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:
“[As we say], “When you get to be 18 you’re out of here.” No wonder teenagers start to distance themselves from us.”
—Jerry Tello (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)
“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)