1

πThe circle has 360° and that fact is connected to π in a peculiar way. Look at the 360th decimal position of π: the number 3 is in the 359th place, the number 6 is in the 360th place, and the number 0 is in the 361st place. This places 360 centered in the 360th place.

2

1 – Numeral Systems

The student will learn about

Numeral systems from primitive time up to modern base systems.

3

Cultural ConnectionThe Hunters of the Savanna

The Stone Age – ca. 5,000,000 – 3000 B.C.

Student led discussion.

4

§1-1 Primitive Counting

Student Discussion.

5

§1-1 Primitive Counting

One-to-one correspondences with sticks, stones, scratches or notches in bone or wood.

6

§1-2 Number Bases

Student Discussion.

7

§1-2 Number Bases

Quinary scale base 5

Duodecimal scale base 12

Vigesimal scale base 20

Sexagesimal scale base 60

Decimal scale base 10

8

§1-3 Finger Numbers and Written Numbers

Student Discussion.

Numbers to 10,000 still used today.

Not good for calculations.

9

§1-4 Simple Grouping Systems

Student Discussion.

10

§1-4 Simple Grouping Systems

Additive Grouping as by the Egyptian Hieroglyphics.

Subtraction added by the Babylonians.

Grouping as by the Greek

234 On the board!

234

234

234Grouping as by the Greek or Roman systems.

11

§1-5 Multiplicative Grouping Systems

Student Discussion.

12

§1-5 Multiplicative Grouping Systems

A base b is selected and then symbols are chosen for 1, 2, 3, …, b – 1, b, b2, b3, …

Chinese-Japanese system 234

13

§1-6 Ciphered Number System

Student Discussion.

14

§1-6 Ciphered Number System

A base b is selected and then symbols are chosen for 1, 2, 3, …, b – 1, b, 2b, …, (b – 1)b, b2, 2b2, …, (b – 1) b2, …

Greek number system. 234

15

§1-7 Positional Numeral Systems

Student Discussion.

16

§1-7 Positional Numeral Systems

N = anbn + an – 1bn – 1 + . . . + a2b2 + a1b + a0 where N is then written as an an = 1 . . . a2 a1 a0.

234 in Hindu-Arabic.

Babylonian using base 60. 234

Mayan base 20*. Third place was 20 x 18 not 20 2 .

234

Using cuneiform and base 60.

Use modern notation.

17

Zits By Jerry Scott and Jim Borgman

18

Zits By Jerry Scott and Jim Borgman

19

Zits By Jerry Scott and Jim Borgman

20

§1-8 Early Computing

Student Discussion.

21

§1-8 Early Computing

Addition and subtraction in all systems was easy. Remember we are unfamiliar with their systems.

Physical difficulties with clay tablets, papyrus, parchment, sand trays, waxed boards, abaci, etc.

Do you understand the addition on the top of page 23? Subtraction was similar.

22

§1-9 Hindu-Arabic Number System

Student Discussion.

23

§1-9 Hindu-Arabic Number System

Hindus invented and Arabs transmitted about 250 B.C.

Disagreement between the algorists and the abacists with the present system (algorists) winning in about 1500 A.D.

24

§1-10 Arbitrary Bases

Student Discussion.

25

§1-10 Arbitrary Bases

Base 7.

Count.

26

§1-10 Base 7 Add/Subtract

Homework - Addition table with problems including subtraction.

235 seven

+ 121 seven

235 seven

+ 354 seven

435 seven

- 121 seven

235 seven

- 156 seven

Check your work! Check your work!

27

§1-10 Base 7 Multiply/Divide

Homework - Multiplication table with problems including division.

235 seven

x 121 seven

235 seven

x 354 seven

121 seven 4 seven

Check your work!

15321 seven 13 seven

Check your work!

28

Assignment

Read Chapter 2 and work on your first paper.