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 (18031882)
“Like the water, the Walden ice, seen near at hand, has a green tint, but at a distance is beautifully blue, and you can easily tell it from the white ice of the river, or the merely greenish ice of some ponds, a quarter of a mile off.”
—Henry David Thoreau (18171862)
“Let me approach at least, and touch thy hand.
[Samson:] Not for thy life, lest fierce remembrance wake
My sudden rage to tear thee joint by joint.
At distance I forgive thee, go with that;
Bewail thy falsehood, and the pious works
It hath brought forth to make thee memorable
Among illustrious women, faithful wives:
Cherish thy hastnd widowhood with the gold
Of Matrimonial treason: so farewel.”
—John Milton (16081674)