[Today’s class: Early human history and earliest mathematical artifacts, including Ishango bone activity,

discussion of very earliest counting: how would you “mark” the days, seasons, etc.?,

how would you construct rectangles and circles, measure fields? (Sticks, Stones, and Rope activity),

number systems in various (early) civilizations, mainly via worksheets,

review/introduction of base 5, 10, 20, and 60 number systems,

worksheet on zero as placeholder]



Mathematics 115

Homework Assignment #1

Due Tuesday, January 8, 2002



Prof. Beery's office hours this week:           Tuesday 1/8   11:30 a.m.-12:30 p.m., 4-5 p.m.

                                                            Wednesday 1/9   10:30 a.m.-12:30 p.m., 4-5 p.m.

                                                              Thursday 1/10   10:30 a.m.-12:30 p.m., 4-5 p.m.

                                                                  Friday 1/11   1:30 - 3:30 p.m.

and by appointment, Hentschke 203D, x3118



Read:   "Historical Background" from Africa Counts:  Number and Pattern in African

               Culture, by Claudia Zaslavsky (handout)

            "Bodily Mathematics" from From Five Fingers to Infinity, by Frank Swetz


            "Counting and Recording of Numbers" from Number Theory and its History,

               by Oystein Ore (handout)

            "Zero:  The Exceptional Number," by Stephanos Gialamas and Miriam K. McCann



Do:    Answer the following questions about the readings above on a separate sheet of


1.      In Africa Counts, Claudia Zaslavsky writes that the markings on the Ishango bone

   comprise a period of almost six months.  What evidence does she have for this

   assertion?  (Hint:  How many marks are in each of the three columns?  What is 

   the total number of marks?)


         2.   Claudia Zaslavsky reports that the ancient Greeks generally are regarded as the

               "fathers" of Western civilization, but she points out that the Greeks relied on the

               two earlier civilizations of ­­­_____ and _____ for much of their knowledge.  She

               also points out that, among ancient African civilizations, the civilization of _____

               has been studied extensively, while the neighboring civilization of _____ (to the

               south) and the civilizations along the _____ River need and deserve further study.


         3.   Write the number 7,528 using      the base 5 number system,

                                                                  the base 20 number system,

                                                                  early Egyptian numerals (right to left, or

                                                                        left to right, whichever you prefer),

                                                                  Chinese-Japanese numerals,

                                                                  alphabetic Greek numerals

                                                                  Babylonian (Mesopotamian) numerals,

                                                                  Maya numerals (calendar:  18x20),

                                                                  Maya numerals (ordinary:  20x20), and

                                                                  Gobar (Western Arabic) numerals.


         4.   Oystein Ore claims that one advantage of a positional number system, such as our

               Indo-Arabic (or Hindu-Arabic) number system, is that the "numerical notation is

               very compact."  Give an example supporting his claim.  That is, give an example

               (perhaps from Exercise 2) showing that, in at least one other type of number

               system, the notation is less compact than in the Indo-Arabic system.


         5.   Oystein Ore claims that another advantage of a positional number system, such as

               our Indo-Arabic number system, is that we may "express arbitrarily large

               numbers only by the digits in the basic group."  Give an example (perhaps from

               Exercise 2) showing that, in at least one other type of number system, it is not

               possible to write large numbers using only the digits from the basic group.  Be

               sure to identify the digits from the basic group in your example.


         6.   Write the number  1206  using Chinese stick numerals, as illustrated in the article

               by Gialamas and McCann (that is, with 0).  


         7.   Give an example showing why  0  is important as a placeholder in a positional

               number system.  (Hint:  What could go wrong without zero?)

               Note:  Gialamas and McCann make a distinction between zero as a placeholder

               and zero as a number.  Here's an example illustrating what they mean:

               In the number  1206,  0  is a placeholder; whereas

               in the calculation  3 + 0 = 3,  0  is a number. 

               In the calculation  1206 + 820 = 2026,  0  serves both as a placeholder

               and---during the actual calculation---as a number.


8.      According to Gialamas and McCann, what other numbers besides  0  did

      humans have a hard time accepting?


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