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 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 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. © 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

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 Units (Ones) Hundreds Tens 7, 5 4 1, 7 2 5 . © 2008 Pearson Addison-Wesley. All rights reserved

Section 4.1: Historical Numerical Systems A mathematical system has a) Elements b) Operations c) Relations d) All of the above

Section 4.1: Historical Numerical Systems 2. Our numeration system is an example of a simple grouping system multiplicative grouping system positional system d) complex grouping

Arithmetic in the Hindu-Arabic System Chapter 1 Section 4-2 Arithmetic in the Hindu-Arabic System

Arithmetic in the Hindu-Arabic System Expanded Form Historical Calculation Devices

Expanded Form By using exponents, numbers can be written in expanded form in which the value of the digit in each position is made clear.

Example: Expanded Form Write the number 23,671 in expanded form.

Distributive Property For all real numbers a, b, and c, For example,

Example: Expanded Form Use expanded notation to add 34 and 45.

Decimal System Because our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem, meaning ten.

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?

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. The method is shown in the next example.

Example: Lattice Method Find the product by the lattice method. 7 9 4 3 8

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. The rods are pictured on the next slide.

Napier’s Rods See figure 2 on page 174

Russian Peasant Method Method of multiplication which works by expanding one of the numbers to be multiplied in base two.

Nines Complement Method Step 1 Align the digits as in the standard subtraction algorithm. Step 2 Add leading zeros, if necessary, in the subtrahend so that both numbers have the same number of digits. Step 3 Replace each digit in the subtrahend with its nines complement, and then add. Step 4 Delete the leading (1) and add 1 to the remaining part of the sum.

Example: Nines Complement Method Use the nines complement method to subtract 2803 – 647. Solution Step 1 Step 2 Step 3 Step 4

Section 4.2: Arithmetic in the Hindu-Arabic System 1. Which of the following is an example of expanded form? 205 b) c) 5(40 + 1)

Section 4.2: Arithmetic in the Hindu-Arabic System 2. Which of the following is an example of the distributive property? a) b) c) 5(3 + 4) = 5(7)

Conversion Between Number Bases Chapter 1 Section 4-3 Conversion Between Number Bases

Conversion Between Number Bases General Base Conversions Computer Mathematics

General Base Conversions We consider bases other than ten. Bases other than ten will have a spelled-out subscript as in the numeral 54eight. When a number appears without a subscript assume it is base ten. Note that 54eight is read “five four base eight.” Do not read it as “fifty-four.”

Powers of Alternative Bases Fourth Power Third power Second Power First Power Zero Power Base two 16 8 4 2 1 Base five 625 125 25 5 Base seven 2401 343 49 7 Base eight 4096 512 64 Base sixteen 65,536 256

Example: Converting Bases Convert 2134five to decimal form.

Calculator Shortcut for Base Conversion To convert from another base to decimal form: Start with the first digit on the left and multiply by the base. Then add the next digit, multiply again by the base, and so on. The last step is to add the last digit on the right. Do not multiply it by the base.

Example: Use the calculator shortcut to convert 432134five to decimal form.

Example: Converting Bases Convert 7508 to base seven.

Converting Between Two Bases Other Than Ten Many people feel the most comfortable handling conversions between arbitrary bases (where neither is ten) by going from the given base to base ten and then to the desired base.

Computer Mathematics There are three alternative base systems that are most useful in computer applications. These are binary (base two), octal (base eight), and hexadecimal (base sixteen) systems. Computers and handheld calculators use the binary system.

Example: Convert Binary to Decimal Convert 111001two to decimal form. Solution 111001two

Example: Convert Hexadecimal to Binary Convert 8B4Fsixteen to binary form.

Section 4.3: Conversion Between Number Bases Which of the following is not a valid base 8 number? a) 456 b) 781 c) 0

Section 4.3: Conversion Between Number Bases The following is a way to convert what base to base 10? a) 7 b) 5 c) 2

Section 4-4 Chapter 1 Clock Arithmetic and Modular Systems 4-4-55 55 © 2008 Pearson Addison-Wesley. All rights reserved 55

Clock Arithmetic and Modular Systems Finite Systems and Clock Arithmetic Modular Systems 4-4-56 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved 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. 4-4-57 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved 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}. 4-4-58 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Clock Arithmetic As an operation for this clock system, addition is defined as follows: add by moving the hour hand in the clockwise direction. 1 11 2 10 5 + 3 = 8 3 9 4 8 5 7 6 4-4-59 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Clock Sums by Hand Rotation Find the sum: 8 + 7 in 12-hour clock arithmetic Plus 7 hours Solution Start at 8 and move the hand clockwise through 7 more hours. Answer: 3 1 11 2 10 3 9 4 8 5 7 6 4-4-60 © 2008 Pearson Addison-Wesley. All rights reserved

12-Hour Clock Addition Table 4-4-61 © 2008 Pearson Addison-Wesley. All rights reserved

