Properties of Real Numbers. Closure Property Commutative Property.
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Transcript of Properties of Real Numbers. Closure Property Commutative Property.
Closure Property
If a and b are real numbers, then
We should think of any combination of addition as a single, real number
If a and b are real numbers, then
We should think of any combination of multiplication as a single, real number
Commutative Property
If a and b are real numbers, then
We can change the order of addition without changing the result
If a and b are real numbers, then
We can change the order of multiplication without changing the result
Associative Property
If a, b, and c are real numbers, then
Under addition, we can place parentheses wherever we please, or choose not to use parentheses
If a, b, and c are real numbers, then
Under multiplication, we can place parentheses wherever we please or choose not to use parentheses
Identity Property
There exists a unique number called zero (0) such that, for any number a
There exists a unique number called one (1) such that, for any number a
Identity Property
There exists a unique number called zero (0) such that, for any number a
If we ever end up with zero plus a number, we can drop the zero
There exists a unique number called one (1) such that, for any number a
If we ever end up with one times a number, we can drop the 1
Inverse Property
For every non-zero real number a, there exists the number such that
For every non-zero real number a, there exists the number such that
Inverse Property
For every non-zero real number a, there exists the number such that
The “canceling” property for addition
For every non-zero real number a, there exists the number such that
The “canceling” property for multiplication
Distributive Property
If a, b, and c are real numbers, then
and
The top equation is multiplication. The bottom is factoring.
Examples
Use the:
a) Commutative Property for Addition
b) Commutative Property for Multiplication
c) Associative Property for Addition
d) Associative Property for Multiplication
Definitions of Subtraction and Division
DEFINITION:
For real numbers a and b, we define subtraction to be
For real numbers a and b, with , we define division to be
Examples
Show that each equation is a true statement. Justify each step using the number properties.
a)
b)
c)
d)
e)
Examples
• Justification
• Definition of division
• Distributive Property
• Commutative Property for Multiplication
• Associative Property for Multiplication
• Multiplication
• Commutative Property for Addition
Examples
• Justification
• Definition of subtraction
• Commutative Property for Addition
• Associative Property for Addition
• Inverse Property of Addition
• Identity Property of Addition
Examples
• Justification
• Definition of Division
• Commutative Property for Multiplication
• Associative Property for Multiplication
• Inverse Property for Multiplication
• Identity Property for Multiplication
Examples
• Justification
• Definition of Division
• Distributive Property
• Commutative Property for Multiplication
• Associative Property for Multiplication
• Inverse Property for Multiplication
• Identity Property for Multiplication
• Addition