A horizontal beam separates the frame into an upper deck and a lower deck, and a place notation is used to represent numbers and calculations by sliding the beads into various positions on the two decks. The abacus illustrated is the Chinese or "2/5" abacus (having 2 beads on each rod on the upper deck and 5 beads on each rod on the lower deck); the Japanese abacus is "1/4". Abel, Henrik Neils|(1802 – 1829)|biograph|none|none|

Abel

Norwegian algebraist who proved the unsolvability of the general quintic (fifth degree) equation by radicals. He published this landmark result when he was 22, but had great difficulty in getting contemporary establishment mathematicians (including Gauss and Cauchy) to notice his work. Suffering from tuberculosis, he returned to Norway from Paris, but continued to publish papers in Crelle’s Journal. He died in 1829, just as the mathematical world was beginning to wake up to his achievements. Abelian group| n. |algebr|none|none| A group in which the group operation is commutative. abscissa| n. |calculu|none|none| The first element of an ordered pair. Compare: ordinate. absolute geometry| n. |geometr|none|none| Euclidean geometry without the parallel postulate. absolute value| n. |none|none|none| For a real number x, the absolute value of x, denoted abs(x) or I x I, is defined by I x I = x if x >= 0, and I x I = –x if x <= 0. Equivalently, the absolute value of x is the distance between x and 0 on the real number line. If z is a complex number, then I z I denotes the modulus of z, which is the distance between z and 0 in the Argand plane. absolutely continuous| adj. |analysi|none|none| (Of measures:) Given a measurable space (X, M) and two signed measures m and n on X, then n is said to be absolutely continuous with respect to m, denoted by

if for any E in M such that m(E) = 0 we have n(E) = 0. (Of functions:) If f is a function with domain the set of real numbers and range the set of complex numbers, then f is absolutely continuous if for any positive e there is some d greater than zero such that for any finite set of disjoint intervals we have:

It is a theorem that f is absolutely continuous if and only if its derivative is integrable and the fundamental theorem of calculus holds for f. abundant number| n. |numthry|none|none| A natural number whose proper factors have a sum greater than the number. For example, the proper factors of 12 are 1, 2, 3, 4, and 6, which sum to 16, so 12 is abundant. A number whose proper factors sum to less than the number is called deficient. Compare: perfect number. accumulation point| n. |topolog|none|none| Given a set X, an accumulation point of X is a point p (which may or may not be in X) such that every neighborhood of p contains points of X different from p. actual infinite| n.|none|none|none| An infinite set considered as a completed whole, e.g., the natural numbers in modern set theory. Aristotle distinguished the actually infinite from the potential infinite, and considered that only the latter was permissible to thought. a.e.| adj. |analysi|none|none| See: almost everywhere. algebra| n. |algebr|none|none| The term algebra has broadened enormously in modern mathematics from its original meaning as the abstract study of the laws of arithmetic, in which letters or other symbols are used in place of specific numbers in equations or other arithmetic statements. The term algebra should now be understood to denote any set of objects together with a collection of finitary operations defined on it. Thus, we may have an algebra of sets, an algebra of numbers, an algebra of functions, and so on. See the following entries for many particular uses of the words "algebra" and "algebaic." algebra of functions| n. |analysi|none|none| Given a topological space X, an algebra of functions on X is a subset A of the space of all continuous functions on X such that for all functions f and g we have:

fg is in A,
f + g is in A, and
for all constants c, cf is in A

Archimedes

Greek geometer and applied mathematician who lived at Syracuse, a Greek settlement on the coast of Sicily. He is famous both for his profound mathematical work and for his ingenious mechanical inventions, which at the end of his life he used to defend his city against the Romans during the Second Punic War. When he discovered the hydrostatic principle – that a floating body of a given weight displaces a volume of liquid having an equal weight – he is reputed to have jumped from his bath and run naked through the streets shouting “Eureka!” (I have found it!) Archimedes is credited with many brilliant mathematical discoveries, especially in properties of geometrical figures, areas and volumes of conics, and principles of trigonometry. (He is thought to have known Heron's formula – several centuries before Heron.) He found areas of plane figures by the method of exhaustion, prefiguring modern integral calculus. He was killed by a Roman soldier during the sack of Syracuse. Aristotle|(384 – 322 b.c.)|biograph|none|none|

