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Chapters 1 - 4 Review

Chapters 1 - 4 Review. The student will learn more about. Some of the ancient numeration systems. Sexigesimal System. Note: one can have but not. Ancient Notation versus Modern Notation. 1, 56 ;. Let a, b, . . . Be integers  0 and < than 60, then

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Chapters 1 - 4 Review

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  1. Chapters 1 - 4 Review The student will learn more about Some of the ancient numeration systems.

  2. Sexigesimal System Note: one can have but not . Ancient Notation versus Modern Notation 1, 56 ; Let a, b, . . . Be integers  0 and < than 60, then a, b, c;d,e = a · 60 2 + b · 60 + c + d · 60 –1 + e · 60 -2

  3. Sexigesimal Conversion One may convert from base 10 to base 60 and vice-a-versa. We will not do that at this time since I want you to have a feeling for base 60 and how the Babylonians did their calculations and of course they did not use base 10.

  4. Sexigesimal Addition Base 10 21ten + 34ten Babylonian Modern Notation 27 + 45 12 1, 12 55 ten

  5. Sexigesimal Addition All Modern Notation 1 1 1 12, 32 25, 41 11, 00 + 00, 45 2, 34, 56 ; 23, 15 + 25, 52 ; 14, 27 3, 00, 48 ; 37, 42 49, 58

  6. Sexigesimal Subtraction Base 10 45 ten - 27 ten Babylonian Modern Notation 45 - 27 18 18 ten

  7. Sexigesimal Subtraction All Modern Notation 0, 87 24 92 2, 34, 56 ; 23, 15 - 15, 52 ; 14, 27 1, 27 - 45 25, 32 - 12, 41 42 12, 51 2, 19, 04 ; 08, 48

  8. Duplation Review Duplation method of Multiplication 17ten· 42ten 17 · 42 1 42 2 84 4 168 8 336 16 672 17 714

  9. Duplation Review · 6 of these carry 10 of these carry Duplation method of Multiplication Babylonian - 13 · 21

  10. Duplation Review Duplation method of Multiplication 13 · 21 in modern sexigesimal notation 13 · 21 1 21 Try 28 · 35. 1 2 42 4 1, 24 Try 27 · 42 in Babylonian, Modern 60, Egyptian, Greek, Roman, and Mayan! 8 2, 48 13 4, 33

  11. Mediation Review Mediation method of Division 534ten 37ten 534  37 1 37 Sum too great 2 74 518 4 148 444 8 296 14 Stop – next too big. Quotient = 14 Remainder 534 – 518 = 16

  12. Mediation Review  Babylonian Mediation method of Division Final answer?

  13. Mediation Review 11 Quotient Mediation method of Division 7, 11  38 in modern sexigesimal notation 7, 11  38 7, 11 - 6, 58 6, 58 1 38 6, 20 2 1, 16 13 the remainder 4 2, 32 7, 36 8 5, 04 Try 12, 34  56 ! Stop – too big. Try 534  37 in Babylonian, Modern 60, Egyptian, Greek, Roman, and Mayan!

  14. Unit Fractions as Decimals 1/n Base 10 Base 60 ½ 0.5 ; 30 1/3 0.333… ; 20 1/4 0.25 ; 15 * 1/5 0.2 ; 12 1/6 0.166… ; 10 * 1/7 0.142856…; 08, 34, 17, … 1/8 0.125 ; 07, 30 * 1/9 0.11… ; 06, 40 * 1/10 0.1 ; 06 Decimals in red repeat. * Indicates numbers that are one half of previous numbers.

  15. Fractions 13  9 in modern sexigesimal notation 13  9 1 9 2 18 12, 54 6 short of 13, 00! 4 36 12, 36 8 1, 12 16 2, 24 12, 00 32 4, 48 1, 04 9, 36 1; 26 + 6/9 But 6/9 is ;40 so the answer is 1 ; 26, 40

  16. 2 by Babalonian Methods For ease of understanding I will use base 10 fractions. The ancients knew that if 2 < x then 2/x < 2 . First iteration: 2 < 2 so 2/2 = 1 < 2 For a better approximation average these results: continued

  17. 2 by Babalonian Methods With basically two iterations we arrive at 577 / 408 In decimal form this is 1.414212963 In base sixty notation this is 1 ; 24, 51, 10, 35, . . . To three decimal places 1 ; 24, 51, 10 is what the Babylonians used for 2 ! Accuracy to 0.0000006 or about the equivalency of 2 and 1/4 inches between Baltimore and York!!

  18. Ptolemy’s Armagest The “Almagest” c. 150 A.D. was a table of chords by ½ degree. Ptolemy used a circle of 60 unit radius In his table he gave the chord of 24 as 24; 56, 58 in base 60 of course. Let’s examine how accurate he was. continued

  19. Ptolemy’s Armagest The chord of 24 = 24; 56, 58 Chord 24 = 2 · 60 · sin 12 sin 12 = chord 24 / 120 60 x sin 12 = 24; 56, 58 / 120 12  12  sin 12 = 12; 28, 29 / 60 sin 12 = 00 ; 12, 28, 29 sin 12 = 0.207912037ten Too large by 0.000000346 or 1 5/16 inches from York to Baltimore.

  20. Assignment Read chapter 5. Work on paper 2.

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