Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
1) Commutative Properties of Addition and Multiplication:
 The order in which you add or multiply does not matter.
a + b = b + a and
a∙b=b∙a
 Examples:
2 + 4 = 6 and 4 + 2 = 6
5 ∙ 6 = 30 and 6 ∙ 5 = 30
2) Symmetric Property:
 If a + b = c, then c = a + b
 If , then
3) Reflexive Property:
 a+b=a+b
 Nothing changes
4) Associative Properties of Addition and Multiplication.
 The grouping of addition and multiplication does not matter. (Parenthesis)
 Examples:
2 + (4 + 7) = 13 and (2 + 4) + 7 = 13
4∙(6 ∙ 2) = 48 and (4 ∙ 6) ∙ 2 =
48
5) Transitive Property:
 If a = b and b = c, then a = c.
 If, and, then
 If 8 ∙ 2 = 16 and 16 = 42, then 8 ∙ 2 = 42 .
6) Distributive Property:
 a (b + c) = ab + ac
and
a(b – c) = ab – ac
 Examples:
3(7 + 2) = 3(7) + 3(2) = 21 + 6 = 27
5(9 – 6) = 5(9) – 5(6) = 45 – 30 = 15
Combining Like Terms
Like terms are terms such as 4x and 5x that differ only in their numerical coefficients.
To combine 4x and 5x, use the reverse of the distributive property:
 4 x  5 x  x(4  5)  9 x
To add or subtract like terms, simply combine their numerical coefficients:
 9 y  7 y  y (9  7)  2 y
 7b  b  7b  1b  b(7  1)  6b
 2 xy  3xy  4 xy  xy(2  3  4)  9 xy
Chapter 1, Section 1.5
Page 1
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
7) Additive Identity:
 When zero is added to any number or variable, the sum is the number
or variable.
 a+0=a
8) Multiplicative Identity:
 When any number or variable is multiplied by 1, the product is the
number or variable.
 a∙1=a
9) Multiplicative Property of Zero:
 When any number or variable is multiplied by zero, the product is
0.
 a∙0=0
10) Closure Property
Closure is a property (or characteristic) of a set of numbers, such as integers, and an
operation, such as addition, subtraction, multiplication or division.
Closed
Same Output Set
Given
Operation
Given Input Set
A set and an operation are
closed if the inputs and the
outputs of the operation are
from the same set.
Example: The set of
integers is closed to the
operation of addition.
et
Same S
Not Closed
Different Output Set
Given
Operation
nt Sets
Differe
Chapter 1, Section 1.5
Page 2
Given Input Set
A set and an operation are
not closed if the inputs and
the outputs of the operation
are from the different sets.
Example: The set of
integers is not closed to the
operation of division.
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
Sample Regents Problem
Which expression is an example of the associative property?
(1) (x + y) + z = x + (y + z)
(2) x + y + z = z + y + x
(3) x(y + z) = xy + xz
(4) x • 1 = x
One Solution
Choice (1) is an example of the associative property,
Choice (2) is an example of the commutative property.
Choice (3) is an example of the distributive property.
Choice (4) is an example of the identity property of multiplication.
Another Math A Regents Problem
The operation * for the set {p,r,s,v} is defined in the accompanying table. What is the
inverse element of r under the operation *?
(1) p
(3) s
(2) r
(4) v
One Solution
These little tables can be tricky to understand. The key in this one is to understand that
the inverse properties always yield the identity elements as outputs. The way to read this
table is as follows:
 The operation is the symbol *
 The input set is {p,r,s,v)
 The outputs are the cells in the matrix.
p*p results in s
p*r results in v
p*s results in p (either p or s must be the identity element)
r*s results in r (either r or s must be the identify element)
s*s results in s (either s or s must be the identify element – not a difficult choice)
The inverse of an element is the element which, under the given operation, results in the
identify element. We want to know what element works with r under the operation * to
produce s, so we look at the column beneath r until we find an s, and we see that v
operates with r to yield s. The answer is v.
Chapter 1, Section 1.5
Page 3
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
Which set is closed under division?
(1) {1}
(3) integers
(2) counting numbers
(4) whole numbers
One Solution
1
(1)  1 This is a closed set under division. This is the answer. Verify by checking the
1
other answers to make sure they are wrong.
counting number
(2)
 fraction Fractions and counting numbers are not the same set, so
counting number
this cannot be the answer.
integer
(3)
 fraction Same problem as choice 2. This is not the answer.
integer
whole number
 fraction Same problem as choices 2 and 3. This is not the answer.
