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:
“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 (1817–1862)
“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)
“1946: I go to graduate school at Tulane in order to get distance from a “possessive” mother. I see a lot of a red-haired girl named Maude-Ellen. My mother asks one day: “Does Maude-Ellen have warts? Every girl I’ve known named Maude-Ellen has had warts.” Right: Maude-Ellen had warts.”
—Bill Bouke (20th century)