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Chapter 2 Working with Real Numbers

2-1 Basic Assumptions

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

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

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

Properties of Equality

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

2-2 Addition on a Number Line

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

 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

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)

2-3 Rules for Addition

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

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

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

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

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

2-4 Subtracting Real Numbers

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

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

2-5 The Distributive Property

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

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)

2-6 Rules for Multiplication

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

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

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

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

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

2-7 Problem Solving: Consecutive Integers

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

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

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

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

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

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

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

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.

2-8 The Reciprocal of a Real Number

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

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

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.

2-9 Dividing Real Numbers

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

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

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

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

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

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