Problem (Page 88). In a group of nine coins, eight weigh the same ...

CHAPTER 3Whole Numbers: Operations and PropertiesProblem (Page 88). In a group of nine coins, eight weigh the same and theninth is either heavier or lighter. Assume the coins are identical in appearance.Using a pan balance, what is the smallest number of balancings needed toidentify the counterfeit coin?Strategy 8 – Use Direct Reasoning.This strategy man be appropriate when:(a) a proof is required.(b) a statement of the form “If . . . , then . . . ” is involved.(c) you see a statement that you want to imply from a collection of knownconditions.Here, separate the coins into three groups of three: A, B, C.(1) Balance A against B.A = B =) counterfeit coin is in C.(2) Balance A against C to find if C is lighter or heavier.A 6= B =) counterfeit coin is in A or B and we know which is heavier.(2) Balance A against C.A = C =) counterfeit coin is in B.A 6= C =) counterfeit coin is in A.28

3.1. ADDITION AND SUBTRACTION 29Name the three coins in the chosen group a, b, c.(3) Balance a against b.a = b =) counterfeit coin is c.a 6= b and chosen group heavier =) counterfeit coin is heavier of a and b.a 6= b and chosen group lighter =) counterfeit coin is lighter of a and b.Two models of addition:3.1. Addition and Subtraction(1) The set model – uses the union of disjoint sets.Definition (Addition of Whole Numbers). Let A and B be any two wholenumbers. If A and B are disjoint sets with a = n(A) and b = n(B), thena + b = n(A [ B).“a” and “b” are called addends or summands.Addition is called a binary operation since two numbers are combined to producea unique number.

30 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIES(2) The measurement model – uses directed arrows on a number line.Properties of whole number addition:Closure propertyThe sum of any two whole numbers is a whole number.Example. The even whole numbers are closed under addition.The odd whole numbers are not closed under addition. Why?Commutative propertyLet a and b be any whole numbers. Then a + b = b + a.Associative propertyLet a, b, and c be any whole numbers. Then(a + b) + c = a + (b + c).

3.1. ADDITION AND SUBTRACTION 31Identity propertyThere is a unique whole number, namely 0, such that for all whole numbers a,a + 0 = a = 0 + a.The number 0 is called the additive identity or the identity for addition.Thinking strategies for learning the addition facts:There are 100 places to be filled in the following table. We illustrate 4 + 1 = 5.

32 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIES(1) Commutativity – once the lower left of the table is completer, the upperright is obtained by commutativity.(2) Adding zero –using a + 0 = a fills the leftmost column.(3) Counting on by 1 and 2 – e.g., for 7 + 1, 7 + 2, think 7, 8, 9. This gives thenext two columns.

3.1. ADDITION AND SUBTRACTION 33(4) Combinations to 10 – using fingers, 7 + 3, 6 + 4, 5 + 5, etc. all equal 10.(5) Adding doubles –down the main diagonal, 1 + 1 = 2, 2 + 2 = 4, 3 + 3 = 6,etc., or counting by two’s: 2, 4, 6, 8, . . .The pattern for the remaining numbers is clear, or just use (6) - (8) below.(6) Adding 10 – 4 + 10 = 14, 8 + 10 = 18, etc. Pretty clear(7) Associativity –9 + 7 = 10 + 6 = 16 since 9 + 7 = 9 + (1 + 6) = (9 + 1) + 6 = 10 + 68 + 5 = 10 + 3 = 13 since 8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3(8) Doubles ±1 and ±2 –8 + 9 = 8 + 8 + 1 = 16 + 1 = 178 + 7 = 8 + 8 1 = 16 1 = 15

34 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIES7 + 9 = 7 + 7 + 2 = 14 + 2 = 168 + 6 = 8 + 8 2 = 16 2 = 14Problem (Page 101 # 8).Subtraction – two approaches:(1) The take-away approach –The set model:(2) The measurement model:

3.1. ADDITION AND SUBTRACTION 35Definition (Subtraction – Take-Away Approach).Let a and b be any whole numbers and A and B be sets such that a = n(A),b = n(B), and B ✓ A. Thena b = n(A B).“a b” is the di↵erence of a and b, “a” is the minuend, and “b” is the subtrahend.Subtraction – the missing addend approachDefinition (Subtraction - Missing Addend Approach).Let a and b be any whole numbers. Then a b = c if and only if ( () )a = b + c for some whole number c.This approach is useful for learning subtraction facts by relating them to additionfacts via four-fact families:

36 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIESMany subtraction facts can be obtained from an addition facts table.Example (A Two Set, Comparison Model, Using Matching). If Larry has$7 and Judy has $3, how much more money does Larry have?7 3 = 4After matching, we can use either the take-away or the missing addend approach.Note. Subtraction is not closed, for 3 5 does not have a whole number asan answer. What about the other properties we have for addition?Example (Four fact family in base 5).3 5 + 4 5 =12 54 5 + 3 5 =12 512 5 3 5 =4 512 5 4 5 =3 5

