1.
*(a + b) / c = (a / c) + (b / c)*: and (a − b) / c = (a / c) − (b / c)
2.
*ab + ac = a(b + c)*: ba + ca = (b + c)a
3.
*ab − ac = a(b − c)*: ba − ca = (b − c)a
4.
*additive inverses*: A number and its opposite.
5.
*associative properties*: For all real numbers a, b, and c,
(a + b) + c = a + (b + c)
(ab)c = a(bc)
When you add or multiply
any three real numbers,
the grouping (or association)
of the numbers
does not affect the result.
6.
*average*: The average of a set of numbers
is the sum of the numbers
divided by
the number of numbers.
7.
*closure properties*: For all real numbers a and b,
a + b is a unique real number.
ab is a unique real number.
The sum and product
of any two real numbers
are also real numbers.
Moreover, they are unique.
(This means there is one and only one possible answer
when you add or multiply two real numbers.)
8.
*commutative properties*: For all real numbers a and b,
a + b = b + a
ab = ba
The order
in which you add or multiply
two numbers
does not affect the result.
9.
*consecutive even integers*: Integers obtained by
counting by twos,
beginning with
any even integer.
10.
*consecutive integers*: Numbers
obtained by
counting by ones
from any number
in the set of integers.
11.
*consecutive odd integers*: Integers obtained by
counting by twos,
beginning with
any odd integer.
12.
*Definition of Division*: For every real number a
and every nonzero real number b,
the quotient a ÷ b, or a/b,
is defined by
a ÷ b = a⋅1/b.
To divide by a nonzero number,
multiply by its reciprocal.
13.
*Definition of Subtraction*: For all real numbers a and b,
the difference a − b
is defined by
a − b = a + (−b).
To subtract b,
add the opposite of b.
14.
*distributive property*
*(of multiplication*
*with respect to addition)*: For all real numbers a, b, and c,
a(b + c) = ab + ac
and
(b + c)a = ba + ca.
15.
*distributive property*
*(of multiplication*
*with respect to subtraction)*: For all real numbers a, b, and c,
a(b − c) = ab − ac
and
(b − c)a = ba − ca.
16.
*equivalent expressions*: Expressions that represent
the same number
for all values
of the variable
that they contain.
17.
*even integer*: An integer
that is the product of
2 and any integer.
18.
*factors*: Numbers
that are multiplied together
to produce a product.
19.
*identity element for addition*: Zero (0).
20.
*identity element for multiplication*: One (1).
21.
*identity property*
*of addition*: There is
a unique real number 0
such that
for every real number a,
a + 0 = a and 0 + a = a.
22.
*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.
23.
*multiple*: The product of
any real number
and an integer
is a multiple
of the real number.
24.
*multiplicative property of −1*: For every real number a,
a(−1) = −a and (−1)a = −a.
25.
*multiplicative property of zero*: For every real number a,
a⋅0 = 0 and 0⋅a = 0.
26.
*natural order*
*of integers*: Order from least to greatest.
27.
*Objective* 2-1: To use number properties
to simplify expressions.
28.
*Objective* 2-2: To add real numbers using
a number line or
properties about opposites.
29.
*Objective* 2-3: To add real numbers using
rules for addition.
30.
*Objective* 2-4: To subtract real numbers and
to simplify expressions involving differences.
31.
*Objective* 2-5: To use the distributive property
to simplify expressions.
32.
*Objective* 2-6: To multiply real numbers.
33.
*Objective* 2-7: To write equations
to represent relationships
among integers.
34.
*Objective* 2-8: To simplify expressions
involving reciprocals.
35.
*Objective* 2-9: To divide real numbers and
to simplify expressions
involving quotients.
36.
*odd integer*: An integer
that is not even.
37.
*Properties of Equality*: For all real numbers a, b, and c:
38.
*property of opposites in products*: For all real numbers a and b,
(−a)(b) = −ab,
a(−b) = −ab,
and (−a)(−b) = ab.
39.
*property of opposites*: For every real number a,
there is
a unique real number −a
such that
a + (−a) = 0 and (−a) + a = 0.
40.
*property of reciprocals*: For every nonzero real number a,
there is
a unique real number 1/a
such that
a⋅1/a = 1 and 1/a⋅a = 1.
