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:
“Midway the lake we took on board two manly-looking middle-aged men.... I talked with one of them, telling him that I had come all this distance partly to see where the white pine, the Eastern stuff of which our houses are built, grew, but that on this and a previous excursion into another part of Maine I had found it a scarce tree; and I asked him where I must look for it. With a smile, he answered that he could hardly tell me.”
—Henry David Thoreau (1817–1862)
“Remember? We sat on a slab of rock.
From this distance in time,
it seems the color
of iris, rotting and turning purpler,
but it was only
the usual gray rock”
—Robert Lowell (1917–1977)
“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)