Chapter 4 Numeration and Mathematical Systems © 2008 Pearson Addison-Wesley. All rights reserved

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 © 2008 Pearson Addison-Wesley. All rights reserved

Section 4-1 Chapter 1 Historical Numeration Systems © 2008 Pearson Addison-Wesley. All rights reserved

Historical Numeration Systems Mathematical and Numeration Systems Ancient Egyptian Numeration – Simple Grouping Traditional Chinese Numeration – Multiplicative Grouping Hindu-Arabic Numeration - Positional © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved 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. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

Ancient Egyptian Numeration © 2008 Pearson Addison-Wesley. All rights reserved

Example: Egyptian Numeral Write the number below in our system. Solution 2 (100,000) = 200,000 3 (1,000) = 3,000 1 (100) = 100 4 (10) = 40 5 (1) = 5 Answer: 203,145 © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chinese Numeration © 2008 Pearson Addison-Wesley. All rights reserved

Example: Chinese Numeral Interpret each Chinese numeral. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

Example: Chinese Numeral Solution a) b) 7000 200 400 0 (tens) 1 80 Answer: 201 2 Answer: 7482 © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

Hindu-Arabic Numeration Hundred thousands Millions Ten thousands Thousands Decimal point Hundreds Tens Units 7, 5 4 1, 7 2 5 . © 2008 Pearson Addison-Wesley. All rights reserved