Math 137 notes  FALL 2019: September 4  September 20
Ian Payne
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September 4
Absolute Values and Distances
For a real number
x
, what is meant by

x

?
•
x
without the negative.
•
The size of
x
.
•
The distance from
x
to 0.
•
The absolute value of
x
.
•
The modulus of
x
.
All of these are correct. The last two are terminology and the first three are
definitions. Here are a few examples:
• 
2

= 2.
•  
33
.
945

= 33
.
945.
• 
π

=
π
.
• 
3

4

= 1.
Keep a close eye on the last one.
In mathematics, we need formal definitions, lest all of our work fall apart on us
later in life. Here is a more mathie looking definition:
Definition.
For a real number
x
, we define

x

by the piecewise function

x

=
x
if
x
≥
0

x
if
x <
0
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Not to be posted on any website without permission from the author
It may look like we have complicated things unnecessarily, but those feeling will
pass with time. At least, have a careful read of this definition. It says that when
x
is nonnegative, the absolute value function doesn’t do anything, and when
x
is
negative, it negates it (negating a negative makes something positive, remember.)
Here is a proof. It will probably be the easiest proof in the course.
Proposition.
For any real number
x
,

x

=
 
x

.
Sure, you already knew this. Here is a formal proof.
Proof.
FIrst, suppose
x
= 0. Then

x
=
x
, so

x

=
 
x

because all functions
do the same thing to the same input. If
x >
0, then

x

=
x
by the definition of
the absolute value function. As well,

x <
0, so
 
x

=

(

x
) =
x
=

x

by the
definition of the absolute value function. Finally, if
x <
0, then

x >
0. Again
using the definition of the absolute value function gives

x

=

x
and
 
x

=

x
,
so

x

=
 
x

.
Up to this point in your life, you have probably used the absolute value function
as another example of a function you can graph and on which you can perform