Intermediate Algebra
Tutorial 5: Properties of Real Numbers

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

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. 
Tutorial
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. 
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 such
that

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.
Example
1: Write the opposite (or additive inverse) of 3. 
The opposite of 3 is 3, since 3 + 3 = 0. 
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. 
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. 
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. 
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.

Example
6: 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
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).

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

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

*Distribute the 3 to EVERY term
*Multiply

Example
11: Simplify (hint use the distributive property):
2(6 x  5)  3(5 x + 4). 
Let's first apply the distributive property and see what we get: 

*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: 

*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. 
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

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. 
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
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
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
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
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
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
Let's put everything together going left to right:
The difference of x and 5 is not equal to 10.

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. 
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

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

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

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. 
Practice
Problems 1a  1b: Write the opposite (additive inverse) and the reciprocal
(multiplicative inverse) of each number.
Practice
Problems 2a  2b: Use a commutative property to write an equivalent expression.
Practice
Problems 3a  3b: Use an associative property to write an equivalent
expression.
Practice
Problems 4a  4b: Use the distributive property to find the product.
Practice
Problem 5a: Simplify the expression.
Practice
Problems 6a  6d: Write each statement using mathematical symbols.
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) 
Practice
Problems 8a  8c: Insert <, > or = to form a true statement.
Need Extra Help on these Topics?
Last revised on June 11, 2011 by Kim Seward.
All contents copyright (C) 2002  2011, WTAMU and Kim Seward. All rights reserved.
