Download Rational Numbers (Q) Irrational Numbers

9.1 Symbols and Sets of Numbers
Real Numbers
The set of real numbers is the set of all numbers that correspond to points on the number line.
Rational Numbers (Q)
Irrational Numbers
Numbers that can be written as a quotient of
integers (with a nonzero denominator).
Irrational numbers cannot be expressed as a
quotient of integers.
Some examples are shown below:
This set includes:
1
= 0.3 ,
3
1
= 0.25 ,
4
repeating decimals
terminating decimals
integers
−
4
= − 4,
1
13
= −2.16
6
1
− = −0.1
10
0
3
= 0,
=3
2
1
−
Non Perfect Roots: ( x must be used )
7,
Multiples of pi ( π )
π,
Integers (I)
Whole numbers and their opposites
{. . .,-3, -2 ,-1, 0, 1, 2, 3, …}
3
5
, −
2
3
2π
, − 7π , 2π
3
any decimal number that does not
terminate or repeat, such as:
Whole numbers (W)
Zero and the Natural Numbers
{ 0, 1, 2, 3, …}
0.12123123412345…
-0.242462468…
Natural Numbers (N)
Counting Numbers
{1, 2, 3, …}
0.1298753496…
Besides Real Numbers, tell which sets of numbers each of the following belong to:
1.6
-3
8
3
1

0, 1.112233..., 3.14, − 5 ,
4

list the numbers that belong to the set of:
Given the set
5
12, − 100, 2.18, − 25, 245, 600,
16
π
4
}
Integers:
Irrational numbers:
1
Numbers can be pictured on a number line. The number to the right of another number is larger.
The following symbols are used to show the order of numbers.
Ordering Symbols
= “is equal to”
≠ “is not equal to”
Inequality symbols
<
“is less than”
>
“is greater than”
≤
”is less than or equal to”
≥
“is greater than or equal to”
Place the following numbers on the number line provided.
{ -π, , -1.9, -1.1, 0, , 1.1, π, 2.75 }
Insert <, >, or = in the appropriate space to make each statement true.
1.
-π
6. −6
-3
2. -1.1
5
7. −5
-1.9
3. 0
25
5
8. − −8
−
4. -1.9
-3
−9
9.
5. 1.1
Are the mathematical statements below true or false?
1.
-3 ≤ -3.0
2.
4.
−3.3 ≥ − 3.33
5.
-4 ≤ -3.0
3.
6.
-3.3 ≤ -3.33
Translate each sentence into a mathematical statement
1. Eight is greater than or equal to two __________________________________________
2. Negative eleven is less than zero ____________________________________________
3. Three is not equal to negative three __________________________________________
Rewrite:
2
-6 < -5 _________________
2 ≥ -1 _____________________
9.1 State if true or false and give a reason if the statement is false.
1.
{ 0, 1, 2, 3, … } is the set of Integers
T
F
2.
All Natural Numbers are Whole Numbers.
T
F
3.
{ 1, 2, 3, … } is the set of Nonnegative Integers.
T
F
4.
81 is an Irrational Number.
T
F
5.
The set of Integers is a subset of the set of Rational Numbers.
T
F
6.
{ 0, 1, 2, 3, … } is the set of Positive Integers.
T
F
7.
All Real Numbers are Rational Numbers.
T
F
8.
|a| is always a positive number.
T
F
9.
All Natural numbers are positive
T
F
10.
3.14 is an Irrational Number.
T
F
.
3
9.2 Properties of Real Numbers
Commutative Property of Addition
Commutative Property of Multiplication
a+b =b+a
a ⋅b = b⋅ a
Associative Property of Addition
Associative Property of Multiplication
( a + b ) + c = a + (b + c )
( a ⋅ b) ⋅ c = a ⋅ (b ⋅ c )
Distributive Property (of Multiplication Over Addition or Over Subtraction)
a ( b + c ) = ab + ac
or
( b + c ) a = ba + ca
a ( b − c ) = ab − ac
or
( b − c ) a = ba − ca
or
a ( b + c + d ) = ab + ac + ad
Identity Property of Addition
a + 0 = a or
0+a = a
Inverse Property for Addition
a + ( −a ) = 0
Multiplication Property of Zero
a ⋅ 0 = 0 or
4
0⋅a = 0
Identity Property of Multiplication
a ⋅1 = a or 1 ⋅ a = a
Inverse Property for Multiplication
a⋅
1
=1
a
1. Use the commutative property to complete each statement.
a. yx =
b. 14 + 5x =
2. Use the associative property to complete each statement.
a. −7 ( ab ) =
b.
( 2x + y ) + 4 =
3. Use the commutative and associative properties to simplify each expression.
a. (12 + x ) + 5 =
b.
78 
 r =
87 
4. Use the distributive property to write each expression without parentheses. Then simplify.
a. 6 ( x + 2 ) =
d.
−(2 + x − y) =
b. −4 ( −7 + 2 y ) =
c, −2 ( −3 y − 5 ) =
e. 3 ( 4 x − y ) + 7 =
f. 3 ( 4 x − y + 7 ) =
5. Use the distributive property to write the sum as a product.
a. 3 ⋅ 2 + 3 ⋅ y = ____________________
b 8 x + 8 y = ____________________
6. Fill in the table.
Expression
Opposite
(Additive inverse)
Reciprocal
(Multiplicative inverse)
5
3
4
1
4
5
−
−3x
5
Number Property Homework
Name the property illustrated by each true statement.
a.
1.
–3(2 + 1) = –3(1 + 2)
2.
–3(2 + 1) = (2 + 1)( –3)
4.
(–3 + 4) + 5 = –3 + (4 + 5)
5.
( −2 x ) x = −2 ( x ⋅ x )
7.
9 x − 18 = 9( x − 2)
8.
( −3.2 ) ⋅1 ⋅ 0 = ( −3.2 )  ⋅ 0
3.
–3(2 + x) = –6 + –3x
6.
(–8 + 8) + 2 = 0 + 2
9.
–3.5 + 0 = –3.5
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Harder: Name the properties illustrated by each true statement.
b.
[(− 6.2) ⋅ 1] + 0 = (− 6.2) ⋅ 1 ___________________________________________________
c.
[(− 6.2) ⋅ 1] + 0 = (− 6.2) + 0 __________________________________________________
d.
[(− 6.2) ⋅ 1] + 0 = 0 + [(− 6.2) ⋅1] _________________________________________________
e.
( −4 x ) y = −4 ( xy )
f.
− 3(2 + 1) = −3(1 + 2) ________________________________________________________
__________________________________________________________
g. −3 ( x + 1) = ( x + 1) ⋅ ( −3 ) ______________________________________________________
h. −4 ( 2 − r ) = −8 + 4r _________________________________________________________
i.
( y + 4 ) + 5 = y + ( 4 + 5 ) ______________________________________________________
j.
( −2 + 2 ) + x = 0 + x
_________________________________________________________
 1
k.  2 ⋅  ⋅1 = 1⋅1 ______________________________________________________________
 2
l.
1
 1
 2 ⋅  ⋅1 = 2 ⋅ _____________________________________________________________
2
 2
m. 4 x + 20 = 4 ( x + 5 ) ___________________________________________________________
Answers for Number Property HW.
1. commutative property of addition
4. associative property of addition
7. distributive property
6
2. commutative property of multiplication
5. associative property of multiplication
8. identity property for multiplication
3. distributive property
6. inverse property of addition
9. identity property of addition