Babylonian Mathematics
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Theorem of Pythagoras
The Most Famous Theorem The Theorem of Pythagoras feature Understanding History – Who, When and Where Does mathematics have a history? In this article the author studies the tangled and multi-layered past of a famous result through the lens of modern thinking, look- ing at contributions from schools of learning across the world, and connecting the mathematics recorded in archaeological finds with that taught in the classroom. Shashidhar Jagadeeshan magine, in our modern era, a very important theorem being Iattributed to a cult figure, a new age guru, who has collected a band of followers sworn to secrecy. The worldview of this cult includes number mysticism, vegetarianism and the transmigration of souls! One of the main preoccupations of the group is mathematics: however, all new discoveries are ascribed to the guru, and these new results are not to be shared with anyone outside the group. Moreover, they celebrate the discovery of a new result by sacrificing a hundred oxen! I wonder what would be the current scientific community’s reaction to such events. This is the legacy associated with the most ‘famous’ theorem of all times, the Pythagoras Theorem. In this article, we will go into the history of the theorem, explain difficulties historians have with dating and authorship and also speculate as to what might have led to the general statement and proof of the theorem. Vol. 1, No. 1, June 2012 | At Right Angles 5 Making sense of the history Why Pythagoras? Greek scholars seem to be in Often in the history of ideas, especially when there agreement that the first person to clearly state PT has been a discovery which has had a significant in all its generality, and attempt to establish its influence on mankind, there is this struggle to find truth by the use of rigorous logic (what we now call mathematical proof), was perhaps Pythagoras out who discovered it first. -
Abstract of Counting Systems of Papua New Guinea and Oceania
Abstract of http://www.uog.ac.pg/glec/thesis/ch1web/ABSTRACT.htm Abstract of Counting Systems of Papua New Guinea and Oceania by Glendon A. Lean In modern technological societies we take the existence of numbers and the act of counting for granted: they occur in most everyday activities. They are regarded as being sufficiently important to warrant their occupying a substantial part of the primary school curriculum. Most of us, however, would find it difficult to answer with any authority several basic questions about number and counting. For example, how and when did numbers arise in human cultures: are they relatively recent inventions or are they an ancient feature of language? Is counting an important part of all cultures or only of some? Do all cultures count in essentially the same ways? In English, for example, we use what is known as a base 10 counting system and this is true of other European languages. Indeed our view of counting and number tends to be very much a Eurocentric one and yet the large majority the languages spoken in the world - about 4500 - are not European in nature but are the languages of the indigenous peoples of the Pacific, Africa, and the Americas. If we take these into account we obtain a quite different picture of counting systems from that of the Eurocentric view. This study, which attempts to answer these questions, is the culmination of more than twenty years on the counting systems of the indigenous and largely unwritten languages of the Pacific region and it involved extensive fieldwork as well as the consultation of published and rare unpublished sources. -
2 1 2 = 30 60 and 1
Math 153 Spring 2010 R. Schultz SOLUTIONS TO EXERCISES FROM math153exercises01.pdf As usual, \Burton" refers to the Seventh Edition of the course text by Burton (the page numbers for the Sixth Edition may be off slightly). Problems from Burton, p. 28 3. The fraction 1=6 is equal to 10=60 and therefore the sexagesimal expression is 0;10. To find the expansion for 1=9 we need to solve 1=9 = x=60. By elementary algebra this means 2 9x = 60 or x = 6 3 . Thus 6 2 1 6 40 1 x = + = + 60 3 · 60 60 60 · 60 which yields the sexagsimal expression 0; 10; 40 for 1/9. Finding the expression for 1/5 just amounts to writing this as 12/60, so the form here is 0;12. 1 1 30 To find 1=24 we again write 1=24 = x=60 and solve for x to get x = 2 2 . Now 2 = 60 and therefore we can proceed as in the second example to conclude that the sexagesimal form for 1/24 is 0;2,30. 1 One proceeds similarly for 1/40, solving 1=40 = x=60 to get x = 1 2 . Much as in the preceding discussion this yields the form 0;1,30. Finally, the same method leads to the equation 5=12 = x=60, which implies that 5/12 has the sexagesimal form 0;25. 4. We shall only rewrite these in standard base 10 fractional notation. The answers are in the back of Burton. (a) The sexagesimal number 1,23,45 is equal to 1 3600 + 23 60 + 45. -
On the Shoulders of Hipparchus a Reappraisal of Ancient Greek Combinatorics
