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Intermediate Algebra
Tutorial 5: Properties of Real Numbers
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deskLearning Objectives

 
After completing this tutorial, you should be able to:
  1. Identify and use the addition and multiplication identity properties.
  2. Identify and use the addition and multiplication inverse properties.
  3. Identify and use the addition and multiplication commutative properties.
  4. Identify and use the addition and multiplication associative properties.
  5. Identify and use the distributive property.
  6. Know the key words that translate into an equal sign.
  7. Know the symbol for 'not equal to'.
  8. Know the symbol for and the meaning of  'less than', 'greater than', 'less than or equal to', and 'greater than or equal to'.
  9. Write mathematical expressions that have an equal, less than, greater than, less than or equal to, or greater than or equal to sign. 




desk 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. 
 

We will start with the properties for real numbers and then look at writing out equalities and inequalities in mathematical statements.

 

 

desk Tutorial

 
 
 
Identity Properties

 
 
Addition

The additive identity is 0

a + 0 = 0 + a = a


 
In other words, when you add 0 to any number, you end up with that number as a result.

 
 
Multiplication

Multiplication identity is 1

a(1) = 1(a) = a


 
And when you multiply any number by 1, you wind up with that number as your answer.

 
 
 
 
The Inverse Properties

 
Additive Inverse (or negative)

For each real number a, there is a unique real number, denoted -a,
such that 

a + (-a) = 0.


 
In other words, when you add a number to its additive inverse, the result is 0. Other terms that are synonymous with additive inverse are negative and opposite.

 
 
Multiplicative Inverse
(or reciprocal)

For each real number a, except 0, there is a unique real number reicprocal such that

reciprocal 2


 
 
In other words, when you multiply a number by its multiplicative inverse the result is 1.  A more common term used to indicate a  multiplicative inverse is the reciprocal.  A multiplicative inverse or reciprocal of a real number a (except 0) is found by "flipping" a upside down.  The numerator of a becomes the denominator of the reciprocal of a and the denominator of a becomes the numerator of the reciprocal of a.
 

These two inverses will come in big time handy when you go to solve equations later on.  Keep them in your memory bank until that time.
 

notebook Example 1:   Write the opposite (or additive inverse) of -3.

 
The opposite of -3 is 3, since -3 + 3 = 0.

 
 
 
notebook Example 2:  Write the opposite (or additive inverse) of 1/5.

 
The opposite of 1/5 is -1/5, since 1/5 + (-1/5) = 0.

 
 
 
notebook Example 3:   Write the reciprocal (or multiplicative inverse) of -3.

 
The reciprocal of -3 is -1/3, since -3(-1/3) = 1.

When you take the reciprocal, the sign of the original number stays intact.  Remember that you need a number that when you multiply times the given number you get 1.  If you change the sign when you take the reciprocal, you would get a -1, instead of 1, and that is a no no.

 
 
notebook Example 4:   Write the reciprocal (or multiplicative inverse) of 1/5.

 
The reciprocal of 1/5 is 5, since 5(1/5) = 1.

 
 
 
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.

 
 
notebook Example 5:  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.


 
 
notebook Example 6:  Use the commutative property to write an equivalent expression to example 6a.

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

example 6b


 
 
 
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.
 

notebook Example 7:   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).


 
 
notebook Example 8:   Use the associative property to write an equivalent expression  to  (1.5x)y.

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

(1.5x)y = 1.5(xy)


 

Distributive Properties

a(b + c) = ab + ac 
or 
(b + c)a = ba + ca


 
In other words, when you have a term being multiplied times two or more terms that are being added (or subtracted) in a ( ), multiply the outside term times EVERY term on the inside.  Remember terms are separated by + and -.  This idea can be extended to more than two terms in the ( ).
 

notebook Example 9:  Use the distributive property to find the product  - (5x + 7).

 
example 9

*A - outside a ( ) is the same as times (-1)
*Distribute the (-1) to EVERY term inside ( )
*Multiply

 
Basically, when you have a negative sign in front of a ( ), like this example, you can think of it as taking a -1 times the ( ).  What you end up doing in the end is taking the opposite of every term in the ( ).

 
 
notebook Example 10:  Use the distributive property to find the product  3(2a + 3b + 4c).

 
As mentioned above, you can extend the distributive property to as many terms as are inside the ( ).  The basic idea is that you multiply the outside term times EVERY term on the inside.

 
example 10

*Distribute the 3 to EVERY term
*Multiply

 
notebook Example 11:   Simplify (hint use the distributive property): 2(6x - 5) - 3(5x + 4).

 
Let's first apply the distributive property and see what we get:

 
example 11
*Dist. 2 to EVERY term of 1st ( )
*Dist. -3 to EVERY term of 2nd ( )
*Multiply

 
You can use any of these properties forwards or backwards, and that includes the distributive property.  When it says to simplify, that means we want to write it in equivalent form that is more compact - get it down to as few terms as possible.

Of course, we can combine the -10 and -12.  But with the help of the distributive property in reverse, we can also combine 12x and -15x.

Let's check it out:

 
example 11b

*x is distributed to the 1st 2 terms
*Reverse Dist. Prop with x
*Subtract

 
 
Now the above properties will all eventually help you when you are solving equations and inequalities.  Since that is the case, this is a good time to introduce the concept of equality and inequality and look at them in mathematical statements.

 
 
 
Equality 
=

 
It makes sense that an equality involves an equal sign. 

Here are some key words that translate into an  =  when writing out mathematical statements:

Equals, is, represents, is the same as, 
gives, yields, amounts to, is equal to.

FYI, when you put an = between two mathematical expressions, you have yourself an equation.

