"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:]
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.)
The absolute value is greater than or equal to zero.
The absolute value of a product is the product of the absolute values.
The square of the absolute value is the square of the number.
The absolute value of a sum is less than or equal to the sum of the absolute values.
The absolute value of a difference is greater than or equal to the aboslute value of the difference of absolute values.
Converted [ in part ] by Mathematica
December 4, 2007