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Egyptian_BabylonianMath - Egyptian and Babylonian Egyptian...

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Unformatted text preview: Egyptian and Babylonian Egyptian and Babylonian Mathematics Egyptian Mathematics Egyptian Mathematics 3100 B.C. ­ On a royal Egyptian mace are several numbers in the millions and hundred thousands written in Egyptian hieroglyphics. 2600 B.C. ­ The Great Pyramid at Gizeh was built. Involved some mathematical and engineering problems. 1850 B.C. – Approximate date of Moscow papyrus. Contains 25 mathematical problems. 1650 B.C. – Approximate date of Rhind papyrus. Rich source of ancient Egyptian mathematics. Rhind Papyrus Rhind Papyrus Describes the Egyptian methods of multiplying and dividing. Use of unit fractions. Solution of the problem of finding the area of a circle. Method of false position. Many practical applications of mathematics Egyptian algorithm for division Egyptian algorithm for division Ex. 534/26 1 26 4 104 8 208 16 416 Now, since 2 52 Double the divisor 26 up to the point where the next doubling would exceed the dividend 534 534 = 416 + 118 = 416 + 104 + 14 Then 534/26 = 16 + 4 + 14/26 = 20 remainder 14 Geometric problems Geometric problems 26 of the 110 problems in the Moscow and Rhind papyri are geometric. Area of a circle is taken as equal to area of the square on 8/9 of the diameter. Volume of right circular cylinder. Area of triangle ­ 1/2(base)(height) Volume of a truncated square pyramid Unit Fractions Unit Fractions To avoid some of the computational difficulties with fractions, Egyptians expressed all fractions, except 2/3, as the sum of unit fractions – 1/n. Possible by representing fractions of the form 2/n. Table of 2/n for all odd n from 5 to 101 in the Rhind Papyrus. Pythagorean Theorem Pythagorean Theorem Egyptians architects used a clever devise for making right angles. Tie 12 equally long segments of rope into a loop. Stretching 5 consecutive segments in a straight line from B to C, pulling the rope taut at A, formed a right triangle ABC. Pythagorean theorem implicit in construction Egyptian Right Triangle Egyptian Right Triangle C 4 5 A 3 B Solved Linear Equations Solved Linear Equations On the Moscow papyrus, find the number such that if it is taken 1 and ½ times and then 4 is added, the sum is 10. Problem 31 of the Rhind papyrus asks to find a quantity such that the sum of itself, its 2/3, its ½ and its 1/7 becomes 33. The answer given is 14 28/97. Method of False Position Method of False Position Problem 26 of the Rhind papyrus seeks to find a quantity such that when it is added to ¼ of itself, the result is 15. This problem is solved by the method of false position. Assume a convenient but incorrect answer and adjust appropriately. Considered an algorithm for solving linear equations. Babylonian Mathematics Babylonian Mathematics Discovery of half a million inscribed clay tablets in Mesopotamia. 400 tablets identified as strictly mathematical tablets containing tables and mathematical problems. Range over many periods of Babylonian history and all phases of daily life. Time Period Time Period 2100 B.C. ­ oldest tablets 2100 B.C. – 1600 B.C. – First Babylonian Dynasty of King Hammurabi – large collection 600 B.C. – 300 A.D. ­ third group of tablets Show a high level of computational ability Used sexagesimal (base 60) positional system Mathematical Tablets Mathematical Tablets Tablets show that ancientBabylonians were familiar with bills, receipts, account, simple and compound interest, mortgages. Out of 400 tablets, 200 contain tables such as multiplication tables, tables of reciprocals (reduce division to multiplication), tables of squares and cubes, tables of exponentials. Used for problems on compound interest. 2000 B.C. – 1600 B.C. 2000 B.C. – 1600 B.C. Familiar with general rules for finding areas of rectangles, right and isosceles triangles, trapezoids having one side perpendicular to the parallel sides. Volume of a rectangular box. Circumference of a circle was taken as 3 times the diameter and the area as 1/12 the square of the circumference. Babylonian Geometry Babylonian Geometry Volume of right circular cylinder Pythagorean Theorem A tablet was recently discovered in which 3 1/8 was used as an estimate for pi. Main characteristics of Babylonian geometry is its algebraic character. The more intricate geometrical problems are nontrivial algebra problems. Babylonian Algebra Babylonian Algebra By 2000 B.C. Babylonian arithmetic had developed into algebra. Considered problems that led to quadratic equations or to systems of equations. Quadratic equations were solved either by completing the square or by substituting in a general formula. Also cubic and biquadratic equations. Babylonian Algebra Babylonian Algebra A tablet was found with tables for the squares and cubes of the integers from 1 to 30 and also of the combination n + n A number of problems are given that lead b to cubics of the form x 3 +x 2 = Tablets of about 1600 B.C. listing hundreds of unsolved problems involving systems of equations. 3 2 Interesting Problems Interesting Problems Sums of series (geometric etc) Approximations to square roots Remarkable tablet called Plimpton 322, contains Pythagorean triples. A set of three positive integers which can be the sides of a right triangle is known as a Pythagorean triple. Primitive Pythagorean triple. Examples Examples (3, 4, 5) – primitive triple (6, 8, 10) – not a primitive triple (5, 12, 13) – primitive triple (8, 15, 17) – primitive triple All primitive Pythagorean triples (a, b, c) are given by a = 2uv, b=u −v , 2 2 c=u +v 2 2 Babylonian Mathematics Babylonian Mathematics Looking at all of these tablets, we can conclude that Babylonians were highly skilled at computing and very strong in algebra. Remarkable how diverse and deep were the problems they considered. ...
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