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Chapter 1 Section 4-1 Historical Numeration Systems Symbols See word doc for complete list of symbols used for ancient numeration systems. 2 Roman Numeration Roman numerals are written as combinations of the seven letters in the table below. The letters can be written as capital (XVI) or lower-case letters (xvi). Roman Numerals I=1 C = 100 V=5 D = 500 X = 10 M = 1000 L = 50 Roman Numeration Roman Numeral Table 1I 14 XIV 27 XXVII 150 CL 2 II 15 XV 28 XXVIII 200 CC 3 III 16 XVI 29 XXIX 300 CCC 4 IV 17 XVII 30 XXX 400 CD 5V 18 XVIII 31 XXXI 500 D 6 VI 19 XIX 40 XL 600 DC 7 VII 20 XX 50 L 700 DCC 8 VIII 21 XXI 60 LX 800 DCCC 9 IX 22 XXII 70 LXX 900 CM 10 X 23 XXIII 80 LXXX 1000 M 11 XI 24 XXIV 90 XC 1600 MDC 12 XII 25 XXV 100 C 1700 MDCC 13 XIII 26 XXVI 101 CI 1900 MCM Roman Numeral Calculator http://www.novaroma.org/via_romana/numbers.html Note: see “Roman Numerals” on website for additional information Roman Numeration (Additional HW Problems) Change to Roman Numerals 1. 215 2. 379 3. 1995 Change to Decimal Numbers 4. DCIV 5. CDXXIX 6. MCMXCVII 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. Ancient Egyptian Numeration 7 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 8 Example: Egyptian Numeral Convert 427 to Egyptian. Symbol Inventory: 9 Example: Egyptian Numeral Convert 427 to Egyptian. Solution 10 Egyptian Numerals See word doc from website 11 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. 4-1-12 Chinese Numeration 13 Example: Chinese Numeral Interpret each Chinese numeral. a) b) 14 Example: Chinese Numeral Solution a) 7000 400 80 2 Answer: 7482 b) 200 0 (tens) 1 Answer: 201 15 Example: Convert to Chinese Numeral Symbol Inventory 2018 16 Example: Convert to Chinese Numeral 2018 2000 0 hundreds 10 8 17 Chinese Numerals See word doc from website 18 Positional Numeration The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral. 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. 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. 20 Hindu-Arabic Numeration 7, 5 4 1, 7 2 5 . 21 Hindu-Arabic Numeration In Expanded Notation 45, 414 = 4 x 10000 + 5 x 1000 + 4 x 100 + 1 x 10 + 4 x 1 = 4 x 104 + 5 x 103 + 4 x 102 + 1 x 101 + 4 x 100 5, 014 = 5 x 103 + 0 x 102 + 1 x 101 + 4 x 100 = 5 x 103 + 1 x 101 + 4 x 100 22 Hindu-Arabic Numeration – Scientific Notation Scientific Notation: Number written in powers of ten such that one digit is left of the decimal place. 45,414 = 4.5414 x 104 .045414 = 4.5414 x 10-2 23 Chapter 1 Section 4-2 More Historical Numeration Systems Babylonian Numeration The ancient Babylonians used a modified base 60 numeration system. The digits in a base 60 system represent the number of 1s, the number of 60s, the number of 3600s, and so on. The Babylonians used only two symbols to create all the numbers between 1 and 59. ▼ = 1 and ‹ =10 Example: Babylonian Numeral Interpret each Babylonian numeral. a) ‹‹‹‹▼▼ b) ‹ ‹ ‹‹▼▼‹ ‹ ▼▼▼▼ c) ▼ ▼ ‹ ‹ ‹‹▼▼‹ ‹ ▼▼▼▼ 26 Example: Babylonian Numeral Convert to Babylonian numeral. a) 47, 094 b) 7,241 27 Mayan Numerals See word doc from website 28 Mayan Numeration The ancient Mayans used a base 20 numeration system, but with a twist. Normally the place values in a base 20 system would be 1s, 20s, 400s, 8000s, etc. Instead, the Mayans used 360s as their third place value but multiplied by 18 in one case. 1, 20, 20 x 18 = 360, 360 x 20 = 7200, 7200 x 20 = 144,000, and so on 29 Mayan Numeration Mayan numerals are written from top to bottom. 30 Example: Mayan Numeral Write the number below in our system. 31 Example: Mayan Numeral Write the number below in our system. Solution 10 360 0 20 Answer: 3619 19 1 32 Greek Numeration The classical Greeks used a ciphered counting system. They had 27 individual symbols for numbers, based on the 24 letters of the Greek alphabet, with 3 Phoenician letters added. Multiples of 1000 are indicated with a small stroke next to a symbol. Multiples of 10000 are indicated by the letter M. 33 Greek Numerals See word doc from website 34 Greek Numeration Table 2 Table 2 (cont.) 35 Example: Greek Numerals Interpret each Greek numeral. a) ma b) cpq 36 Example: Greek Numerals Solution a) ma b) cpq Answer: 41 Answer: 689 37 Chapter 1 Section 4-3 Arithmetic in the Hindu-Arabic System Historical Calculation Devices One of the oldest devices used in calculations is the abacus. It has a series of rods with sliding beads and a dividing bar. The abacus is pictured on the next slide. Abacus Reading from right to left, the rods have values of 1, 10, 100, 1000, and so on. The bead above the bar has five times the value of those below. Beads moved towards the bar are in “active” position. Example: Abacus Which number is shown below? Solution 104 103 102 101 100 1000 + (500 + 200) + 0 + (5 + 1) = 1706 Example: Abacus Which number is shown below? Solution 104 103 102 101 100 10000 + (5000 + 1000) + (500 + 200) + 30 + (5 + 2) = 16737 Example: Abacus Use an abacus to show a number? Lattice Method The Lattice Method was an early form of a paper-and-pencil method of calculation. This method arranged products of single digits into a diagonalized lattice. Example: Lattice Method Find the product 32 x 741 by the lattice method. Solution Set up the grid to the right. 