Problem. Prove that the square of any whole number n is a multiple of 4 or one more than a multiple of 4.

Transcription

1 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 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

2 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.

3 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.

4 Addition of Integers 8.1. ADDITION AND SUBTRACTION 145

5 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 = ( 4) = 11 4 = = (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.

6 Note. 1) If a is positive, a is negative. 2) If a is negative, a is positive. 3) If a = 0, a = 0 also 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

7 INTEGERS Subtraction of Integers 1) Viewed as a Pattern. 5 2 = = = 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 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).

9 INTEGERS 4) Viewed as Missing Addend. 3 ( 5) = = = 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 = , 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 : 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.

10 8.2. MULTIPLICATION, DIVISION, AND ORDER 151 2) As an extension of patterns: 2 4 = = = = = 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 = = = = 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

11 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

12 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

13 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.

14 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

15 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

16 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 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 = = = = 6 4

17 INTEGERS Theorem (Exponential Properties). For any nonzero numbers a and b and integers m and n, a m a n = a m+n = = 3 6 a m b m = (ab) m = (4 5) 2 (a m ) n = a mn (6 2 ) 3 = = 6 6 a m a = n am n = = 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 move decimal 7 places to the right to get 32, 500, move decimal 4 places to the right to get move decimal 4 places to the left to get = = 45, 940. ( )( ) = ( )( ) = = =

18 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 = 3. 3 < 0 since = 0. 2 < 4 since = 4. 0 < 5 since = 5. 2 < 7 since = 7. a + p = b.

19 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.