1. Properties of Real Numbers
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
Additive Identity Property
Multiplicative Identity Property
Additive Inverse Property
Multiplicative Inverse Property
Distributive Property
Zero Property

2. Subtraction is NOT Commutative Division is NOT Commutative
Commutative Property of Addition
When two numbers are added, the order can be switched and the sum will still be the same.
Subtraction is NOT Commutative
Commutative Property of Multiplication
When two numbers are multiplied, the order can be switched and the product will still be the same.
Division is NOT Commutative

3. Subtraction is NOT Associative Division is NOT Associative
Associative Property of Addition
When three or more numbers are added, any two or more can be grouped together and the sum will still be the same.
Subtraction is NOT Associative
Associative Property of Multiplication
When three or more numbers are multiplied, any two or more can be grouped together and the product will still be the same.
Division is NOT Associative

4. Additive Identity Property
When zero is added to any number, the sum is the original number.
Zero is the Identity Element of Addition
Multiplicative Identity Property
When any number is multiplied by one, the product is the original number.
One is the Identity Element of Multiplication

5. Additive Inverse Property
When the opposite of a number is added to it the sum is zero.
Zero is the Identity Element of Addition
Multiplicative Inverse Property
When any number is multiplied by its reciprocal the product is one.
One is the Identity Element of Multiplication

6. When any number is multiplied by zero the product is zero.
Distributive Property
Any number outside parenthesis can be distributed to the numbers inside the parenthesis.
Zero Property
When any number is multiplied by zero the product is zero.

7. Non-Zero Rational Numbers
Binary Operations
In a Binary Operation, two elements from a set are replaced by exactly one element from the same set.
Property of Closure
A set is Closed under a binary operation when every pair of elements from the set, under the given operation, yields an element from that set.
The following sets of numbers are closed under the indicated operation.
Division
Addition
Subtraction
Multiplication
Natural Numbers
Natural Numbers
Non-Zero Rational Numbers
Integers
Whole Numbers
Rational Numbers
Whole Numbers
Integers
Real Numbers
Integers
Non-Zero Real Numbers
Rational Numbers
Rational Numbers
Real Numbers
Real Numbers