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The Mathematics of Ancient Mesopotamia the Mathematics Of The Mathematics of Ancient Mesopotamia Background • Mesopotamia: Greek , “between the rivers,” specifically the Tigris and Euphrates. This area occupies most of what is present-day Iraq, and parts of Syria, Turkey, Lebanon, and Iran. Background • Thought to be the (or at least a) “cradle of civilization.” • Delta region extremely fertile – The “Fertile Crescent” • Semi-arid climate required extensive irrigation projects Four Empires • Four civilizations flourished here, from 3100 BCE to 539 BCE. These included the early Sumerian (3100 – 2400 BCE) and Akkadian (2400-2100 BCE) empires, and the later Old Babylonian (1800-1200 BCE) and Assyrian (1200 -612 BCE; Ashurbanipal) empires. There followed a brief Neo-Babylonian period from 612 – 539 BCE. Then Persia. Then Alexander the Great. Then…. Timeline Archaic Old Kingdom Int Middle Kingdom Int New Kingdom EGYPT 3000 BCE 2500 BCE 2000 BCE 1500 BCE 1000 BCE Sumaria Akkadia Int Old Babylon Assyria MESOPOTAMIA Some names you might recognize • Hammurabi, founder of the Old Babylonian Empire • Code of Hammurabi - 232 laws, lex talionus, an eye for an eye – If anyone strikes the body of a man higher in rank than he, he shall receive sixty blows with an ox-whip in public. Some names you might recognize • The Epic of Gilgamesh – Poem relates story of Gilgamesh, ruler of Uruk, who seeks out survivor of great flood in quest of immortality. • Ur of the Chaldees, Birthplace of Abraham. • King Nebuchadnezzar (Neo- Babylonian Empire) Sources • Most of what we know about Mesopotamian mathematics comes from several hundred clay tablets belonging to the Old Babylonian kingdom, around roughly 1800-1600 BCE. • Tablets are of two kinds: – Table texts – Problem texts But Before We Go There… • We need to understand a little about the number system used in that Old Babylonian era. The theories about how it evolved the way it did are interesting in themselves. Babylonian Number System • A base-60 positional system with individual numbers formed by two different wedge-shaped marks: a horizontal wedge worth 10 and a vertical wedge worth 1. • Numbers less than 60 were written using these two symbols in a purely additive fashion. Babylonian Number System Notational Aside: • Notice that the marks from the previous table don’t look exactly like the and the that I used a while ago. They look even less like the marks I’ll end up using from here on because they are easier: ‹ and ˅. There is considerable variation in both the original texts and the modern interpretations. Babylonian Number System • These 59 symbols would be written in a place value system based on powers of 60. Powers of 60 increased from right to left, just as powers of 10 increase from right to left in our system. Babylonian Number System • Thus, writing ‹˅ ‹‹‹˅˅˅˅ ‹‹˅˅˅ would most likely represent ଶ or 41,663. The First of Two Problems • There was no “0” or placeholder so we really can’t be sure which power of 60 is being used. Thus, ‹˅˅ ˅˅‹ could represent either: , or ଶ , or ଶ , or many other possibilities. The Second of Two Problems • Even though the Babylonians used this system to write fractions as sexagecimals, there was no “sexagecimal point” or other way of marking where the fractional part began. So, again, ‹˅˅ ˅˅‹ could mean any of: , or ିଵ , or ିଵ ିଶ Resolution of Problems • These two problems were usually quite easily resolved by the context of the arithmetic being done, so it bothers us much more than it did the Babylonians. Also, there were very frequently units attached. For example, any ambiguity in writing 1 1 is resolved if we say $1 1₵. Resolution of Problems • In about 300 BC there was a placeholder symbol invented and used, but only between symbols, never at the end. • In our notation, it used for 604 but never 640 or 6400. Our Babylonian Notation • We will use a comma to separate place values, use a 0 when we need it, and use a semicolon as a “sexagecimal point.” Thus, ଵ But Why 60? Why? Why? Some suggested reasons: • Lots of non-repeating sexagecimals, since 60 as more divisors than 10 (btw, how do you tell if one of our fractions will terminate or repeat when converted to a decimal?). • Sacred or Mystical numbers • Combination of two number cultures. Why 60? • Well, actually, we aren’t sure. • But we’ll talk about one suggested solution. • According to Peter Rudman in his book How Mathematics Happened: The First 50,000 Years, it’s probably more like 6’s and 10’s than 60. Example: 60x60 60 1 1/60 › ٧ › ٧ (carrying row) ٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number) ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number) ٧ ›› ٧٧٧٧ ›› ٧٧ ›› (Sum) 60x60 60 1 1/60 › ٧ › ٧ (carrying row) ٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number) ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number) ٧ (Sum) 60x60 60 1 1/60 › ٧ › ٧ (carrying row) ٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number) ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number) ٧ ›› (Sum) 60x60 60 1 1/60 › ٧ › ٧ (carrying row) ٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number) ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number) ٧ ›› ٧٧٧٧ (Sum) 60x60 60 1 1/60 › ٧ › ٧ (carrying row) ٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number) ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number) ٧ ›› ٧٧٧٧ ›› (Sum) 60x60 60 1 1/60 › ٧ › ٧ › (carrying row) ٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number) ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number) ٧ ›› ٧٧٧٧ ›› ٧٧ (Sum) 60x60 60 1 1/60 › ٧ › ٧ › (carrying row) ٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number) ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number) ٧ ›› ٧٧٧٧ ›› ٧٧ ››› (Sum) Alternating 10-for-1 and 6-for -1 Exchanges Ok, so….. • We can understand using groups of 10. But we have to ask: “Why the freak are there groups of 6?” • Well, let’s look at Ancient Sumer: • First, realize that these folks used different measures for different things, and that these measures had different “exchanges” from larger to smaller units. • We did this too: Weight 16 ounces = 1 pound 14 pounds = 1 stone 8 stone = 1 hundredweight 20 hundredweight = 1 ton (except for us 100 pounds = 1 hundredweight, and 20 hundredweight = 1 ton = 2000 pounds) Capacity 8 (fluid) ounces = 1 cup 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon Length: 12 inches = 1 foot 3 feet = 1 yard 22 yards = 1 chain 10 chains = 1 furlong 8 furlongs = 1 mile 3 miles = 1 league And then you have rods and links and thous. Land Measures • Originally, communal plots of land were laid out in rectangular plots of 1 furlong by 1 chain (660 by 66 feet), or 10 chains by 1 chain (= 1 acre). The furrows ran in the long direction, so the plots were a “furrow long.” So actually furlongs were an agricultural measure that were independent of feet, which was a body-part measure. • By the way, a cricket pitch is still 66 feet long, or 1 chain, or a tenth of a furlong. Moving on…. • Eventually small measures based on body parts had to be reconciled with large agricultural measures like furlongs, so things were shifted and fudged in the measures so that everything was an integral multiple of everything else. The Same Thing Happened in Babylon: • A body-part measure called a kush, about 1 2/3 feet, was the basis for a nindan, which needed to be reconciled with two agricultural measures, the eshe and the USH. The eshe and the USH came pre- loaded with a 6-to-1 exchange, and the nindan and the eshe became an easy 10-to-1 exchange. 10-for-1 and 6-for-1 • Because units of both length and area were exchanged for larger units in both groups of 10 and groups of 6, using counters that reflected those exchanges greatly facilitated calculations with lengths and areas. And the number system went along for the ride. 10-for-1 and 6-for-1 • So Rudman claims that units of length and area that came pre-loaded with exchanges gave rise to a system of arithmetic with alternating 10-to-1 and 6-to-1 exchanges, and then to a base 60 system. In any event, there was real genius in moving to a place-value system. • Now, back to Babylon: Babylonian Tablet Texts: • Table Texts • Problem Texts Table Texts • Multiplication tables, of which about 160 are known. • Single tables have the form: p a-rá 1 p a-rá 2 2p a-rá 3 3p And so on, up to 20p, then: a-rá 30 30p a-rá 40 40p a-rá 50 50p Table Texts • Combined tables (of which there are about 80) have several single tables included on one tablet. • One of them (A 7897) is a large cylinder containing an almost complete set of tables written in 13 columns. There is a hole through the center of the cylinder so that it could be turned on some kind of peg. Table Texts • Reciprocal Tables had reciprocals of numbers from 2 to 81 (provided their sexagecimal representations did not repeat. • These were used to divide, which they did by multiplying by reciprocals. Table Texts • There are a few tables of squares, square roots, cube roots, powers, sums of squares and cubes, … . • Also some conversions and a few special tables used for particular business transactions (finding market rates). Table Texts • It is likely that many of the table texts we have are “exercises” from students learning to be scribes, or perhaps tables copied and made by students for use in computations. Problem Texts • Also probably intended for educational purposes. • Story problems aimed at computing a number. • Often contrived or “tricky” problems: – If camel A leaves Phoenicia travelling at nindan per day. • Kinda like our modern story problems or recreational math problems.
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