Arch. Hist. Exact Sci. 57 (2003) 465–502 Digital Object Identifier (DOI) 10.1007/s00407-003-0067-0 On the Shoulders of Hipparchus A Reappraisal of Ancient Greek Combinatorics F. Acerbi Communicated by A. Jones 1. Introduction To write about combinatorics in ancient Greek mathematics is to write about an empty subject. The surviving evidence is so scanty that even the possibility that “the Greeks took [any] interest in these matters” has been denied by some historians of mathe- matics.1 Tranchant judgments of this sort reveal, if not a cursory scrutiny of the extant sources, at least a defective acquaintance with the strong selectivity of the process of textual transmission the ancient mathematical corpus has been exposed to–afactthat should induce, about a sparingly attested field of research, a cautious attitude rather than a priori negative assessments.2 (Should not the onus probandi be required also when the existence in the past of a certain domain of research is being denied?) I suspect that, behind such a strongly negative historiographic position, two different motives could have conspired: the prejudice of “ancient Greek mathematicians” as geometri- cally-minded and the attempt to revalue certain aspects of non-western mathematics by tendentiously maintaining that such aspects could not have been even conceived by “the Greeks”. Combinatorics is the ideal field in this respect, since many interesting instances of combinatorial calculations come from other cultures,3 whereas combina- torial examples in the ancient Greek mathematical corpus have been considered, as we have seen, worse than disappointing, and in any event such as to justify a very negative attitude. -
Babylonian Numerals the Units
Babylonian numerals the units. For example the decimal 12345 represents 1 104 + 2 103 + 3 102 + 4 10 + 5. Babylonian index History Topics Index If one thinks about it this is perhaps illogical for we read from left to right so when we read the first digit we do not know its value until we have read the complete number to find out how many powers of 10 The Babylonian civilisation in Mesopotamia replaced the Sumerian civilisation and the Akkadian are associated with this first place. The Babylonian sexagesimal positional system places numbers with civilisation. We give a little historical background to these events in our article Babylonian mathematics. the same convention, so the right most position is for the units up to 59, the position one to the left is for Certainly in terms of their number system the Babylonians inherited ideas from the Sumerians and from 60 n where 1 n 59, etc. Now we adopt a notation where we separate the numerals by commas so, the Akkadians. From the number systems of these earlier peoples came the base of 60, that is the for example, 1,57,46,40 represents the sexagesimal number sexagesimal system. Yet neither the Sumerian nor the Akkadian system was a positional system and this advance by the Babylonians was undoubtedly their greatest achievement in terms of developing the 1 603 + 57 602 + 46 60 + 40 number system. Some would argue that it was their biggest achievement in mathematics. which, in decimal notation is 424000. Often when told that the Babylonian number system was base 60 people's first reaction is: what a lot of special number symbols they must have had to learn. -
The Use of Similarity in Old Babylonian Mathematics
The Use of Similarity in Old Babylonian Mathematics By Palmer Rampell Harvard University (composed at Phillips Academy in the 12th grade under the supervision of Donald T. Barry) 237 Ridgeview Drive Palm Beach, FL 33480 [email protected] 2 1 0 ; 05 53 53 39 50 24 0 ; 05 11 02 24 02 11 05 ; 0 0 ; 45 0 ; 03 19 56 09 36 1 ; 15 0 ; 08 06 Diagram #1 The above diagram is a representation and transliteration of the ancient Babylonian tablet, IM 55357 from Tell Harmal. IM 55357, which was likely written during the First Babylonian Empire (1900-1600 B.C.E.), has been used by historians of mathematics to prove that the Babylonians understood similarity. On what grounds can they make this claim? Is their assertion valid? Let us examine the tablet. In Eleanor Robson’s translation, the problem is stated as follows: A wedge. The length is 1, the long length 1;15, the upper width 0;45, the complete area 0;22 30. Within 0;22 30, the complete area, the upper area is 0;08 06, the next area 0;05 11 02 24, the third area 0;03 19 03 56 09 36, the lower area 0;05 53 53 39 50 24. What are the upper length, the middle length, the lower length, and the vertical? [10] Since the Babylonians used a base 60 number system, 12 20 could represent or , among other possibilities. The Babylonians did not use decimal places in 3 their mathematics because the results would still be consistent no matter where the decimal place happened to fall. -
The Sumerian Ternary System and the Concept of Number António José Gonçalves De Freitas Centro De Estudos Humanísticos Universidade Do Minho Portugal
