MAT 300-02L                                     REVIEW SHEET #2

SPRING 2006

 

The exam will be drawn from the questions given below, questions I will be posting on the website, and from possible combinations of them.  Not all of them will actually be on the exams, and even then, most of them will be in questions in which you will be able to choose an alternative.  I reserve the right to make the essay questions take-home and to give take-home questions other than the ones listed here.  I also reserve the right to give extra credit questions on material not specifically listed on this sheet.  Some take-home essay questions may count both as test questions and papers.

 

1.         IDENTIFICATIONS.  For each of the following mathematicians, be able to state the approximate dates of his life and either his birthplace or the place in which he did most of his/her work.  Further, be able to give two different mathematically significant facts about each of the following.  The mathematical facts should be fairly specific.  If you cite a work, be able to write a sentence describing its contents or why it was historically or mathematically significant.  Dates within 75 years will be accepted, although incorrect sequencing will be penalized.

 

a.                   Pythagoras

b.                  Eudoxus

c.                   Euclid

d.                  Apollonius

e.                   Archimedes

f.                    Diophantus

 

 

2. TIMELINE.  Given a timeline from approximately 600 BCE to 500 CE, be able to place on that time line the mathematicians in #1 above and the following mathematicians/philosophers..  You should be able to place them within 75 years of the accepted dates for their respective lives, and incorrect sequencing will be penalized.

.

a.                   Hippocrates of Chios

b.                  Plato

c.                   Heron

d.                  Hipparchus

e.                   Claudius Ptolemy

f.                    Hypatia

 

3.         MATCHING.  There will be a question on the exam in which you will need to match Greek mathematicians with something significant about their mathematics and their lives.  I will choose things that I think are significant and that I would have discussed in class; this question is not meant to be a trivia question.  I will also probably give you some sort of choice; for example, I might say to match 10 out of 15 options.  For this question you will be responsible for the mathematicians/philosophers in #1 and #2 above, along with the following mathematicians.

 

g.                   Pappus

h.                   Eudemus

i.                     Theon of Alexandria

 

 

4.         I may give you a matching question in which you would need to match the Books of Euclid’s  Elements with their respective contents.

5.         IDENTIFICATIONS.  Be able to define or state or be able to write a three-four sentence description or explanation of each of the following, including perhaps something about its historical significance.  I might also use these in a matching question. 

 

a.                   the difference between actually infinite and potentially infinite

b.                  method of exhaustion

c.                   figurate numbers (triangle numbers, square numbers, etc.)

d.                  commensurability

e.                   chord tables

f.                    the conics (in the context in which we have discussed them in class)

g.                   geometric mean

h.                   Euclid’s Fifth Postulate

           

6.                  SHORT ESSAYS.  Be able to answer each of the following and/or combinations of them.  The answer should fill approximately one to two pages in a blue book.

 

a.                   Discuss the nature of Pythagorean mathematics, giving specific examples.  The Pythagorean Theorem should only be a small part of this discussion, except as it may be used to illustrate characteristics of Pythagorean   mathematics.

b.                  Discuss the Three Construction Problems of Antiquity.  Be able to state the problems, and discuss their importance in the history of mathematics.  Be as specific as you can.

 

d.         Discuss the works of Archimedes.  Your answer should include something about the effect and/or influence of his works.  You should name several specific works and discuss their content.

 

 

7.         LONG ESSAYS.  Be able to answer each of the following and/or combinations of them.  The answers to these questions should fill approximately three-four pages in a bluebook

 

            Discuss the commensurability crisis in Greek mathematics, and its origins, its relation to the theory of proportion, its "resolution", and its effects (up to the time of Archimedes). Identify the major mathematicians involved.

 

 

 

8.         DOING MATHEMATICS.  You should be able to do each of the following:

 

a.         Be able to give a geometric statement of the Pythagorean Theorem and then be able to give a geometric proof of the Pythagorean Theorem.

 

b.         Be able to give a geometric interpretation, using dots, of the odd, even, triangle, oblong, and square numbers.   An oblong number is one that can be represented by an n×(n+1) rectangular array of numbers.

 

c.         Be able to give geometric demonstrations of the following.  Also be able
to explain in words what the picture shows.

 

 i.   A square number is the sum of two consecutive triangle numbers.

 

 ii.  Twice any triangle number is an oblong number.

 

 iii. The sum of the first n odd numbers is a square number.

 

 d.        Give geometric interpretations of each of the following:

 i.    a(b+c)=ab+ac

 ii.   (a+b)² = a² + 2ab + b²

 

e.         For the following proposition from Euclid's Elements, interpret the statement, if appropriate, and indicate its algebraic or modern equivalent.  I will provide the original statement.  You should also be able to interpret similar statements using this language.

 

            Book II, Proposition 1
Book II, Proposition 4
Book IX, Proposition 20

 

f.          Be able to give an example discussed in this course of something proven using the method of reductio ad absurdum (proof by contradiction), other than the fact that the square root of 2 is not rational.  You need not be able to actually do the proof, but you should be able to state what in the proof would be contradicted.

 

 g.        Be able to explain how the method of exhaustion is used by Archimedes in the proof of Proposition 1 in “On the Measurement of the Circle”.

 

 

THE END!