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Chapters 1 - 4 Review

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The student will learn more about

Some of the ancient numeration systems.

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

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.

Base 10

21ten

+ 34ten

Babylonian

Modern Notation

27

+45

12

1, 12

55 ten

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

Base 10

45 ten

- 27 ten

Babylonian

Modern Notation

45

-27

18

18 ten

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

Duplation method of Multiplication

17ten· 42ten

17 · 42

1 42

2 84

4 168

8 336

16 672

17

714

·

6 of these carry

10 of these carry

Duplation method of Multiplication

Babylonian - 13 · 21

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

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

Babylonian

Mediation method of Division

Final answer?

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!

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.

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

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

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!!

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

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.

Read chapter 5.

Work on paper 2.