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. 4-4-62 © 2008 Pearson Addison-Wesley. All rights reserved

Inverses for 12-Hour Clock Addition Clock value a 1 2 3 4 5 6 7 8 9 10 11 Additive Inverse -a 4-4-63 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Subtraction on a Clock If a and b are elements in clock arithmetic, then the difference, a – b, is defined as a – b = a + (–b) 4-4-64 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Clock Differences Find the difference 5 – 9. 4-4-65 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Finding Clock Products Find the product 4-4-66 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Modular Systems In this area the ideas of clock arithmetic are expanded to modular systems in general. 4-4-67 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Congruent Modulo m The integers a and b are congruent modulo m (where m is a natural number greater than 1 called the modulus) if and only if the difference a – b is divisible by m. Symbolically, this congruence is written 4-4-68 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Truth of Modular Equations Decide whether each statement is true or false. 4-4-69 © 2008 Pearson Addison-Wesley. All rights reserved

Criterion for Congruence if and only if the same remainder is obtained when a and b are divided by m. 4-4-70 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solving Modular Equations Solve the modular equation below for the whole number solutions. 4-4-71 © 2008 Pearson Addison-Wesley. All rights reserved

Section 4.4: Clock Arithmetic and Modular Systems In 12-hour clock addition, find the additive inverse of 3. a) 8 b) 9 c) –3

Section 4.4: Clock Arithmetic and Modular Systems 2. Is it true that a) Yes b) No

Properties of Mathematical Systems Chapter 1 Section 4-5 Properties of Mathematical Systems

Properties of Mathematical Systems An Abstract System Closure Property Commutative Property Associative Property Identity Property Inverse Property Distributive Property

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

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. ☺ a b c d This system is closed.

Commutative Property ☺ a b c d 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. ☺ a b c d This system has the commutative property. The symmetry with respect to the diagonal line shows this property

Associative Property ☺ a b c d 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. ☺ a b c d This system has the associative property. There is no quick check – just work through cases.

Identity Property ☺ a b c d 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 b c d a is the identity element of the set.

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.

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.

Example: Identifying Properties Consider the system shown with elements {0, 1, 2, 3, 4} and operation Which properties are satisfied by this system? 1 2 3 4

Distributive Property 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 Is addition distributive over multiplication on the set of whole numbers?

Section 4.5: Properties of Mathematical Systems For elements a and e and the operation represents which property? a) Commutative b) Inverse c) Identity

Section 4.5: Properties of Mathematical Systems 2. When a system has two operations, which property would we look for? a) Commutative b) Distributive c) Associative

Chapter 1 Section 4-6 Groups

Groups Groups Symmetry Groups Permutation Groups

Group A mathematical system is called a group if, under its operation, it satisfies the closure, associative, identity, and inverse properties.

Example: Checking Group Properties Does the set {–1, 1} under the operation of multiplication form a group?

Example: Checking Group Properties Does the set {–1, 1} under the operation of addition form a group?

Symmetry Groups A group can be built on sets of objects other than numbers. Consider the group of symmetries of a square. Start with a square labeled below. Front Back 4 1 1' 4' 3 2 2' 3'

Symmetries - Rotational M rotate 90° N rotate 180° 3 4 2 3 2 1 1 4 P rotate 270° Q original 1 2 4 1 4 3 3 2

Symmetries - Flip Flip about horizontal line through middle. R 4 1 3' 2' 3 2 4' 1' Flip about vertical line through middle. S 4 1 1' 4' 3 2 2' 3'

Symmetries - Flip Flip about diagonal line upper left to lower right. 4 1 4' 3' 3 2 1' 2' Flip about diagonal line upper right to lower left. V 4 1 2' 1' 3 2 3' 4'

Symmetries of the Square □ M N P Q R S T V

Example: Verifying a Subgroup Form a mathematical system by using only the set {M, N, P, Q} from the group of symmetries of a square. Is this new system a subgroup? Solution □ M N P Q The operational table is given and the system is a group. The new group is a subgroup of the original group.

Permutation Groups A group comes from studying the arrangements, or permutations, of a list of numbers. The next slide shows the possible permutations of the numbers 1-2-3.

Arrangements of 1-2-3 A*: 1-2-3 2-3-1 B*: 1-2-3 2-1-3 C*: 1-2-3 1-2-3 D*: 1-2-3 1-3-2 E*: 1-2-3 3-1-2 F*: 1-2-3 3-2-1

Example: Combining Arrangements Find D*E*. Solution 1-2-3 1-3-2 Rearrange according to D*. 3 E* replaces 1 with 3. 3 1 E* replaces 2 with 1. 3-2-1 E* replaces 3 with 2.

Section 4.6: Groups 1. A mathematical group does not have to satisfy which property? a) Commutative b) Closure c) Associative

Section 4.6: Groups Does {a, b} satisfy a group under the operation shown below? Yes No ¤ a b