The presentation given today by Dr. Barsky, after the course introduction, held to the main concept that mathematics has existed since the beginning of civilization, and because of the difference in time, we as Math History students should try to interpret the information as opposed to merely absorbing it. In this lecture, the topics covered consisted of the median age of mathematics, Ishango Bone, the ancient civilizations, and Egyptian ways of counting, addition, and multiplication.
Among these topics, the Egyptian section interested me the most. Dr. Barsky began this section introducing us to the Hieroglyphic numbering system used on stone, and he briefly mentioned the Hieratic and Demotic systems used on papyri. After he covered the ways of counting, addition, and multiplication in class, I went home and did some counting and calculations on my own, based on his explanation. If I wanted to count to the number seventy seven in Hieroglyphic, I would write down seven staffs/strokes and seven yokes/heel bones. I will note that the smaller units go on the left of the larger units probably because they could easily count them from left to right in this fashion. If I wanted to add seventy seven to 700, I would first write down seven staffs and seven yokes, and directly underneath it I would write down seven scrolls. To do this calculation, I could simply count up all of the units collectively and write them down underneath the previous work,(777 or seven staffs, seven yokes, and seven scrolls). I performed the following multiplication first according to text by Victor Katz. If I wanted to multiply seventeen by seven, I would use the doubling technique by using seven as the so called multiplicand. I would write down one staff as the multiplier and seven staffs to the right of it for the first doubling, (ex. in Arabic 1 7). For simplicity, I will now demonstrate the rest of this problem in Arabic numerals. I would then double seven which would give me the product as fourteen. I would then continue to double the multiplier until just before the multiplier would exceed the first multiplier seventeen.
ex. Multiplier Multiplicand
1 ... 7
2 ... 14
4 ... 28
8 ... 56
16 .. 112
I would then take the extreme multipliers, which add to seventeen, and add they're corresponding multiplicands to come up with the correct answer,(ex. 112+7+119). I then tried Dr. Barsky's explanation ((1+16)*7=(7*1)+(16*7)=7+112=119), which used the distributive property. In my attempted interpretation of all of this, I believe the Egyptians used doubling when multiplying because they needed to be able to see all of the units in front of them so they could count them up. They linked counting with addition and multiplication.
The topic of today's discussion was an introduction to the class as well as an opening lecture on ancient civilizations and the first encounters with mathematics. The class tried to guess the median year for mathematics knowledge. Although I had heard the answer before, I am still surprised that the year is so recent. After learning that the year was 1960, we tried to come up with reasons that could explain the year. One of the biggest is the many advances in technology. Another is that in ancient times, math was used as a tool of power (Katz, 1), and therefore only certain people had access to such knowledge. Now, barriers such as economic status and gender have been broken and many more people are entering the field.
The Ishango Bone fascinates me. We discussed the patterns of numbers that were found on the bone as well as possible interpretations of those numbers. I tend to agree with Alexander Marshack's view. He proposes that the number patterns had to to with a measurement of time based on a lunar calendar. Many civilizations based their lives on the moon and its cycles. However, although I find Marshack's interpretation more plausible, Jean de Heinzelin's interpretation is a valid one. It would not surprise me to find that these people were doing computations at that point in time.
My favorite topic of discussion was the Egyptian method of counting. Although the hieroglyphic method is simple, I like the hieratic method. Although there are more symbols to remember, it makes reading the numbers much easier. It, in some ways, reminds me of our own method of writing numbers.
Dr. Barsky began the lecture with a brief reason as to why half of the mathematics known today has been discovered within the last thirty-plus years. He then did an introduction of the Ishango bone and gave some possibilities of its meaning and purpose. Finally, Dr. Barsky named some ancient civilizations and covered some of the Egyptian hieroglyphic numbering system.
I think the interpretation of the Ishango bone provoked my interest the most. It fascinates me how many different ways carved notches on a piece of bone can represent so many possibilities. Jean deHeinzelin suggested that the 11, 13, 17, 19 column represents the prime numbers from 10 to 20. Other interesting suggestions were Alexander Marshack's lunar count or that the notches represented a game of some sort.
In addition to the regular formalities of the first day of class, the material covered included; early civilizations, counting, the Ishango bone, hieroglyphic numbers. My attention was caught by the Ishango bone and the two very different interpetations of what the marks might have represented. Both of the explanations made sense to me, but later I wondered about another possibilitiy. The time period of the bone 3000BC, is about the same time women were still the spiritual leaders of the clans. A woman's cycle is very similar to the phases of the moon. I wondered if this could be another explanation for the markings on the bone?
The presentation on August 22, 1996 was a quick overview of very ancient mathematics. We started with the very first evidence of mathematics, the Ishango Bone. We discussed the Ishango Bone, the people who found it, what others had to say about it and near the end of class we discussed Egyptian mathematics.
One of the topics that most interested me was the Egyptian mathematics. We discussed how the Egyptians wrote their numbers and words, by using the hieroglyphic, hieratic and demotic ways of writing. They used the hieratic and the demotic for paper writing. I think they did this because the characters used in hieratic and demotic were less intricate than those used in hieroglyphic. These less intricate characters are easier and faster to write, also they would probably be easier on the fragile papyrus. The hieroglyphics were much more ornate and decorative so they were used only for the Pharaohs. I found this very interesting because there is the same thing today. We have printing and handwriting just like the Egyptians had Hieroglyphics and demotic.
In his lecture on 22 August 1996, Dr. Barsky started off with an overview of the course. He gave a brief description of what would be covered during the semester, and an explanation of what would be required of the students. Following this, Dr. Barsky posed a question to the class about when the "median point" in math occurred, the "median point" being the point in time at which man had acquired approximately half of the mathematical knowledge we have today. Somewhat to my surprise, the answer was somewhere around 1960. The purpose for this discussion was to point out how much more rapidly our knowledge is expanding today than ever before. Several reasons for this were discussed, most of which centered around technological progress in communications, printing, and publishing, and, of course, computers.
Dr. Barsky then went on to discuss why people in ancient civilizations studied math, and how it was used. The main purposes for studying math were counting and representation of numbers on the algebraic side, and shapes and their uses on the geometric side. Other uses for math included taxation, calendars, and building.
The last topic of discussion was representation of numbers and methods of arithmetic in ancient civilizations, focusing on Egypt. I found this especially interesting for a number of reasons. For one thing, the ancient Egyptians had a different symbol for each power of ten, showing a leaning toward a base ten number system. For multiplication they used a doubling technique that essentially wound up representing one of the factors as a sum of powers of two, which is the basis for a base two representation. In late twentieth century America, we use a base ten number system for most things, and computing is usually done with some form of base two, either binary or two's complement. According to Katz, base ten systems were the norm, even in ancient times, with the notable exception of the Babylonians, who used a combination of base ten and base sixty.
Katz goes on to talk about how the Egyptians did division. They would treat division as the inverse operation of multiplication, and look for a number which, when multiplied by the divisor, would yield the dividend. Since division doesn't always come out even, the Egyptians used fractions. But the only fractions they used were the "natural" fractions, or those of the form 1/n, where n is an integer, with the single exception of 2/3. I would be interested in how they devised these methods. Did they just come up with ideas that worked, or was it a drawn out process of trial and error? Katz says repeatedly that no records have been found lending an answer to this question.