Math 629: History of Mathematics
Frank Sottile

• Book Review: Please see the information page . Get cracking on it!
• Opening Remarks: While surviving Egyptian and Babylonian mathematical texts are roughly contemporaneous, it is likely that Egyptian mathematics was developed earlier. This week's reading begins with the Mayans, who were much later than either, and we believe much less advanced in their mathematics. I am including them because of their vigesimal (base 20) numbering system and use of a symbol for zero, to help reinforce that decimal has not always been the only game in town, as we saw with the Babylonians and their sexagesimal system.
      Our focus will be in the Egyptian system, and their method of representing fractions, as well as their method of `false position' for solving linear equations.
  Warning: There are a significant number of typographic and other errors in Allen's notes.
• Reading:
  • Read the St. Andrew's pages on Mayan Mathematics
  • Allen's Egyptian Mathematics
  • Read the St. Andrew's pages on Egyptian Mathematics.
  • To get start on Greek Mathematics, read Allen's The Origins of Greek Mathematics. You might also dip your toe into Chapter 1 of Bell's Men of Mathematics, for a completely different take on our Greek Colleagues
• Assignment: Due Monday 29 January (HW 3)
  To hand in: Email a .pdf to Mehrzad Monzavi Mehrzad@math.tamu.edu
  
  
  1. The decimal expansion of 1/11 has the form 0.090909090.... It is a repeating decimal with period 2.
     Express the common fraction 1/7 as a repeating vigesimal. (E.g. 0;a,b,c... = a/20 + b/20² + c/20³ + ...). Compare the period of the repeats for this same number in decimal and in sexigesimal. Can you explain the relation between the different periods in the different bases?
  2. Compare and contrast the different methods used to represent whole numbers used by Babylonians, Egyptians, Mayans, and by us in our decimal positional system. For each ancient system give an example of a computation or representation for which is was superior to the others, and one where it was inferior. You can include fractions for this second question.
  3. The Egyptians used a different algorithm for multiplying than we do. Explain their method, and use it to compute the following products: 7*17, 17*7 (this should be a different procedure than 7*17), and 61*28.
     Compare this to multiplying numbers represented in binary (base 2). Is this Egyptian method primitive or modern?
  4. The Egyptian algorithm for division is a modification of their algorithm for multiplication. Use it to divide 13 into 1664. Now divide 17 into 5865. Compare this to the familiar algorithm that we use for long division.
  5. You have 288 heqats of grain to distribute to your workers, in proportions 2/3 (for the scribe), and then 1/4, 1/5, and 1/12 for the others. How many heqats do you distribute to each? Do your calculations only using arithmetic that the ancient Egyptians would use.
  6. Solve x+x/2 = 16 using the method of false position. Use this method to solve 3x+x/5 = 24. Express your answers in the method Egyptians would use.