polyhedron regular polygon
A closed, solid figure consisting of four or more planar faces, pairs of which meet at edges. The faces form polygons, and the points where three or more edges meet are called vertices. The number of faces, edges, and vertices of a polyhedron are related by the Euler polyhedron formula.
A polyhedron with four faces is called a tetrahedron, a pentahedron has five faces, a hexadron six, a heptahedron seven, an octahedron eight, a dodecahedron twelve, and an icosahedron twenty. If all the faces of a polyhedron are congruent, it is called a regular solid or Platonic solid.
General: A function of the form
p(x) = a0 + a1x + a2x2 + ... + anxn
where the ai (called coefficients) are real numbers or complex numbers, and the exponents are all natural numbers. The highest exponent is called the degree of the polynomial, and the coefficient an on the highest degree term is called the leading coefficient. Polynomials of degree two are called quadratic polynomials, of degree 3 cubic, of degree 4 quartic, and those of degree 5 are called quintic.
More generally, a polynomial may be in several variables x1, ... ,xk, and may be thought of as a sum of the form
where all but finitely many of the ai are 0. In this case, the degree of the polynomial is the highest sum of exponents appearing in any term. For example, 2x4 and 3xy2z are both fourth-degree terms.
Abstract Algebra (formal polynomial): A formal sum of the form
The ai are called coefficients, and are elements of some commutative ring R. p is said to be a polynomial over R, or with coefficients in R. The x is just a formal symbol. Only finitely many of the ai can be non-zero, and if an is the last non-zero coefficient, n is called the degree of p.
Polynomials can be added and multiplied in the natural way, by using the cummutative and distributive laws of addition and multiplication in R. Explicitly, if p and q are polynomials over the same ring, and the ai and bi are the coefficients of p and q, respectively, then
These operations allow us to define R[x], the polynomial ring over R.
The Substitution Principle allows us evaluate a polynomial p on any element of R, and we can use this to define a function corresponding to p, thereby capturing the informal notion of a polynomial.
Cf. Diophantine equation, quadratic formula.
A statement in a mathematical theory that is assumed without proof. Essentially synonymous with “axiom.”
Given a set X, the power set of X, denoted P(X), is the collection of all subsets of X. If X is a finite set with n elements, then P(X) is a finite set with 2n elements.
Cf. power set axiom.
In a structure with an order relation defined upon it, the predecessor of an element b is the greatest element less than b.
Any natural number greater than 1 that is evenly divisible only by itself and 1. There are infinitely many prime numbers. The number of primes less than a given number n is denoted p(n), and approaches the value n/lnn for sufficiently large n.
The result of applying a multiplication operation to two or more numbers or quantities.
In Euclidean geometry, the sum of the areas of the squares on the legs of any right triangle is equal to the area of the square on the hypotenuse. This is arguably the most important theorem of classical mathematics, and perhaps of all time.
An ordered triple (a,b,c) of natural numbers satisfying a2 + b2 = c2. The triples (3,4,5) and (5,12,13) are the first of infinitely many examples.
Given a quadratic function, i.e., a polynomial function of second degree y = ax 2 + bx + c, the zeros of the function are given by
The expression under the radical is called the determinant. If the determinant is positive, both solutions are real; if negative, both solutions are complex; and if zero, there is a single solution of multiplicity two.
A closed, plane figure with four straight sides.
The number that results from dividing one number by another.
Cf. division algorithm.
A dimensionless unit of measure of angles. An angle of one radian is given by the central angle of a circle subtending an arc of length equal to the radius of the circle. Consequently, 360 degrees is the same as 2p radians. See the related article for a more extensive exposition.
The set of elements to which a function maps the elements of its domain set.
An exponent of the form p/q, with p and q integers and q not zero. Evaluated as the qth root of the base, raised to the pth power, or equivalently, as the qth root of the pth power of the base. For a negative base, this operation is not defined except when q is odd. Irrational roots may be considered as limits of sequences of rational roots.
Cf. laws of exponents.
An element of the set Q consisting of ordered pairs (p, q) of integers, with q not 0, and with the order relation (p, q) < (r, s) if and only if ps < rq as integers. (The ordered pairs are usually written p/q, i.e., as a fraction (ratio) with integer numerator and denominator.) The rational numbers are countably infinite.
Cf. natural number, real number.
An element of the set R consisting of all of the rational numbers together with all of the irrational numbers. Sometimes called the continuum. Usually defined formally by a Dedekind cut of rational numbers. The real numbers form (uniquely) a complete ordered field, but are not algebraically complete.
It is a famous theorem of Georg Cantor that the real numbers are not countable.
Cf. complex number.
real number line
A geometrical line graphically representing the set of real numbers, in which every real number corresponds to a unique point on the line, and every point on the line corresponds to a unique real number.
A relation “ ~ ” on a set X is reflexive if for every element x in X we have x ~ x. The relation “ ~ ” is called irreflexive if for every x we have that x ~ x is false. Note that a relation may be neither reflexive nor irreflexive.
Cf. symmetric relation, transitive relation.
A polygon all of whose sides are equal in length and all of whose interior angles are equal.