MATH 390  Mid-Term Exam                                                   Name________________

Instructions:  Answer all the questions on the paper provided and show all your work.  If you have work on scrap paper be sure to put your name on the paper.

 

Part I.  Identification – Briefly identify each of the following:                   (20)

1.      Hippocrates of Chios

 

 

 

 

2.      Amicable Numbers

 

 

 

 

 

3.      Doubling the Cube

 

 

 

 

 

4.      Ahmes the Scribe

 

 

 

 

 

Part II.  Problems.

1.  Perform the division 3¸19 using the Egyptian Method, starting with .          (14)

 

 

 

 

 

 

 

 

 

 

 

2.  Express  5/54 in our version of the Mesopotamian number system.               (10)

 

3.   Express the following proposition from Euclid’s Elements as an algebraic equation.  Draw a diagram and carefully label the parts to show how you arrived at the equation. Book II, Proposition 7.

     If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the other segment.                                                    (12)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.   One of the rows of the Mesopotamian “Plimpton Tablet” contains the primitive Pythagorean triple  1,12   1,5    1,37.  Show that this is a primitive Pythagorean triple and find its generators,  p and q.                                                                                     (14)

5.   It can be shown that  is a prime number.  Based on this fact, what number does Euclid prove is a perfect number?  (Your expression may contain powers of 2.)                                                                                                 (12)

 

 

 

 

 

 

 

 

Part III  Questions.                                                                                           (18)

  Answer each of the following in a few sentences.

1.  A Mesopotamian table of reciprocals of the numbers 1-12 exists. However, the reciprocals of 7 and 11 are left out.  Why?

 

 

 

 

 

 

 

 

2.  How did Archimedes’ book,  The  Method, differ from all of his other works and the works of Euclid?

 

 

 

 

 

 

 

 

 

3.  We say today “ The Pythagoreans discovered that the square root of two is irrational.”  The ancient Greeks would have stated the result quite differently.  How would that have stated it?  Explain briefly the terms you use.