ETHNOMATHEMATICS. Dr. Eduardo Jesús Arismendi-Pardi. The Study of People, Culture, and Mathematical Anthropology. Orange Coast College

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1 ETHNOMATHEMATICS The Study of People, Culture, and Mathematical Anthropology Dr. Eduardo Jesús Arismendi-Pardi Orange Coast College The field of ethnomathematics examines the epistemology and genesis of mathematical ideas from a global and cultural perspective. ethno refers to an identifiable cultural group and their jargon, codes, symbols, myths, and specific ways of reasoning and making inferences about the world around them. 1. How are ad hoc practices and solutions to problems developed into methods?. How are methods developed into theories? 3. How are theories developed into scientific invention?

2 Ethnomathematical knowledge demonstrates and proposes that Greek foundations of European knowledge are founded upon the ancient Black Egyptian civilization. The mathematical traditions of many non-western cultures focus on the applications and usefulness of mathematics. Srinivasa Ramanujan ( ) style of mathematics was different from conventional Western tradition of deductive axiomatic method of proof. Ethnomathematics literature: Mesopotamia, Egypt, China, pre-colombian America (Aztec, Maya, and Inca), India, Arab- Islamic (Iran, Turkey, Afghanistan, and Pakistan).

3 African Americans that have contributed to the mathematical sciences include: Benjamin Banneker ( ) Marjorie Lee Browne ( ) David Blackwell (1919-) Evelyn Boyd Granville (194-). Anatomy of Eurocentric Bias 1. The Classical Eurocentric Trajectory. The Modified Eurocentric Trajectory 3. The Alternative Trajectory for the Dark Ages The Critic s Argument (1 of ) Eurocentric argument: You see they had no proofs, as we know them in modern mathematics. Were their contributions really mathematics? Design APR,Inc

4 The Critic s Argument ( of ) The fact is that the notion of what a proof is has changed over time and there is no consensus on what constitutes proof. Ancient Ancient Babylonian Babylonian Tablet Tablet (1 (1 of of 3) 3) The length of a rectangle exceeds width by 7 units. width Its area by 7 is units. 60 Its square area units. is 60 Find square the length units. and Find the width the (c. length 3,500 and years the ago). width (c. 3,500 years ago). The length of a rectangle exceeds its x 7x60 In modern algebraic notation: where x is the width of the rectangle. 1. Halve the quantity by which the length exceeds, that is, Square 3.5. To this result add the area, that is, Find the square root of this sum. 4. Subtract 3.5 from this square root, that is, 8.5 to get the width as 5 units. 5. Add 3.5 to the square root to get the length as 1 units.

5 Ancient Babylonian Tablet (3 of 3) The modern symbolic form of the solution for b b is x c x bx c. Can one claim that the Babylonians were not aware of this general form of the solution, even if they did not express it in symbolic terms? Design APR, In. Hypothesis It is plausible that a non-symbolic, rhetorical argument or proof can be quite sophisticated and rigorous when given a particular value of the variable. The condition for the sophistication or rigor is that the particular value of the variable should be typical and the generalization of the result to any value should be immediate. Design APR, In. of the Aztec ( of 3) The Mathematics of the Navajo (1 of ) The Navajo mathematics is a creative process visualized in the mind of the weaver. The results of such art can be beautifully contemplated in a completed rug.

6 The Mathematics of the Navajo ( of ) Creative process of weaving rugs: Counting, limit, circle, symmetry, slope of the line. The Mathematics of the Aztec (1 of 3) The Aztecs also had a special symbol for zero that was represented as a corn glyph. Geometrical patterns used by the Aztecs: micocoli field and tlahuelmantlii area. Micocoli field is a rectangular or quadrilateral region whose sides are measured in quahuitls (.5 meters). The Mathematics of the Aztec ( of 3) A tlahuelmantlii area with a constraint of at most 400 square quahuitls. Area < 400 square quahuitls Subject to: A 0 y x 0 x 0 0 y 0 z 0

