Teachable Moments
Prepared by: Sa’diyya Hendrickson
Understanding Absolute Value
I. The Geometric Definition
Recall that “|a|” is used to represent ”the distance of a from zero.” Suppose, for example, that
a = 3. Then, “|3|” means ”the distance of 3 from zero,” which we know is 3 units! i.e. |3| = 3
Similarly, |
3| is the ”distance of
3 from zero.” Again, that distance is 3 units. i.e. |
3| = 3
We know that the numbers 3 and 3 are opposites. Is it a coincidence that their absolute values
are the same? Can we make a generalization that for any a 2 R, the absolute value of the opposites
a and a will always be the same? Take a moment to convince yourself that the answer is yes!
II. Think Cases!
Let a 2 R. Determine |a| under each of the following cases:
C1
a>0
(i.e. a is positive)
If a is positive, then its distance from zero is just itself! Symbolically: |a| = a.
C2
a=0
If a = 0, then its distance from zero is zero units! Symbolically: |a| = |0| = 0.
C3
a<0
(i.e. a is negative)
If a < 0, then its distance from zero is its opposite, since its opposite is positive!
Question: Can you think of a simple way to create the opposite of a?
Answer: Multiply a by 1, which reflects it over zero! Symbolically, |a| = a.
III. The Algebraic Definition
By convention, we combine the cases C1 and C2 in the algebraic definition.
8
>
< a
if a 0
(Case 1)
|a| =
>
: a
if a < 0
(Case 2)
Making Math Possible
1 of 2
c Sa’diyya Hendrickson
Teachable Moments
IV. Exercise
Use the algebraic definition of absolute value to simplify the following expression:
|e
Making Math Possible
|e
p
⇡| + ⇡
p
2| + 2
2 of 2
c Sa’diyya Hendrickson