Download 223 Reference Chapter

223 Reference Chapter
Section R2: Real Numbers and Their Properties
Natural Numbers: {1, 2, 3, 4, …}
Whole Numbers: {0, 1, 2, …}
Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}
Rational Numbers: {p/q | p and q are both integers and q ≠0
Examples of Rational Numbers: 9, -17, 3/5, -21/9, √4
Irrational Numbers: {x | x is real but not rational}
Examples of Irrationals: √2, -√10, π
Real Numbers: {x | x corresponds to a point on a number line}
Example: Let set V = {-12, -√5, -4π, 0, 2/5, 3, √7, 10}
List the elements from set V that belong to each set:
a) Natural numbers
b) Whole numbers
c) Integers
d) Rational
e) Irrational
f) Real
223 Reference Chapter
Section R2: Real Numbers and Their Properties
Natural Numbers: {1, 2, 3, 4, …}
Whole Numbers: {0, 1, 2, …}
Integers: {…, -3, -2, -1, 0, 1, 2, 3, …}
Rational Numbers: {p/q | p and q are both integers and q ≠0
Examples of Rational Numbers: 9, -17, 3/5, -21/9, √4
Irrational Numbers: {x | x is real but not rational}
Examples of Irrationals: √2, -√10, π
Real Numbers: {x | x corresponds to a point on a number line}
Example: Let set V = {-12, -√5, -4π, 0, 2/5, 3, √7, 10}
List the elements from set V that belong to each set:
a) Natural numbers 3, 10
b) Whole numbers 0, 3, 10
c) Integers -12, 0, 3, 10
d) Rational -12, 0, 2/5, 3, 10
e) Irrational -√5, -4π, √7
f) Real -12, -√5, -4π, 0, 2/5, 3, √7, 10
223 Reference Chapter
Section R2: Real Numbers and Their Properties
Properties of Real Numbers
Closure Property of Addition
Closure Property of Multiplication
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
Identity Property of Addition
Identity Property of Multiplication
Inverse Property of Addition
Inverse Property of Multiplication
Distributive Property
a + b is a real number
ab is a real number
a+b=b+a
ab = ba
(a + b) + c = a + (b + c)
(ab)c = a(bc)
a+0=a
a ∙1 = a
a + -a = -a + a = 0
a ∙1/a = 1/a∙a = 1
a(b + c) = ab + ac
223 Reference Chapter
Section R2: Real Numbers and Their Properties
Properties of Real Numbers
Write which property is being illustrated in each statement:
a) 3 + 4 = 4 + 3 __________________________________________
b) 6∙ 1= 6 _______________________________________________
c) 5x + 5y = 5(x + y) _______________________________________
d) (7-y)∙1/(7-y) = 1 ________________________________________
e) (6 + y) + 2 = 6 + (y + 2) __________________________________
f) 7 + ¾ is a real number ___________________________________
g) 15 + -15 = 0 ___________________________________________
223 Reference Chapter
Section R2: Real Numbers and Their Properties
Properties of Real Numbers
Write which property is being illustrated in each statement:
a) 3 + 4 = 4 + 3 commutative property of addition
b) 6∙ 1= 6 identity property of multiplication
c) 5x + 5y = 5(x + y) distributive property
d) (7-y)∙1/(7-y) = 1 inverse property of multiplication
e) (6 + y) + 2 = 6 + (y + 2) associative property of addition
f) 7 + ¾ is a real number closure property of addition
g) 15 + -15 = 0 inverse property of addition
223 Reference Chapter
Section R2: Real Numbers and Their Properties
The Absolute Value of number is the distance on the number line
from that place to 0.
Example: | 5 | = 5
Example: | -8| = 8
Example: |0| = 0
Properties of Absolute Value
1.
|a| ≥ 0
2.
|-a| = |a|
3.
|a|∙|b| = |ab|
4.
|a|/|b| = |a/b| (b ≠ 0)
5.
|a + b| ≤ |a| + |b| (this is the triangle inequality)
223 Reference Chapter
Section R2: Real Numbers and Their Properties
Order of Operations (left to right for each)
Parentheses
Exponents
Multiplication and Division
Addition and Subtraction
Example: Evaluate the following expressions
a) 5(3+1)^2-(9+10/2)
b)
12/3 + (5-2)(4+1)
(9-7)^3 - 7∙2
223 Reference Chapter
Section R2: Real Numbers and Their Properties
Order of Operations (left to right for each)
Parentheses
Exponents
Multiplication and Division
Addition and Subtraction
Example: Evaluate the following expressions
a) 5(3+1)^2-(9+10/2)
5(4)^2-(9+5) = 5*16 – 14 = 80 – 14 = 66
b)
12/3 + (5-2)(4+1)
(9-7)^3 - 7∙2
4 + (3)(5) = 4 + 15 = 19 = -19/6
(2)^3 – 14
8 – 14
-6