1 – Numeral Systems
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1 π The circle has 360° and that fact is connected to π in a peculiar way. Look at the 360 th decimal position of π: the number 3 is in the 359 th place, the number 6 is in the 360 th place, and the number 0 is in the 361 st place. This places 360 centered in the 360 th place.
description
π. The circle has 360 ° and that fact is connected to π in a peculiar way. Look at the 360 th decimal position of π : the number 3 is in the 359 th place, the number 6 is in the 360 th place, and the number 0 is in the 361 st place. This places 360 centered in the 360 th place. - PowerPoint PPT Presentation
Transcript of 1 – Numeral Systems
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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.
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1 – Numeral Systems
The student will learn about
Numeral systems from primitive time up to modern base systems.
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Cultural ConnectionThe Hunters of the Savanna
The Stone Age – ca. 5,000,000 – 3000 B.C.
Student led discussion.
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§1-1 Primitive Counting
Student Discussion.
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§1-1 Primitive Counting
One-to-one correspondences with sticks, stones, scratches or notches in bone or wood.
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§1-2 Number Bases
Student Discussion.
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§1-2 Number Bases
Quinary scale base 5
Duodecimal scale base 12
Vigesimal scale base 20
Sexagesimal scale base 60
Decimal scale base 10
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§1-3 Finger Numbers and Written Numbers
Student Discussion.
Numbers to 10,000 still used today.
Not good for calculations.
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§1-4 Simple Grouping Systems
Student Discussion.
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§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.
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§1-5 Multiplicative Grouping Systems
Student Discussion.
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§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
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§1-6 Ciphered Number System
Student Discussion.
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§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
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§1-7 Positional Numeral Systems
Student Discussion.
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§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.
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Zits By Jerry Scott and Jim Borgman
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Zits By Jerry Scott and Jim Borgman
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Zits By Jerry Scott and Jim Borgman
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§1-8 Early Computing
Student Discussion.
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§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.
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§1-9 Hindu-Arabic Number System
Student Discussion.
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§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.
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§1-10 Arbitrary Bases
Student Discussion.
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§1-10 Arbitrary Bases
Base 7.
Count.
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§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!
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§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!
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Assignment
Read Chapter 2 and work on your first paper.