7 4 1 3 2 Example: Lattice Method Fill in products 7 4 2 1 1 1 1 0 2 0 4 3 0 8 2 3 2 Example: Lattice Method Add diagonally right to left and carry as necessary to the next diagonal. 1 2 2 1 1 1 2 0 4 3 7 0 3 0 8 1 2 2 Example: Lattice Method 1 2 2 1 1 1 2 0 4 3 0 7 3 0 8 1 Answer: 23,712 See Lattice Template on Website 2 2 Napier’s Rods (Napier’s Bones) John Napier’s invention, based on the lattice method of multiplication, is often acknowledged as an early forerunner to modern computers. Refer to figure 2 on page 155 Russian Peasant Method Similar to the Egyptian Method of multiplication but dividing one column by 2 instead of doubling. Chapter 1 Extension Clock Arithmetic and Modular Systems Clock Arithmetic and Modular Systems • Finite Systems and Clock Arithmetic • Modular Systems Finite Systems Because the whole numbers are infinite, numeration systems based on them are infinite mathematical systems. Finite mathematical systems are based on finite sets. 12-Hour Clock System The 12-hour clock system is based on an ordinary clock face, except that 12 is replaced by 0 so that the finite set of the system is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. Clock Arithmetic As an operation for this clock system, addition is defined as follows: add by moving the hour hand in the clockwise direction. 11 0 1 10 2 9 8 3 4 7 6 5 5+3=8 Example: Finding Clock Sums by Hand Rotation Find the sum: 8 + 7 in 12-hour clock arithmetic Solution Start at 8 and move the hand clockwise through 7 more hours. Answer: 3 11 0 1 10 2 9 8 3 4 7 6 5 Example: Finding Clock Sums by Hand Rotation Find the sum: 8 + 7 in 12-hour clock arithmetic Solution Start at 8 and move the hand clockwise through 7 more hours. Answer: 3 11 0 1 10 2 9 8 3 4 7 6 5 Example: Clock Arithmetic If is 3 o’clock, what time will it be 94 hours from now? Solution Add 3 and 94. Divide by 12. The remainder is the answer. Answer: 1 o,clock 11 0 1 10 2 9 8 3 4 7 6 5 12-Hour Clock Addition Table Note the 12-hour is symmetric therefore commutative under addition Properties of Real Numbers Let a, b, and c be real numbers. Addition Closure a + b is in the set Multiplication ab is in the set Commutative a+b=b+a ab = ba Associative a + (b + c) = (a + b) + c a(bc) = (ab)c Identity a+0=a a1 = a Inverse a + (-a) = 0 a (1/a) = 1 Notes: A set is closed if any possible combination of elements under addition or multiplication must be in the set 12-Hour Clock Addition Properties Closure The set is closed under addition. Commutative For elements a and b, a + b = b + a. Associative For elements a, b, and c, a + (b + c) = (a + b) + c. Identity The number 0 is the identity element. Inverse Every element has an additive inverse. Inverses for 12-Hour Clock Addition Clock value a 0 Additive Inverse -a 0 11 10 9 8 7 6 5 4 3 1 2 3 4 5 6 7 8 9 10 11 2 Note these pairs are inverses because they add to the identity which is 0. 1 Modular Systems In this area the ideas of clock arithmetic are expanded to modular systems in general. Example: Modular Arithmetic Work each modular arithmetic problem. a) (16 14)(mod 7) b) (82 45)(mod 3) c) (4 x14)(mod 4) Solution a) 2 b) 1 c) 0 © 2008 Pearson Addison-Wesley. All rights reserved 4-4-64 Example: Truth of Modular Equations Decide whether each statement is true or false. a) 12 4 (mod 2) b) 35 4 (mod 7) c) 11 44 (mod 3) Solution a) True. 12 – 4 = 8 is divisible by 2. b) False. 35 – 4 = 31 is not divisible by 7. c) True. 11 – 44 = –33 is divisible by 3. © 2008 Pearson Addison-Wesley. All rights reserved 4-4-65 Example: Mod 4 Multiplication Table Determine which are satisfied by the system. x 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1 Solution Closed? Yes Commutative? Yes Identity = 1 Inverses: 1 & 3 are own inverses © 2008 Pearson Addison-Wesley. All rights reserved 4-4-66 Chapter 1 Extension Properties of Mathematical Systems An Abstract System The focus will be on elements and operations that have no implied mathematical significance. We can investigate the properties of the system without notions of what they might be. Operation Table Consider the mathematical system with elements {a, b, c, d} and an operation denoted by ☺. The operation table on the next slide shows how operation ☺ combines