Aristotle

Greek philosopher and scientist who developed syllogistic logic as a formal scholastic discipline. He defined the syllogism as a “discourse in which certain things having been stated, something else follows of necessity from their being so.” Aristotle’s philosophy of the infinite, in which he held that only the potential infinite, and not any collection which was actually infinite, could be entertained by the reason, held great authority for 2,000 years, until Georg Cantor introduced his theory of transfinite sets in the 19th century. Aristotle wrote widely on every field of learning, interpreted and disagreed with the writings of his teacher Plato, and served as tutor to Alexander the Great. Ascoli-Arzelà Theorem| n. |topolog|none|none| If {X} is a compact Hausdorff space and F is an equicontinuous, pointwise bounded subset of the space C(X) of continuous functions on X, then F is totally bounded in the uniform metric and the closure of F in C(X) is compact. Also, if X is a s-compact, locally compact Hausdorff space and if (f_n} is an equicontinuous, pointwise bounded sequence in C(X), then there exists a subsequence and an f in C(X) such that the subsequence converges to f uniformly on compact sets. associative| adj. |algebr|none|none| An operation * on a set A is associative if for all a, b, and c in A, (a*b)*c = a*(b*c). Cf. commutative, distributive. associative property| n. |algebr|none|none| A property of numbers which states that the operations of addition and multiplication are associative. axiom| n. |foundtns|none|none| In formal logic, a formula or schema of formulas stipulated as true in the system, i.e., not standing in need of proof. Axioms are the counterpart in formal logic of suppositions, assumptions, or premises in ordinary syllogistic logic. axiom of choice| n. |foundtns|none|none| An axiom of set theory, particularly ZFC, which asserts that from any (infinite) family of sets a new set may be created containing exactly one element from each set in the family. This axiom has been shown to be independent of the other axioms of ZFC (ZF). It is equivalent to: 1) Zorn's Lemma; 2) the Hausdorff Maximality Theorem; 3) the well-ordering principle; and 4) the statement that the cartesian product of an infinite family of sets is non-empty. Because the axiom of choice permits non-constructive proofs, it is rejected by intuitionism, and careful mathematicians refer explicitly to its use in proofs which require it. Baire Category Theorem| n. |analysi|none|none| If X is a complete metric space, then the intersection of a countable collection of dense open subsets of X is also dense. Equivalently, the union of a countable collection of closed, nowhere dense subsets of X is nowhere dense. Banach space| n. |analysi|none|none| A complete normed space. Banach-Tarski Paradox| n. |foundtns|none|none| Given two bounded subsets of 3-dimensional Euclidean space, provided each has interior points, then the first may be decomposed into finitely many pieces and translated by rigid motions into a subset congruent to the second subset. For instance, a ball of unit radius can be decomposed into finitely many pieces, and then reassembled into two balls of unit radius. This shocking result is a consequence of the axiom of choice. base| n. |topolog|none|none| Given a topological space X, a base is a class B of sets such that, for every x in X and every neighborhood U of x, there is a set b in B such that x is contained in b and b is contained in U. bijective| adj. |analysi|none|none| A function is bijective if it is both injective and surjective, i.e., both “one-to-one” and “onto.” Bolzano-Weierstrass property| n. |topolog|none|none| A topological space X is said to have the Bolzano-Weierstrass property if every infinite sequence in X has a convergent subsequence. It is a theorem that every compact space has the Bolzano-Weierstrass property. Borel measure| n. |analysi|none|none| A measure m on a metric space X defined on a s-algebra of sets which contains the Borel sets is called a Borel measure. See also: Lebesgue measure. Borel set| n. |analysi|none|none| If X is a topological space, then the s-algebra of sets generated by the open sets of X is called the Borel s-algebra, and its members are called Borel sets. The Borel subsets of the real line are generated by the open intervals (equivalently, by the closed intervals, half-open intervals, and rays). bound| n. |none|none|none| A lower bound of any subset of a linear order (linearly ordered set) is an element which is less than or equal to every element of the subset. The greatest lower bound is the largest of its lower bounds. An upper bound of any subset of a linear order is an element which is greater than or equal to every element of the subset. The least upper bound is the smallest of the upper bounds. See also: infimum, supremum. bounded| adj. |none|none|none| A sequence of values is called bounded if there is a value M such that the values are never bigger than M and never smaller than –M. A function is bounded if its values are bounded. A sequence of functions is pointwise bounded if at every domain element x the sequence of values of the functions at x is bounded. A subset E of a locally compact topological space is bounded if there exists a compact set C such that E is contained in C. Such a subset is called s-bounded if there exists a sequence of compact sets C_i such that E is contained in their union. See also: totally bounded. bounded variation| n. |analysi|none|none| A function on the real numbers with range in the complex numbers is said to be of bounded variation if its total variation is finite. The space of all such functions is denoted by BV. If in addition f is right continuous and its limit at negative infinity is zero, then f is said to be of normalized bounded variation. The space of such functions is denoted by NBV. Bourbaki, Nicolas| (1935 – ) |biograph|none|none|Although the myth of an actual French mathematician named Bourbaki was maintained for many years, it became generally known in 1952 that “Bourbaki” is the nom de plume of a private congress of young French mathematicians which formed in 1935, with the purpose of writing a comprehensive exposition of modern mathematics in many volumes, called Elements. Bourbaki's original members included André Weil, Henri Cartan, Claude Chevalley, and Jean Dieudonné. Other mathematicians have been recruited into Bourbaki in the years since, including G. de Rham, A. Grothendieck, S. Lang, and R. Thom. A principle motivation for the activity of the Bourbaki group is the felt need to unify the entire corpus of mathematical work in the face of accelerating diversification and specialization of mathematics. Cantor, Georg| (1845 – 1918) |biograph|none|none|

