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Euclidean geometry – Fibonacci sequence

Euclidean geometry   The theory of geometry arising from the five postulates (axioms) of Euclid:
1. That any two points determine a line;
2. That any line may be extended indefinitely;
3. That a circle may be drawn with any center and radius;
4. That all right angles are equal to one another;
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
Euclidean geometry is characterized by the 5th postulate, also called the “parallel’s postulate,” which may be restated as; given a line and a point not on the line, exactly one line may be drawn through the given point parallel to the given line. Theories of geometry which reject the 5th postulate form the subject of non-Euclidean geometry. Euclid recorded his theory of geometry in his 13-book opus titled Elements, c. 300 b.c., and his theory was considered the only “true” theory of geometry until the 18th century, when non-Euclidean geometry was discovered independently by Nikolai Lobachevsky, Janos Bolyai, and Karl Friedrich Gauss. Euclid’s axioms were found to be deficient in the 19th century, and Euclidean geometry was re-axiomatized by David Hilbert with the addition of a between-ness axiom, and the relegation of the concepts of point and line, which Euclid had defined, to the status of undefined terms.

Euclidean space   A space satisfying the axioms of Euclidean geometry. Specifically, a space in which the parallel postulate holds; equivalently, a space in which the distance between points is given by a straight line.

Euclid number   See perfect number.

Euler number   The transcendental number e, approximately 2.71828.... It may be defined as the limit

or as the infinite sum

The function ex is called the exponential function, and has the property that it is its own derivative.

Euler phi function   Given a natural number n, this function returns the number of integers between 1 and n which are relatively prime to n.

Euler Polyhedron Formula   V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces of a polyhedron or a planar graph.

even function   A real-valued function y = f(x) is even if f(x) = f(–x) for all x in the domain of f. The graph of an even function in the Cartesian plane is symmetric with respect to the y-axis.
Cf. odd function.

existential quantifier

exp

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   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.

 Fibonacci
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   The sequence discovered by the medieval mathematician Fibonacci, and described in his book Liber Abaci. See the article for a complete description.

Euclidean geometry – Fibonacci sequence

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