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
extreme and mean ratio
greatest common measure
regular sexigesimal number
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
Area of a circle
Dawning of age of Aquarius
Mathematical Treatise in 9 sections
Existence of Incommensurables
Chinese remainder Theorem
Precession of earth's Axes
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.