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Working with Real Numbers

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a b and ab are unique. 7 5 = 12. 7 x 5 = 35. COMMUTATIVE PROPERTIES. a b = b a ... If a = b, and b = c, then a = c. 2-2. Addition on a Number Line ... – PowerPoint PPT presentation

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Title: 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
5
ASSOCIATIVE PROPERTIES

(a b) c a (b c) (ab)c a(bc)
(5 15) 20 5 (15 20) (515)20 5(1520)
6
  • Properties of Equality

7
  • Reflexive Property - a a
  • Symmetric Property
  • If a b, then b a
  • Transitive Property
  • If a b, and b c, then a c

8
  • 2-2
  • Addition on a Number Line

9
IDENTITY PROPERTIES

There is a unique real number 0 such that a 0
0 a a
-3 0 0 -3 -3
10
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

11
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

13
Addition Rules
  • If a and b are both positive, then
  • a b ?a? ?b?
  • 3 7 10

14
Addition Rules
  • If a and b are both negative, then
  • a b -(?a? ?b?)
  • -6 (-2) -(6 2) -8

15
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

16
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

17
Addition Rules
  • If a and b are opposites, then a b 0
  • 2 (-2) 0

18
  • 2-4
  • Subtracting Real Numbers

19
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

21
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
22
DISTRIBUTIVE PROPERTY
  • For all real number a ,b, and c
  • a(b - c) ab ac
  • and
  • (b c)a ba - ca

23
  • 2-6
  • Rules for Multiplication

24
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
25
MULTIPLICATIVE PROPERTY OF 0

For every real number a, a 0 0 and 0 a 0
26
MULTIPLICATIVE PROPERTY OF -1

For every real number a, a(-1) -a and (-1)a
-a
27
PROPERTY of OPPOSITES in PRODUCTS

For all real number a and b, -ab
(-a)(b) and -ab a(-b)
28
  • Rules for Multiplication

29
  • The product of two positive numbers or two
    negative numbers is a positive number.
  • (5)(9) 45 or (-5)(-9) 45

30
  • The product of a positive number and a negative
    number is a negative number.
  • (-5)(9) -45 or
  • (5)(-9) -45

31
  • The product of an even number of negative numbers
    is positive.
  • (-5)(-9) 45
  • (-2)(-3)(-1)(-4) 24

32
  • The product of an odd number of negative numbers
    is negative.
  • (-5)(-9)(-2) -90
  • (-2)(-3)(-1)(-4)(-2) -48

33
  • 2-7
  • Problem Solving Consecutive Integers

34
Consecutive Integers
  • Integers that are listed in natural order, from
    least to greatest
  • ,-2, -1, 0, 1, 2,

35
EVEN INTEGER

An integer that is the product of 2 and any
integer. -6, -4, -2, 0, 2, 4, 6,
36
ODD INTEGER

An integer that is not even. -5, -3, -1, 1,
3, 5,
37
CONSECUTIVE EVEN INTEGER

Integers obtained by counting by twos beginning
with any even integer. 12, 14, 16
38
CONSECUTIVE ODD INTEGER

Integers obtained by counting by twos beginning
with any odd integer. 5,7,9
39
  • 2-8
  • The Reciprocal of a Real Number

40
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

41
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

42
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

44
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

45
  • The quotient of two positive numbers or two
    negative numbers is a positive number
  • -24/-3 8 and 24/3 8

46
  • The quotient of two numbers when one is positive
    and the other negative is a negative number.
  • 24/-3 -8 and -24/3 -8

47
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

48
  • The End
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