1

Chapters 1 - 4 Review

The student will learn more about

Some of the ancient numeration systems.

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

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

+ 34 ten

Babylonian Modern Notation

27

+ 45

12 55 ten 1, 12

5

Sexigesimal Addition

2, 34, 56 ; 23, 15

+ 25, 52 ; 14, 27

12, 32

25, 41

11, 00

+ 00, 4542

58

All Modern Notation

37, 48 ; 00,

3,

49,

111

6

Sexigesimal Subtraction

Base 10

45 ten

- 27 ten

Babylonian Modern Notation

45

- 27

18 18 ten

7

Sexigesimal Subtraction

1, 27

- 45

2, 34, 56 ; 23, 15

- 15, 52 ; 14, 27

25, 32

- 12, 4142

48

51

All Modern Notation

12,

9224

08, 04 ; 19, 2,

0, 87

8

Duplation Review

17ten · 42ten

1 42

Duplation method of Multiplication

17 · 42

4 1682 84

8 33616 672

17 714

9

Duplation Review Duplation method of Multiplication

Babylonian - 13 · 21 ·

10 of these carry

6 of these carry

10

Duplation Review

13 · 21 in modern sexigesimal notation

1 21

Duplation method of Multiplication

13 · 21

4 1, 242 42

8 2, 48

13

1

334,

Try 27 · 42 in Babylonian, Modern 60, Egyptian, Greek, Roman, and Mayan!

Try 28 · 35.

11

Mediation Review

534ten 37ten

1 37

Mediation method of Division

534 37

4 1482 74

8 296

14

444

Quotient = 14 Remainder 534 – 518 = 16

518Sum too great

Stop – next too big.

12

Mediation Review Mediation method of Division Babylonian

Final answer?

13

Mediation Review

7, 11 38 in modern sexigesimal notation

1 38

Mediation method of Division

7, 11 38

4 2, 322 1, 16

8 5, 047, 366, 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

Unit Fractions as Decimals1/n Base 10 Base 60

½ 0.5 ; 301/3 0.333… ; 201/4 0.25 ; 15 *1/5 0.2 ; 121/6 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

1 9

13 9

4 362 18

8 1, 12

1; 26

But 6/9 is ;40 so the answer is 1 ; 26, 40

6 short of 13, 00!

+ 6/9

16 2, 2432 4, 48

1, 04 9, 36

12, 00

12, 3612, 54

16

2 by Babalonian MethodsFor 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:

x 2/x Average

2 1 3/2

3/2 4/3 17 / 12

17 / 12 24 / 17 577 / 408

continued

17

2 by Babalonian MethodsWith 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

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 / 60sin 12 = 00 ; 12, 28, 29

sin 12 = 0.207912037ten

12 12

x60

20

Assignment

Read chapter 5.

Work on paper 2.