Distance
See also: Metric spaceThe absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.
The standard Euclidean distance between two points
and
in Euclidean n-space is defined as:
This can be seen to be a generalization of | a − b |, since if a and b are real, then by equation (1),
While if
and
are complex numbers, then
The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclidean spaces respectively.
The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows:
A real valued function d on a set X × X is called a metric (or a distance function) on X, if it satisfies the following four axioms:
-
Non-negativity Identity of indiscernibles Symmetry Triangle inequality
Read more about this topic: Absolute Value
Famous quotes containing the word distance:
“The rage for road building is beneficent for America, where vast distance is so main a consideration in our domestic politics and trade, inasmuch as the great political promise of the invention is to hold the Union staunch, whose days already seem numbered by the mere inconvenience of transporting representatives, judges and officers across such tedious distances of land and water.”
—Ralph Waldo Emerson (1803–1882)
“Her personality had an architectonic quality; I think of her when I see some of the great London railway termini, especially St. Pancras, with its soot and turrets, and she overshadowed her own daughters, whom she did not understand—my mother, who liked things to be nice; my dotty aunt. But my mother had not the strength to put even some physical distance between them, let alone keep the old monster at emotional arm’s length.”
—Angela Carter (1940–1992)
“You say I’m small? I certainly can relate, although it is a matter of perspective. The distance is deceptive, my friend, you stand too low.”
—Franz Grillparzer (1791–1872)