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Beginning Algebra
Tutorial 8:
Properties of Real Numbers

Learning Objectives
After completing this tutorial, you should be able to:
  1. Identify and use the addition and multiplication commutative properties.
  2. Identify and use the addition and multiplication associative properties.
  3. Identify and use the distributive property.
  4. Identify and use the addition and multiplication identity properties.
  5. Identify and use the addition and multiplication inverse properties.

Introduction
It is important to be familiar with the properties in this tutorial.  They lay the foundation that you need to work with equations, functions, and formulas all of which are covered in later tutorials, as well as, your algebra class.   In some cases, it isn't very helpful to rewrite an expression, but in other cases it helps to write an equivalent expression to be able to continue with a  problem and solve it.  An equivalent expression is one that is written differently, but has the same value.  The properties on this page will get you up to speed as to how you can write expressions in equivalent forms.

Tutorial

The Commutative Properties of 
Addition and Multiplication

a + b = b + a       and      ab = ba
The Commutative Property, in general, states that changing the ORDER of  two numbers either being added or multiplied, does NOT change the value of it. 

The two sides are called equivalent expressions because they look different but have the same value.

Example 1:  Use the commutative property to write an equivalent expression to 2.5x + 3y.

Using the commutative property of addition (where changing the order of a sum does not change the value of it) we get 

2.5x + 3y = 3y + 2.5x.

Example 2:  Use the commutative property to write an equivalent expression to 

Using the communicative property of multiplication (where changing the order of a product does not change the value of it), we get

The Associative Properties of 
Addition and Multiplication

a + (b + c) = (a + b) + c     and    a(bc) = (ab)c

The Associative property, in general, states that changing the GROUPING of numbers that are either being added or multiplied does NOT change the value of it.  Again, the two sides are equivalent to each other.

At this point it is good to remind you that both the commutative and associative properties do NOT work for subtraction or division.

Example 3:   Use the associative property to write an equivalent expression to   (a + 5b) + 2c.

Using the associative property of addition (where changing the grouping of a sum does not change the value of it) we get 

(a + 5b) + 2c = a + (5b + 2c).