Egyptian Mathematics Timeline
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EGYPTIAN MATHEMATICS TIMELINE Archaic Period (3100 - 2650 BCE) Old Kingdom (2650 - 2134 BCE) – Large pyramids built; rich and productive period 1st Intermediate Period (2200 - 2050) BCE – Chaotic Middle Kingdom (2050 - 1640 BCE) – “Golden Age” Moscow and Ahmes Papyri. 2nd Intermediate Period (1640 – 1550 BCE) New Kingdom (1550 – 1070 BCE) – temple building, empire building. Ramses, Tutankamon, Moses. 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 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 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. HIERATIC TEXT 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 BCE. DEMOTIC TEXT Evolved from an even more cursive form of hieratic and became the standard for the administration from the 25th or 26th Dynasty on (around 600 BCE – 450 CE). 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 one key to our understanding hieroglyphics. Jean-François Champollion did the first successful translation, building on the work of Thomas Young. 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 And thus we come to: “How do we know about Egyptian Mathematics?” Mostly from various papyrus scrolls that have been found and translated. EGYPTIAN MATHEMATICS Two major sources: Moscow Papyrus (1850 BCE): volume of a frustrum, other methods for finding areas, volumes. Rhind Mathematical (Ahmes) Papyrus (1650 < 2000 < 2650 BCE): multiplication and division by doubling and halving, use of unit fractions, fraction doubling tables, approximation of π, linear equations, use of cotangent, attempt to square the circle. EGYPTIAN MATHEMATICS A few minor sources: Berlin Papyrus (1300 BCE): a quadratic equation and its solution; simultaneous equations. Reisner Papyri (1800 BCE): volumes of temples. Kahun (Lahun) Papyrus (1800 BCE): 6 fragments; much is still not translated. Egyptian Mathematical Leather Roll (1850 BCE): Table of decompositions into unit fractions. EGYPTIAN NUMERATION This is covered pretty well in the text… DON’T KNOW MUCH ABOUT HISTORY….. A word about history – There are lots of versions of things. The internet is a dangerous and wonderful thing. “Wikipedia is the best thing ever. Anyone in the world can write anything they want about any subject. So you know you are getting the best possible information.” –Michael Scott DON’T KNOW MUCH ABOUT HISTORY….. Books are also wonderful and dangerous things, for that matter. Scholarship is about differences in opinion. Frog, tadpole, bird, burbot fish Scroll, snare, coil of rope. Astonished man, kneeling figure, scribe, prisoner, Heh, the god of the unending. Heel, rope, hobble for cattle. EGYPTIAN NUMERATION - FRACTIONS For reasons unknown, the ancient Egyptians worked only with unit fractions, that is, fractions with a numerator of 1. The single exception was having a symbol for 2/3*. All other fractions were written as a sum of unit fractions: . So tables were provided to help with this task. *(Maybe ¾, too. That’s history for you.) EGYPTIAN NUMERATION - FRACTIONS Fractions were symbolized by putting the hieroglyph above the denominator. This may stand for “r” meaning an “open mouth.” No, I don’t know why that has anything to do with fractions. The direction of the little man’s feet determines whether we add or subtract. (Sometimes just the feet were used for this symbol.) EGYPTIAN NUMERATION - FRACTIONS Eventually, as hieroglyphic moved to hieratic, the open mouth became a line. Some suggest this was the beginning of our fraction bar. 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 . EGYPTIAN ARITHMETIC: ADDITION We’re not sure how it was done. Tables? Memorization? Counting? Perhaps something like regrouping, based on the base-10 nature of the number system. But it got done, and done pretty accurately, in the problems found in the Papyri. Same for subtraction. DIVISION USING FRACTIONS “Reckon with 8 so as to get 19.” In other words, “What, times 8, will give you 19?” 81 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?” 81 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 1 1 2 Or, 2 4 8 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 3 1 what? 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 3 1 what? 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 1 1 1 2 3 6 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.