Chapter 8 Integers 8.1 Addition and Subtraction

Transcription

1 Chapter 8 Integers 8.1 Addition and Subtraction

2 Negative numbers Negative numbers are helpful in: Describing temperature below zero Elevation below sea level Losses in the stock market Overdrawn checking accounts.

3 Definition Integers The set of integers is the set I={,-3,-2,-1,0,1,2,3, } The numbers 1,2,3, are called positive integers and the numbers -1,-2,-3, are called negative integers. Zero is neither a positive nor a negative integer.

4 Representing Integers Chip Model One black chip represents a credit of 1 One red chip represents a debit of 1 Then one black chip and one red chip will cancel each other out (or make zero).

5 Representing Integers Integer number line Integers are equally spaced and arranged symmetrically.

6 The opposite on an Integer The opposite of the integer a, written a or (-a) is defined: Set Model the opposite of a is represented by the same number of chips as a (but of the opposite color) Measurement Model The opposite of a is the integer that is its mirror image about 0.

7 Addition and Its Properties Definition Let a and b be any integers. 1. Adding zero: 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 (hence a and b are negative), then (-a) + (-b) = -(a + b), where a + b is the wholenumber sum of a and b.

8 Addition and Its Properties- cont 4. Adding a positive and a negative: a. If a and b are positive and a>=b, then a + (-b)= a b, where a b is the whole-number difference 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 difference of a and b.

9 Addition Set Model addition means to put together or form the union of two disjoint sets Adding two positives.

10 Addition Set Model addition means to put together or form the union of two disjoint sets Adding two negatives.

11 Addition Set Model addition means to put together or form the union of two disjoint sets Adding a positive and a negative:.

12 Addition Measurement Model Addition means to put directed arrows end to end starting at zero. Positive integers are represented by arrows pointing to the right and negative integers by arrows pointing to the left. Adding two positives.

13 Addition Measurement Model Addition means to put directed arrows end to end starting at zero. Positive integers are represented by arrows pointing to the right and negative integers by arrows pointing to the left. Adding two negatives:.

14 Addition Measurement Model Addition means to put directed arrows end to end starting at zero. Positive integers are represented by arrows pointing to the right and negative integers by arrows pointing to the left. Adding a positive and a negative:.

15 Properties of Integer Addition Let a, b, and c be any integers. Closure Property for Integer Addition a + b is an integer Commutative Property for Integer Addition a + b = b + a Associative Property for Integer Addition (a + b) + c = a + (b + c).

16 Properties of Integer Addition Let a, b, and c be any integers. Identity Property for Integer Addition 0 is the unique integer such that a + 0 = a = 0 + a for all a Additive Inverse Property for Integer Addition For each integer a there is a unique integer, written a, such that a + (-a) = 0 The integer a is called the additive inverse of a.

17 Additive Cancellation for Integers Let a, b, and c be any integers. If a + c = b + c then a = b.

18 Theorem Let a be any integer Then (-a) = a.

19 Subtraction Subtraction of integers can be viewed in several ways. -4 (-1) Take-Away 6-2

20 Subtraction Take-Away -2 (-3)

21 Subtraction Adding the Opposite inserting an equal number of red and black chips before performing the operation 2 5 OR

22 Definition Subtraction of Integers: Adding the Opposite Let a and b be any integers. Then a b = a + (-b) Adding the opposite is perhaps the most efficient method for subtracting integers replacing a subtraction problem with an equivalent addition problem.

23 Alternative Definition Subtraction of Integers: Missing-Addend Approach Let a, b, and c be any integers. Then a b = c if and only if a = b + c.

24 Summary of Subtraction Methods Three equivalent ways to view subtraction of integers 1. Take-away 2. Adding the opposite 3. Missing addend.

25 Summary Find 4 (-2) using all three methods Take-Away

26 Chapter 8 Integers 8.2 Multiplication, Division, and Order

27 Multiplication and Its Properties Integer multiplication can be viewed as extending whole-number multiplication thus: 3 x 4 = = 12 If you were selling tickets and you accepted three bad checks worth $4 each then: 3 x (-4) = (-4) = (-4) = (-4) = -12 Number line.