Georg Cantor

German mathematician and founder of modern set theory. Cantor's concern with the set-theoretic nature of the real numbers began while he was working on certain properties of trigonometric series during the early 1870’s. Inspired by the work of his friend Dedekind, Cantor proved in 1873 that the rational numbers are countable, and later the same year proved that the real numbers themselves are uncountable. These proofs introduced important methods, including the notion of a one-to-one correspondence and the method of diagonalization. (See the Infinity Minitext for a full treatment of these ideas.) Cantor's results were counterintuitive and broke with established dogma about the nature of infinity. Consequently, he became a controversial figure, opposed by some mathematicians (e.g., Kronecker) who used their influence to stifle Cantor's career as much as possible. This opposition exacerbated Cantor's predisposition to severe depression, and he ultimately died, of a heart attack, in a sanitorium. Cantor set| n. |foundtns|cantset|none| A subset of the unit interval (on the real number line) which is a perfect set, nowhere dense, and uncountable. See the article for a complete exposition. Cantor's Theorem| n. |foundtns|cantor_theorem|none| Given any set, a new set may be constructed which is cardinally larger, i.e., whose size is represented by a larger cardinal than the size of the initial set. More specifically, for no set X is there a bijection of X onto its power set. This theorem establishes both that there are different sizes of infinity, and that there is no greatest such “size.” Cantor's proof was the first use of a diagonalization argument to derive a contradiction. Cantor-Bendixson Theorem| n. |topolog|none|none| Every closed subset of the real numbers is the disjoint union of a perfect set and a countable set. See: derived set. Carathéodory's Theorem| n. |analysi|none|none| See: outer measure. cardinal| n. |foundtns|none|none| A cardinal number represents the size of a set irrespective of the order or structure of its elements. A cardinal is an initial ordinal, i.e., an ordinal for which there does not exist a bijection onto any lesser ordinal. Thus, all finite ordinals are cardinals, but most transfinite ordinals are not cardinals. cardinality| n. |foundtns|none|none| Two sets X and Y have the same cardinality if there exists a bijection between them. Specifically, the cardinality of X may be understood as a canonical object meant to represent the proper class of sets to which X is bijective. If the axiom of choice is assumed, then the cardinality of X is the unique cardinal number having the same cardinality as X Carroll’s Paradox| n. |foundtns|carroll|none| Paradox published by Lewis Carroll as “What the Tortoise Said to Achilles.” See the article for a full exposition. See also: Dodgson, Charles Lutwidge. Cartesian plane| n. |calculu|none|none| The Cartesian product R², represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.

Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane. Cartesian product| n. |analysi|none|none| For any collection {A_i}, i = 1, 2, 3, ..., n, of sets, the Cartesian product

is the set of ordered n-tuples (a₁,a₂, ... ,a_n) with a₁ an element of A₁, a₂ an element of A₂, etc. The Cartesian product R² of the set of real numbers is called the Cartesian plane, and in general n-dimensional real space is the Cartesian product Rⁿ. The assertion that the Cartesian product of an infinite collection of non-empty sets is non-empty is equivalent to the axiom of choice. Cauchy sequence| n. |analysi|none|none| A sequence (x₁, x₂, x₃, ... ) of elements of a metric space X with metric d(x, y) is Cauchy if for any e greater than zero there is some natural number N such that