Working with Real Numbers

1
Chapter 2

• Working with Real Numbers

2

• 2-1
• Basic Assumptions

3
CLOSURE PROPERTIES

• a b and ab are unique
• 7 5 12
• 7 x 5 35

4
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) (515)20 5(1520)
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• Properties of Equality

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• Reflexive Property - a a
• Symmetric Property
• If a b, then b a
• 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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PROPERTY OF OPPOSITES

• 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

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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)
12

• 2-3
• Rules for Addition

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Addition Rules

• If a and b are both positive, then
• a b ?a? ?b?
• 3 7 10

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Addition Rules

• If a and b are both negative, then
• a b -(?a? ?b?)
• -6 (-2) -(6 2) -8

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Addition Rules

• 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

• 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

• If a and b are opposites, then a b 0
• 2 (-2) 0

18

• 2-4
• Subtracting Real Numbers

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DEFINITION of SUBTRACTION

• For all real number a and b,
• a b a (-b)
• To subtract any real number, add its opposite

20

• 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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DISTRIBUTIVE PROPERTY

• For all real number a ,b, and c
• a(b - c) ab ac
• and
• (b c)a ba - ca

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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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• Rules for Multiplication

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• The product of two positive numbers or two
  negative numbers is a positive number.
• (5)(9) 45 or (-5)(-9) 45

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• The product of a positive number and a negative
  number is a negative number.
• (-5)(9) -45 or
• (5)(-9) -45

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• The product of an even number of negative numbers
  is positive.
• (-5)(-9) 45
• (-2)(-3)(-1)(-4) 24

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• The product of an odd number of negative numbers
  is negative.
• (-5)(-9)(-2) -90
• (-2)(-3)(-1)(-4)(-2) -48

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• 2-7
• Problem Solving Consecutive Integers

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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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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 EVEN INTEGER

Integers obtained by counting by twos beginning
with any even integer. 12, 14, 16
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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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• 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
• 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,
• 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,
• 1/ab 1/a 1/b
• The reciprocal of the product of two nonzero
  numbers is the product of their reciprocals.

43

• 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
• ab a1/b
• To divide by a nonzero number, multiply by its
  reciprocal

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• 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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• 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
• a - b a - b
• c c c

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• The End