7 The Mathematics of the Aztec (3 of 3) A tlahuelmantlii area with a constraint of least 400 square quahuitls. Area > 400 square quahuitls Subject to: A 0z x 0 x 0 z 1 y 0 The Mathematics of the Maya (1 of 4) Let n be a fixed integer greater than 1. If a and b are integers such that a b is divisible by n, we say that a is congruent to b modulo n. a b(m od n) The Mathematics of the Maya ( of 4) Let n = 5, then we have 18 since 18 3 is divisible by 5. 3(mod5)

8 The Mathematics of the Maya (3 of 4) Speculative algorithm: If a specified date is denoted in vigesimal (i.e., base-0) calendrical notation by m, n, p, q, and r, where 0 mnpqr,,,, 19 and 0 q 17,, and an initial date is given by ( t0, v0, y 0), then a new date (, tvy, ) which is m, n, p, q, and r days later. The Mathematics of the Maya (4 of 4) t t0 mn4p7 qr(mod13) v v0 r(mod 0) yy0 190m100n5p0 qr(mod365) If the given Maya date is given by (4, 15, 10) then the new date (t, v, y) which is 0,, 5, 11, and 18 days later is (10, 13, 133). Design APR, Inc Euclidean Proposition 1: Two points determine a line. Maya Proposition 1: A point is used for infinity; a line is used to indicate five units; the Canamayté Quadrivertex is formed by line segments. Design APR, Inc

9 Euclidean Proposition : A line can be extended from each end. Proposition : Maya The lines that calculate the solstice angles can be extended. Euclidean Proposition 3: Given a point and a center it is possible to draw a circle. Maya Proposition 3: For the Maya, the Earth is the center of a circle; the center of the universe is the sun; the radius of such circle was the segment given by the distance from the Earth to the Moon; The circle was observed in the phases of the Moon indicated in the Canamayté Quadrivertex. Euclidean Proposition 4: All right angles are congruent. Maya Proposition 4: In the cross inserted on the Canamayté Quadrivertex allangles are right angles; the four interior squares and all the interior angles are congruent.

10 Euclidean Proposition 5: Two lines are parallel if they do not intersect as they approach infinity. Maya Proposition 5: The Canamayté Quadrivertex is a geometrical structure with opposite sides parallel, such that if they are extended toward infinity, they will never intersect. Design APR,Inc Mathematics of the Inca (1 of 5) The quipu was used to carry messages that had to be clear, concise, and portable from one region to another region that may have been separated by hundreds or even thousands of miles. Quipu makers were responsible for coding and de-coding messages. Mathematics of the Inca ( of 5) Mathematics of the Inca ( of 5) A single knot represents powers of 10, that is 10 n. A figure eight knot denotes the number 1 as well as a 5- turn slip knot. The absence of knots denotes zero.

11 Mathematics of the Inca (3 of 5) In combination these knots represented the numbers, 3, 4, 5, 6, 7, 8, and 9. Main Cord, Pendant Cord, Top Cord, Subsidiary Cords (various hierarchical levels), and Dangle End Cord. Mathematics of the Inca (4 of 5) Three sheds are built. They are different in size but each has walls made of cinder block; a floor and roof made of wooden boards. The materials used for the first shed are 84 cinder blocks, 100 pounds of mortar, 8 boards, and 00 pounds of nails. Mathematics of the Inca (5 of 5) For the second and third shed respectively the materials consist of 44 cider blocks, 85 pounds of mortar, 4 boards, 170 pounds of nails; and 364 cinder blocks, 150 pounds of mortar, 51 boards, and 400 pounds of nails.

12 Mathematics of Egypt (1 of 7) Problem 14 of the Moscow Papyrus: A truncated pyramid of six cubits; vertical height of 4 cubits and base of cubits. The Egyptian approach to solving the problem is equivalent to the modern symbolic representation of the formula given by 1 ( V h a ab b ) 3 Design APR, Inc Mathematics of Egypt ( of 7) Berlin Papyrus: It is said to thee that the area of a square of 100 square cubits is equal to that of two smaller squares. The side of one is of the other. Let us find the sides of the two smallest squares. Mathematics of Egypt (3 of 7) x y 100 4x3y0 (6,8) or (-6,-8).

