EGYPTIAN MATHEMATICS

TIMELINE Archaic Period (3100-2650 BC) Old Kingdom (2650-2134 BC) Large pyramids built; rich and productive period 1 st Intermediate Period (2200-2050) BC Chaotic Middle Kingdom (2050-1640 BC) Golden Age Moscow and Rhind Papyri. 2 nd Intermediate Period (1640 1550 BC) New Kingdom (1550 1070 BC) temple building, empire building. Ramses, Tutankamon, Moses.

TIMELINE Archaic Old Kingdom Int Middle Kingdom Int New Kingdom EGYPT 3000 BC 2500 BC 2000 BC 1500 BC 1000 BC Sumaria Akkadia Int Old Babylon Assyria MESOPOTAMIA

EGYPT S GEOGRAPHY Relatively isolated, hard to attack, and therefore stable. Populated along the Nile, which makes life in Egypt possible. The Nile floods predictably every July (when Sirius rises), which provides silt and nutrients to the rich soil along the banks.

EGYPT S GEOGRAPHY Egypt subsisted on organized and centralized farming in the area flooded annually by the Nile. Tracking and managing the allocation of land required extensive record-keeping, measuring, and written language. And, in particular, a calendar.

A SIDE NOTE ABOUT ASTRONOMY Ancient peoples of both Mesopotamia and Egypt could easily track the movement of the celestial sphere as it revolved around the earth every year. They could also track the movement of the sun in the ecliptic against the celestial sphere. Finally, they could plot the changes in the moon. This gave them four cycles to keep track of: years, seasons, months, and (of course) days.

EGYPTIAN CALENDARS The beginning of the Egyptian calendar year was when the Nile was predicted to flood, July on our calendars. When Sirius was visible just above the horizon at dawn, the flood was imminent. Like most calendars, there was some coordination of the cycle of the sun and the moon.

EGYPTIAN CALENDARS The earliest Egyptian calendar had 12 months, alternating 29 days and 30 days. The actual cycle of the moon is about 29 ½ days. The year was therefore 354 days. So, every 2 or 3 years, an additional month was added.

EGYPTIAN CALENDARS The second Egyptian calendar had a 365-day year. All 12 months were 30 days long. Then an extra 5 days was added at the end. This calendar worked better for tracking the solar year, but the coordination with the moon cycle was lost.

THE EGYPTIAN SEASONS The year was divided into three seasons, as suited what was important: Inundation (the flooding of the Nile) Emergence (of the crops) Harvest

BACK TO RECORD-KEEPING, MEASURING, AND WRITTEN LANGUAGE We all know that Egypt developed a pictorial writing system known as hieroglyphics. A more cursive form of hieroglyphic, the result of quickly drawing signs by hand on a sheet of papyrus with a reed brush. Here the original hieroglyphic signs were reduced to their simplest form. Used from around 3000 BC.

WRITTEN LANGUAGE The Rosetta stone was discovered in 1799 as part of the building material used in the construction of Fort Julien near the town of Rashid (Rosetta) in the Nile Delta. It provided the key to our understanding hieroglyphics.

PAPYRUS Egyptians developed a sort of paper made from the pith of the papyrus reeds growing on the side of the Nile. These were made into long strips and then rolled and unrolled for use.

EGYPTIAN MATHEMATICS A few minor sources: Berlin Papyrus (1300 BC): a quadratic equation and its solution; simultaneous equations. Reisner Papyri (1800 BC): volumes of temples. Kahun (Lahun) Papyrus (1800 BC): 6 fragments; much still not translated. Egyptian Mathematical Leather Roll (1850 BC): Table of decompositions into unit fractions.

EGYPTIAN NUMERATION OUR VERSION When we write Egyptian numbers, we ll use our regular notation. For fractions, we ll follow the Egyptian practice of writing them as sums of unit fractions. is how we will write.

DIVISION USING FRACTIONS Reckon with 8 so as to get 19. In other words, What, times 8, will give you 19? 8 1 Next double > 19 16 2 Half of 8 4 1 2 Half... 2 1 4 until you get to 1 1 1 8

DIVISION USING FRACTIONS Reckon with 8 so as to get 19. In other words, What, times 8, will give you 19? 8 1 Next double > 19 16 2 x Half of 8 4 1 2 Half... 2 1 x 4 until you get to 1 1 1 x 8 19 2 1 4 1 8 Or, 2 Find the numbers in the left column that add to 19; add the corresponding numbers in the right column to get the quotient.

MORE DIVISION WITH FRACTIONS Reckon with 12 so as to obtain 31 12 1 The usual doubling 24 2 Half of 12 6 1 2 Half of 6 now what? 3 1 4

MORE DIVISION WITH FRACTIONS Reckon with 12 so as to obtain 31 12 1 The usual doubling 24 2 Half of 12 6 1 2 Half of 6 now what? 3 1 4 Dag, yo.

