absolute value

The absolute value of a number strips away its sign and reports only its size, its distance from zero on the number line. Whether you walk three steps left or three steps right, you have walked a distance of three.

Formally, the absolute value of x, written |x|, equals x when x is at least 0 and equals minus x when x is negative; it is always nonnegative. The quantity |x minus y| is then the distance between x and y, and the condition |x minus a| less than delta describes exactly the open interval of radius delta centered at a.

Absolute value obeys a few key rules used constantly in analysis: it is multiplicative, so |xy| equals |x| times |y|; it is zero only when x is zero; and it satisfies the triangle inequality, |x plus y| at most |x| plus |y|. These make absolute value the prototype of a norm and the source of the standard distance on the real line.

Because epsilon–delta arguments are fundamentally statements about distances, almost every definition of limit, continuity, and convergence is written using absolute value. Mastering its manipulations is a prerequisite for everything that follows.