2
Chapter 3: Numeration Systems and Whole-Number Computation 3.1 Numeration Systems 3.1.1. Vocabulary 3.1.1.1. numerals – written symbols used to represent a number like, 2 or 5 3.1.1.2. numeration system – collection of properties and symbols agreed upon to represent numbers systematically 3.1.1.3. place value – assigns a value dependent on the placement in a numeral 3.1.1.4. face value – the amount of the digit without regard to placement 3.1.1.5. expanded forms o place value: 5984 = 5000 + 900 + 80 + 4 o denominate: 5984: 5 thousands 9 hundreds 8 tens 4 units o exponential: 5984 = 5 10 3 + 9 10 2 + 8 10 1 + 4 10 0 3.1.1.6. factor – multiplier to obtain a product: factor x factor = product 3.1.1.7. tally numeration system – uses single strokes to represent each item counted | , || , ||| , |||| , |||| , |||| | , |||| || , … 3.1.1.8. additive property – used with Egyptian numeration the value of the number is the sum of the face values 3.1.1.9. subtractive property – used with Roman numeration the value of a smaller symbol is subtracted from a larger symbol when placed to the immediate left of the larger symbol: see table 3-7 p. 161 3.1.1.10. multiplicative property – used in Roman numeration a bar placed over the top of a number means to multiply it by 1000 for each bar 3.1.1.11. binary system – used by some Australian tribes and computers it has two digits, 0 and 1 3.1.2. Hindu-Arabic Numeration System 3.1.2.1. Formal name for the number system we use in USA 3.1.2.2. All numerals are constructed from the ten digits 3.1.2.3. Place value is based on powers of ten 3.1.2.4. Place value vs. face value 3.1.2.5. expanded form 5984 = 5 10 3 + 9 10 2 + 8 10 1 + 4 10 0 3.1.2.6. Definition of a n : If a is any number and n is any natural number, then 4 43 4 42 1 L factors n n a a a a a = 3.1.2.7. Many types of numeration systems – see table 3-2 p. 154 3.1.2.7.1. Babylonian 3.1.2.7.2. Egyptian 3.1.2.7.3. Mayan 3.1.2.7.4. Greek 3.1.2.7.5. Roman 3.1.2.7.6. Hindu 3.1.2.7.7. Arabic 3.1.2.7.8. Hindu-Arabic 3.1.3. Mayan Numeration System 3.1.3.1. Mayans first to use zero

Chapter 3: Numeration Systems and Whole-Number Computationmypages.valdosta.edu/plmoch/MATH3180/Spring 2009/3-1.pdf · Chapter 3: Numeration Systems and Whole-Number Computation 3.1

Embed Size (px)

Citation preview

Page 1: Chapter 3: Numeration Systems and Whole-Number Computationmypages.valdosta.edu/plmoch/MATH3180/Spring 2009/3-1.pdf · Chapter 3: Numeration Systems and Whole-Number Computation 3.1

Chapter 3: Numeration Systems and Whole-Number Computation 3.1 Numeration Systems

3.1.1. Vocabulary 3.1.1.1. numerals – written symbols used to represent a number like, 2 or 5 3.1.1.2. numeration system – collection of properties and symbols agreed

upon to represent numbers systematically 3.1.1.3. place value – assigns a value dependent on the placement in a

numeral 3.1.1.4. face value – the amount of the digit without regard to placement 3.1.1.5. expanded forms –

o place value: 5984 = 5000 + 900 + 80 + 4 o denominate: 5984: 5 thousands 9 hundreds 8 tens 4 units

o exponential: 5984 = 5 ⋅ 103 + 9 ⋅ 102 + 8 ⋅ 101 + 4 ⋅ 100 3.1.1.6. factor – multiplier to obtain a product: factor x factor = product 3.1.1.7. tally numeration system – uses single strokes to represent each

item counted | , || , ||| , |||| , |||| , |||| | , |||| || , …

3.1.1.8. additive property – used with Egyptian numeration the value of the number is the sum of the face values

3.1.1.9. subtractive property – used with Roman numeration the value of a smaller symbol is subtracted from a larger symbol when placed to the immediate left of the larger symbol: see table 3-7 p. 161

3.1.1.10. multiplicative property – used in Roman numeration a bar placed over the top of a number means to multiply it by 1000 for each bar

3.1.1.11. binary system – used by some Australian tribes and computers it has two digits, 0 and 1

3.1.2. Hindu-Arabic Numeration System 3.1.2.1. Formal name for the number system we use in USA 3.1.2.2. All numerals are constructed from the ten digits 3.1.2.3. Place value is based on powers of ten 3.1.2.4. Place value vs. face value

3.1.2.5. expanded form 5984 = 5 ⋅103 + 9 ⋅102 + 8 ⋅101 + 4 ⋅100 3.1.2.6. Definition of an : If a is any number and n is any natural number,

then 4434421 L

factors n

n aaaaa ⋅⋅⋅⋅=

3.1.2.7. Many types of numeration systems – see table 3-2 p. 154 3.1.2.7.1. Babylonian 3.1.2.7.2. Egyptian 3.1.2.7.3. Mayan 3.1.2.7.4. Greek 3.1.2.7.5. Roman 3.1.2.7.6. Hindu 3.1.2.7.7. Arabic 3.1.2.7.8. Hindu-Arabic

3.1.3. Mayan Numeration System 3.1.3.1. Mayans first to use zero

Page 2: Chapter 3: Numeration Systems and Whole-Number Computationmypages.valdosta.edu/plmoch/MATH3180/Spring 2009/3-1.pdf · Chapter 3: Numeration Systems and Whole-Number Computation 3.1

3.1.3.2. only used 3 symbols see table 3-5 p. 160 3.1.3.3. wrote numbers vertically with greatest place value on top 3.1.3.4. base twenty system of sorts: 1, 20, 20 x 18, 202 x 18, … 3.1.3.5. 360 shows up here too

3.1.4. Roman Numeration System 3.1.4.1. used from 3rd century BC 3.1.4.2. use additive and subtractive properties 3.1.4.3. see table 3-6 p. 161 and 3-7 p. 161 3.1.4.4. added multiplicative property in Middle Ages 3.1.4.5. combined and repeated as necessary to form a number 3.1.4.6. NO more than 3 of any symbol are used in a numeral

3.1.4.7. A bar over a letter represents multiples of 1000: V = 5000 3.1.4.8. A symbol representing a smaller number placed in front of number

representing a larger number reduces the number by that amount, i.e. IV = 4; IX = 9; XL = 40; etc.

3.1.5. Other Number Bases 3.1.5.1. base 2 – binary system

3.1.5.1.1. digits {0,1} 3.1.5.2. base 5

3.1.5.2.1. digits {0, 1, 2, 3, 4} 3.1.5.3. base 12

3.1.5.3.1. digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E} 3.1.5.4. base anything

3.1.6. Ongoing Assessment p. 166 3.1.6.1. Homework: 1ab, 2ac, 11, 12, 16ac, 20ac, 30