MORE DIVISION WITH FRACTIONS Reckon with 12 so as to obtain 31 12 1 The usual doubling 24 2 Two-thirds of 12 8 2 3 Half of 8 4 1 3 Half 2 1 6 Done 1 1 12

MORE DIVISION WITH FRACTIONS Reckon with 12 so as to obtain 31 12 1 The usual doubling 24 2 x Two-thirds of 12 8 2 3 Half of 8 4 1 x 3 Half 2 1 x 6 Done 1 1 x 12 31 2 1 3 1 6 1 12

A LITTLE BIT ABOUT THESE FRACTIONS The Ahmes Papyrus 2/n table: 2/3 = 1/2 + 1/6 2/5 = 1/3 + 1/15 2/7 = 1/4 + 1/28 2/9 = 1/6 + 1/18 2/11 = 1/6 + 1/66 2/13 = 1/8 + 1/52 + 1/104 2/15 = 1/10 + 1/30 2/17 = 1/12 + 1/51 + 1/68 2/19 = 1/12 + 1/76 + 1/114 2/21= 1/14 + 1/42 2/23 = 1/12 + 1/276 2/25 = 1/15 + 1/75 2/27 = 1/18 + 1/54 2/29 = 1/24 + 1/58 + 1/174 + 1/232 2/31 = 1/20 + 1/124 + 1/155 2/33 = 1/22 + 1/66 2/35 = 1/25 + 1/30 + 1/42 2/37 = 1/24 + 1/111 + 1/296 2/39 = 1/26 + 1/78 2/41 = 1/24 + 1/246 + 1/328 2/43 = 1/42 + 1/86 + 1/129 + 1/301 2/45 = 1/30 + 1/90 2/47 = 1/30 + 1/141 + 1/470 2/49 = 1/28 + 1/196 2/51 = 1/34 + 1/102 2/53 = 1/30 + 1/318 + 1/795 2/55 = 1/30 + 1/330 2/57 = 1/38 + 1/114 2/59 = 1/36 + 1/236 + 1/531 2/61 = 1/40 + 1/244 + 1/488 + 1/610 2/63 = 1/42 + 1/126 2/65 = 1/39 + 1/195 2/67 = 1/40 + 1/335 + 1/536 2/69 = 1/46 + 1/138 2/71 = 1/40 + 1/568 + 1/710 2/73 = 1/60 + 1/219 + 1/292 + 1/365 2/75 = 1/50 + 1/150 2/77 = 1/44 + 1/308 2/79 = 1/60 + 1/237 + 1/316 + 1/790 2/81 = 1/54 + 1/162 2/83 = 1/60 + 1/332 + 1/415 + 1/498 2/85 = 1/51 + 1/255 2/87 = 1/58 + 1/174 2/89 = 1/60 + 1/356 + 1/534 + 1/890 2/91 = 1/70 + 1/130 2/93 = 1/62 + 1/186 2/95 = 1/60 + 1/380 + 1/570 2/97 = 1/56 + 1/679 + 1/776 2/99 = 1/66 + 1/198 2/101 = 1/101 + 1/202 + 1/303 + 1/606

WHY THIS PARTICULAR DECOMPOSITION? Can you express every proper fraction as a sum of unit fractions? Yes Proved by Leonardo of Pisa (Fibonacci) in 1202. The Greedy Algorithm works every time.

THE GREEDY ALGORITHM ILLUSTRATED Turn into a sum of unit fractions (UFs): Find the largest UF that less than so we want. Subtract: Do it again with So,.... Notice the intermediate numerators: 7>4>1. Fibonacci proved these always form a decreasing sequence, ending with 1.,

JUST ONE PROBLEM: The Egyptians didn t use the greedy algorithm. What did they use? We don t know. Some algorithms have been found that explain many, but not all, of the choices for how fractions are broken up. So if you need a topic for your PhD in Egyptology or History of Mathematics....

AHMES (RHIND) PAPYRUS Giving Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets. "This book was copied in regnal year 33, month 4 of Akhet, under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre (?). The scribe Ahmes writes this copy."

AHMES (RHIND) PAPYRUS 2/n and n/10 fraction decomposition tables Division of loaves (1 6) Multiplication of fractions (7 20) Completion & Aha (quantity) problems (21 29) Division of fractions (30 34) More division of loaves and of hekats of grain (and beer) arithmetic sequences (35 40) Geometry problems; volumes, areas, pyramids, seked (41 60) Other Miscellaneous problems, including pesu problems and one on geometric progressions (61 84)

N/10 TABLE 1 divided by 10 yields 1/10 2 divided by 10 yields 1/5 3 divided by 10 yields 1/5 +1/10 4 divided by 10 yields 1/3 +1/15 5 divided by 10 yields 1/2 6 divided by 10 yields 1/2 +1/10 7 divided by 10 yields 2/3 +1/30 8 divided by 10 yields 2/3 +1/10 + 1/30 9 divided by 10 yields 2/3 +1/5 + 1/30

DIVIDING LOAVES AMONG 10 MEN (#3) The answers to these are the same as the answers in the n/10 table, and the computations show that it is correct. Divide 6 loaves among 10 men. 1 1/2 + 1/10 2 x 1 + 1/5 4 2 + 1/3 + 1/15 8 x 4 + 2/3 + 1/10 + 1/30 10 6 (1/5 + 1/10 + 1/30 = 1/3, from a table)

MULTIPLICATION OF FRACTIONS (#7) Problems 7 20 show how to multiply the expressions 1 + 1/2 + 1/4 and 1 + 2/3 + 1/3 by different fractions.

