In mathematics, the Fibonacci numbers denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 2, for n > 1. One has F2 = 1. In some books, in old ones, F0, the "0" is omitted, the Fibonacci sequence starts with F1 = F2 = 1; the beginning of the sequence is thus: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … Fibonacci numbers are related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are named after Italian mathematician Leonardo of Pisa known as Fibonacci, they appear to have first arisen as early as 200 BC in work by Pingala on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.
Fibonacci numbers appear unexpectedly in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, graphs called Fibonacci cubes used for interconnecting parallel and distributed systems, they appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts. Fibonacci numbers are closely related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. Lucas numbers are intimately connected with the golden ratio; the Fibonacci sequence appears in Indian mathematics in connection with Sanskrit prosody, as pointed out by Parmanand Singh in 1985. In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long syllables of 2 units duration, juxtaposed with short syllables of 1 unit duration.
Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm + 1. Knowledge of the Fibonacci sequence was expressed as early as Pingala. Singh cites Pingala's cryptic formula misrau cha and scholars who interpret it in context as saying that the number of patterns for m beats is obtained by adding one to the Fm cases and one to the Fm−1 cases. Bharata Muni expresses knowledge of the sequence in the Natya Shastra. However, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala: Variations of two earlier meters... For example, for four, variations of meters of two three being mixed, five happens.... In this way, the process should be followed in all mātrā-vṛttas. Hemachandra is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before the last is the number... of the next mātrā-vṛtta."
Outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. Using it to calculate the growth of rabbit populations. Fibonacci considers the growth of a hypothetical, idealized rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field. Fibonacci posed the puzzle: how many pairs will there be in one year? At the end of the first month, they mate. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair, making 5 pairs. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month; this is the nth Fibonacci number. The name "Fibonacci sequence" was first used by the 19th
In mathematics, more in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit, within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Banach spaces grew out of the study of function spaces by Hilbert, Fréchet, Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are Banach spaces. A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers, equipped with a norm ‖ ⋅ ‖ X and, complete with respect to the distance function induced by the norm, to say, for every Cauchy sequence in X, there exists an element x in X such that lim n → ∞ x n = x, or equivalently: lim n → ∞ ‖ x n − x ‖ X = 0.
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space X is a Banach space if and only if each convergent series in X converges, ∑ n = 1 ∞ ‖ v n ‖ X < ∞ implies that ∑ n = 1 ∞ v n converges in X. Completeness of a normed space is preserved if the given norm is replaced by an equivalent one. All norms on a finite-dimensional vector space are equivalent; every finite-dimensional normed space over R or C is a Banach space. If X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B. In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space X to another normed space is continuous if and only if it is bounded on the closed unit ball of X. Thus, the vector space B can be given the operator norm ‖ T ‖ = sup. For Y a Banach space, the space B is a Banach space with respect to this norm. If X is a Banach space, the space B = B forms a unital Banach algebra.
If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T: X → Y such that T and its inverse T −1 are continuous. If one of the two spaces X or Y is complete so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, i.e. ||T|| = ||x|| for every x in X. The Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ; every normed space X can be isometrically embedded in a Banach space. More for every normed space X, there exist a Banach space Y and a mapping T: X → Y such that T is an isometric mapping and T is dense in Y. If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Z Z is isometrically isomorphic to Y; this Banach space Y is the completion of the normed space X. The underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y.
The completion of X is denoted by X ^. The cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ‖ ‖ 1 = ‖ x ‖ + ‖ y ‖, ‖ ‖ ∞ = max and give rise to isomorphic normed spaces. In this sense, the product X × Y is only if the two factors are complete. If M is a closed linear subspace of a normed space X, there is a natural norm on the quotient space X / M, ‖ x + M ‖ = inf m ∈ M ‖ x + m ‖; the quotient X / M is a Banach space
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. It is an irrational number, a solution to the quadratic equation x 2 − x − 1 = 0, with a value of: φ = 1 + 5 2 = 1.6180339887 …. The golden ratio is called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, golden number. Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio; the golden ratio has been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.