13 Mathematics of Egypt (4 of 7) In Egypt Fibonacci learned about the series 1, 1,, 3, 5, 8, 13, 1, 34,, He popularized the mystical qualities of the series that now bears his name by bringing to Europe the famous rabbit problem during the 1th century. Mathematics of Egypt (5 of 7) The quotient resulting from the ratio of any term divided by the previous term approaches a k lim k a k Mathematics of Egypt (6 of 7) There is also architectural and archaeological evidence that the Egyptians had worked out a relationship that exists between the values of the 6 constants and in 5

14 Mathematics of Egypt (7 of 7) The following limit problem may have been used by Imhotep among the many calculations that he carried out in his design efforts for the building of the Step Pyramid n1 lim ( ) n n n n n n (the answer is ½) Mathematics of Arabia (1 of 5) al-khwarizmi wrote Hisab al-jabar w almukabala (85 A.D.), that is, Calculation by Restoration and Reduction was the subject of algebra. al-khwarizmi also wrote Algorithmi de Numero Indorum, that is,calculation with Indian Numerals whose original Arabic version no longer exists. The general forms of equations outlined by al- Khawarizmi are ( of 5) : 1. Roots equal squares, bx ax. Roots equal numbers, bx c 3. Squares equal numbers, ax 4. Squares and roots equal numbers, ax bx c 5. Roots and numbers equal squares bx c ax 6. Squares and numbers equal roots, c ax c bx

15 Mathematics of Arabia (3 of 5) Omar Khayyam s solution to the cubic equation of the form x 3 cx d is what he referred to as the equation for the case in which a cube and its sides equals a number. Mathematics of Arabia (4 of 5) Nasir al-din al-tusi also provided the following rule, here in modern notation, for the solution of any triangle with angles ABC and sides a, b, c. b sin r B c r sin C Mathematics of Arabia (5 of 5) Development of the sine tables for various angles. 3 For any angle sin(3 ) 3sin4sin (al-kashi s identity)

16 Mathematics of China (1 of 3) The Chinese also made impressive contributions in trigonometry. Their work was probably based on the work of the Arabs. Mathematics of China ( of 3) The Chinese used red colored counting rods to represent positive and black colored rods to represent negative quantities to solve systems of linear equations. Problem from the Chiu Chang Suan Shu x 3 y 8 z 3 6 x y z 6 3 x 1 y 3 z 0 Mathematics of China (3 of 3) Yang Hui s book the Chiu Chang or The Nine Chapters on the Mathematical Arts: Land Surveying, Percentages and Proportions, Distributions by Proportions, Extraction of Square and Cube Roots, Engineering Mathematics, Fair Taxation, Excess and Deficiency (topics in linear algebra and determinants), Solutions of Simultaneous Equations and the Method of Rectangular Arrays, Right-angle Triangles

17 Mathematics of India (1 of 9) Rama who lived in the middle of the fifteenth century provided an approximation for square root of on three different sulbasutras (3)(4) (3)(4)(34) (3)(4)(34)(33) (3)(4)(34)(34) Mathematics of India (Amuyoga Dwara Sutra) ( of 9) a,( a ),(( a ) ),... a, a, a, a,... Mathematics of India (Bakhshali Manuscript) (3 of 9) r ( ) r a a r ( a ) a A a r a where a is the perfect square nearest to A and raa

18 Mathematics of India (4 of 9) 5 ( ) (6 ) 1 Mathematics of India (Bakhshali Manuscript) (5 of 9) x b 4acb a is a solution to: ax bx c 0 Mathematics of India (6 of 9) (Bhaskara I) sin 16 ( ) 5 4 ( ) (Aryabhata I) 1 sin( n1) sin( n) sin( n) sin( n1) (sin( n)) 5

19 Mathematics of India (7 of 9) cos sin( ) (Varahamihira) sin cos 1 (Varahamihira) Mathematics of India (8 of 9) (Varahamihira) sin (sin ( ) ver sin ( )) (1 cos( )) (Brahmagupta) 1sin cos sin ( ) Mathematics of India (9 of 9) (Arybhata II) 1 sin sin( ) 4 (Bhaskaracharya) sin( ) sin coscossin

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