Egypt Egyptian mathematics

1680 B.C. Egypt (“Middle Kingdom”) Civil War Hyksos  “Rulers of Foreign Lands” invaded Joseph (Coat of Many Colors) set against this background Turning point for Egypt Invasion

Located near major rivers Slave class for Manual Labor Babylonian Math and Egyptian Math Located near major rivers Slave class for Manual Labor Agriculture and Farming Motivated Math Theocratic Society Comparing

contrasting Babylonian Egyptian Constant Invasions by neighbors Trade more prevalent Tigris and Euphrates required more engineering Math problems were more difficult Remained Secluded and in peace Semi-isolation didn’t require trade Nile flood seasons required simple irrigation More regular and welcome Emphasis on Astronomy to predict seasons contrasting

Egyptian Pyramids 2500 BC: Great Pyramid at Gizeh 13 Acres 2 million stone blocks Each 2.5 tons 54 ton granite blocks – roof Dragged 600 miles Set 200 ft above ground Right angles within 1/27000 of 90 degrees Sides of equal length up to 1/14000

Astronomical instruments found in Great pyramids

Egyptians are believed to have the first base-10 system. As early as 2700 BCE (probably earlier) Numeration

Writing System Egyptians wrote on papyrus, leather, cloth, and stone. Hieroglyphics Lead to Tally System Hieratic Cursive form of hieroglyphics Writing System

The Moscow Papyrus Student text covering practical problems and the application of geometric formula Older than the Rhind Papyrus

Rhind Papyrus: 1650 B.C. Written by A’h-mose Found by Henry Rhind Illustrates Egyptian Method of Multiplication Math was taught by example Applied arithmetic

Egyptian Method of Multiplication Method of Doubling All numbers can be written in base 2

Multiply (27)(19) using only Egyptian methods 27 1 54 2 108 4 216 8 432 16 19 = 16 + 2 + 1 (27)(19) = 432 + 54 + 27 Multiply (27)(19) using only Egyptian methods

Egyptian Division Division worked using the same method in reverse. Called Halving For example, let’s divide 1495 by 65. Double 65 until we are about to pass 1495

Egyptian Division Tick off the numbers on the right column that add to a number between 1495 and 1495 – 65.

Egyptian Division What if the numbers didn’t add up perfectly

Egyptian Division What if the numbers didn’t add up perfectly

12 1 24 2 48 4 96 8 192 16 192 + 96 + 24 + 12 = 324 1 + 2 + 8 + 16 = 27 Your turn: Divide 324 by 12

Fractions Egyptians only allowed unit fractions: Symbol for fraction was oval Same symbol for mouth Common used fractions – Horus –eye 2/3 and 3/4 were only exceptions

Egyptians are given credit for the earliest known formula for the area of a circle in terms of the diameter D: Area of a Circle