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 particular source of frustration of women observing their own self-study and measuring their worth as women by the distance they kept from men necessitated that a distance be kept, and so what vindicated them also poured fuel on the furnace of their rage. One delight presumed another dissatisfaction, but their hatefulness confessed to their own lack of power to please. They hated men because they needed husbands, and they loathed the men they chased away for going.”
—Alexander Theroux (b. 1940)
“Why does the past look so enticing to us? For the same reason why from a distance a meadow with flowers looks like a flower bed.”
—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)