Absolute Value

 

Motivation:  

                       For two numbers 8 and -9,  we know that  8 is larger than -9 in our usual concept. But consider  the distance between 0 and 8 and the distance between 0 and -9 in the following diagram

 

                

 

            0 and -9 are 9 units away;  0 and 8 are 8 units away.   In this case,  positive sign (+) and minus sign (-) in front of the numerals are just used to indicate the direction.  The positive sign or negative sign could stand for different meanings in different occasions. Sometimes we just want to know the “magnitude” associated with an expression. That is the reason why we define “Absolute value” ..

 

 

Definition ( Absolute Value )

              |s| is used to stand for the “absolute value” of   s :

              if  s > 0,  then  |s| = s  ;

              if  s  0, then  |s| = -s .

 

Example:

                    |5| = 5

                    |-8| = 8

 

Example:

                   |x| = 5 ,  what is the possible value of x ?

                   The answer is:  x could be 5 or -5.

 

Question:   |x-2| = 5,  what is the possible value of x ?

                    The answer is: x could be -3 or 7 .

 

                  What is the actual meaning of the question?  It can be interpreted as:

      In the following diagram, find all the points with the distance 5 units to point 2 on the line.