MULTIPLICATION OF FRACTIONS (#7) Multiply by. 1 1 2 1 14 1 1 2 4 1 28 1 1 4 8 1 56 1 1 2 1 4 1 2 1 4 1 8 1 14 1 28 1 56 1 Answer

MULTIPLICATION OF FRACTIONS (#7) The question is, how does this long string of fractions turn into 1? The fact that was actually used quite frequently in the records we have. From there, hard. wouldn t be too So: addition facts for unit fractions were probably partly memorized, partly taken from tables, partly determined by red auxiliaries.

COMPLETION PROBLEMS (#21) Complete to 1 In other words, subtract, or find the difference. Of course, we would do this with common denominators. Ahmes did it with red auxiliaries.

COMPLETION PROBLEMS (#21) Ahmes thought like this (we think):. Use 15 as a reference number and multiply all the fractions by this: Then has to be 4, and the question is, what fraction of 15 is 4? We would say, but Ahmes figured it this way:

COMPLETION PROBLEMS (#21) 1 15 1/10 1 + ½ 1/5 3 X 1/15 1 X 1/5 + 1/15 4 So the final answer is 1/5 + 1/15. Red auxiliaries are the Egyptian version of common denominators. They are called this because they were written in red ink.

AHA PROBLEMS (#26) A quantity, its ¼ added to it so that 15 results. What is the quantity? (A quantity and its quarter add to 15; what is the quantity?) If the answer were 4, then the sum of 4 and its ¼ would be 5. To get 15, I would have to multiply by 3. So 4 x 3 =12 is the answer. Called the method of false position.

DIVISION OF FRACTIONS (#34) A quantity, its ½ and its ¼ becomes 10. What is the quantity? 1 1 1 2 1 4 2 3 1 7 4 7

DIVISION OF FRACTIONS (#34) A quantity, its ½ and its ¼ becomes 10. What is the quantity? 1 1 1 2 1 4 2 3 1 2 Since 4 x (1+ ½ + ¼) is 7, then (1/7) x (1+ ½ + ¼) is ¼. 4 7 1 7 1 4

DIVISION OF FRACTIONS (#34) A quantity, its ½ and its ¼ becomes 10. What is the quantity? X 1 1 1 2 1 4 2 3 1 2 X 4 7 X 1 7 1 4 1 4 1 1 28 2 x 1 2 1 1 14 5 1 2 1 7 1 14 5 1 2 1 7 1 14 9 1 2 1 4 1 4 10

AREA (#50) A round field has diameter 9 khet. What is its area? Take away 1 / 9 of the diameter, namely 1; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore it contains 64 setat of land. Thus the Egyptian formula for the area of a circle with diameter d is: 2 2 1 8 64 256 A d d d d r 9 9 81 81 2 2

AREA (#50) Note that this implicitly uses the value 256 81 3.16049 Using this produces an error of about 0.6%

PESU PROBLEMS Pesu is a measure of the strength or as we would think of it, the weakness of beer or bread in terms of the grain it is made from. Pesu =

PESU PROBLEMS (#76) Then one hekat will make 24, or 12 loaves of each kind. The quantity of wedyet flour in the 1000 loaves of pesu 10 is 100 hekats. Multiply 100 by 12; the result is 1200, which is the number of loaves of each kind for the exchange.

EGYPTIANS SUMMING UP Like Babylonians, focused on practical problems. No general rules stated. No proofs beyond demonstrations that the number they found satisfies the conditions. We only have a small snapshot of their mathematics, so we could be way off in our assessments.

EGYPTIANS SUMMING UP Solved linear equations, often by false position. A couple of simultaneous equations, one of second degree, are solved in the Berlin papyrus: and. In general, not as sophisticated as Babylonian algebraic methods. Did some squares and square roots.

A LITTLE BIT OF HISTORICAL BIAS: Morris Kline, Mathematics, A Cultural Approach, 1962. 700 pages, 3 of which focus on Egypt and Babylonia. [Egyptian] contributions to mathematics were almost insignificant. Compared with the Greeks, the mathematics of the Egyptians and Babylonians, is the scrawling of children just learning to write, as opposed to great literature. they barely recognized mathematics as a distinct subject and over a period of 4,000 years hardly any progress was made in the subject.

A LITTLE BIT OF HISTORICAL BIAS: Dirk Struik, A Concise History of Mathematics, 1987..Oriental mathematics, in this respect, never seems to have been emancipated from the millennial influence of the problems in technology and administration, for the use of which it had been invented. stagnant cultural atmosphere leading to the relative ossification of mathematics.

NOT TO GET ALL PC ON YOU, BUT: You do have to be careful. Everything is written with a point of view and a cultural context. It is sort of natural for every age of human beings to believe they are the pinnacle of progress, knowledge, understanding. But they are pretty much proven wrong by each successive generation. It behooves us to be humble.