1. Chapter 4 Numeration and Mathematical Systems
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2. Chapter 4: Numeration and Mathematical Systems
4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems 4.5 Properties of Mathematical Systems 4.6 Groups
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3. Section 4-1 Chapter 1 Historical Numeration Systems
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4. Historical Numeration Systems
Mathematical and Numeration Systems
Ancient Egyptian Numeration – Simple Grouping
Traditional Chinese Numeration – Multiplicative Grouping
Hindu-Arabic Numeration - Positional
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5. Mathematical and Numeration Systems
A mathematical system is made up of
three components:
1. a set of elements;
one or more operations for combining
the elements;
3. one or more relations for comparing the
elements.
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6. Mathematical and Numeration Systems
The various ways if symbolizing and working with the counting numbers are called numeration systems. The symbols of a numeration system are called numerals.
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7. Example: Counting by Tallying
Tally sticks and tally marks have been used for a long time. Each mark represents one item. For example, eight items are tallied by writing
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Counting by Grouping
Counting by grouping allows for less repetition of symbols and makes numerals easier to interpret. The size of the group is called the base (usually ten) of the number system.
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9. Ancient Egyptian Numeration – Simple Grouping
The ancient Egyptian system is an example of a simple grouping system. It used ten as its base and the various symbols are shown on the next slide.
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10. Ancient Egyptian Numeration
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11. Example: Egyptian Numeral
Write the number below in our system.
Solution
2 (100,000) = 200,000
3 (1,000) = 3,000
1 (100) =
4 (10) =
5 (1) =
Answer: 203,145
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12. Traditional Chinese Numeration – Multiplicative Grouping
A multiplicative grouping system involves pairs of symbols, each pair containing a multiplier and then a power of the base. The symbols for a Chinese version are shown on the next slide.
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Chinese Numeration
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14. Example: Chinese Numeral
Interpret each Chinese numeral.
a) b)
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15. Example: Chinese Numeral
Solution
a) b)
7000
200
400
0 (tens) 1 80
Answer: 201 2
Answer: 7482
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16. Positional Numeration
A positional system is one where the various powers of the base require no separate symbols. The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral.
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17. Positional Numeration
In a positional numeral, each symbol (called a
digit) conveys two things:
1. Face value – the inherent value of the
symbol.
2. Place value – the power of the base which
is associated with the position that the digit
occupies in the numeral.
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18. Positional Numeration
To work successfully, a positional system must have a symbol for zero to serve as a placeholder in case one or more powers of the base are not needed.
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19. Hindu-Arabic Numeration – Positional
One such system that uses positional form is our system, the Hindu-Arabic system.
The place values in a Hindu-Arabic numeral, from right to left, are 1, 10, 100, 1000, and so on. The three 4s in the number 45,414 all have the same face value but different place values.
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20. Hindu-Arabic Numeration
Hundred thousands
Millions
Ten thousands
Thousands
Decimal point
Hundreds
Tens
Units
7, ,
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