 Definition 1: Geometric Definition of Absolute Value
The absolute value of a number a, denoted
a
a
, is the distance from
a
a
to zero on the number line.
The absolute value of any nonzero real number will always be positive since it represents the distance to the origin.
10
=
10
and
10
=
10

10
=
10
and
10
=
10
Both 10 and 10 are ten units from zero on the number line. It is also of interest to note that:
0
=
0
0
=
0
.
Absolute values within an expression have the same level of precedence as parenthesis. Apply the innermost absolute values first as we would with parenthesis in respect to the order of operations.
Simplify:
5−

7

5

7
5−

7

=
5−7
=
2
=
2
5

7
=
5
7
=

2
=
2
 Definition 2: Algebraic Definition of Absolute Value
The absolute value of a number
a
a
is
a
=
a
if
a ≥ 0

a
if
a < 0
a
=
a
if
a
0

a
if
a
0
.
If the variable
a
a
is positive
a ≥ 0
a
0
then the absolute value will be that number. If the variable
a
a
is negative
a < 0
a
0
then the absolute value will the the opposite of that number.
10
=


10
=
10
and
10
=
10
10
=


10
=
10
and
10
=
10
Since 10 is less than zero the absolute value will be the opposite of that number, (10) = 10.