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A Very Brief History of Mathematics of the Mesopotamians and Egyptians

A Very Brief History of Mathematics of the Mesopotamians and Egyptians. Michael Flicker Spring 2018. Egyptian and Mesopotamian. 3300 BCE to 1500 BCE Early number systems Simple arithmetic, practical geometry Mesopotamian cuneiform tablets & Egyptian papyri Mathematical tables

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A Very Brief History of Mathematics of the Mesopotamians and Egyptians

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  1. A Very Brief History of Mathematics of the Mesopotamians and Egyptians Michael Flicker Spring 2018

  2. Egyptian and Mesopotamian • 3300 BCE to 1500 BCE • Early number systems • Simple arithmetic, practical geometry • Mesopotamian cuneiform tablets & Egyptian papyri • Mathematical tables • Collections of mathematical problems

  3. Uruk 3350 – 3200 BCE • Five thousand clay tablets, reused as building rubble in the central temple precinct of the city of Uruk, constitute the world’s assemblage of written records. • 10% were written by trainee administrators as they learned to write. • Most of those exercises are standardized lists of nouns used within the book-keeping system, but one tablet (W 19408,76) contains two exercises on calculating the areas of fields. • It is the world’s oldest piece of recorded mathematics.

  4. What the Tablet tells us

  5. Oldest Datable Mathematical Table c. 2600 BCE Shuruppag North of Uruk (VAT12593)

  6. Area Units • 1 square rod = 1 sar • 1 ubu = 50 sar • 1 iku = 2 ubu = 1 40 sar (100 in sexagesimal) • 1 eshe = 6 iku = 10 ; sar • 1 bur = 3 eshe = 30 ; sar • 5 ; rods x 5 ; rods = 25 ; ; sar • 1 bur = 30 ; sar • 2 bur = 60 ; sar = 1 ; ; 50 bur = 25 ; ; sar • 5 ; rods x 5 ; rods = 50 bur

  7. VAT 12593 Table comes from the Sumerian city of Shuruppag to the north of Uruk c 2600 BCE

  8. Earliest Known Mathematical DiagramIM 58045 2350 – 2250 BCE

  9. Some problems from 24th Century BCE • HS 815 -- Given a rectangle with area 1 iku and long side 60 + 7 1/2 rods (decimal notation), what is the short side? • Answer 1 rod, 5 cubits, 2 double-hands, 3 fingers, and 1/3 finger • The solution provided is correct but the method is not given. • AO11409 -- The long side is 1 03 ½ rods The short side by the watercourse is 22 ½ rods The lower long side is 1 30 rods The short side (reached by) irrigation is 22 ½ rods The harrowed area is 2 eshe 1 ½ iku Foxy the felter

  10. Mesopotamian Mathematics • Babylonian civilization (2000 BC – 600 BC) replaced Sumerian and Akkadian civilizations • Base 60 positional number system • Numbers system built on symbol for 1 and 10 • No character to indicate a position was vacant until 300 BC • No Zero • More general treatment of fractions than Egyptians • Great advance was that their number system was extended to fractions and positional notation.

  11. Symbols for numbers 1 – 59

  12. Number System Examples • Use ‘I’ for the Babylonian one and ‘<‘ for the Babylonian 10 • <<II = 22, <<<IIII = 34 • << << = 20*60 + 20 = 1220 • << << = 20*602 + 20*60 = 73200 • II II <I = 2*60 + 2 + 11/60 • I II could be 62 or 3602 In approximately 300 BC the Babylonians added a symbol like “ to show if a position was empty. It was only used for intermediate position and not end positions (no true zero) • I “ II means 1*602 + 0*60 + 2 or 60 + 0 + 2/60

  13. Multiplication • Babylonians could multiply in their base 60 system the same way we multiply in our base 10 system 139 2,19 x721,12 12x19 = 3,48 (from table) 278 27,48 973 2,19___ 10008 2,46,48 = 2x602 + 46x60 + 48 = 10008

  14. The Old Babylonian Period (c 2000 – 1600 BCE) • Trainee scribes memorized a large number of reciprocal and multiplicative relationships as a normal part of their elementary education, writing them down as part of the memorization process. • Memorizing reciprocals 2 05 28 48 8 53 20 6 45 4 10 14 24 1 06 40 54 17 46 40 3 22 30 • Large tables for converting units of weight, length, area • Tables of squares (1 – 59) and cubes

  15. An Algebra Problem in Two Variables

  16. √2 Yale Collection 7289 (1800 – 1600 BCE) 30 1; 24; 51; 10 42; 25; 35 Observation 30 x 1; 24; 51; 10 = 42; 25; 35 If side of square is 1, tablet says diagonal is 1; 24; 51; 10 = 1.41421296 √2 = 1.41421356 … 1; 24; 51; 11 = 1.41427593 How did they calculate the approximation to √2?