Paper delivered at 1st CLE Colloquium for Philosophy and Formal Sciences Campinas, 21-23 March 2013 Accepted for publication in the Proceedings of the Conference DRAFT &1 The Sumerian ternary system and the concept of number António José Gonçalves de Freitas Centro de Estudos Humanísticos Universidade do Minho Portugal Abstract It is well known that Sumerians and Babylonians used a numeration system of base 12 and 60. We still have influence of that system in our nowadays counting of the hours of a day, twelve plus twelve, each hour has 60 minute and each minute 60 seconds. Also the circle has 360 degrees. What is not so well known is that the Sumerians in an earlier period used a ternary system of numeration; the first notice about that system is in Thureau- Dangin (1928). Later Diakonoff (1983) described it in good details and recently, Balke (2010) studied the system and described it with more precision. Still the use of this system and the concept of number involved are open questions. I will answer to those problems making a formalization of the system and showing how it could be related to a cosmogonic design. DRAFT &2 1. Numeral systems. DRAFT &3 ��� 1 11 21 31 41 51 � �� ��� �� �� � ��� 2 12 22 32 41 52 � �� ��� �� �� � � �� ��� � 3 13 23 33 42 53 � �� � � � � � �� ��� � 4 14 24 34 43 54 � �� � � � � � �� ��� � 5 15 25 35 44 55 � �� � � � � � �� ��� � 6 16 26 36 45 56 � �� �DRAFT� � � � �� ��� � 7 17 27 37 46 57 � �� � � � � &4 � �� � 8 18 28 38 47 58 � �� �� � � � � �� � � 9 19 29 39 48 59 � �� � � � � 10 20 30 ��� 40 50 � �� � � The main Babylonian and Sumerian counting system was a sexagesimal system. -
On Old Babylonian Mathematical Terminology and Its Transformations in the Mathematics of Later Periods*
On Old Babylonian Mathematical Terminology and its Transformations in the Mathematics of Later Periods* J ens H 0 yrup ABSTRACT Third-millennium (bce) Mesopotamian mathematics seems to have possessed a very restricted technical terminology. However, with the sudden flourishing of supra-utilitarian mathematics during the Old Babylonian period, in particular its second half (1800-1600 bce) a rich terminology unfolds. This mostly concerns terms for operations and for definition of a problem format, but names for mathematical objects, for tools, and for methods or tricks can also be identified. In particular the terms for operations and the way to structure problems turn out to allow distinction between single localities or even schools. After the end of the Old Babylonian period, the richness of the terminology is strongly reduced, as is the number of known mathematical texts, but it presents us with survival as well as innovations. Apart from analyzing the terminology synchronically and diachronically, the article looks at two long-lived non-linguistic mathematical practices that can be identified through the varying ways they are spoken about: the use of some kind of calculating board, and a way to construct the perimeter of a circle without calculating it - the former at least in use from the 26th to the 5th century bce, the later from no later than Old Babylonian times and surviving until the European 15th century ce. Keywords: Terminology, Mathematical, Old Babylonian mathematics, continuity, Mesopotamian mathematics. First presented as a contribution to Seminar SAW "History of Mathematics, History of Economical and Financial Practices", session of 15 June 2012: Names of operations : Meaning of the terms and sociolinguistic analysis". -
The Sumerians , the Sexagesimal System and the Babylonian Legacy
THE SUMERIANS , THE SEXAGESIMAL SYSTEM AND THE BABYLONIAN LEGACY TO ASTRONOMY by Paul Coffre The Sexagesimal System The Sumerians who lived in Mesopotamia (nowadays Iraq) 5000 years ago, developed a sexagesimal ComputaDon system. The sexagesimal system is a base 60 ComputaDon system. Why? Why not base 10 (deCimal), base 2 (Computers), base 12 (EgypDans), base 20 (Mayans)? The sexagesimal system is sDll in use today aLer 5000 years: CloCks, Coordinates of maps, Trigonometry… Why was the number 60 so valued in Mesopotamia? 60 is Countable on the fingers of both hands, 60 is a highly Composite number, 60 has an astronomiCal signifiCanCe. Strange enough that 60 is the first number between two prime numbers (59 and 61) For more details go to: hTps://en.wikipedia.org/wiki/Sexagesimal If this finger-CounDng method was the reason for the Mesopotamian sexagesimal system, why was its sub-base 10 instead of 12? It is inConsistent that the CounDng system is 12 × 5 = 60, while the deCimal notaDon system is 10 × 6 = 60. Note that Egypt uses this duodeCimal finger-CounDng method (base 12). The AnCient EgypDans had a deCimal system and did not acCept a sexagesimal system. They did not need to Count to 60, but they divided day and night into 12 hours each and they needed only to Count 12. Therefore we Can safely assume that this finger-CounDng method was originally developed to Count 12 and later Convert- ed to Count 60 and that Sumerians developed the sexagesimal notaDon independent of this CounDng method and then was passed on the Babylonians. -