(4)
whole number
Homework Questions
1 Which expression is an example of the associative property?
(1) (x + y) + z = x + (y + z)
(2) x + y + z = z + y + x
(3) x(y + z) = xy + xz
(4) x • 1 = x
2 Which equation illustrates the associative property of addition?
(1) x + y = y + x
(2) 3(x + 2) = 3x + 6
(3) (3 + x) + y = 3 + (x + y)
(4) 3 + x = 0
3 Which equation illustrates the associative property?
(1) a (1)  a
(2) a  b  b  a
(3) a (b  c)  (ab)  (ac)
(4) (a  b)  c  a  (b  c)
4
Which equation illustrates the distributive property of multiplication over
Chapter 1, Section 1.5
Page 4
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
addition?
(1) 6(3a + 4b) = 18a + 4b
(2) 6(3a + 4b) = 18a + 24b
(3) 6(3a + 4b) = (3a + 4b)6
(4) 6(3a + 4b) = 6(4b + 3a)
5
Which equation illustrates the distributive property?
(1) 5(a + b) = 5a + 5b
(2) a + b = b + a
(3) a + (b + c) = (a + b) + c
(4) a + 0 = a
Homework Solutions
1 Which expression is an example of the
(1)
associative property?
(2) is associative, (3) is distributive and
(1) (x + y) + z = x + (y + z)
(4) is multiplicative identity.
(2) x + y + z = z + y + x
(3) x(y + z) = xy + xz
(4) x • 1 = x
2 Which
equation
illustrates
associative property of addition?
(1) x + y = y + x
(2) 3(x + 2) = 3x + 6
(3) (3 + x) + y = 3 + (x + y)
(4) 3 + x = 0
the
(3)
(1) is commutative, (2) is distributive and
(4) is additive inverse.
3 Which
equation
illustrates
associative property?
(1) a (1)  a
(2) a  b  b  a
(3) a (b  c)  (ab)  (ac)
(4) (a  b)  c  a  (b  c)
the
(4)
(1) is multiplicative identity, (2) is
commutative and (3) is distributive.
Chapter 1, Section 1.5
Page 5
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
4
Which
equation
illustrates
the
distributive property of multiplication
over addition?
(1) 6(3a + 4b) = 18a + 4b
(2) 6(3a + 4b) = 18a + 24b
(3) 6(3a + 4b) = (3a + 4b)6
(4) 6(3a + 4b) = 6(4b + 3a)
5
Which
equation
illustrates
distributive property?
(1) 5(a + b) = 5a + 5b
(2) a + b = b + a
(3) a + (b + c) = (a + b) + c
(4) a + 0 = a
the
(2)
(1)
(2) is commutative, (3) is associative and
(4) is additive identity
More Regents Questions
1
2
(1)
(2) is associative, (3) is distributive and
Which expression is an example of the
(4) is multiplicative identity.
associative property?
(1) (x + y) + z = x + (y + z)
(2) x + y + z = z + y + x
(3) x(y + z) = xy + xz
(4) x • 1 = x
060424a
010428a
Which
equation
illustrates
associative property of addition?
(1) x + y = y + x
(2) 3(x + 2) = 3x + 6
(3) (3 + x) + y = 3 + (x + y)
(4) 3 + x = 0
Chapter 1, Section 1.5
Page 6
(3)
(1) is commutative, (2) is distributive and
the
(4) is additive inverse.
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
3
080725a
Which
equation
illustrates
associative property?
(1) a (1)  a
(2) a  b  b  a
(3) a (b  c)  (ab)  (ac)
(4) (a  b)  c  a  (b  c)
4
(4)
(1) is multiplicative identity, (2) is
the
commutative and (3) is distributive.
(2)
080413a
Which
equation
illustrates
the
distributive property of multiplication
over addition?
(1) 6(3a + 4b) = 18a + 4b
(2) 6(3a + 4b) = 18a + 24b
(3) 6(3a + 4b) = (3a + 4b)6
(4) 6(3a + 4b) = 6(4b + 3a)
5
060503a
Which
equation
illustrates
distributive property?
(1) 5(a + b) = 5a + 5b
(2) a + b = b + a
(3) a + (b + c) = (a + b) + c
(4) a + 0 = a
(1)
(2) is commutative, (3) is associative and
the
(4) is additive identity
6
(4)
(1) is commutative, (2) is additive identity
Which
equation
illustrates
the
and (3) is associative.
distributive property for real numbers?
