BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    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 number – perpendicular number   There is no precise mathematical definition of the word “number.” There are however precise definitions of the terms “natural number,” “rational number,” “real number,” “complex number,” and other less commonly used kinds of number. When a mathematician speaks about numbers she usually has one of these cases in mind and she should, at the outset, make it clear to which type of number she is referring. The naive, inborn concept of number that is shared to some degree by all humans is a matter for philosophical rather than strictly mathematical inquiry, and it may be noted that there has historically been strong opposition to the introduction of new generalizations of established concepts of number. numeral   Graphical symbol representing a number. obtuse   An angle is called obtuse if it is greater than a right angle, that is, if its measure is greater than 90° (p/2 radians). A triangle is called obtuse if one of its angles is obtuse.Cf. acute. octahedron   A polyhedron having eight faces.The faces of a regular octahedron are congruent, equilateral triangles.Cf. Platonic solid, polyhedron. open disk   The interior of a circle.Cf. neighborhood, disk. open interval   An interval of the real number line (or any other totally ordered set) which does not include its endpoints. An interval containing only one of its endpoints is called half-open.Cf. closed interval. ordered pair   An ordered tuple (a,b), the first element of which is called the abscissa, and the second element the ordinate, and for which (a,b) = (b,a) if and only if a = b. Functions, graphs of functions, and binary relations are represented as sets of ordered pairs. In standard set theory, the ordered pair (a,b) is defined to be the set { {a}, {a,b} }.Cf. flat pair. order of operations   As a matter of convention, in any given expression involving arithmetic and/or algebraic operations, operations within parentheses (or other grouping symbols) are evaluated first. Within this constraint, exponentiation precedes multiplication and division, and the latter precede addition and subtraction. Within these constraints, operations are evaluated from left to right. order relation   A relation R on a set S is an order relation exactly if it is reflexive, transitive and antisymmetric. Order relations are usually denoted by “ < ” or “ ”.Cf. partial order, total order ordinal   The class of ordinals is defined by:0 is an ordinal;if a is an ordinal, then a + 1 = a union {a}) is an ordinal;if A is a collection of ordinals, then union(A) is an ordinal;nothing else is an ordinal.The class of ordinals is transitive, and is a well-founded, linear ordering. An ordinal of the form a + 1 is called a successor ordinal, and is otherwise called a limit ordinal.Cf. Von Neumann Heirarchy. ordinate   The second element of an ordered pair.Cf. abscissa. parabola   The locus of points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are equal.Like the ellipse and hyperbola, the parabola is a conic section. See the related article for a full exposition. Related article: Conics paradox   A seemingly 'necessary' contradiction or absurdity. A paradox arising logically out of formal axioms is called an antinomy. True paradoxes may be broadly classified as paradoxes of logic, paradoxes of infinity, paradoxes of knowledge, paradoxes of language, and paradoxes of self-reference. See:Antistrephon ParadoxBanach-Tarski ParadoxBerry ParadoxBoundary ParadoxFinitude ParadoxFirst Boring Number ParadoxGrelling's ParadoxGrue-Bleen ParadoxLiar ParadoxQuine's ParadoxPrisoner's DilemmaRussell ParadoxSanta Sentence ParadoxSid's ParadoxSorites ParadoxUnexpected Hanging ParadoxZeno's Paradox of the ArrowZenos Paradox of the Moving RowsZeno's Paradox of the Tortoise and Achilles parallelogram   A quadrilateral with opposite sides (and opposite angles) equal. parallel postulate   The 5th postulate of Euclidean geometry. partially ordered set   A set with a partial order defined on it. partition   General: A partition of a set X is a collection of subsets of X such that every element of X is in exactly one of the subsets. Such a partition is given by (and gives rise to) an equivalence relation on X. For example, division modulo 3 partitions the set of natural numbers into three subsets, each containing all those numbers leaving remainders of 0, 1, or 2 respectively when divided by 3.Algebra: A partition of a matrix is a division of the matrix into conformable submatrices.Analysis: A partition of a space is a collection of pairwise disjoint regions of the space whose union is the entire space. For example, a partition of an interval [a, b] of the real line is given by a finite set of points {xi} such that a = x1 < x2 < . . . < xn = b which divide the interval into disjoint subintervals.Number Theory: Given a positive integer n, a partition of n is a set of positive integers whose sum is n. For example, 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 are the four possible partitions of the number 4. Peano axioms   The axiom system developed by Giuseppe Peano to formalize arithmetic. The key to his method is the introduction of a “successor operation” S on numbers; if a is a number, then Sa is the successor of that number. In this way Peano reduced arithmetic to the conceptually primitive operation of counting.The axioms are:0 is a number.If a is a number, then Sa is a number.If a and b are numbers, then a + 0 = a, and a + Sb = S(a + b).If a and b are numbers, then a × 0 = 0, and a × Sb = a × b + a.(Induction Principle.) For any formula f, if we have f(0) and if f(a) always implies f(Sa), then we have f(a) for all numbers a. perfect number   A natural number n whose distinct divisors, including 1 but not including n itself, sum to n.Example: 6 is perfect since its divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. The first three perfect numbers are 6, 28, and 496. Any number of the form 2n – 1(2n – 1) is perfect provided that 2n – 1 is a Mersenne prime. Such numbers are called Euclid numbers. Euler proved that all even perfect numbers are Euclid numbers. It is not known whether there are infinitely many perfect numbers, or if there are any odd perfect numbers. permutation   A permutation on a finite set is a reordering of its elements. On a set of n elements there are n! (n-factorial) permutations, and together they form a group, the symmetric group on n elements.The number of permutations of a subselection of r elements from a set of n elements is usually denoted nPr, and is given byA cyclic permutation, also called a cycle, of an ordered collection of n objects is obtained by replacing each kth object with the (k - 1)th object, and replacing the first object with the nth object. In other words, each object advances one place in the order, and the last object “wraps around” to the first position. The number of objects n is called the degree (sometimes length) of the cycle. A cycle of length 2 is called a transposition. Every permutation can be factored as a product of transpositions, and is called an even (or odd) permutation if it factors into an even (resp. odd) number of transpositions.The subgroup of the symmetric group on n elements consisting of all the even permutations is called the alternating group on n elements.Cf. combination. perpendicular   Two lines are perpendicular if they intersect in a right angle. In particular, two lines in the Cartesian plane are perpendicular if their respective slopes m1 and m2 satisfy m 1 × m2 = –1.Two curves intersect perpendicularly if their tangent lines at the point of intersection are perpendicular.Perpendicularity of planes and surfaces is defined analogously. Cf. normal. number – perpendicular
 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.