Slide 1Chapter 4: Numeration and Mathematical SystemsSlide 3Historical Numeration SystemsMathematical and Numeration SystemsSlide 6Example: Counting by TallyingCounting by GroupingAncient Egyptian Numeration – Simple GroupingAncient Egyptian NumerationExample: Egyptian NumeralTraditional Chinese Numeration – Multiplicative GroupingChinese NumerationExample: Chinese NumeralSlide 15Positional NumerationSlide 17Slide 18Hindu-Arabic Numeration – PositionalHindu-Arabic NumerationChapter 4Numeration and Mathematical Systems© 2008 Pearson Addison-Wesley.All rights reserved© 2008 Pearson Addison-Wesley. All rights reserved4-1-2Chapter 4: Numeration and Mathematical Systems4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems4.5 Properties of Mathematical Systems4.6 Groups© 2008 Pearson Addison-Wesley. All rights reserved4-1-3Chapter 1Section 4-1Historical Numeration Systems© 2008 Pearson Addison-Wesley. All rights reserved4-1-4Historical 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 reserved4-1-5Mathematical and Numeration SystemsA mathematical system is made up ofthree components:1. a set of elements;2. one or more operations for combining the elements;3. one or more relations for comparing the elements.© 2008 Pearson Addison-Wesley. All rights reserved4-1-6Mathematical and Numeration SystemsThe various ways of 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 reserved4-1-7Example: Counting by TallyingTally 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 reserved4-1-8Counting by GroupingCounting 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 reserved4-1-9Ancient Egyptian Numeration – Simple GroupingThe 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 reserved4-1-10Ancient Egyptian Numeration© 2008 Pearson Addison-Wesley. All rights reserved4-1-11Example: Egyptian NumeralWrite the number below in our system.Solution2 (100,000) = 200,000 3 (1,000) = 3,000 1 (100) = 100 4 (10) = 40 5 (1) = 5Answer: 203,145© 2008 Pearson Addison-Wesley. All rights reserved4-1-12Traditional Chinese Numeration – Multiplicative GroupingA 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 reserved4-1-13Chinese Numeration© 2008 Pearson Addison-Wesley. All rights reserved4-1-14Example: Chinese NumeralInterpret each Chinese numeral.a) b)© 2008 Pearson Addison-Wesley. All rights reserved4-1-15Example: Chinese Numerala) b)Solution7000400802Answer: 74822000 (tens)1Answer: 201© 2008 Pearson Addison-Wesley. All rights reserved4-1-16Positional NumerationA 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 reserved4-1-17Positional NumerationIn a positional numeral, each symbol (called adigit) 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 reserved4-1-18Positional NumerationTo 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 reserved4-1-19Hindu-Arabic Numeration – PositionalOne 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 reserved4-1-20Hindu-Arabic NumerationHundredsThousandsTen thousandsMillionsHundred thousandsTensUnitsDecimal point7, 5 4 1, 7 2 5