exponential function hypotenuse
The function ex. We say that an expression is expontiated when it is used as an exponent on e.
More generally, any function which operates on its argument by placing it as an exponent may be called an exponential function. Example: f(x) = 2x.
Cf. Euler number.
The factors of a natural number n are those whole numbers which divide it evenly. Since every number is divisible by itself and 1, neither 1 nor n is considered a proper factor of n.
More generally, if an expression can be written as a product of other expressions, then those other expressions are its factors. For instance, a polynomial with rational coefficients may always be factored into a product of first and/or second degree polynomial factors.
A number (or other expression) with no proper factors is called prime.
Graph Theory: A spanning subgraph of a given graph having at least one edge. In many contexts, it is interesting to determine whether some graph G can be decomposed into the edge-disjoint union of factors with some prescribed property. Such a decomposition is called a factorization of G. Often, the property in question is regularity of degree k. In this case, the factors are called k-factors, and the factorization a k-factorization. If G has a k-factorization, it is called k-factorable.
An operation on natural numbers, denoted n! and given by n! = 1 × 2 × 3 × ... × n. Thus 3! = 1 × 2 × 3 = 6, and 4! = 24. By convention, 0! = 1 and 1! = 1.
The process of factoring, that is, of finding proper factors.
An unjustified step in logic, or incorrect form of reasoning, leading to an invalid conclusion. See the article for a complete description.
General: A drawing, picture, or illustration, usually accompanying a text description. Also synonymous with digit or numeral.
Geometry: A graphical (visual) representation of points, lines, curves, surfaces, solids, or regions. The word “figure” may also be used to refer to the abstract object or set of points thus represented.
Cf. plane figure.
A set X is finite if there is a natural number n (possibly 0) such that X can be said to have exactly n elements. More formally, a set is finite if it is not bijective with any proper subset of itself.
Given a binary relation R on sets A and B, we say that the R is a function if each element of A is paired with at most one element of B. In this case we call A the domain set, and we call B the range set. If f is such a function and the ordered pair (x,y) is an element of the function, we typically write f(x) = y.
More generally, if R is an n-place relation, then R is a function of n-1 variables if each (n-1)-tuple is matched with at most one element of the range set (i.e., the nth set of the Cartesian product on which the relation is defined). If in addition there is at most one element of the domain set matched with any given element of the range set, then the function is called “one-to-one” or injective. If the function maps at least one element of the domain set to every element of its range set, then it is called “onto” or surjective. A function which is both injective and surjective is called bijective. Functions with range in the real numbers or complex numbers are called real-valued or complex-valued respectively.
Fundamental Theorem of Algebra
The field of complex numbers is
algebraically closed, that is, any polynomial with complex coefficients has a complex root.
Fundamental Theorem of Arithmetic
Every natural number is either prime or may be decomposed uniquely into prime factors.
Fundamental Theorem of Calculus
Intitively, that the integral and derivative are inverse operatorations on functions. The fundamental theorem is commonly expressed in either one of two formal guises:
Cf. Riemann integral.
Abbreviation for greatest common divisor.
An infinite series such that the ratio of consecutive terms is constant. Such a series may be denoted
where r is the common ratio. A geometric series converges if and only if the common ratio is less than 1, in which case the sum of the series is given by
Cf. arithmetic series, harmonic series.
Abbreviation for greatest lower bound.
General: A picture or diagram representing one or more relationships among certain objects or quantities.
Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples.
Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph.
Cf. Konigsberg Bridge Problem.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
greatest lower bound
A lower bound which is greater than or equal to every other lower bound.
Cf. Least upper bound.
Letters of the Greek alphabet are commonly used in mathematics. See the article for a full description.
The infinite series whose terms are the reciprocals of the natural numbers
This series diverges, i.e., the sum does not exist.
The locus of points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Like the ellipse and parabola, the hyperbola is a conic section.
On a right triangle, the side opposite the right angle.