BROWSE
ALPHABETICALLY

LEVEL:
Elementary
Both

INCLUDE TOPICS:
Basic Math
Algebra
Analysis
Biography
Calculus
Comp Sci
Discrete
Economics
Foundations
Geometry
Graph Thry
History
Number Thry
Phys Sci
Statistics
Topology
Trigonometry

polyhedron – Pythagorean theorem

polyhedron   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.

polynomial   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

and

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.

polynomial ring   The ring, denoted by R[x], of all formal polynomials over a given commutative ring R. If we choose to consider multivariate polynomials, we can define the ring R[x1, x2, ... xn] in an analagous manner. A worthwhile observation, which is a corollary of the Substitution Principle, is that R[x1, x2, ... xn] is isomorphic to R[x1][x2] ... [xn].

poset   A partially ordered set, i.e., a set with a partial order defined on it.

positive set   Given a signed measure m on a measure space X, a measurable set A in X is called a positive set if the measure of all measurable subsets of A is greater than or equal to zero.
Cf. negative set, null set.

postulate   A statement in a mathematical theory that is assumed without proof. Essentially synonymous with “axiom.”

potential infinite   A distinction made by Aristotle: a set is potentially infinite if it cannot be finitely completed, e.g., our naive or given conception of the natural numbers. Aristotle admitted potentially infinite sets, but denied the logical possiblity of the actual infinite, that is, infinite totalities considered as completed entities.

Related MiniText: Infinity -- You Can't Get There From Here...

power series   An infinite series of the form

See the related article for a complete description.

Related article: Series

power set   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.

power set axiom   An axiom of set theory which states that, for any given set X, the power set, i.e., the collection of all subsets of X, exists and is a set.

precompact   Given a topological space X, a subset E of X is called precompact if its closure is compact.

predecessor   In a structure with an order relation defined upon it, the predecessor of an element b is the greatest element less than b.
Cf. successor

predicate calculus   A system of symbolic logic which augments the propositional calculus with quantification over variables. The two forms of quantification are existential and universal, and are denoted by

respectively. This permits the construction of sentences such as

which could be read, “There exists an x such that for all y, x times y is equal to y.” (Such a sentence would be true in arithmetic or group theory, for instance.) When quantification is permitted only over variables, the logic is first-order. If quantification is permitted over classes of variables or over predicates, the logic is second-order.

prime number   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.

Related article: Fundamental Theorem of Arithmetic

product   The result of applying a multiplication operation to two or more numbers or quantities.

proper class   A collection of elements that is not a set. For example, the collection of all sets must be thought of as a proper class in order to avoid the Russell paradox.

proper factor   See factor.

propositional calculus   The formal system of symbolic logic in which sentences are treated as objects related by the logical connectives:

The last two symbols are called the conditional and the biconditional respectively, but they are not essential; indeed all the connectives are fully expressible by the use of the two connectives for “or” and “not.” For example, “p implies q” may be equivalently expressed as “not p or q.” In the notation of propositional calculus, this equivalence may be written,

Cf. predicate calculus.

p-series   An infinite series of the form

with p a positive real number. See the related article for details.

Related article: Series

 Pythagoras
Pythagoras
(born 580 bce)   What we know of this ancient Greek mystic is mostly legend, for of the true facts of his life almost nothing is known. However, it is believed that he started a religious cult, the Pythagoreans, among whose beliefs was that number was the highest reality, and that all other aspects of reality could be understood in terms of integers and ratios of integers.

Pythagorean theorem   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.

polyhedron – Pythagorean theorem

 HOME | ABOUT | CONTACT | AD INFO | PRIVACYCopyright © 1997-2013, Math Academy Online™ / Platonic Realms™. Except where otherwise prohibited, material on this site may be printed for personal classroom use without permission by students and instructors for non-profit, educational purposes only. All other reproduction in whole or in part, including electronic reproduction or redistribution, for any purpose, except by express written agreement is strictly prohibited. Please send comments, corrections, and enquiries using our contact page.