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Chapter 2 Working with Real Numbers

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2-1 Basic Assumptions

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CLOSURE PROPERTIES a + b and ab are unique 7 + 5 = 12 7 x 5 = 35

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COMMUTATIVE PROPERTIES a + b = b + a ab = ba 2 + 6 = 6 + 2 2 x 6 = 6 x 2

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ASSOCIATIVE PROPERTIES (a + b) + c = a + (b +c) (ab)c = a(bc) (5 + 15) + 20 = 5 + (15 +20) (5·15)20 = 5(15 · 20)

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Properties of Equality

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Reflexive Property - a = a Reflexive Property - a = a Symmetric Property – Symmetric Property – If a = b, then b = a Transitive Property – Transitive Property – If a = b, and b = c, then a = c

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2-2 Addition on a Number Line

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IDENTITY PROPERTIES There is a unique real number 0 such that: a + 0 = 0 + a = a - 3 + 0 = 0 + -3 = -3

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For each a, there is a unique real number – a such that: a + (-a) = 0 and (-a)+ a = 0 (-a) is called the opposite or additive inverse of a PROPERTY OF OPPOSITES

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Property of the opposite of a Sum For all real numbers a and b: -(a + b) = (-a) + (-b) The opposite of a sum of real numbers is equal to the sum of the opposites of the numbers. -(8 +2) = (-8) + (-2)

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2-3 Rules for Addition

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Addition Rules 1. If a and b are both positive, then a + b = a + b 3 + 7 = 10

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Addition Rules 2. If a and b are both negative, then a + b = -( a + b ) (-6) + (-2) = -(6 +2) = -8 (-6) + (-2) = -(6 +2) = -8

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Addition Rules 3. If a is positive and b is negative and a has the greater absolute value, then a + b = a - b 6 + (-2) = (6 - 2) = 4

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Addition Rules 4. If a is positive and b is negative and b has the greater absolute value, then a + b = -( b - a ) 4 + (-9) = -(9 -4) = -5

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Addition Rules 5. If a and b are opposites, then a + b = 0 2 + (-2) = 0

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2-4 Subtracting Real Numbers

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DEFINITION of SUBTRACTION For all real numbers a and b, a – b = a + (-b) To subtract any real number, add its opposite

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Examples 1. 3 – (-4) 2. -y – (-y + 4) 3. -(f + 8) 4. -(-b + 6 – a) 5. m – (-n + 3)

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2-5 The Distributive Property

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DISTRIBUTIVE PROPERTY a(b + c) = ab + ac (b +c)a = ba + ca 5(12 + 3) = 512 + 5 3 = 75 (12 + 3)5 = 12 5 + 3 5 = 75

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Examples 1. 2(3x + 4) 2. 5n + 7(n – 3) 3. 2(x – 6) + 9 4. 8 + 3(4 – y) 5. 8(k + m) - 15(2k + 5m)

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2-6 Rules for Multiplication

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IDENTITY PROPERTY of MULTIPLICATION There is a unique real number 1 such that for every real number a, a · 1 = a and 1 · a = a

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MULTIPLICATIVE PROPERTY OF 0 For every real number a, a · 0 = 0 and 0 · a = 0

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MULTIPLICATIVE PROPERTY OF -1 For every real number a, a(-1) = -a and (-1)a = -a

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PROPERTY of OPPOSITES in PRODUCTS For all real number a and b, -ab = (-a)(b) and -ab = a(-b)

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Examples 1. (-1)(3d – e + 8) 2. -6(7n – 6) 3. -[-4(x – y)]

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2-7 Problem Solving: Consecutive Integers

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EVEN INTEGER An integer that is the product of 2 and any integer. …-6, -4, -2, 0, 2, 4, 6,…

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ODD INTEGER An integer that is not even. …-5, -3, -1, 1, 3, 5,…

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Consecutive Integers Integers that are listed in natural order, from least to greatest …,-2, -1, 0, 1, 2, …

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Example Three consecutive integers have the sum of 24. Find all three integers.

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CONSECUTIVE EVEN INTEGER Integers obtained by counting by twos beginning with any even integer. 12, 14, 16

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Example Four consecutive even integers have a sum of 36. Find all four integers.

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CONSECUTIVE ODD INTEGER Integers obtained by counting by twos beginning with any odd integer. 5,7,9

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Example There are three consecutive odd integers. The largest integer is 9 less than the sum of the smaller two integers. Find all three integers.

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2-8 The Reciprocal of a Real Number

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PROPERTY OF RECIPROCALS For each a except 0, there is a unique real number 1/a such that: For each a except 0, there is a unique real number 1/a such that: a · (1/a) = 1 and (1/a)· a = 1 1/a is called the reciprocal or multiplicative inverse of a

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PROPERTY of the RECIPROCAL of the OPPOSITE of a Number For each a except 0, For each a except 0, 1/-a = -1/a The reciprocal of –a is -1/a

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PROPERTY of the RECIPROCAL of a PRODUCT For all nonzero numbers a and b, For all nonzero numbers a and b, 1/ab = 1/a ·1/b The reciprocal of the product of two nonzero numbers is the product of their reciprocals.

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2-9 Dividing Real Numbers

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DEFINITION OF DIVISION For every real number a and every nonzero real number b, the quotient is defined by: a÷b = a·1/b a÷b = a·1/b To divide by a nonzero number, multiply by its reciprocal

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1. The quotient of two positive numbers or two negative numbers is a positive number -24/-3 = 8 and 24/3 = 8

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2. The quotient of two numbers when one is positive and the other negative is a negative number. 24/-3 = -8 and -24/3 = -8

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PROPERTY OF DIVISION For all real numbers a, b, and c such that c 0, a + b = a + b and c c c c c c a - b = a - b c c c c c c

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Examples 1. 4 ÷ 16 2. 8x 16 16 3. 5x + 25 5

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The End

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