files:
Here is a listing with comments on their significance:
Files for Unit 1.
history01c.pdf  This is the basic file describing the Method of False Position.    
FALSE1.xls  This is a spreadsheet file which traces the method through successive approximations.   The example studied in class was the last one:  Find the value of  x  such that  x2 - 5  =  0.
history01c.program.pdf  Here is an analysis of the general procedure which can be used to set up a computer program for using the method.
Files for Unit 2.
history02.pdf  This is the main file for the unit and should be read from beginning to end.  After doing so one should read the cited sections of Burton.
historical-maps1.pdf  The first map gives the locations of many places mentioned in the history of Greek mathematics, and the second is focused on Greece, the Aegean Islands, and the western portion of Asia Minor (Turkey in Asia).  If these are difficult to view, try increasing the magnification to 200% or more.
historical-maps1a.pdf   There are separate maps, one for ancient Greece and nearby areas and another for ancient Italy.  These are more detailed and may be easier to read.
orthocenter.pdf   Example of a geometric result showing that deductive methods can play an important role in discovering new mathematical facts.  This was not covered in the lecture, but it might be illuminating.
vert-angles1.pdf  This a traditional geometrical proof of the Vertical Angle Theorem, reputed to be one of the earliest theorems in Greek geometry.   A proof like this (more or less along classical Greek lines) is said to be synthetic.
vert-angles2.pdf  This a proof of the Vertical Angle Theorem using vectors.  Many calculus textbooks have a few pages on vector proofs of geometric results, which can be viewed as a streamlining of proofs based upon coordinate geometry.  Proofs of this sort may seem needlessly redundant or awkward, but currently vectors are the main tools of choice for analyzing geometric problems and proving geometric theorems (however, there are some noteworthy exceptions).  
circleright.pdf         AND
circleright2.pdf  These are analogs of the preceding two files for another result attributed to Thales:  An angle inscribed in a semicircle is a right angle.  This material was not covered in the lecture, but it is worth reading because it shows the sorts of geometrical techniques that will be used througout the course. 
pythagorean-thm.pdf   This gives a simple picture which suggests one proof for the Pythagorean Theorem.  Of course there are many other proofs and many other approaches to proving this result, some which are highly geometric and others which use mainly algebraic techniques.  
history02e.pdf   Polygonal numbers (triangular, square, pentagonal etc.) were families of positive integers which the Pythagoreans found very interesting.  This file contains definitions, geometrical illustrations, and some curious results about such numbers.  This material was not covered in the lecture. but  it should be read and understood  because it is a good source of problems; there is additional discussion in Burton.
zeno.pdf   An illustration of some paradoxes stated by Zeno of Elea, ending with a potentially bizarre consequence.  
 Misprint alert:
In the file http://math.ucr.edu/~res/math153-2020/exercises/math153exercises01a.pdf there is a misrprint in the definition of the function.  The corrected description is given by f(x)  =  8x3 - 6x - 1; a factor of  8  is missing from the third degree term.  This is consistent with the discussion in the solutions file for these exercises.
Historical questions:
There was a request for an example, particularly about the four historical periods of Greek mathematics.   Here are two possibilities:
(1)  Describe one similarity and one difference between the mathematical work done during the Ionian and Post-Hellenistic periods.  
(2)  What were the differences between the approach to irrational numbers in Egyptian and Babylonian mathematics and the approach in Post-Pythagorean Greek mathematics? 
Currently we have not covered enough material to give totally informed answers to these questions, but here is one that we can answer:
(3)  For each of the Egyptian, Babylonian and Greek schools of ancient mathematics, determine whether or not their descriptions for fractions always gave the precise values.  
One valid response to the third question.  Egyptian fractions were precise but a little awkward for computational purposes, Babylonian expressions for fractions were in general only approximations to within 1/216,000 (a specific example in history02.pdf is 1/7), and Greek expressions in turns of ratios were precise and very close to the modern concept.
Homework files:
The first of these will be posted by Monday's class.