n. 1. (Euclidean geometry) a measure of the extent of a line segment that is zero only when its endpoints coincide and is otherwise positive, that is unaffected by changing the orientation of the line. In a Cartesian coordinate system, the length of the straight line segment AB, where A is the point (a₁, a₂) and B is (b₁, b₂), is as is made clear by the figure part a, in which the sides of the right-angled triangle are parallel to the axes; in an n-dimensional space, this generalizes to Length (a) sense 1. (b) sense 2 2. arc-length. a measure of an arc of a curve, equal to the length of the straight line segment obtained by deforming it into that shape without dilatation; if the line is the graph of the function y = f(x) between a and b, and f'(x) is continuous, then its length is the integral As is clear from part b of the figure above, the length of an element of the curve approximates to the limit of the sum of such elements as their number tends to infinity and each dx tends to zero is thus the length of the curve. For example, the arc length of a semi-circle of unit radius is 3. a non-negative number associated with a vector and indicating its magnitude without regard to direction, written | v |, or sometimes v. If v has orthogonal components (x, 0, 0), (0, y, 0) and (0, 0, z) in the directions of the respective coordinate axes, then Two vectors and with resultant zero have the same length. Compare measure, norm, metric, distance. 4. the norm in a normed space, X, of functions defined by
|| f ||_X = sup_x∈X || f(x)||
for a function f into a compact normed space. 5. (of a polynomial) the sum of the absolute values of the coefficients. Compare height. 6. (of a cycle) the number of elements moved by the given permutation. A transposition has length two.