41.
*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.
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.
*property of*
*the reciprocal of*
*the opposite of a number*: For every nonzero number a,
1/−a = −1/a.
Read, "The reciprocal of -a is -1/a."
44.
*reciprocals*: Two numbers
whose product is 1;
also called multiplicative inverses.
45.
*reflexive property*
*of equality*: If a is a real number,
then a = a.
46.
*Rules for Addition*
1. If a and b are both positive,: then a + b = |a| + |b|.
47.
*Rules for Addition*
2. If a and b are both negative,: then a + b = −(|a| + |b|).
48.
*Rules for Addition*
3. If a is positive and b is negative
and a has the greater absolute value,: then a + b = |a| − |b|.
49.
*Rules for Addition*
4. If a is positive and b is negative
and b has the greater absolute value,: then a + b = −(|b| − |a|).
50.
*Rules for Addition*
5. If a and b are opposites,: then a + b = 0.
51.
*Rules for Division*: If two numbers have the same sign,
their quotient is positive.
If two numbers have opposite signs,
their quotient is negative.
52.
*Rules for Multiplication* 1.: If two numbers have the same sign,
their product is positive.
If two numbers have opposite signs,
their product is negative.
53.
*Rules for Multiplication* 2.: The product
of an even number
of negative numbers is positive.
The product
of an odd number
of negative numbers is negative.
54.
*simplifying an expression*: Replacing an expression
containing variables
by an equivalent expression
with as few terms as possible.
55.
*symmetric property*
*of equality*: If a and b are real numbers,
and a = b,
then b = a.
56.
*terms*: In the sum a + b,
a and b are called terms.
57.
*transitive property*
*of equality*: For all real numbers a, b, and c,
if a = b and b = c,
then a = c.
58.
−1.25 and −0.8 are reciprocals because: −1.25·(−0.8) = 1
59.
0 has no reciprocal because: 0 times any number is 0, not 1.
60.
5 and ¹/₅ are reciprocals because: 5 ·¹/₅ = 1
61.
a − b is the number
to add to b
to obtain: a
62.
By applying the symmetric property of equality,: the distributive properties of multiplication
can also be written in the following forms:
63.
Can you divide zero
by any number
other than zero?: Yes.
64.
Caution:
Is subtraction commutative?: No.
65.
Chapter *2*: Working with Real Numbers
66.
Chapter 2
Section 2: Multiplication
67.
Chapter 2
Section 3: Division
68.
Chapter 2
Section 1: Addition and Subtraction
69.
Consecutive multiples of a are: multiples of a
by consecutive integers.
70.
From the properties of addition,
prove the property of the opposite of a sum:
(a + b) + [ −(a + b) ] = 0
(a + b) + [ (−a) + (−b) ] = 0: −(a + b) = (−a) + (−b)
71.
Is division associative?: No.
72.
Is division commutative?: No.
73.
Is subtraction associative?: No.
74.
Lesson *2-1*: Basic Assumptions
75.
Lesson *2-2*: Addition on a Number Line
76.
Lesson *2-3*: Rules for Addition
77.
Lesson *2-4*: Subtracting Real Numbers
78.
Lesson *2-5*: The Distributive Property
79.
Lesson *2-6*: Rules for Multiplication
80.
Lesson *2-7*: Problem Solving: Consecutive Integers
81.
Lesson *2-8*: The Reciprocal of a Real Number
82.
Lesson *2-9*: Dividing Real Numbers
83.
Multiplying any real number by −1 produces: the opposite of the number.
84.
Properties of division: For all real numbers, a, b, and c
such that c ≠ 0,
85.
The reciprocal of the reciprocal of
a nonzero number is: the number
86.
Using the definition of subtraction,
you may replace
any difference with: a sum.
87.
When one (or at least one) of
the factors of a produce is zero,: the product itself is zero.
88.
When you find the opposite of
a sum or a difference,: you change the sign of
each term of
the sum or difference.
89.
Why can you never divide by zero?: Dividing by 0
would mean
multiplying by the reciprocal of 0.
But 0 has no reciprocal
(because 0 times any number is 0,
not 1).
Therefore, division by zero
has no meaning
in the set of real numbers.
90.
You can use the definition of division
to express any quotient as: a product.
