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:
“We have been told over and over about the importance of bonding to our children. Rarely do we hear about the skill of letting go, or, as one parent said, “that we raise our children to leave us.” Early childhood, as our kids gain skills and eagerly want some distance from us, is a time to build a kind of adult-child balance which permits both of us room.”
—Joan Sheingold Ditzion (20th century)
“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 hast’n’d widowhood with the gold
Of Matrimonial treason: so farewel.”
—John Milton (1608–1674)
“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)