REAL NUMBERS, R

7. ABSOLUTE VALUE

The absolute value of a real number x, denoted by ÷ x÷ is defined by the formula

|x|.  = x if x > 0

         -x if x < 0

that is, if x is positive or zero then |x| equals x, and if x is negative then

|x| equals – x. Consequently, the absolute value of any number is always nonnegative, i.e. |x| > 0 for every x € R.

Geometrically speaking, the absolute value of x is the distance between the point x on the real line and the origin, i.e. the point 0. Moreover, the distance between any two points, i.e. real numbers, a and b is |a - b| = |b - a|.

Example 2.1: |-2| = 2,  |7| = 7. |-p| = p

Example 2.2: The statement |x| < 5 can be interpreted to mean that the distance between x and the origin is less than 5, i.e. x must lies between -5 and 5 on the real line. In other words,

           |x| < 5 and -5 < x < 5

have identical meaning. Similarly,

            |x| < 5 and -5 < x < 5

have identical meaning.