2Decimal and Sexagesimal System
I.E.S. “Andrés de Vandelvira” - Sección Europea Mathematics 2 Decimal and sexagesimal system Keywords Regular numbers repeating decimals period irrational number rounding number line sexagesimal system 1 Decimal numbers Remember that to express numbers that are not whole numbers we use decimal numbers as 75.324 in which every digit has a value which is divided by ten when we move to the right. So 7 is seventy units 5 is five units 3 is three tenths of a unit 2 two hundredths of the unit 4 is four thousandths of a unit And we continue like that if there are more digits. This is the decimal system that is commonly use nowadays except, sometimes, for time and angles. We read these numbers naming the whole part then “point” and then the decimals digits one by one Example The number 75.324 is read as seventy five, point, three, two, four When the numbers express money or length can be read on a different way, for example 5.24€ is read as five point twenty four euros or five euros and twenty five cents or the number 5.36 m can be read as five point thirty six metres. 2 Types of decimal numbers As a result of some operations we can get different types of decimal numbers: Regular numbers: Are decimal numbers with a limited quantity of decimal digits and from them all could be zeros. 2-1 I.E.S. “Andrés de Vandelvira” - Sección Europea Mathematics 14 Example = 8.2 we find an exact division 5 Repeating decimals: There is a group of digits that are repeated forever. -
Much Ado About Zero
Much ado about zero Asis Kumar Chaudhuri Variable Energy Cycltron Centre Kolkata-700 064 Abstract: A brief historical introduction for the enigmatic number Zero is given. The discussions are for popular consumption. Zero is such a well known number today that to talk about it may seem "much ado about nothing." However, in the history of mathematics, the concept of zero is rather new. Homo sapiens or humans appeared on the earth approximately 200,000 years ago and only around 3000 BCE, men learned mathematics and only around 628 CE, zero was discovered in India. There is a saying that “necessity is the mother of inventions”. Out of necessity only, men learned mathematics. Over the ages, initially nomadic men learned to light the fire, learned to herd animals, learned agriculture and settled around the fertile river valleys, where the essential ingredient for farming, water; was available in plenty. All the ancient civilizations were river bank civilizations e.g. Mesopotamian civilization (part of modern Iraq) between Tigris and Euphrates, Indus valley civilization (now in Pakistan) along the river Indus, Egyptian civilization along the Nile river and Huan He civilization along the river Huan He in China. Men settled with agriculture and husbandry needed to learn mathematics. It was required to count their domesticated animals, to measure the plot of land, to fix taxation, etc. It is true that intuitively, from the very beginning, men could distinguish between, say one horse and two horses, but could not distinguish between say 25 horses and 26 horses. Even now, in several tribal societies, counting beyond three or four is beyond their ability. -
Learning Mathematics and Science in the Ancient Middle East I Am Not A
Learning mathematics and science in the ancient Middle East Eleanor Robson I am not a specialist in science education in the modern world; rather, I look at the very beginnings of science education in the Middle East, 5000–2000 years ago. I cannot cover all three thousand of those years in this paper, so instead I shall present two examples of my work, and explain why I think it is important for today's students of science to understand the very ancient history of science and mathematics in this region. The role of science and mathematics in the medieval Islamic world, starting in the 'Abbasid court of Baghdad in the 9th century CE, is already well known, but science and mathematics have a much, much older history in the Middle East, starting from the origins of culture, civilisation and cities five thousand years ago. I have a rather strange job: I am an archaeologist of mathematics and science. Archaeology essentially is the study of rubbish—the things that other people have abandoned and left behind—in order to discover how those people lived and died, how and why they made and used objects to work with and think with. Like any other human activity, mathematics and science leave trails of rubbish behind. I study the mathematical and scientific rubbish of the ancient Babylonians (and Sumerians and Assyrians), who lived in Iraq many millennia ago. These cultures happen to have left a particularly full trail of physical evidence for archaeologists to excavate. Historians like me, who work on the languages and the thought, can then use that evidence to re-capture the thinking and practices behind some of the world's oldest science and mathematics.