Lattices, Dust Boards, and Galleys

Transcription

1 Lattices, Dust Boards, and Galleys J. B. Thoo Yuba College 2012 CMC3-South Conference, Orange, CA

2 References Jean-Luc Chabert (editor) et al. A History of Algorithms: From the Pebble to the Microchip. Springer-Verlag, Berlin, Translator of the English Edition: Chris Weeks. Sir Thomas Heath. A History of Greek Mathematics Volume II: From Aristarchus to Diophantus. Dover Publications, Inc., New York, Annette Imhausen. Egyptian mathematics. In Victor J. Katz, editor, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, Princeton, Victor J. Katz. A History of Mathematics: An Introduction. Pearson Education, Inc., Boston, 3rd edition, Leonardo of Pisa (Fibonacci). Fibonacci s Liber Abaci: A Translation into Modern English of Leonardo Pisano s Book of Calculation. Springer, New York, Translated L. E. Sigler.

3 Multiplication

4 Lattice Also called grating method, gelosia, and multiplication using a tableau, or grid, or net, or the jalousie. India: Ganesa s commentary (16th c.) on Lilavati by Bhaskara II (12th c.). Arabic: arithmetic book Talkhis by Ibn al-banna (13th c.). China: Jiuzhang suanfa by Wu Jing (1450). Europe: earliest known in a Latin ms. ca in England. Also later in Renaissance period arithmetic books like the Treviso (anonymous; 1478) and the Summa by Pacioli (1494).

5 Draw a lattice 2 5

6 Draw a lattice Find the partial products

7 Draw a lattice Find the partial products Add down that diagonals

8 You Try Try found in the 16th century Indian astronomer Ganesa s commentary on the 12th century Indian book Lilavati by Bhaskara II. 1 2

9 You Try Try found in the 16th century Indian astronomer Ganesa s commentary on the 12th century Indian book Lilavati by Bhaskara II

10 Chabert et al., A History of Algorithms, p. 24.

11 Chabert et al., A History of Algorithms, p. 26.

12 Chabert et al., A History of Algorithms, p. 26.

13 Egyptian Hieratic Hieroglyphic Rhind Papyrus

14 Base 10. Not positional. Additive: each symbol repeated as many times as needed, with ten of one symbol replaced by one of next higher value. No more than four of the same symbol grouped together, and when more than four were needed the larger group would be written to the left of above the smaller group Four thousand, three hundred seventy-nine

15 2 3 4 (staff or stroke) (heel bone or loaf) (coiled rope or snake) (lotus flower) one ten hundred thousand 5 6 (bent or pointed finger) (burbot or tadpole) ten thousand hundred thousand 7 (astonished man) (rising sun) million ten million

16 2 3 4 (staff or stroke) (heel bone or loaf) (coiled rope or snake) (lotus flower) one ten hundred thousand 5 6 (bent or pointed finger) (burbot or tadpole) ten thousand hundred thousand 7 (astonished man) (rising sun) million ten million Three hundred twenty-four Thirteen million, one hundred one thousand, twenty-three

17 34 25 \. 2 4 \ 8 \ 16

18 34 25 \ \ \

19 34 25 \ \ = = 850 \

20 You Try

21 You Try = = 1620 \ \

22 Chabert et al., A History of Algorithms, p. 16.

23 Chabert et al., A History of Algorithms, p. 17.

24 ⁹ ₁₀ ⁹ ₁₀ 1⁴ ₅ 3³ ₅ 7¹ ₅ Chabert et al., A History of Algorithms, p. 17.

25 Exercise Show that every counting number can be expressed as a sum of powers of 2.

26 Ibn Labban Kushyar ibn Labban (fl. ca. 1000). Indian mathematician. In Principles of Hindu Recokning shows one method of multiplication. The method was most likely carried out on a dust board or some other easily erasable surface.

27 Positional. Base 10. Uses ten symbols of numeration, corresponding to the base (ten) of the system. Place-holder symbol that is also a number (zero). Separatrix (decimal point) that separates the integer of a numeral from the fraction part.

28 The evolution of the Indo-Arabic numerals from India to Europe. Image from Karl Menninger, Zahlwort und Ziffer, Vandenhoeck & Reprect, Gottingen (1958), Vol. II p <

29 Pierre Simon de Laplace ( ) It is from the Indians that there has come to us the ingenious method of expressing all numbers, in ten characters, by giving them, at the same time, an absolute and a place value; an idea fine and important, which appears indeed so simple, that for this very reason we do not sufficiently recognize its merit. But this very simplicity, and the extreme facility which this method imparts to all calculation, place our system of arithmetic in the first rank of the useful inventions. How difficult it was to invent such a method one can infer from the fact that it escaped the genius of Archimedes and of Apollonius of Perga, two of the greatest men of antiquity.

