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Ch5 Notes (revised pres F11)

# Ch5 Notes (revised pres F11) - Section 5-1 Integers and the...

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1 Section 5-1 – Integers and the Operations of Addition and Subtraction The INTEGERS are made up of the natural numbers, negative natural numbers, and zero. I = { …–4, –3, –2, –1, 0, 1, 2, 3, 4, …} Integers are usually represented by drawing a number line (either horizontal or vertical). The negative integers are the opposites of the positive integers. Examples : The opposite of 4 is ____. The opposite of –10 is _____. The opposite of 0 is ____. Absolute Value: “The distance from zero” Definition : ,0 xx x Examples : a) │10│ = b) 5│ = c) –│ 4│ = d) 2 + 6│ = e) x │ = 2 What is x ? Integer Addition Methods and Models: Using Absolute Value : (useful when adding a positive integer and a negative integer) Find the difference of the two numbers and keep the sign of the number that is farther away from zero (ie, has a larger absolute value). Examples : a) 3 + 5 = b) 8 + 4 = c) 10 + ( 7) =

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2 Chip or Charged-Field Model for Addition: black dot (or a +) = positive number white dot (or a –) = negative number a) 3 + 6 = b) 5 + 3 = c) 2 + –4 = d) 4 + 6 =
3 Number Line Model for Addition: Always start at 0. If an integer is positive, walk forward. If an integer is negative, walk backward. a) 3 + 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 b) 2 + (–6) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 c) (–7) + 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 d) (–1) + (–4) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Addition Properties for Integers: The Closure Property : a + b is a unique integer The Commutative Property : a + b = b + a The Associative Property : ( a + b ) + c = a + ( b + c ) The Identity Property : a + 0 = 0 + a = a (zero is called the identity element because when added to an integer, it leaves it unchanged) Other Properties of Integers: 1. –(– a ) = a 2. a + – b = –( a + b )

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4 Examples : Find the additive inverse (opposite) of each expression: a) ( x + 1) b) 4 + 5 c) ( x + 2) Integer Subtraction Methods and Models: “Adding the Opposite” Method: Theorem : For all integers a and b , a b = a + – b Examples : Work the following by rewriting them as addition a) 6 – 7 b) 9 – 8 c) –8 – 9 d) 4 – (–10) Number Line Model for Subtraction: Always start at 0. If an integer is positive, walk forward. If an integer is negative, walk backward. The subtraction sign means “turn around”. a) 8 – 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 b) 2 – (–6) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
5 c) (–7) – 3 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 d) (–1) – (–4) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Missing Addend Method: To solve 10 – 6 = n , rewrite it as n + 6 = 10. n = 4.

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Ch5 Notes (revised pres F11) - Section 5-1 Integers and the...

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