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:
“A solitary traveler whom we saw perambulating in the distance loomed like a giant. He appeared to walk slouchingly, as if held up from above by straps under his shoulders, as much as supported by the plain below. Men and boys would have appeared alike at a little distance, there being no object by which to measure them. Indeed, to an inlander, the Cape landscape is a constant mirage.”
—Henry David Thoreau (1817–1862)
“One sees qualities at a distance and defects at close range.”
—Victor Hugo (1802–1885)
“Her personality had an architectonic quality; I think of her when I see some of the great London railway termini, especially St. Pancras, with its soot and turrets, and she overshadowed her own daughters, whom she did not understand—my mother, who liked things to be nice; my dotty aunt. But my mother had not the strength to put even some physical distance between them, let alone keep the old monster at emotional arm’s length.”
—Angela Carter (1940–1992)