28 Multiplication and Its Properties Modeling integer multiplication with chips 1. 4 x -3 combine 4 groups of red chips 2. Take-away -4 x 3 add an equal number of black and red chips and then take away the black.

29 Multiplication and Its Properties Modeling integer multiplication with chips 2. Take-away -4 x 3 add an equal number of black and red chips and then take away the black.

30 Multiplication of Integers Let a and b be any integers. 1. Multiplying by 0: a x 0 = 0 = 0 x a 2. Multiplying two positives: If a and b are positive, they are multiplied as whole numbers.

31 Multiplication of Integers Let a and b be any integers. 3. Multiplying a positive and a negative: If a is positive and b is positive (thus b is negative), then a(-1) = -(ab) 4. Multiplying two negatives: if a and b are positive, then (-a)(-b) = ab when ab is the whole-number product of a and b. That is, the product of two negatives is positive.

32 Properties of integer Multiplication Let a, b, and c be any integers. Closure Property for Integer Multiplication ab is an integer. Commutative Property for Integer Multiplication ab = ba Associative Property for Integer Multiplication (ab)c = a(bc) Identity property for integer Multiplication 1 is the unique integer such that a x 1 = a = 1 x a for all a.

33 Properties of integer Multiplication Distributivity of Multiplication over Addition of Integers Let a, b, c be any integers. Then a(b + c) = ab + ac.

34 Theorem Let a be any integer. Then a(-1) = -a the product of negative one and any integer is the opposite (or additive inverse) of that integer On the integer number line, multiplication by -1 is equivalent geometrically to reflecting an integer about the origin.

35 Theorem Let a and b be any integers. Then (-a)b = -(ab) Let a and b be any integers. Then (-a)(-b) = ab for all integers a,b.

36 Multiplicative Cancellation Property Let a, b, c be any integers with. If ac = bc then a = b. c 0

37 Zero Divisors Property Let a and b be integers. Then ab = 0 if and only if a = 0 or b = 0 or a and b both equal zero.

38 Division Division of integers can be viewed as an extension of whole-number division using the missing-factor approach. Division of Integers Let a and b be any integers, where b 0. a b = c a = b c Then if and only if for a unique integer c.

39 Following generalizations about the division of integers: Assume that b divides a; that is, that b is a factor of a 1. Dividing by 1: a 1 = a 2. Dividing two positives (negatives): If a and b are both positive (or both negative) then a b is positive.

40 Following generalizations about the division of integers: Assume that b divides a; that is, that b is a factor of a 1. Dividing a positive and a negative: If one of a or b is positives and the other is negative, then is negative a 1 = a a b = 0 1. Dividing zero by a nonzero integer: where b 0, since 0 = b 0 AS with whole numbers, division by zero is undefined for integers.

41 Ordering Integers The concepts of less than and greater than in the integers are defined to be extensions of ordering in the whole numbers Number-Line Approach the integer a is less than the integer b, written a<b, if a is to the left of b on the integer number line.

42 Ordering Integers 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 a + p = b. Thus -5<-3, since = -3 And -7<2, since = 2 The integer a is greater than the integer b, written a>b, if and only if b<a.

43 Properties of Ordering Integers Let a, b, and c be any integers, p a positive integer, and n a negative integer. Transitive Property for Less than If a < b and b < c then a < c Property of Less than and Addition If a < b, then a + c < b + c.

44 Properties of Ordering Integers Property of Less Than and Multiplication by a Positive If a < b, then ap < bp Example -2 < 3 and 4 > 0 then (-2) x 4 < 3 x 4

45 Properties of Ordering Integers Property of Less Than and Multiplication by a Negative If a < b, then an > bn -2 < 3 and -4 < 0 then (-2)(-4) > 3(-4)

46 Properties of Ordering Integers Property of Less Than and Multiplication by a Negative If a < b, then an > bn remember multiplying an integer a by -1 is geometrically the same as reflecting a across the origin on the integer number line. Applying this idea a < b then (-1)a > (-1)b