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

2. 2 Sexigesimal System Ancient Notation versus Modern Notation 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 1, 56 ; Note: one can have but not.

3. 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. 4 Sexigesimal Addition Base 10 21 ten + 34 ten BabylonianModern Notation 27 +45 12 55 ten 1, 12

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

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

7. 7 Sexigesimal Subtraction 1, 27 - 45 2, 34, 56 ; 23, 15 - 15, 52 ; 14, 27 25, 32 - 12, 41 42 48 51 All Modern Notation 12, 9224 08,04 ;19,2, 0, 87

8. 8 Duplation Review 17 ten · 42 ten 1 42 Duplation method of Multiplication 17 · 42 4 168 2 84 8 336 16 672 17714

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

10. 10 Duplation Review 13 · 21 in modern sexigesimal notation 1 21 Duplation method of Multiplication 13 · 21 4 1, 24 2 42 8 2, 48 13 1 334, Try 27 · 42 in Babylonian, Modern 60, Egyptian, Greek, Roman, and Mayan! Try 28 · 35.

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

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

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

14. 14 Unit Fractions as Decimals 1/nBase 10Base 60 ½0.5; 30 1/30.333…; 20 1/40.25 ; 15 * 1/50.2; 12 1/60.166…; 10 * 1/70.142856…; 08, 34, 17, … 1/80.125; 07, 30 * 1/90.11…; 06, 40 * 1/100.1; 06 Decimals in red repeat. * Indicates numbers that are one half of previous numbers.

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

16. 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: x2/xAverage 213/2 4/317 / 12 24 / 17577 / 408 continued

17. 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. 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. 19 Ptolemy’s Armagest The chord of 24  = 24; 56, 58 Chord 24  = 2 · 60 · sin 12  sin 12  = chord 24  / 120 Too large by 0.000000346 or 1 5/16 inches from York to Baltimore. sin 12  = 24; 56, 58 / 120 sin 12  = 12; 28, 29 / 60 sin 12  = 00 ; 12, 28, 29 sin 12  = 0.207912037 ten 12  x 60

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