CHAPTER 8 Integers Problem. Prove that the square of any whole number n is a multiple of 4 or one more than a multiple of 4. Strategy 13 Use cases. This strategy may be appropriate when A problem can be separated into several distinct cases. A problem involves distinct collections of numbers such as odds and evens, primes and composites, and positives and negatives.. Investigations in specific cases can be generalized. Case 1 n is even. Then n = 2x =) n 2 = 4x 2, which is a multiple of 4. Case 2 n is odd. Then n = 2x + 1 =) n 2 = 4x 2 + 4x + 1, which is one more than a multiple of 4. 8.1. Addition and Subtraction Whole numbers and fractions are insu cient for expressing and solving many common problems. (1) At 8:00 am the temperature was 15 below zero, but had risen 20 by 4:00 pm. What was the temperature at 4:00 pm. (2) A submarine is 200 ft below sea level. If it first dives 300 ft, then comes back up 150 ft, what is its current depth? 142
8.1. ADDITION AND SUBTRACTION 143 (3) We would like an equation such as x + 5 = 2 to have a solution. For all of the above, we need negative numbers. Definition. The set of integers is the set {..., 3, 2, 1, 0, 1, 2, 3,... }. The numbers 1, 2, 3,... are the positive integers. The numbers 1, 2, 3,... are the negative integers. Zero is neither a positive nor negative integer. Representations: (1) Set model we use for positive integers and for negative integers (the text uses black chips for positive and red chips for negative integers - just like accounting). represents +1 and represents 1. Thus each cancels out an and vice versa. Set representations for 4. Integer number line. Note the symmetric arrangement to the right and left of 0.
144 8. INTEGERS Each integer a has an opposite, written as a or ( a), as follows: (1) Set model. +5 and 5 are opposites of each other. (2) Number line: Note. (1) If a is positive, a is negative. (2) If a is negative, a is positive.
Addition of Integers 8.1. ADDITION AND SUBTRACTION 145
146 8. INTEGERS Definition (Addition of Integers). Let a and b be any integers. 1. (Adding 0) a + 0 = 0 + a = a. 2. (Adding two positives) If a and b are positive, they are added as whole numbers. 3. (Adding two negatives) If a and b are positive, so that a and b are negative, then ( a) + ( b) = (a + b), where a + b is the whole number sum of a and b. 4. (Adding a positive and a negative) a. If a and b are positive and a b, then a+( b) = a b, the whole number di erence of a and b. b. If a and b are positive and a < b, then a + ( b) = (b a), where b a is the whole number di erence of a and b. 0 + ( 5) = 5 ( 3) + ( 6) = (3 + 6 = 9 11 + ( 4) = 11 4 = 7 5 12 = (12 5) = 7 Properties of Integer Addition Let a, b, and c be any integers. (Closure) a + b is an integer. (Commutative) a + b = b + a (Associative) (a+b)+c=a+(b+c) (Identity) 0 is the unique integer such that a + 0 = 0 + a = a for all a (Additive inverse) For each integer a, there is a unique integer, written as a, such that a + ( a) = 0 The integer a is called the additive inverse of a.
Note. 1) If a is positive, a is negative. 2) If a is negative, a is positive. 3) If a = 0, a = 0 also. 8.1. ADDITION AND SUBTRACTION 147 Theorem (Additive Cancellation for Integers). Let a, b, and c be any integers. If a + c = b + c, then a = b. Proof. a + c = b + c =)(addition) a + c + ( c) = b + c + ( c) =) (associative) a + c + ( c) = b + c + ( c) =) (additive inverse) a + 0 = b + 0 =) (additive identity) a = b. Theorem. Let a be any integer. Then ( a) = a. Proof. a + ( a) = 0 and ( a) + ( a) = 0 =) a + ( a) = ( a) + ( a) =) (cancellation) a = ( a). 5 + ( 11) = 5 + ( 5) + ( 6) = 5 + ( 5) + ( 6) = 0 + ( 6) = 6
148 8. INTEGERS Subtraction of Integers 1) Viewed as a Pattern. 5 2 = 3 5 1 = 4 5 0 = 5 We see a pattern developing and just keep it going. 2)Viewed as Take-away. 5 ( 1) = 6 5 ( 2) = 7 5 ( 3) = 8 = 5
8.1. ADDITION AND SUBTRACTION 149 3) Viewed as Adding the Opposite. Definition (Subtraction of Integers: Adding the Opposite). Let a and b be any integers. Then a b = a + ( b).
