Foundations
Properties of Real Numbers
Learning Objectives
By the end of this section, you will be able to:
- Use the commutative and associative properties
- Use the properties of identity, inverse, and zero
- Simplify expressions using the Distributive Property
A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra chapter, Foundations.
Use the Commutative and Associative Properties
The order we add two numbers doesn’t affect the result. If we add or the results are the same—they both equal 17. So, The order in which we add does not matter!
Similarly, when multiplying two numbers, the order does not affect the result. If we multiply or the results are the same—they both equal 72. So, The order in which we multiply does not matter!
These examples illustrate the Commutative Property.
When adding or multiplying, changing the order gives the same result.
The Commutative Property has to do with order. We subtract and , and see that Since changing the order of the subtraction does not give the same result, we know that subtraction is not commutative.
Division is not commutative either. Since changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!
Addition and multiplication are commutative.
Subtraction and division are not commutative.
When adding three numbers, changing the grouping of the numbers gives the same result. For example, since each side of the equation equals 17.
This is true for multiplication, too. For example, since each side of the equation equals 5.
These examples illustrate the Associative Property.
When adding or multiplying, changing the grouping gives the same result.
The Associative Property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.
We saw that subtraction and division were not commutative. They are not associative either.
When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the Commutative Property of addition to write the like terms together.
Simplify:
Simplify:
Simplify:
When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative Property or Associative Property first.
Simplify:
Simplify:
Simplify:
Use the Properties of Identity, Inverse, and Zero
What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity. The Identity Property of Addition that states that for any real number and
What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity. The Identity Property of Multiplication that states that for any real number and
We summarize the Identity Properties here.
What numb