distributive property exponent
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.
A polyhedron having twelve faces.
The faces of a regular dodecahedron are regular pentagons.
Cf. Platonic solid.
General: A universe of discourse, that is, the class of objects under consideration. Functions and relations: The domain of a function (relation) is the set of elements which the function (relation) maps to its range set.
See scalar product.
See Euler number.
General: A line formed by the intersection of two planes. In a 3-dimensional figure (such as a polyhedron), the line or curve where two faces or surfaces meet.
Graph Theory: One of two kinds of entities in a graph. Restricted to being incident on exactly two vertices.
The set of edges of some graph. For a graph G, the edge set of G is denoted by E(G), or, if there is no ambiguity as to the graph in question, simply by E.
The locus of points in the plane, the sum of whose distances from two fixed points, called the foci, remains constant.
Like the hyperbola and parabola, the ellipse is a conic section.
The unique set having no elements, generally denoted by a circle with a forward slash through it, or by an empty pair of braces.
A morphism f from X to Y is called an epimorphism when it is surjective, that is, when to each element y of Y there corresponds an x in X such that f(x) = y.
A triangle with equal sides and equal angles.
The theory of geometry arising from the five postulates (axioms) of Euclid:
Euclidean geometry is characterized by the 5th postulate, also called the “parallel’s postulate,” which may be restated as; given a line and a point not on the line, exactly one line may be drawn through the given point parallel to the given line. Theories of geometry which reject the 5th postulate form the subject of non-Euclidean geometry. Euclid recorded his theory of geometry in his 13-book opus titled Elements, c. 300 b.c., and his theory was considered the only “true” theory of geometry until the 18th century, when non-Euclidean geometry was discovered independently by Nikolai Lobachevsky, Janos Bolyai, and Karl Friedrich Gauss. Euclid’s axioms were found to be deficient in the 19th century, and Euclidean geometry was re-axiomatized by David Hilbert with the addition of a between-ness axiom, and the relegation of the concepts of point and line, which Euclid had defined, to the status of undefined terms.
- That any two points determine a line;
- That any line may be extended indefinitely;
- That a circle may be drawn with any center and radius;
- That all right angles are equal to one another;
- That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
A space satisfying the axioms of Euclidean geometry. Specifically, a space in which the parallel postulate holds; equivalently, a space in which the distance between points is given by a straight line.
The transcendental number e, approximately 2.71828.... It may be defined as the limit
or as the infinite sum
The function ex is called the exponential function, and has the property that it is its own derivative.
Euler Polyhedron Formula
V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces of a polyhedron or a planar graph.
A real-valued function y = f(x) is even if f(x) = f(–x) for all x in the domain of f. The graph of an even function in the Cartesian plane is symmetric with respect to the y-axis.
Cf. odd function.
See exponential function.
Also called index, a number or expression applying to another number or expression, called the base, and indicating that the base is to be “raised to the power of” the exponent. For integers, this corresponds to the operation of repeated self-multiplication the number of times specified by the exponent.
The exponent is written to the right of, and superscripted to, the base.
Cf. rational exponent, laws of exponents.