150 8. INTEGERS 4) Viewed as Missing Addend. 3 ( 5) = 3 + 5 = 2. 4 6 = 4 + ( 6) = 10. Definition (Subtraction of Integers: Adding the Opposite). Let a, b, and c be any integers. Then Find 7 ( 4). a b = c if and only if a = b + c. 7 ( 4) = c if and only if 7 = 4 + c. But 7 = 4 + 11, so 7 ( 4) = 11. Note. We have 3 di erent meanings for. 1) negative : 8 means negative 8. 2) opposite or additive inverse of : -6 is the opposite or additive inverse of 6. 3) minus : 7 3. 8.2. Multiplication, Division, and Order Multiplication viewed as an extension of whole number multiplication: 1) As repeated addition: John has borrowed $4.00 from his sister Terri each of the last 3 days. 3 ( 4) = ( 4) + ( 4) + ( 4) = 12.
8.2. MULTIPLICATION, DIVISION, AND ORDER 151 2) As an extension of patterns: 2 4 = 8 2 3 = 6 2 2 = 4 2 1 = 2 2 0 = 0 We see each step results in 2 less. So we continue the pattern: 2 ( 1) = 2 2 ( 2) = 4 2 ( 3) = 6 2 ( 4) = 8 Now using the results from above plus commutivity, which we want: 2 3 = 6 2 2 = 4 2 1 = 2 2 0 = 0 Noticing that each step results in 2 more, we continue the pattern 2 ( 1) = 2 2 ( 2) = 4 2 ( 3) = 6 2 ( 4) = 8
152 8. INTEGERS 3) Chips model: 4 ( 2) 4 ( 2) = 8 The sign of the second number determines the kind of chips used. ( 2) 4 Use the above model with commutivity. ( 2) 4 ( 2) 4 = 4 ( 2) = 8 Take away (the minus sign) two groups of 4. ( 2) 4 = 8
8.2. MULTIPLICATION, DIVISION, AND ORDER 153 ( 2) ( 4) As above, but take away two groups of 4. ( 2) ( 4) = 8 Definition (Multiplication of Integers). Let a and b be any integers. 1. a 0 = 0 = 0 a. 2. If a and b are positive, they are multiplied as whole numbers. 3. If a and b are positive (thus b is negative), a( b) = (ab). 4. If a and b are positive, then ( a) ( b) = ab. (1) 5 0 = 0 (2) 5 7 = 35 (3) 3 ( 4) = (3 4) = 12 (4) ( 4) ( 8) = 4 8 = 32
154 8. INTEGERS Properties of Integer Multiplication Let a, b, and c be any integers. (Closure) a b is an integer. (Commutative) a b = b a. (Associative) (a b) c = a (b c). (Identity) 1 is the unique integer such that a 1 = a = 1 a. (Distributive of Multiplication over Addition) a (b + c) = a b + a c Theorem. Let a be any integer. Then a( 1) = a. Proof. We know a 0 = 0 and a + ( a 0 = a 1 + ( 1) Then so by additive cancellation. a) = 0. But = a 1 + a ( 1) = a + a( 1) = 0. a + a( 1) = a + ( a) a( 1) = a Multiplying an integer by -1 reflects it about the origin.
