Math 629: History of Mathematics
Frank Sottile

• Book Review: Please see the information page .
• Opening Remarks:
  Many believe that mathematics started with numbers; counting, measuring, planning, settling accounts, etc. Without entering that debate (despite teaching this course, I am not an historian, but rather a practicing mathematician), number undoubtedly is both fundamental and old. The earliest civilizations that left records, Egyptians, Babylonians, Chinese, Maya, ..., all left records involving numbers. Some even had privileged classes of scribes who performed clerical duties, including many impressive computations. We will begin our course looking at some of this material, particularly the different number systems of the Maya and Babylonians. The Mayans used a vigesimal (base 20) numbering system and use of a symbol for zero (this latter is quite modern, and shows that zero arose twice in the history of mathematics, as sort of a convergent evolution of mathematical concepts). The Babylonians used a sexagesimal system. My goal in this beginning is to reinforce in your minds that the decimal system which we use has not always been the only game in town.
  
• Print-vs.-web
  The availability of electronic resources allows us one to reach a large audience quite easily. This has greatly increased the quantity of information available, but in many cases dramatically decreased the quality of that information. The ancient admonition that the buyer should beware is all the more true in this age. While I will assign and use on-line resources for this course, we should all keep in mind that they are not as carefully edited as print material. Thus they will contain typographic errors, as well as factual errors, and may have more biases than found in a typical book.
• Reading:
  • I would like you to read Allen's The Origins of Mathematics, which is about the origins of counting. You should appreciate how difficult it is to delve into this question. I note that this source become less complete nearer to its end, more like talking points in a lecture.
  • Please read Professor Allen's A General View of Mathematics Before 1000 BCE (BCE:= Before the Common Era, which is the synonym for BC that Historians use). This focuses on Chinese and Indian Mathematics.
  • Read also Allen's Babylonian Mathematics. This has been observed to have a typo on page 6: vv vv <v is equal to 2×60²+ 2×60 + 11 = 7331 (Allen has +21).
  • Read the St. Andrew's pages on Babylonian Mathematics.
  • Finally, read the St. Andrew's pages on Mayan Mathematics


• Assignment: (HW 2) Note that this is based on the reading assignment.
  It is OK to discuss this among yourselves. We all have something to learn from each other.
  To hand in: We are using E-campus for homework submission. While I have not used this before, our capable grader, Elise Walker, prefers it, and she has convinced me that this is the way to go, at least for homeworks. Due Monday 20 January.
  1. Write one or two paragraphs answering the following questions: What are the two kinds of numbers? How are they used in common life? What evidence is there that non-Human animals use/understand either type? How about babies or very young children? Which type of number lends itself to arithmetic?
  2. Allen gives the formula (n³+n)/2 for the (common) sum of the rows/columns of a magic square with side length n. Explain why this is the sum of any row/column of a magic square of side length n.
  3. In Allen's text, the Method of the Mean is discussed as a way to approximate the square root of a number n: Start with some initial guess, e.g a=1. Then, replace a by (a + n/a)/2, the average of a and n/a (which together multiply to n). On a calculator (or better) a computer carry out this procedure for a few steps (3-5), for some natural number n that is not a perfect square. Record and hand in the (base ten, digital) steps that you compute. If you have access to a computer and can do this with many digits of precision, by all means use it, and report your answers to higher precision (such as 100 digits). If anyone gets a particularly interesting answer, share it in a post on Piazza. This problem is very relevant to modern computation; I'll explain this after we share our answers.
  4. I'd like us all to practice some calculations using the sexagesimal system of the Mesopotamians. For this, let us use the notation that `;' represents the `sexagesimal point' and `,' is the delimiter between `places'. Thus `2,22' is one-hundred and forty two, while `1;45' is 1.75. Do these using base 60 and show or explain your work. The purpose of this is to appreciate what it is like to compute in base 60. To that end, do not simply convert to base ten, do the computations, and convert back. Do them purely in base 60, employing the usual algorithms you know.
     1. Warm-up: Express the (decimal) numbers in sexagesimal: 45   150   3253   17589   100,000.
     2. Simpler: 20 + 50 = W     7*17 = X     3,9 - 1,40 = Y     1,24*1,24 = Z.
     3. How about some division: 1/2 = V     1/3 = W     2/5 = X     7/4 = Y     2,16/3 = Z.
     4. Repeating sexagesimals: Why is 1/59 = ;1,1,1,.... ? My favorite decimal fraction is 1/7. What is this in sexagesimal (multiply your answer by 7 to check) ?
     5. 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 sexagesimal. Can you explain the relation between the different periods in the different bases?
     6. Compare and contrast the different methods used to represent whole numbers used by Babylonians,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.
     7. Challenge (only if you have stamina and like this stuff; this is not required): Try to verify that (1;24,51,10)^2 is pretty close to 2, as recorded on YBC 7289. I found this interesting.
  5. Explain how the Babylonians used tables of squares (n²) to facilitate multiplication of whole numbers. Compare this to the method we commonly use for calculations by hand. Which do you think makes more sense for sexagesimal calculations? Why?
     Table of Babylonian squares.
  6. While Babylonian mathematics is fascinating—my favorite story is how they solved quadratic equations, so that the familiar quadratic formula is just a reworking of method—I have already assigned enough to keep us busy.