1 1 1 1
(1)   
3 2 2 3
(2) 3  0  3
(3) (13
.  0.07)  0.63  13
.  (0.07  0.63)
(4) –3(5 + 7) = (–3)(5) + (–3)(7)
7
(2)
(1) is multiplicative identity, (3) is
Which equation is an illustration of the
additive inverse and (4) is multiplicative
additive identity property?
060108a
089907a
Chapter 1, Section 1.5
Page 7
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
(1) x • 1 = x
(3) x – x = 0
1
(4) x   1
x
(2) x + 0 = x
8
inverse.
(4)
Which statement best illustrates the (1) and (2) are commutative and (3) is
additive inverse.
additive identity property?
(1) 6 + 2 = 2 + 6
(3) 6 + (-6) = 0
(2) 6(2) = 2(6)
(4) 6 + 0 = 6
060624a
010314a
9
Which
equation
illustrates
multiplicative identity element?
1
(1) x + 0 = x
(3) x •
=1
x
(2) x – x = 0
(4) x • 1 = x
(4)
(1)
is
additive
identity,
(2) is additive
the
inverse and (3) is multiplicative identity.
(3)
010207a
10
What is the additive inverse of the
expression a - b?
(1) a + b
(3) -a + b
(2) a - b
(4) -a - b
(4)
010207a
11
Which expression must be added to
3x  7 to equal 0?
(1) 0
(3) –3x – 7
(2) 3x + 7
(4) –3x + 7
(1)
060315a
12
What is the additive inverse of
(1) 
(2)
2
3
1
3
Chapter 1, Section 1.5
Page 8
(3) 
(4)
3
2
3
2
2
3
?
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
(1)
010821a
13
The additive inverse of
1
a
(2) a
(1) 
14
1
is
a
(3) 0
(4) a
(3)
060011a
1
is 0,
a
If a ≠ 0 and the sum of x and
then
1
a
(4) x = 1 – a
(3) x = 
(1) x = a
(2) x = –a
15
(2)
060815a
The reciprocal of 5 is
1
(1) 1
(3) 
5
1
(2)
(4) -5
5
16
(2)
010516a
What is the multiplicative inverse of
4
3
3
(4) 
4
(3) 
(1) –1
(2)
4
3
17
(4)
010730a
The multiplicative inverse of 
(1)
3
?
4
1
3
Chapter 1, Section 1.5
Page 9
1
is
3
(3) 3
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
(2) 
18
1
3
(4) -3
010630a
Which
equation
illustrates
multiplicative inverse property?
(1) 1 x  x
(3) 1  0  0
1
(2) x   1
(4) 1 x  x
x
19
(2)
(1) and (3) are the multiplicative identity
the
and (4) is not a property.
080115a
(1)
Using the transitive property of inequality:
If a + b is less than c + d, and d + e is
less than a + b, then e is
If d  e  a  b and a  b  c  d , then
(1) less than c
(3) less than d
d  e  c  d.
(2) equal to c
(4) greater than d
Using the subtraction property of inequality,
subtract d from each side of the inequality:
d ecd
ec
20
080810a
(1)
The sum of two negative numbers
always has to be
(1) negative
(3) zero
(2) positive
(4) an integer
21
060207b
(4)
Which statement is true for all real (1) not true if x = 1
(2) not true if x = 1
number values of x?
(3) not true if x = -1
(1) |x – 1| 0
(3) x 2  x
(4) always true
2
(2) |x – 1| (x – 1)
(4) x  x
Complete the Matching Column (put the corresponding letter next to the number)
Chapter 1, Section 1.5
Page 10
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
1)
26 +0 = 26
a) Reflexive
2)
22 · 0 = 0
b) Additive Identity
3)
3(9 + 2) = 3(9) + 3(2)
c) Multiplicative identity
4)
If 32 = 64 ¸2, then 64 ¸2 = 32
d) Associative Property of Mult.
5)
32 · 1 = 32
e) Transitive
6)
9 + 8 = 8+ 9
f) Associative Property of Add.
7)
If 32 + 4 = 36 and 36 = 62, then 32 + 4 = 62
g) Symmetric
8)
16 + (13 + 8) = (16 +13) + 8
h) Commutative Property of Mult.
9)
6 · (2 · 12) = (6 · 2) · 12
i) Multiplicative property of zero
10) 6 ∙ 9 = 6 ∙ 9
j) Distributive
Complete the Matching Column (put the corresponding letter next to the number)
11)
If 5 + 6 = 11, then 11 = 5 + 6
a) Reflexive
12) 22 · 0 = 0
b) Additive Identity
13) 3(9 – 2) = 3(9) – 3(2)
c) Multiplicative identity
14) 6 + (3 + 8) = (6 +3) + 8
d) Associative Property of Mult.