8.2. MULTIPLICATION, DIVISION, AND ORDER 155 Theorem. Let a and b be any integers. Then ( a)b = (ab). Proof. ( a)b = ( 1)a b = ( 1)(ab) = (ab) Theorem. Let a and b be any integers. Then ( a)( b) = ab. Proof. ( a)( b) = ( 1)a ( 1)b = ( 1)( 1) (ab) = 1(ab) = ab Theorem (Multiplicative Cancellation Property). Let a, b,and c be any integers with c 6= 0. If ac = bc, then a = b. Why must we say c 6= 0? 5 0 = 8 0, but 5 6= 8. Theorem (Zero Divisors Property). Let a and b be any integers. Then ab = 0 if and only if a = 0 {z or b = 0}. or a=b=0
156 8. INTEGERS Division of Integers viewed as an extension of whole number division using the missing factor approach. Definition (Division of Integers). Let a and b be any integers where b 6= 0. Then for a unique integer c. (1) 12 4 = 3 since 12 = 4 3. a b = c if and only if a = bc (2) 10 ( 2) = 5 since 10 = ( 2)( 5). (3) 20 5 = 4 since 20 = 5( 4). (4) 48 ( 6) = 8 since 48 = ( 6)8. Negative Exponents and Scientific Notation a 3 = a a a a 2 = a a a 1 = a a 0 = 1 + a + a + a + a a 1 = 1 a + a a 2 = 1 a 2 + a a 3 = 1 a 3
8.2. MULTIPLICATION, DIVISION, AND ORDER 157 Definition (Negative Integer Exponent). Let a be any nonzero number and n be a positive integer. Then a n = 1 a n. 7 3 = 1 7 3 5 2 = 1 5 2 1 2 = 1 3 1/2 = 3 23 Thus n in the above definition can be any integer. Note. As a base with exponent moves from numerator to denominator or vice-versa, the base remains the same, but the exponent sign changes. 8 4 = 1 8 4 7 3 = 1 7 3 1 8 5 = 85 1 6 4 = 6 4
158 8. INTEGERS Theorem (Exponential Properties). For any nonzero numbers a and b and integers m and n, a m a n = a m+n 3 2 3 4 = 3 2+4 = 3 6 a m b m = (ab) m 4 2 5 2 = (4 5) 2 (a m ) n = a mn (6 2 ) 3 = 6 2 3 = 6 6 a m a = n am n 7 5 7 = 3 75 3 = 7 2 Scientific Notation mantissa% 1 apple a < 10 n is any integer a 10 n.characteristic 32, 500, 000 move decimal 7 places to the left to get 3.25 10 7. 3.25 10 7 move decimal 7 places to the right to get 32, 500, 000. 0.000187 move decimal 4 places to the right to get 1.87 10 4. 1.87 10 4 move decimal 4 places to the left to get 0.000187 6.524 10 7 1.42 10 3 = 6.524 1.42 107 10 3 4.594 104 = 45, 940. (2.17 10 4 )(5.2 10 8 ) = (2.17 5.2)(10 4 10 8 ) 11.284 10 4 = 1.1284 10 1 10 4 = 1.1284 10 3 =.0011284.
8.2. MULTIPLICATION, DIVISION, AND ORDER 159 Ordering Integers less than and greater than are defined as extensions of ordering of the whole numbers. Number Line Approach: the integer a is less than the integer b, written a < b (or b > a) if a is to the left of b on the integer number line. We also have 3 < 2 5 < 3, 3 < 0, 2 < 4, 0 < 5, 2 < 7. Addition Approach: the integer a is less than the integer b, written a < b, if and only if there is a positive integer p such that 5 < 3 since 5 + 2 = 3. 3 < 0 since 3 + 3 = 0. 2 < 4 since 2 + 6 = 4. 0 < 5 since 0 + 5 = 5. 2 < 7 since 2 + 5 = 7. a + p = b.
160 8. INTEGERS Properties of Ordering Integers Let a, b, c be any integers, p a positive integer, and n a negative integer. (Transitive for Less Than) If a < b and b < c, then a < c. (Less Than and Addition) If a < b, then a + c < b + c. (Less Than and Multiplication by a Positive) If a < b, then ap < bp. (Less Than and Multiplication by a Negative) If a < b, then an > bn. Multiplying by a negative changes the direction.