30

31 = = 6

32 = = 6

33 = 15

34 = = 75

35 = = 75

36

37 = = 83

38 = = 83

39 = 20

40 = = 850

41 = = 850

42 You Try

43 You Try

44 You Try

45 You Try

46 You Try

47 You Try

48 You Try

49 You Try

50 You Try

51 Division

52 Egyptian \ \ = = 850 \

53 \ 1080 \

54 \ \

55 = = \ \

56 Leonardo of Pisa ca Better known today as Fibonacci (only since 19th cent.). Well known in history of banking, e.g., credited with introducing present value. Liber Abaci (Book of Calculation; 1202, 1228) on mathematics of trade, valuation, and commercial arbitrage, and one of the first to introduce the Indo-Arabic number system to Europeans and to demonstrate the number system s practical and commercial use with many examples. In math perhaps best known for the Fibonacci sequence : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,. Wikipedia: en.wikipedia.org/wiki/fibonacci

57 Liber Abaci (1) Here Begins the First Chapter (2) On the Multiplication of Whole Numbers (3) On the Addition of Whole Numbers (4) On the Subtraction of Lesser Numbers from Greater Numbers (5) On the Divisions of Integral Numbers (6) On the Multiplication of Integral Numbers with Fractions (7) On the Addition and Subtraction and Division Of Numbers with Fractions and the Reduction of Several Parts to a Single Part (8) On Finding the Value of Merchandise by the Principal Method (9) On the Barter of Merchandise and Similar Things (10) On Companies and Their Members (11) On the Alloying of Monies (12) Here Begins Chapter Twelve (13) On the Method of Elchataym and How with It Nearly All Problems of Mathematics Are Solved (14) On Finding Square and Cubic Roots, and on the Multiplication, Division, and Subtraction of Them, and On the Treatment of Binomials and Apotomes and their Roots (15) On Pertinent Geometric Rules And on Problems of Algebra Wikipedia: en.wikipedia.org/wiki/liber_abaci

58 Liber Abaci (1) Here Begins the First Chapter (2) On the Multiplication of Whole Numbers (3) On the Addition of Whole Numbers (4) On the Subtraction of Lesser Numbers from Greater Numbers (5) On the Divisions of Integral Numbers (6) On the Multiplication of Integral Numbers with Fractions (7) On the Addition and Subtraction and Division Of Numbers with Fractions and the Reduction of Several Parts to a Single Part (8) On Finding the Value of Merchandise by the Principal Method (9) On the Barter of Merchandise and Similar Things (10) On Companies and Their Members (11) On the Alloying of Monies (12) Here Begins Chapter Twelve (13) On the Method of Elchataym and How with It Nearly All Problems of Mathematics Are Solved (14) On Finding Square and Cubic Roots, and on the Multiplication, Division, and Subtraction of Them, and On the Treatment of Binomials and Apotomes and their Roots (15) On Pertinent Geometric Rules And on Problems of Algebra Wikipedia: en.wikipedia.org/wiki/liber_abaci

59 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration.

60 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration

61 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration

62 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration

63 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration

64 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration

65 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration

66 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration

67 365 2 And if one will wish to divide 365 by 2, then...2 he puts beneath the 5, and he begins by dividing the 3 by the 2, namely the last figure, saying ½ of 3 is 1, and 1 remains; he writes the 1 beneath the 3, and the 1 which remains he writes above, as is displayed in the first illustration; and the remaining 1 couples with the 6 that is next to the last given figure, making 16; he takes ½ of the 16 which is 8; 2 he therefore puts the 8 beneath the 6 put before, the 1 the 3, as is displayed in the second illustration; and as there is no remainder in the division of the 16, one divides the 5 by the 2; the quotient is 2 and the remainder 1; he writes the 2 under the 5, and the 1 which remains he writes over the whole; and before the ½ he writes the quotient coming 2 from the division, namely 182, as one shows in the last illustration ½182

68 You Try

69 You Try ³ ₁₁1139

70 Division by a composite number with two digits. If the divisor has a factor that is also a factor of the dividend. Division by prime numbers with three digits. Division by numbers with four or more digits.

71 Galley Piano Regolatore Sociale Comune di Genova:

72 Also called scratch method in England. Originated in India. Popularized in Europe by Luca Pacioli in Summa (1494). Of the four methods of division Pacioli presented, he considered this method the swiftest just as the galley is the swiftest ship. Universität Bayreuth Lehrstul für Mathematik und ihre Didaktik: did.mat.uni-bayreuth.de/mmlu/duerer/lu/bio_pacioli.htm Used in Europe until as late as the 17th century.

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82 ³ ₁₁

83 You Try

84 You Try ¹³² ₁₇₆

85 Square Roots

86 Babylonian (ca BC) Jiuzhang Suan Shu (Chinese, ca. 150 BC) Theon of Alexandria (fl. AD 375) Heron of Alexandria (3rd century AD) Nicolas Chuquet (d. 1487) Bakhshali Manuscript (uncertain, 3rd 12th century)

87 Babylonian (ca BC) Jiuzhang Suan Shu (Chinese, ca. 150 BC) Theon of Alexandria (fl. AD 375) Heron of Alexandria (3rd century AD) Nicolas Chuquet (d. 1487) Bakhshali Manuscript (uncertain, 3rd 12th century)

88 Babylonian a2 + b a + b 2a

89 Babylonian a2 + b a + b 2a a b/a a b/2a a a 2 b a a 2 N = a2 + b = b = N a 2 b/2a

90 Heron It is in Metrica, in the course of finding the area of a triangle using what we now call Heron s formula, area = s(s a)(s b)(s c), that Heron gave a method for approximating the square root of a nonsquare number as well as we please.

