ࡱ> ~   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}Root EntryZ O2LNCONTENTS Object 125YnL -11ContentsRurse. Before we begin any further considerations of the Babylonian number system, it behooves us to take a very brief survey of other earlier methods of counting and using numbers, hence deriving a more complete appreciation for Babylonian conceptions. (A) The earliest use of numbers were extremely simple. Simple counting of ones and by twos is universally regarded by anthropologists as the first example of functional number use. (2) One example of this were the Australian aborigines. Certain other cultures in the Amazonian rainforest region used ones and twos and also two and one, two and two, etc, but not past six. Please note that I do not mean that they were adding, simply that they differentiated between one, two and numbers greater than that. Tally marks seem to be the next most common use of making number sense. There have been wolf bones found, dating tens of thousands of years B.C., which clearly indicate an attempt to keep records regarding matters of everyday life. Speculation abounds in the anthropological community as to the proven use of these tallies, (recording pelts, hunting kills, calendars?) but there is nevertheless, clear evidence that early peoples were making marked strides with their use of numbers, irregardless of the actual meaning of these tally marks. It bears mentioning that these fossil relics of tallies do indicate an understanding of groups of five. A notable advance, even if a small one! But the inevitable advances in Math hardly ended there. All of the more advanced cultures throughout the planet (all of them interestingly enough were agrarian) had contributed certain advances in number sense and applied use. The Mayan peoples, for example, used knots to record their practical matters. They used different lengths of CHNKWKS TEXTTEXT8FDPPFDPP<FDPCFDPC>FDPCFDPC@STSHSTSHBhSTSHSTSHhCSYIDSYID FSGP SGP FINK INK "FBTEPPLC &FBTECPLC >F FONTFONT^FEOBJPLC F4STRSPLC G:MCLDMCLDPGPRNTWNPRFRAMFRAMTITLTITL",DOP DOP N the AmazoniMath 5008 Report / Question 6- Base 10/60 Carl Berg 6/13/06 [Student Challenge- Select two of the bold-faced letters found in this discussion of numbers and be prepared to do both of the following: A) Write a 3 paragraph essay about any of the topics. B) Prepare a five minute oral presentation about a second topic.] Babylonian Number System: The Babylonian culture was an advanced culture, indeed! Blessed with a fertile river valley, nurtured by both the Tigris and Euphrates Rivers, and having learned that agriculture is a much more stable method of survival than hunting and gathering or other such nomadic lifestyles, these amazing people were able to advance their cause and have forever graced us with many evolutionary advances in Math, as well as their application and understanding of numbers. They were the first people to really develop a workable mixture of base 10 and base 60 number system using a marvelously simple combination of two similar wedge shaped ciphers. Their cuneiform numbers were represented as follows: Y = Ones, and groupings of ones. < = Tens, and groups of ten, to 59. I will elaborate upon this newer and easily mastered number system in greater detail, later in this report. This method of communicating number sense was considered so excellent at the time, that even other cultures used it in order to further their own more advanced studies of astronomy and other complex number problems. We have on record today over 400,000 tablets of their cuneiform writings, with approximately 400 of these pertaining to mathematical matters ranging from everyday considerations about crop yields, land parcels, population census information and what some claim might be regarding levied taxes, to much, much more complex arithmetical functions, of cocords, various colors and sizes of the basic knots used to indicate a number. What I find most fascinating is that they skipped a place on their number cord (called quipu) to indicate a zero value. Note that this observation is not proof of a use of a zero place value within a number system, but we are getting closer! (B) Skipping ahead, and closer to home, so to speak, we note that other cultures began using hieroglyphic symbols to represent numbers. One commonly studied culture are the Egyptians, who used their hieroglyphics (they had a cursive system used by the priest class, as well as a more standard form for the common person) to conduct rather complex Math functions, including basic fractions. They wrote extensively upon papyrus paper, indicating a prolific and comprehensive record keeping of all sorts of issues pertinent to their everyday life. Many consider their most advanced contribution to be an established use of specific (and commonly used) numbers within a viable base10 system. So after this whirlwind overview of numbers, and the beginnings of their usage, at long last we arrive once again at the Babylonians. Babylonian numbers were simple to write. (C) The cipher was required to make only two cuneiform symbols in order to symbolize all practical numbers. By a combination of pairings and arrangements of their ciphers (units of one and tens), the Babylonians were able to write large numbers. Here are examples of numbers one-fifty-nine: As we can see, the Babylonian wrote their ones units first, just as do we today. What sets the Babylonian method apart from our method of writing is that they were actually reading their numbers right to left. They also had a distinct way of keeping track of larger numbers equal to, and also greater than, the number sixty. Let s take a look, using our modern numbers: 3 25 4 This actually means 3x60squared, + 25x60 +4, equalling 12, 304. Note that the Babylonians increased the place value simply by shifting their numbers to the left. (3) Later on, the Babylonians began to use a symbol similar to // represent that there was nothing there. This made reading their numbers much easier. (D) The Babylonian base sixty system is used today to measure time (hours equal sixty minutes, minutes equal sixty seconds) , latitude and longitude (360 degrees for both, then being capable of being broken down into minutes and seconds) and even outer space. Their system is often  used to measure, weights and dates. (1) (E) With their advanced abilities and improving number system, the Babylonians made effective use of their base 10/60 sexigesimal system. By 2000 B.C. , the Babylonians had the ability to do not only simple Mathematical operations such as addition subtraction, multiplication and division, but also linear and quadratic equations, geometry, square roots, fractions, and more. They were truly a culture of many mathematical advances. [Study Topics from this Report: (A) Research some early use of numbers. (B) Which culture used zero first? Which one used it as a place value? (C) Practice writing numbers from 1- 500. Explain how you figured it out. (D) Practice writing larger Babylonian numbers, ranging from 1000-10,000. (E) Research any of theses three topics and present your findings to the class.] Report Bibliography: (1) Baumgarten, John K. Historical Topics for the Mathematics Classroom National Council of Teachers of Mathematics, Inc, 1989 (2) Burton, David M. The History of Mathematics: An Introduction 6th Edition New York: McGraw Hill, 2007 (3) Seife, Charles Zero: The Biography of a Dangerous Idea Penguin Books, 2000 r 12, 304. Note that the Babylonians increased the place value simply by shifting thinheir numbers to the left. , using our modern numbers So ---------To summarize,<Tjz|~Z( * n ^& %&)))),,--"/$/155T55266Z7778*8*99:::::::::::(2"'( ) @S 2'((2"'(  \ %*   N'X':)<)))j.p./$/1158\88V9999&:v:::::~\::::" "PSS" $ " "PS" $ " "PS" $   "PS" $  "PS" $ $ "PS" $   "PS" $ :::: "   "PS" $ . 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