any two elements. To use the table to find c ☺ d, locate c on the left and d on the top. The row and column intersect at b, so c ☺ d = b. Operation Table for ☺ ☺ a b c d a a b c d b b d a c c c a d b d d c b a Find a) b ☺ c b) d ☺ a c) (c ☺ a) ☺ b Solutions a) a b) d c) a Potential Properties of a Single Operation Symbol Let a, b, and c be elements from the set of any system, and ◘ represent the operation of the system. Closure a ◘ b is in the set Commutative a ◘ b = a ◘ b. Associative a ◘ (b ◘ c) = (a ◘ b) ◘ c Identity The system has an element e such that a ◘ e = a and e ◘ a = a. Inverse there exists an element x in the set such that a ◘ x = e and x ◘ a = e. Operation Table for ☺ ☺ a b c d a a b c d b b d a c c c a d b d d c b a Closure Property For a system to be closed under an operation, the answer to any possible combination of elements from the system must in the set of elements. This system is closed. ☺ a b c d a a b c d b b d a c c c a d b d d c b a Identity Property For the identity property to hold, there must be an element E in the set such that any element X in the set, X ☺ E = X and E ☺ X = X. a is the identity element of the set. ☺ a b c d a a b c d b b d a c c c a d b d d c b a Inverse Property If there is an inverse in the system then for any element X in the system there is an element Y (the inverse of X) in the system such that X ☺ Y = E and Y ☺ X = E, where E is the identity element of the set. ☺ a b c d You can inspect the table to see that every element has an inverse. a a b c d b b d a c c c a d b d d c b a Commutative Property For a system to have the commutative property, it must be true that for any elements X and Y from the set, X ☺ Y = Y ☺ X. This system has the commutative property. The symmetry with respect to the diagonal line shows this property ☺ a b c d a a b c d b b d a c c c a d b d d c b a Associative Property For a system to have the associative property, it must be true that for any elements X, Y, and Z from the set, X ☺ (Y ☺ Z) = (X ☺ Y) ☺ Z. This system has the associative property. There is no quick check – just work through cases. ☺ a b c d a a b c d b b d a c c c a d b d d c b a Example 1: Identifying Properties Consider the system shown with elements {0, 1, 2, 3} and operation #. Which properties are satisfied by this system? # 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 Example 1: Identifying Properties Solution The system satisfies the closure, associative, commutative, and identity properties, and inverse property. # 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 Example 2: Identifying Properties Consider the system shown with elements {0, 1, 2, 3, 4} and operation . Which properties are satisfied by this system? 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 0 2 3 0 3 0 3 0 4 0 4 2 0 4 Example 2: Identifying Properties Solution The system satisfies the closure, associative, commutative, and identity properties, but not the inverse property. 0 0 0 1 0 2 0 3 0 4 0 1 2 3 0 0 0 1 2 3 2 4 0 3 0 3 4 2 0 4 0 4 2 0 4 Example 3: Identifying Properties Construct a base 5 addition system of remainders. Which properties are satisfied by this system? + 0 1 2 3 4 0 1 2 3 4 Example: Identifying Properties Which properties are satisfied by this system? J m n p m n p n n p m n p n n m Distributive Property (not covered) Let ☺ and ◘ be two operations defined for elements in the same set. Then ☺ is distributive over ◘ if a ☺ (b ◘ c) = (a ☺ b) ◘ (a ☺ c) for every choice of elements a, b, and c from the set. Example: Testing for the Distributive Property (not covered) Is addition distributive over multiplication on the set of whole numbers? Solution We check the statement below: a (b c) (a b) (a c). Notice, it fails when using 1, 2, and 3: 1 (2 3) (1 2) (1 3). This counterexample shows that addition is not distributive over multiplication. Chapter 1 Extension Groups Group A mathematical system is called a group if, under its operation, it satisfies the closure, associative, identity, and inverse properties. Note: the inverse property has to be more than an inverse of itself to be a group. Example 1: Checking Group Properties Does the set {–1, 1} under the operation of multiplication form a group? Solution All of the properties to be a group (closure, associative, identity, inverse) are satisfied as can be seen by the table. x –1 1 –1 1 –1 1 –1 1 Example 2: Checking Group Properties Does the set {–1, 1} under the operation of addition form a group? Solution No, right away it can be seen that closure is not satisfied. + –1 –1 –2 1 0 1 0 2 Example 3: Identifying Group Is the system of integers a group under subtraction? No, there is no identity - -2 -1 0 1 2 -2 0 -1 -2 -3 -4 -1 1 0 -1 -2 -3 0 2 1 0 -1 -2 1 3 2 1 0 -1 2 4 3 2 1 0