For a function f of a single real variable x, the derivative is defined as the limit of the difference quotient
provided this limit exists. In practice, the derivative is interpreted as the instantaneous rate of change of the function at x. Graphically, the derivative returns the slope of the tangent at x.
Cf. differentiation rules.
Given a set X, the derived set of X is the set of accumulation points of X. The second derived set is the derived set of the derived set, and so on.
Cf. Cantor-Bendixson Theorem.
French mathematician who is generally considered to have laid the foundations for modern mathematics. His greatest achievement was the invention of analytic geometry, in which the methods of algebra and those of geometry are used together. He is also a central figure in the history of modern philosophy; his treatises Meditations and Discourse on Method laid the groundwork both for modern rationalism and modern skepticism. Descartes did not actually use the rectangular coordinate system known as the Cartesian plane (this was developed by Leibniz and others), and he permitted only positive values for his variables. Nonetheless, his development of algebraic methods in geometry made possible an explosion of analytic discoveries by his successors, the most important of which was the discovery of the calculus less than a generation after Descartes’ death.
Those statistics used to describe a sample or population.
Cf. inferential statistics.
Geometry: A diameter of a circle (or sphere) is a line containing the center and with endpoints on the perimeter (resp. surface).
Analysis: Given a set X in a metric space, the diameter of X is the supremum of the distances between all pairs of points of X.
Graph Theory: The diameter of a given graph G is the maximum, over all pairs of vertices u, v of G that are in the same connected component of G, of the distance between u and v. In other words, it is the greatest distance between two vertices on the graph.
The difference of two numbers m and n, with n > m, is the number which when added to m yields n. For example, the difference of 3 and 5 is 2.
Set Theory: The difference of two sets A and B, denoted either as AB or as A - B, is the set of elements of A that are not in B.
A function is differentiable at a point of its domain if its derivative exists at that point. A function is said to be (simply) differentiable if its derivative exists at all points of its domain.
A rule permitting easy differentiation of functions having certain forms. See the article for a complete description.
A polynomial equation with integer coefficients. (Named after the 3rd century Greek mathematician Diophantus of Alexandria.)
Cf. Hilbert's Problems (the tenth problem), Fermat's Last Theorem.
A graph whose edges are directed, i.e. have distinguished ends. One end of every directed edge is called the head and the other is called the tail, and the edge is said to be from the tail to the head. In pictorial representations of graphs, directed edges are drawn to end with arrows, pointing to the head. The i, j entry in the adjacency matrix of a directed graph is the number of edges from vertex i to vertex j.
General: So-called “Discrete Mathematics” consists of those branches of mathematics which are concerned with the relations among fixed rather than continuously varying quantities, e.g., combinatorics and probability.
Topology: A topology on a set X is discrete if every subset of X is open, or equivalently if every one-point set of X is open.
Two sets are disjoint if they have empty intersection.
A union of sets which are disjoint.
A set of points consisting of a circle together with its interior points. The set consisting only of the interior points of a circle is called an open disk.
The distance between two points in a space is given by the length of the geodesic joining those two points. In Euclidean space, the geodesic is given by a straight line, and the distance between two points is the length of this line. The distance between two points a and b on a real number line is the absolute value of their difference, i.e., d(a, b) = |a - b|. In two (or more) dimensions, the distance is given by the (generalized) Pythagorean theorem, i.e., in a Cartesian coordinate system of n dimensions, where a = (a1, ... ,an) and b = (b1, ... ,bn), the distance d(a, b) is given by
The concept of distance may be generalized to more abstract spaces – such a distance concept is referred to as a metric.
Graph Theory: The length of the shortest path between two vertices of a graph. If there is no path between two vertices, their distance is defined to be infinite. The distance between two vertices v and u is denoted by d(v, u). In a connected graph, distance is a metric.
See distributive property.
A lattice is called distributive if for all elements x, y, and z of the lattice we have x (y z) = (x y) (x z) and x (y z) = (x y) (x z).
An algebraic property of numbers which states that for all numbers a, b, and c, a(b + c) = ab + ac.
Cf. commutative, associative.
To divide a number a by another number b is to find a third number c such that the product of b and c is a, that is, b × c = a. The number a is called the dividend, the number b is called the divisor, and the number c is called the quotient. The operation of dividing may be denoted by a horizontal or diagonal slash separating the dividend and divisor (with the dividend on top), or by a horizontal dash with a dot above and below it placed between the dividend and divisor.
In the case of whole numbers a and b there may not be a whole number quotient; however, there are always unique whole numbers q and r such that a = b × q + r, with r < b. In this case q is called the quotient and r is called the remainder. If in a particular case r = 0, we say that b divides a, and this is often denoted by b|a.
A number that is being divided.
A number that is dividing another.