15) 54 + 0 = 54
e) Transitive
16) 16 – 5 = 16 – 5
f) Associative Property of Addition
17)
If 12 + 4 = 16 and 16 = 42, then 12 + 4 = 42
g) Symmetric
18) 3 · (22 · 2) = (3 · 22) · 2
Addition
h) Commutative Property of
19) 29 · 1 = 29
i) Multiplicative property of zero
20) 6 +11 = 11+ 6
Chapter 1, Section 1.5
Page 11
j) Distributive
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
21) Which number is a whole number but not a natural number?
a) – 2
b) 3
c) ½
d) 0
22) Which number is an integer but not a whole number?
a) – 5
b) ¼
c) 3
d) 2.5
23) Which number is irrational?
a)
b) 4
c) .1875
d) .33
24) Give an example of a number that is rational, but not an integer.
25) Give an example of a number that is an integer, but not a whole number.
26) Give an example of a number that is a whole number, but not a natural number.
27) Give an example of a number that is a natural number, but not an integer.
Chapter 1, Section 1.5
Page 12
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
Modeling:
Sample Math A Regents Problem
Which expression is an example of the associative property?
(1) (x + y) + z = x + (y + z)
(2) x + y + z = z + y + x
(3) x(y + z) = xy + xz
(4) x • 1 = x
One Solution
Choice (1) is an example of the associative property,
Choice (2) is an example of the commutative property.
Choice (3) is an example of the distributive property.
Choice (4) is an example of the identity property of multiplication.
Another Math A Regents Problem
The operation
p,r,s,v} is defined in the accompanying table. What is the
inverse element of r
(1) p
(3) s
(2) r
(4) v
One Solution
These little tables can be tricky to understand. The key in this one is to understand that
the inverse properties always yield the identity elements as outputs. The way to read this
table is as follows:
 The operation is the symbol *
 The input set is {p,r,s,v)
 The outputs are the cells in the matrix.
p*p results in s
p*r results in v
p*s results in p (either p or s must be the identity element)
r*s results in r (either r or s must be the identify element)
s*s results in s (either s or s must be the identify element – not a difficult choice)
The inverse of an element is the element which, under the given operation, results in the
identify element. We want to know what element works with r under the operation * to
produce s, so we look at the column beneath r until we find an s, and we see that v
operates with r to yield s. The answer is v.
Chapter 1, Section 1.5
Page 13
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
Closure is a property (or characteristic) of a set of numbers, such as integers, and an
operation, such as addition, subtraction, multiplication or division.
Closed
Same Output Set
Given
Operation
Given Input Set
A set and an operation are
closed if the inputs and the
outputs of the operation are
from the same set.
Example: The set of
integers is closed to the
operation of addition.
et
Same S
Not Closed
Different Output Set
Given
Operation
nt
Differe

Sets
Given Input Set
A set and an operation are
not closed if the inputs and
the outputs of the operation
are from the different sets.
Example: The set of
integers is not closed to the
operation of division.
To show that a set of numbers is not closed to an operation, find one example for
which the operation results in a number outside of the given set of numbers.
A Metaphor for Understanding Closed and Open Sets: One way to understand
closure is to imagine a classroom in which the teacher invites all the numbers from a
given set and closes the door. No other numbers are in the room. Then the teacher turns
on an “operations” machine (egs. Add, subtract, multiply, or divide). If the machine can
run forever with just the numbers in the room, the set of numbers is closed for that
operation. If the machine ever creates any number that is not already in the room, the
teacher has to open the door and invite more numbers into the room. The set of numbers
is not closed for that operation.
Chapter 1, Section 1.5
Page 14
Chapter 1: Sets, Operations and Algebraic Language
1.5: Properties of Real Numbers
Another Math A Regents Problem
Which set is closed under division?
(1) {1}
(3) integers
(2) counting numbers
(4) whole numbers
One Solution
1
(1)  1 This is a closed set under division. This is the answer. Verify by checking the
1
other answers to make sure they are wrong.
counting number
(2)
 fraction Fractions and counting numbers are not the same set, so
counting number
this cannot be the answer.
integer
(3)
 fraction Same problem as choice 2. This is not the answer.
integer
whole number
 fraction Same problem as choices 2 and 3. This is not the answer.
(4)
whole number
Chapter 1, Section 1.5
Page 15