91 In Metrica Heron writes, Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26⅔. Add 27 to this, making 53⅔, and take half of this or 26½¹ ₃. The side of 720 will therefore be very nearly 26½¹ ₃. In fact, if we multiply 26½¹ ₃ by itself, the product is 720¹ ₃₆, so that the difference (in the square) is ¹ ₃₆. If we desire to make the difference still smaller than ¹ ₃₆, we shall take 720¹ ₃₆ instead of 729 [or rather we should take 26½¹ ₃ instead of 27], and by proceeding in the same way we shall find that the resulting difference is much less than ¹ ₃₆.

92 In Metrica Heron writes, Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26⅔. Add 27 to this, making 53⅔, and take half of this or 26½¹ ₃. The side of 720 will therefore be very nearly 26½¹ ₃. In fact, if we multiply 26½¹ ₃ by itself, the product is 720¹ ₃₆, so that the difference (in the square) is ¹ ₃₆. If we desire to make the difference still smaller than ¹ ₃₆, we shall take 720¹ ₃₆ instead of 729 [or rather we should take 26½¹ ₃ instead of 27], and by proceeding in the same way we shall find that the resulting difference is much less than ¹ ₃₆. Heron s method amounts to the iteration formula A αn = 1 α n A α n 1, where α 0 is the next square number succeeding A.

93 Newton s method x n = x n 1 f(x n 1) f (x n 1 )

94 Newton s method x n = x n 1 f(x n 1) f (x n 1 ) For A let f(x) =x 2 A, f (x) =2x x n = x n 1 x2 n 1 A 2x n 1 = 1 2 x n 1 + A x n 1

95 Newton s method x n = x n 1 f(x n 1) f (x n 1 ) For A let f(x) =x 2 A, f (x) =2x x n = x n 1 x2 n 1 A 2x n 1 = 1 2 x n 1 + A x n 1 Heron s method A αn = 1 2 α n 1 + A α n 1

96 Nicolas Chuquet French physician (d. 1487) by profession. Wrote but never published Triparty en la science des nombres (Science of Numbers in Three Parts), credited as the earliest work on algebra of the Rennaisance. In Triparty he coined the terms billion for million million (10 12 ) and trillion for million million million (10 18 ), used in England and Germany, but not in France or the U.S. Generalized al-khwarizmi s methods for solving quadratic equations to solving equations of any degree that are of quadratic type. Noted in Triparty that [to] find a number between two fractions, add numerator to numerator and denominator to denominator. This is the key to his method for finding square roots.

97 Square root of 6 2 < 6 < 3

98 Square root of 6 2 < 6 < 3 2 < < 3 and =6 1 4 > 6

99 Square root of 6 2 < 6 < 3 2 < < 3 and =6 1 4 > 6 2 < 6 < 2 1 2

100 Square root of 6 2 < 6 < 3 2 < < 3 and =6 1 4 > 6 2 < 6 < < < and =5 4 9 < 6

101 Square root of 6 2 < 6 < 3 2 < < 3 and =6 1 4 > 6 2 < 6 < < < and =5 4 9 < < 6 < 2 1 2

102 2 1 3 < 6 < < a b < c d = a b < a + c b + d < c d

103 2 1 3 < 6 < < a b < c d = a b < a + c b + d < c d < < and = < < 6 < 2 1 2

104 2 1 3 < 6 < < a b < c d = a b < a + c b + d < c d < < and = < < 6 < < 6 < < 6 < < 6 < < 6 <

105 2 1 3 < 6 < < a b < c d = a b < a + c b + d < c d < < and = < < 6 < < 6 < < 6 < < 6 < < 6 < =

106 2 1 3 < 6 < < a b < c d = a b < a + c b + d < c d < < and = < < 6 < < 6 < < 6 < < 6 < < 6 < =2.45 and 6=

107 Thank you. John Thoo Jean-Luc Chabert (editor) et al. A History of Algorithms: From the Pebble to the Microchip. Springer-Verlag, Berlin, Translator of the English Edition: Chris Weeks. Sir Thomas Heath. A History of Greek Mathematics Volume II: From Aristarchus to Diophantus. Dover Publications, Inc., New York, Annette Imhausen. Egyptian mathematics. In Victor J. Katz, editor, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, Princeton, Victor J. Katz. A History of Mathematics: An Introduction. Pearson Education, Inc., Boston, 3rd edition, Leonardo of Pisa (Fibonacci). Fibonacci s Liber Abaci: A Translation into Modern English of Leonardo Pisano s Book of Calculation. Springer, New York, Translated L. E. Sigler.