  17. Other Features • Developed an accurate process for finding square roots divide & average √2 = 1.41421356 after 4 steps • Constructed tables of squares to assist in multiplication xy = ((x + y)2 – (x – y)2)/4 • Able to solve quadratic equation and linear equations in two variables • Solved cubic equations of the form ax3 + bx2 = c; multiply by a2/b3; (ax/b)3 + (ax/b)2 = ca2/b3 Solutions for y3 + y2 = constant using tables and interpolation

  18. Other Facts Known • They summed arithmetic progressions and geometric progressions in concrete problems 1 + 2 + 22 + … 29 = 210 – 1 • Also, the sum of the squares of the integers from 1 to 10 was given as though they had applied the formula 12 + 22 + … + n2 = (1 x 1/3 + n x 2/3)(1 + 2 + 3 … + n) • In addition to the Pythagorean triplets they also solved x2 + y2 = 2 z2 in integers.

  19. Quadratic Equation Prescription Problem Statement: Find two numbers with sum 20 and product 91 • Find half of the sum : 10 • Subtract the product from the square of result of step 1: 100 – 91 = 9 • Take the square root of the result of step 2: 3 • Add and subtract the result of step 3 to the result of step 1: 10 + 3 = 13 and 10 – 3 = 7 • The numbers are 13 and 7

  20. Line Geometry Problems

  21. Line Geometry Problems

  22. Babylonian GeometryPlimpton 322 Tablet (c 1800 BCE)

  23. Pythagorean Triplets A Pythagorean triple is a set of three integers such that a2 + b2 = c2 Example : 32 + 42 = 52 Plimpton 322 contains 15 sets of three numbers. The 2nd and 3rd numbers are part of triplets c b right triangle a Example (3nd line): 4601 (col 2) & 6649 (col 3) – the third side is 4800 The 1st number is (c/b)^2 where b is the shorter side. Are they doing trig? They may have known the formula for generating the triplets proved by Euclid.

  24. Geometry not a Babylonian Strength • For practical problems they calculated the area of a circle by A = C2/12 where C is the circumference. This implies a value p = 3. • At least one Babylonian tablet states that the ratio of the circumference of a circle to the perimeter of an inscribed hexagon is (in modern notation) 1:0.96, implying a value of p=3.125, a value that is too small by about half a percent. • Babylonian geometry was a collection of rules for the areas of plane figures and the volumes of simple solids.

  25. Observations • Used knowledge of arithmetic and simple algebra to express lengths and weights, to exchange money and merchandise, to compute simple and compound interest, to calculate taxes, and to apportion shares of a harvest to the farmer, church, and state. The division of fields and inheritances led to algebraic problems. • Canals, dams, and other irrigation projects required calculations. The use of bricks raised numerous numerical and geometrical problems. Volumes of granaries and buildings and the areas of fields had to be determined.

  26. Contributions • The Babylonians’ development of a systematic way of writing whole numbers and fractions enabling them to carry arithmetic to a fairly advanced stage. They possessed some numerical and algebraic skill in the solution of special equations of high degree, but on the whole their arithmetic and algebra were very elementary. They understood fractions and the use of a base 60 system made it easy to divide by 2, 3, 4, 5, and 6.

  27. Hieroglyphic & Hieratic Numerals

  28. Egyptian Arithmetic • The Egyptians could add, subtract, multiply and divide • From the papyri we can deduce the methods used for multiplication and division • There are hardly any clues for addition and subtraction of integers • It seems that these operations were performed and checked elsewhere • Did they have an addition table? • Did they add by simply counting?

  29. Hieratic & Hieroglyphic

  30. Egyptian Multiplication • Multiplication of whole numbers used the method of doubling Example: 57 x 117 1 117 2 234 4 468 8 936 16 1872 323744 57 6669

  31. Primary Egyptian Sources • Rhind Mathematical Papyrus (RMP) • About 1650 BCE from writings made 200 years earlier (18 ft x 13 in) • 84 (87) mathematical problems • Recto Table • Moscow Mathematical Papyrus – • 1850 BCE (date somewhat uncertain), (18 ft x 1.5 to 3 in) • 25 problems • Egyptian Mathematical Leather Roll • Date ?, (10 in by 17 in) Bought by Rhind • A collection of 26 sums done in unit fractions • It took 60 years and much work to understand its contents • Big disappointment • Berlin Papyrus • 1800 BCE probably written 150 years before RMP

  32. Recto Table Note: entry for 2/45 is incorrect.

  33. Fractions & Algebra • Except for 2/3 and possibly 3/4 the Egyptian arithmetic notation only permitted fractions with unity in the numerator • 2/n equivalents in the RMP Recto table • Could multiply and divide mixed numbers • Simple Algebra • Simple Geometry • Area and Volume

  34. Introduction to RMP • The entrance into the knowledge of all existing things and all obscure secrets. This book was copied in the year 33, in the 4th month of the inundation season, under the majesty of the king of Upper and Lower Egypt, A-user-Re (probably AweserreApopi), endowed with life, in likeness to writings of old made in the time of Upper and Lower Egypt, Ne-ma-et-Re (NemareAmmenemes III). It is the scribe Ah-mose who copies this writing.