Approaches to Multiplication(1) Repeated addition approach3.2. MULTIPLICATION AND DIVISION 373.2. Multiplication and Division3 + 3 + 3 + 3 + 3 = 15 or 5 ⇥ 3 = 15Definition (Multiplication – Repeated Addition Approach).Let a and b be any whole numbers where a 6= 0. Thenab = b | + b + {z· · · + } b .a addendsIf a = 1, then ab = 1 · b = b; also 0 · b = 0 for all b.a and b are called factors, and ab is called the product.

38 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIES(2) Rectangular array approach3 + 3 + 3 + 3 + 3 = 15 or 5 ⇥ 3 = 15Definition (Multipication - Rectangular Array Approach).Let a and b be any whole numbers. Then ab is the number of elements in arectangular array having a rows and b columns.(3) Cartesian Product Approach2 ⇥ 3 = 6Definition (Multipication - Cartesian Product Approach).Let a and b be any whole numbers. If a = n(A) and b = n(B) for some sets Aand B,ab = n(A ⇥ B).

3.2. MULTIPLICATION AND DIVISION 39An alternative version of (3) is:(4) Tree Diagram Approach2 ⇥ 3 = 6Properties of Whole Number Multiplication:(1) Closure PropertyThe product of two whole numbers is a whole number.Example. The set of odd whole numbers and the set of even whole numbersare both closed under multiplication, but the setis not closed since 3 · 8 /2 A.Commutative PropertyA = {3, 8, 13, 18, 23, 28, . . . }Let a and b be any whole numbers. Thenab = ba.

40 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIES(3) Associative PropertyLet a, b, and c be any whole numbers. Thena(bc) = (ab)c.(4) Identity PropertyThe number 1 is the unique whole number such that for every whole numbera,a · 1 = a = 1 · a.1 is called the multiplicative identity or the identity for multiplication.(5) Distributive Property of Multiplication over AdditionLet a, b, and c be any whole numbers. Thena(b + c) = ab + ac.The 3 is distributed to the 2 and the 4.Note. This property relates the operations of addition and multiplication.

3.2. MULTIPLICATION AND DIVISION 41(6) Distributive Property of Multiplication over AdditionLet a, b, and c be any whole numbers. Then, provided b c,a(b c) = ab ac.Proof.Let b c = n.Then b = c + n (Missing addend).ab = a(c + n).ab = ac + an (distributive)Thus ab ac = an (missing addend)But b c = n.So ab ac = a(b c). ⇤Multiplication Property of ZeroFor every whole number a,a · 0 = 0 · a = 0.

42 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIESThinking Strategies for Learning Multiplication Facts:(1) Commutativity –just as with addition, once we find 55 facts, the others areobtained by commutativity.(2) Multiplication by 0 – since a · 0 = 0, the first column is all 0’s.(3) Multiplication by 1 – since a · 1 = a, the second column is identical to theleft-hand column.(4) Multiplication by 2 – since a · 2 = a + a, the third column is doubles fromaddition.(5) Multiplication by 5 – use counting chant by 5’s : 5, 10, 15, etc.(6) Multiplication by 9 – ten’s digit is one less than number we are multiplyingby 9, and the sum of the digits is 9.9 ⇥ 1 = 9, 9 ⇥ 2 = 18, 9 ⇥ 3 = 27, . . .

3.2. MULTIPLICATION AND DIVISION 43(7) Associativity and Distributivity –Example.8 ⇥ 6 = 8(3 + 3) = 8 · 3 + 8 · 3 = 24 + 24 = 487 ⇥ 4 = 7(2 ⇥ 2) = (7 ⇥ 2) ⇥ 2 = 14 ⇥ 2 = 28Problem (Page 118 # 11a).

44 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIESDivision – two conceptual ways of viewing:Sharing (partative) division – breaking a group of objects into a specified numberof groups (at the beginning, the number of groups is known, but the groupsize isn’t).Measurement division – breaking a group of objects into a number of groupsof a specified size (at the beginning, the size of each group is known, but thenumber of groups isn’t)Missing factor approachDefinition.If a and b are any whole numbers with b 6= 0, then a ÷ b = c if and only ifa = bc for some whole number c.a is called the dividend, b is called the divisor, and c is called the quotient.