 
 
Not Equal 
not equal

 
The above symbol is used when you want to say that two expressions are not equal to each other.

 
 
 
Inequality

Read left to right:

a < b     a is less than b

a < b    a is less than or equal to b

a > b      a is greater than b

a > ba is greater than or equal to b 
 

 
If a is less than b, that means a lies to the left of b on the real number line.

If a is greater than b, that means a lies to the right of b on the real number line.

 
 
notebook Example 12:    Write the statement using mathematical symbols.
The product of 5 and x is the same as 15.

 
As covered in tutorial 2, product translates into multiplication. 

What will we use for the same as?  If you said =, you are correct!!

Let's put everything together going left to right:

The product of 5 and x is the same as 15
example 12


 
 
 
notebook Example 13:    Write the statement using mathematical symbols.
The sum of 3 and y is less than 12.

 
Do you remember what sum translates into?   If you said add, you are doing great.

Is less than will need to be replaced by the symbol <.

Let's put everything together going left to right:

The sum of 3 and y is less than 12.
example 13


 
 
notebook Example 14:   Write the statement using mathematical symbols.
Twice the difference of 4 and a is less than or equal to the reciprocal of 5.

 
Is less than or equal to will need to be replaced by the symbol <.

The reciprocal of 5 is 1/5.

Let's put everything together going left to right:

Twice the difference of 4 and a is less than or equal to the reciprocal of 5.
example 14


 
 
notebook Example 15:   Write the statement using mathematical symbols.
The quotient of x and 2 is greater than the opposite of 1.

 
Do you remember what quotient translates into?  I believe that it is division, don't you agree?

Is greater than will need to be replaced by the symbol >.

What is the opposite of 1?  Why, it is -1.

Let's put everything together going left to right:

The quotient of x and 2 is greater than the opposite of 1.
example 15


 
 
notebook Example 16:   Write the statement using mathematical symbols.
3 times the sum of 2 and x is greater than or equal to 10.

 
Times will translate as multiplication and sum as addition.

Is greater than or equal to will need to be replaced by the symbol >
 

Let's put everything together going left to right:

3 times the sum of 2 and x is greater than or equal to 10.
example 16


 
 
notebook Example 17:   Write the statement using mathematical symbols.
The difference of x and 5 is not equal to 10.

 
Difference is translated into subtraction.

Is not equal to will need to be replaced by the symbol not equal

Let's put everything together going left to right:

The difference of x and 5 is not equal to 10.
example 17


 
 
 
notebook Example 18:   Write the following as an algebraic expression:
The cost of x pizzas, if each pizza costs $8.55.

 
If each pizza costs 8.55, then, in order to find the cost, we would have to multiply the number of pizzas (x) by the cost per pizza (8.55). 

Hence, we would get the algebraic expression 8.55x.

 
 
 
notebook Example 19:   Insert <, > , or = to form a true statement. -5 ?  0.

 
Since -5 is to the left of 0 on the number line, then -5 is less than 0:

-5 < 0


 
 
 
notebook Example 20:   Insert <, > , or = to form a true statement.  -3.5 ?  - 4.5.

 
Since -3.5 is to the right  of - 4.5 on the number line, then -3.5 is greater than - 4.5:

-3.5 > - 4.5


 
 
 
notebook Example 21:   Insert <, > , or = to form a true statement.  10/2 ? 15/3.

 
Since both 10/2 and 15/3 simplify to be 5, then 10/2 equals 15/3:

10/2 = 15/3


 

 
desk Practice Problems
  
These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1b: Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of each number.

 

 

pencil Practice Problems 2a - 2b: Use a commutative property to write an equivalent expression.

 

 

pencil Practice Problems 3a - 3b: Use an associative property to write an equivalent expression.

 
3a. (a + b) + 1.5
(answer/discussion to 3a)
 

 

pencil Practice Problems 4a - 4b: Use the distributive property to find the product.

 

4b.  7(5a + 4b + 3c)
(answer/discussion to 4b)

 

pencil Practice Problem 5a: Simplify the expression.

 

5a.   2(x + 3) - 3x + 4
(answer/discussion to 5a)

 

pencil Practice Problems 6a - 6d: Write each statement using mathematical symbols.

 

6a.  The difference of x and 5 is greater than or equal to 7.
(answer/discussion to 6a)

 
6b.  Twice the sum of y and 3 is the opposite of 10.
(answer/discussion to 6b)

 
6c.  The product of -3 and x is not equal to the reciprocal of 9.
(answer/discussion to 6c)

 
6d.  The quotient of x and 11 is less than - 4.
(answer/discussion to 6d)

 

pencil Practice Problem 7a: Write the following as an algebraic expression.

 

7a.  Two angles are complimentary if the sum of their measures is 90 degrees.  If the measure of one angle is x degrees, represent the measure of the other angle as an expression of x. 
(answer/discussion to 7a)

pencil Practice Problems 8a - 8c: Insert <, > or = to form a true statement.

 

8a.   ½   ?  .5
(answer/discussion to 8a)
8b.  -3.5   ?   1.5
(answer/discussion to 8b)

 
8c.  4.3    ?   4.21
(answer/discussion to 8c)


desk Need Extra Help on these Topics?


 
The following are webpages that can assist you in the topics that were covered on this page:
 

http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#commutativeproperty
This website helps with the commutative property.

http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#associativeproperty
This website helps with the associative property.

http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#distributiveproperty
This website helps with the distributive property.

http://home.earthlink.net/~djbach/basic.html#anchor904011
This website goes over the commutative, associative, and distributive properties.

 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

 


Last revised on June 11, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.