  35. Contents of Rhind Mathematical Papyrus • Problems 1 – 6 • The division of 1,2,6,7,8,9, loaves among 10 men • Problems 7 – 20 • The multiplication of (1 2 4) and ( 1 3 3) by various multipliers containing unit fractions • Problems 21 – 23 • Examples in subtraction with fractions • Problems 24 – 27 • Equations in one unknown of the first degree solved by method of false assumption • Problem 24 A quantity and its seventh, added together give 19. What is the quantity? • Solve on board

  36. Contents of RMP (2) • Problems 28 – 29 • “Think of a number” problems • Problems 30 – 38 • More difficult equations of one unknown of the first degree • Problems 39 – 40 • Arithmetic progressions • Problems 41 – 46 • Volumes and contents of cylindrical and rectangular granaries • Problems 48 – 55 • Areas of triangles, rectangles, trapezia, and circles • Problems 56 – 60 • Sekeds, altitudes, and bases of pyramids • Problems 61 – 87 • Miscellaneous problems

  37. Rhind Papyrus

  38. Problem 56 of RMP Problem 56 says: If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked? Ahmes’s solution

  39. Ahmes’s solution to RMP 56 • Seked – slope of the sides of the pyramid (Δx/Δy) • First find ½ of 360 = 180 • Divide 180 by 250 - equivalent to what must you multiply 250 by to get 180 1 250 1/2 125 1/5 50 1/50 5 250 x (1/2 + 1/5 + 1/50) = 180 implies the seked is 1/2 + 1/5 + 1/50 The Egyptians use units of palms/cubit with 7 palms per cubit. In these units the seked is 7x(1/2 +1/5 + 1/50) = 5 + 1/25 palms per cubit

  40. Moscow Mathematical Papyrus - Problem 14 Find the volume of a truncated right pyramid. The method used is equivalent to the formula V = (h/3)(a2 + ab + b2) where h is the height of the truncated pyramid, and a and b are the edges of the lower and upper edges of the truncated pyramid.

  41. Egyptian Geometry • Area and perimeter • Rectangle, triangle, trapezoid, arbitrary quadrilateral, circle • Rule for arbitrary quadrilateral was incorrect • Area of circle given by the area of a square with side 8D/9 where D is the circle diameter. Implies π ≈ 3.16. • Egyptians knew that the ratio area/perimeter for a circle and a circumscribed square are equal. (some debate) • May have known the surface area of a sphere • Volume • Pyramid (no written evidence but must have known it), truncated pyramid • Volume cylinder • Rectangular

  42. Berlin Papyrus - Equations of the Second Degree • If it is said to thee … the area of a square of 100 (square cubits) is equal to that of two smaller squares. The side of one (x) is 2 4 the side of the other (y). Let me know the sides of the unknown squares. • 2 4 is 3/4 so we would write the problem as x = 3/4y and x2 + y2 = 100 Our solution: 9/16y2 + y2 = 100 25/16y2 = 100 y2 = 64 and y = 8 (no negatives for the Egyptians) x = 3/4y = 6

  43. Berlin Papyrus - Equations of the Second Degree • What would the Egyptians have done? • Let the side of square A be 1. • The side of square B is (2 4). • The area of the square B is (1/2 + 1/4)x(1/2 + 1/4) 1 (2 4) 2 (4 8) 4 (8 16) • The area of the square B is (4 8 8 16) = (2 16) • The area of squares A + B is (1 2 16) • The side of the scaled down large square is the square root of (1 2 16)

  44. What would the Egyptians have done? • The Egyptians knew that the square root of (1 2 16) was (1 4). • We have a large square with area 100 and side 10 and a scaled down version with side(1 4). What is the scale factor? 1 (1 4) 2 (2 2) 4 5 8 10 so the scale factor is 8. The sides of two squares are 8 times 1 or 8 and 8 times (2 4) or 6. Notice the 6, 8, 10 triangle • Did the Egyptians know the Pythagorean theorem?

  45. Egyptian Mathematics • The Egyptian papyri show practical techniques for solving everyday problems • The rules in the papyri are seldom motivated and the papyri may in fact only be a study guide for students. • However, they demonstrated a solid understanding of the operations of addition, subtraction, multiplication and division and enough geometry for their needs. • Did their knowledge remain static for the next 1000 yrs?

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