Absolute Values

"The definition of the absolute value of a real number is most conveniently introduced at this point:  if  a  [is a real number], then the absolute value of  a,  denoted by  |a|, is [defined as follows:]

[Graphics:../Images/index_gr_196.gif]
[Graphics:../Images/index_gr_197.gif]

The absolute value has the following properties:"  (see ita-2-2-17-pg22)

(Rosenlicht describes the first three properties below as trivial consequences of the definition, but I will review the concepts in detail.)

AV 1.   The absolute value is greater than or equal to zero.

AV 2.   The absolute value of a product is the product of the absolute values.

AV 3.   The square of the absolute value is the square of the number.

AV 4.   The absolute value of a sum is less than or equal to the sum of the absolute values.

AV 5.   The absolute value of a difference is greater than or equal to the aboslute value of the difference of absolute values.

On Substitution

Generalization of AV 4

ε Neighborhoods

ita


Converted [ in part ] by Mathematica      December 4, 2007