3.2. MULTIPLICATION AND DIVISION 45This is based on another four-fact family:Division Property of zeroIf a 6= 0, then 0 ÷ a = 0.What about dividing by 0?(1) If a 6= 0 and a ÷ 0 = c, then a = 0 · c = 0, a contradiction.(2) If 0 ÷ 0 = c, then 0 = 0 · c for any whole number c, so there is no uniquequotient. Thus division by 0 is undefined.Division with Remainder – having some objects left over.Theorem – a statement that can be proved based on other results.Theorem (Division Algorithm). If a and b are any whole numbers withb 6= 0, then their exist unique whole numbers q and r such thata = bq + r where 0 apple r < b.a is the dividend, b the divisor, q the quotient, and r the remainder.

46 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIESAlternative ways of viewing the repeated-subtraction approach.Problem (Page 118 # 11b).i) 61 8 ÷ 7 8 = 7 8ii) 17 8 ÷ 3 8 = 5 8iii) 30 8 ÷ 6 8 = 4 8iv) 16 8 ÷ 2 8 = 7 8v) 44 8 ÷ 6 8 = 6 8vi) 25 8 ÷ 7 8 = 3 83.3. Ordering and ExponentsOrdering and Whole Number OperationsDefinition (“Less Than” for Whole Numbers).For any two whole numbers a and b, a < b (or b > a) if and only if there is anonzero whole number n such that a + n = b.apple means less than or equal to.means greater than or equal to.

3.3. ORDERING AND EXPONENTS 47Transitive Property of Less Than for Whole NumbersFor all whole numbers a, b, and c, if a < b and b < c, then a < c.Proof.a < b =) a + n = b for some whole nonzero whole number n.b < c =) b + m = c for some whole nonzero whole number m.Then (a + n) + m = b + m =) a + (n + m) = b + m =) a + (n + m) = c,and n + m is a nonzero whole number, so a < c.⇤Note. The transitive property also holds if >,, or apple are used throughout.Less Than and Addition for Whole NumbersIf a < b, then a + c < b + c.Less Than and Multiplication for Whole NumbersIf a < b and c 6= 0, then ac < bc.

48 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIESExponents – used to simplify multiplication expressions.Definition (Whole Number Exponent).Let a and m be any whole numbers where m 6= 0. Thena m = a · a · · · a | {z }m factorsm is the exponent or power of a, and a is called the base.Example.Theorem.3 5 · 3 3 = (3 · 3 · 3 · 3 · 3) · (3 · 3 · 3) = 3 8Let a, m, and n be any whole numbers where m and n are nonzero. ThenProof.a m · a n = a · a · · · a | {z }m factorsa m · a n = a m+n .· a · a · · · a | {z }n factors= a · a · · · a | {z }m + n factors= a m+n ⇤Example.3 3 · 5 3 = (3 · 3 · 3) = (5 · 5 · 5) = (3 · 5) · (3 · 5) · (3 · 5) = (3 · 5) 3Theorem.Let a, b, and m be any whole numbers where m 6= 0. ThenProof.a m · b m = a · a · · · a | {z }m factorsa m · b m = (ab) m .· b · b · · · b | {z }m factors= (ab) · (ab) · · · (ab)| {z }m factors= (ab) m ⇤

Example.Theorem.3.3. ORDERING AND EXPONENTS 49(4 2 ) 3 = 4 2 · 4 2 · 4 2 = 4 6Let a, m, and n be any whole numbers where m and n are nonzero. ThenThusExample.Theorem.(a m ) n = a mn .6 8 ÷ 6 3 = 6 5 since 6 3 · 6 5 = 6 8 .6 8 ÷ 6 3 = 6 8 3 .Let a, m, and n be any whole numbers where m > n and a, m, and n arenonzero. Thena m ÷ a n = a m n .0 as an exponentFor a 6= 0, consider the pattern:a 3 = a · a · aDivide each side by a.a 2 = a · aDivide each side by a.a 1 = aDivide each side by a.a 0 = 1Definition. a 0 = 1 for all whole numbers a 6= 0.

50 3. WHOLE NUMBERS: OPERATIONS AND PROPERTIESConsider the two following patterns:Pattern 1: 3 0 = 1, 2 0 = 1, 1 0 = 1, 0 0 = ?Pattern 2: 0 3 = 0, 0 2 = 0, 0 1 = 0, 0 0 = ?Each pattern suggests a di↵erent answer to 0 0 , so:Note. 0 0 is not defined.Order of Operations – an agreementExample.11 2 2 ÷ 2 · 5 + (7 (6 2)) =(1) Parentheses (from inside out)11 2 2 ÷ 2 · 5 + (7 4) =11 2 2 ÷ 2 · 5 + 3 =(2) Exponents in order from left to right11 4 ÷ 2 · 5 + 3 =(3) Multiplications and divisions in order from left to right11 2 · 5 + 3 =11 10 + 3 =(4) Additions and subtractions in order fro left to rightNote.1 + 3 =(1) “Please Excuse My Dear Aunt Sally” is a useful pneunomic.(2) Most calculators (including those used at CBU) follow these rules.4