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exponent – finite measure
exponent
Also called index, a number or expression applying to another number or expression, called the base, and indicating that the base is to be “raised to the power of” the exponent. For integers, this corresponds to the operation of repeated self-multiplication the number of times specified by the exponent.
The exponent is written to the right of, and superscripted to, the base.
 Cf. rational exponent, laws of exponents.
exponential function
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.
extended real numbers
The set of real numbers together with a “point(s) at infinity.” If both a positive and a negative infinity are added, then the resulting space is order equivalent and topologically equivalent (i.e., homeomorphic) to the unit interval [0, 1]. If only a single point at infinity is added, the resulting space is topologically equivalent to the unit circle.
factor
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.
factorial
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.
factorization
The process of factoring, that is, of finding proper factors.
fallacy
ARTICLE
An unjustified step in logic, or incorrect form of reasoning, leading to an invalid conclusion. See the article for a complete description.
Fermat’s Last Theorem
The Diophantine equation x n + y n = z n has no non-trivial solutions in integers x, y, and z for any n greater than 2. (There are infinitely many solutions in integers for n = 2, and these are called the Pythagorean triples.) Fermat penned this theorem in the margin of his copy of the Arithmetica of Diophantus of Alexandria, and added, “I have discovered a remarkable proof of this theorem, which unfortunately this margin is too small to contain.” He died without ever writing down the proof, and the theorem remained unproved for 300 years, until Andrew Wiles presented a proof at a Cambridge lecture in 1992.
Fermat’s Little Theorem
For any integer n and prime number p that does not divide n, n p - 1 is congruent to 1 modulo p.
Cf. Euler phi function.
Fermat test
A test for primality based on the converse of Fermat's little theorem:
the Fermat test for n base b is that the congruence bp-1  1 modulo n should hold. For prime n, this is always provided that
b and n are coprime; however there
are also some composite numbers which will pass the test. For example, 341 = 11 × 31 is composite, yet 2340 1 modulo 341, so we call 2 a Fermat pseudoprime to the base 2.
Even worse, Carmichael numbers are pseudoprimes for the Fermat test and any base.
Fortunately, pseudoprimes and Carmichael numbers are rare compared to primes.
Such tests are thus probabilistic, in that numbers passing the test are likely but not certain to be prime.
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Fibonacci
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Fibonacci
(born 1170)
Medieval Italian mathematician, whose real name was Leonardo of Pisa. (“Fibonacci” is a nickname meaning “son of good nature.”) Although best known for the number-sequence which bears his name, his most important contribution to mathematics was introducing the Arabic numerals to Europe through his book Liber Abaci (pub. 1202), a treatise on algebraic methods of arithmetic, and the application of these methods in business. Fibonacci was a businessman and the son of a businessman, educated in North Africa, and it was on his extensive travels in the near east and Africa that he became familiar with the notation which made practical commerce so much easier, since it was a place notation and hence made algorithmic arithmetic calculations possible. It was in the Liber Abaci that the problem which leads to his famous sequence was posed and solved:How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on? The solution gives rise to the sequence beginning 1, 1, ..., and in which each succeeding term is given by adding the previous two.
Cf. Fibonacci sequence.
Fibonacci sequence
ARTICLE
The sequence discovered by the medieval mathematician Fibonacci, and described in his book Liber Abaci. See the article for a complete description.
field
A set together with two binary operations (called addition and multiplication) defined on its elements, and satisfying - the set is an Abelian group under the addition operation,
- the set is a commutative ring with addition as the group operation and multiplication as the ring operation, and
- every element except the additive identity has a multiplicative inverse (i.e., the set is a division ring).
In other words, a field is a commutative division ring. If in addition the field is linearly ordered, it is called an ordered field. An ordered field is complete if every subset of the field with an upper bound has a least upper bound. The real numbers are fully characterized by the fact that they form a complete ordered field, that is, every complete ordered field is isomorphic to the real numbers.
field automorphism
A field isomorphism from a field to itself. That is, a bijective function from a field to itself that preserves addition and multiplication, i.e. f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b).
Cf. field isomorphism.
field homomorphism
A function from one field to another that preserves addition and multiplication, i.e. f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b). All field homomorphisms are actually isomorphisms of the domain with some subfield of the range.
Cf. field isomorphism, field automorphism.
field isomorphism
A field homomorphism that is both “one-to-one” and “onto,” that is, a bijective function from one field to another that preserves addition and multiplication, i.e. f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b).
Cf. field automorphism.
figure
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.
filter
If X is a set (or class) and F is a family of subsets of X, then F is called a filter provided- F is closed under intersections, i.e., for any sets A, B in F their intersection is also in F, and
- F is closed under supersets (upwardly closed), i.e., if A is any set in F and B is any set in X containing A, then B is in F.
If X and F are as above, and if in addition for every subset Y of X either Y or its complement is in F, then F is called an ultrafilter (or maximal filter). Equivalently, an ultrafilter is a filter that is not the proper subset of any filter. A free ultrafilter is an ultrafilter which contains the complement of every finite set. If A is any subset of X, then the collection of supersets of A together with all finite intersections of supersets of A is called the filter generated by A. More generally, if U is any family of subsets of X that is closed unter finite intersections, then the family F of subsets which includes precisely U and all of the intersections of supersets of members of U is the filter generated by U, and U is called its filter base. If a filter F is generated by a singleton set, then it is called a principal filter.
finite graph
A graph whose vertex and edge sets are finite. In all papers, theorems, conjectures, textbooks, discussions, and interactions of graph theory all graphs considered are always finite, unless explicitly specified otherwise.
Cf. infinite graph.
finite intersection property
A collection of sets has the finite intersection property if every finite subcollection has non-empty intersection.
finite measure
See measure.
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