Dr. Byrne
Fall 2010
Math 112
Worksheet R.1: Properties of the Real Numbers
for Addition and Multiplication
Property
Addition
Multiplication
commutativity
a+b=b+a
ab=ba
associativity
a+(b+c)=(a+b)+c
a(bc)=(ab)c
identity
a+0=a
a×1=a
existence of the
inverse
a-a=0
a÷a=1
distributivity
a(b+c)=ab+ac
The 2 properties of commutativity and associativity apply for addition and multiplication,
but not subtraction and division.
Show that commutativity doesn’t work for division:
Show that associativity doesn’t work for subtraction:
The fifth property, distributivity, is about how multiplication distributes over addition. How
does addition distribute over multiplication?
Dr. Byrne
Fall 2010
Math 112
Worksheet R.1: Describing an Interval
Example: Tetia needs to buy a microwave that is less than 22.8 inches wide but at least 10
inches. What is the range of microwave sizes Tetia is interested in buying?
There are three ways of precisely describing this range:
(1) Set Builder Notation:
(2) Interval Notation:
(3) Interval Diagram (graph):
Communicating Whether An Interval is Closed or Open
Example



‘closed’ means the endpoint is
included
‘open’ means the endpoint is not
included
an interval can be half open and
half closed
Interval Graph
[3,7] means 3≤x≤7
(3,7) means 3<x<7
[3,7) means 3≤x<7
Exercise (Fill in.)
verbal description
set builder notation
interval notation
1,4
the interval from -1 to 4,
endpoints included
0.0,3.5
x | 1  x  2
any value less than 3.5
any value greater than or
equal to 10
the entire real line
 ,3.5
interval diagram (graph)