Review guide for Midterm in Mathematics 436 - Spring 2006

1. SPECIFIC KNOWLEDGE. Know how to do the following:

Write a fraction as a sum of Egyptian unit fractions

Solve problems by the Egyptian false position method

Describe the Rhind Papyrus

Convert between Babylonian sexigesimal numbers and decimal numbers

Recognize Babylonian regular sexigesimal numbers

Compute iterative approximations to square roots using the Babylonian method

Describe Plimpton 322

Find Pythagorean triples

Explain the concept of Incommensurability

Discuss the Theatesus and Eudoxus definitions of ratio

Prove that the square root of two is irrational.

Mean and extreme ratio

Discuss the three classical problems of Greek mathematics

Discuss the parallel line postulate.

Find greatest common divisors using the Euclidean algorithm.

Give the proof of Euclid that there are infinitely many primes.

Finnd the solution to simultaneous linear congruences (Chinese remainder theorem)

2. TERMS TO KNOW. Know the defininition of the term and its significance. Give an example of its use. What period of mathematics is the term associated with?

Alexander the Great

Alexandria

analysis

anthyphairesis

Archimedes

arithmos

Athens

commensurable

congruence

Euclid

Euclidean algorithm

Eudoxus

extreme and mean ratio

geometric algebra

greatest common measure

Hellenistic

irrational number

Mesopotamia

Old Babylonian

pentagram

Plimpton 322

Plato

papyrus

prime number

Proclos

Pythagoras

regular sexigesimal number

sexagesimal

Thales

Theatesus

3. CAST OF CHARACTERS - Be able to identify the approximate time period and the relation to the history of mathematics

Ahmmose; Archimedes; Aristotle; Diophantus; Euclid; Eudoxos; Hypatia; Pappus; Plimpton; Ptolemy, Pythagoras; Qin; Thales; Theatesus

4. SAMPLE PROBLEMS FOR EXAMINATION

1. Express 2/19 as the sum of unit fractions.

2. Compute 361/7 as a sexigesimal number ( it has a repeating sexigesimal expansion).

3. Circle the regular numbers below (numbers whose reciprocal is a terminating sexagesimal fraction). Explain your answer.

72 7 80 81

4. What was the Delian Problem of Greek mathematics?

5. The following statements are both historically reasonable:

Greek Mathematicians discovered that the square root of two is an irrational num ber.

Greek mathematicians did not believe that the square root of two was a number.

Discuss the merits of these two statements, explain why they are both reasonable, and evaluate the significance of the evident contradiction between the two points of view.

6. A group of at most 200 people forms rows of 13 with 11 left over and forms rows of 17 with 6 left over. How many people are in the group?

7. Match the terms in the first group with the related terms in the second group

Group I

Area of a circle

Dawning of age of Aquarius

Rhind papyrus

Euclid's elements

Plimpton 322

Mathematical Treatise in 9 sections

Existence of Incommensurables

Group II

Pythagorean triples

Chinese remainder Theorem

Precession of earth's Axes

Pythagoras

Unit fractions

Algorithm for greatest common divisor

Method of exhaustion

8. Explain how the Babylonians could find the sides of a rectangle given that the perimeter is 6 and the area is 1.