The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ. One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a = a a + b a = 1 + b a = 1 + 1 φ. Therefore, 1 + 1 φ = φ. Multiplying by φ gives φ + 1 = φ 2 which can be rearranged to φ 2 − φ − 1 = 0. Using the quadratic formula, two solutions are obtained: 1 + 5 2 = 1.618 033 988 7 … and 1 − 5 2 = − 0.618 033 988 7 … Because φ is the ratio between positive quantities, φ is positive: φ = 1 + 5 2 = 1.61803 39887 … The golden ratio has been claimed to have held a special fascination for at least 2,400 years, although without reliable evidence.
According to Mario Livio: Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, musicians, architects and mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction, surprising Pythagoreans. Euclid's Elements provides several propositions and their proofs employing the golden ratio and contains the first known definition: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
The golden ratio was studied peripherally over the next millennium. Abu Kamil employed it in his geometric calculati
In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope though these entities were quantitatively small; the word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which referred to the "infinite-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object, smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.
The concept of infinitesimals was introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the method of exhaustion; the 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors; the method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus.
He exploited an infinitesimal denoted 1/∞ in area calculations. The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds for the infinite numbers and vice versa; the 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, in defining an early form of a Dirac delta function; as Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both Émile Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955.
The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it is not popular to talk about infinitesimal quantities. Present-day students are not in command of this language, it is still necessary to have command of it. The notion of infinitely small quantities was discussed by the Eleatic School; the Greek mathematician Archimedes, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1... and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains infinitesimal members; the English mathematician John Wallis introduced the expression 1/∞ in his 1655 book Treatise on the Conic Sections.
The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal. In his Treatise on the Conic Sections Wallis discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol ∞; the concept suggests a thought experiment of adding an infinite number of parallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in integral calculus; the conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher Zeno of Elea, whose Zeno's dichotomy paradox was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval. Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.
Prior to the invention of calculus mathematicians were able to calculate tangent lines using Pierre de
Normed vector space
In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of distance in the real world. A norm is a real-valued function defined on the vector space that has the following properties: The zero vector, 0, has zero length. ‖ x ‖ ≥ 0, ‖ x ‖ = 0 if and only if x = 0 Multiplying a vector by a positive number changes its length without changing its direction. Moreover, ‖ α x ‖ = | α | ‖ x ‖ for any scalar α; the triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line. ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ for any vectors x and y. The generalization of these three properties to more abstract vector spaces leads to the notion of norm. A vector space on which a norm is defined is called a normed space or normed vector space.
Normed vector spaces are central to the study of linear algebra and functional analysis. A normed vector space is a pair where V is a vector space and ‖ ⋅ ‖ a norm on V. A seminormed vector space is a pair where V is a vector space and p a seminorm on V. We omit p or ‖ ⋅ ‖ and just write V for a space if it is clear from the context what norm we are using. In a more general sense, a vector norm can be taken to be any real-valued function that satisfies the three properties above. A useful variation of the triangle inequality is ‖ x − y ‖ ≥ | ‖ x ‖ − ‖ y ‖ | for any vectors x and y; this shows that a vector norm is a continuous function. Note that property 2 depends on a choice of norm | α | on the field of scalars; when the scalar field is R, this is taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over Q one could take | α | to be the p-adic norm, which gives rise to a different class of normed vector spaces. If is a normed vector space, the norm ‖·‖ induces a metric and therefore a topology on V.
This metric is defined in the natural way: the distance between two vectors u and v is given by ‖u−v‖. This topology is the weakest topology which makes ‖·‖ continuous and, compatible with the linear structure of V in the following sense: The vector addition +: V × V → V is jointly continuous with respect to this topology; this follows directly from the triangle inequality. The scalar multiplication ·: K × V → V, where K is the underlying scalar field of V, is jointly continuous; this follows from the triangle homogeneity of the norm. For any semi-normed vector space we can define the distance between two vectors u and v as ‖u−v‖; this turns the seminormed space into a pseudometric space and allows the definition of notions such as continuity and convergence. To put it more abstractly every semi-normed vector space is a topological vector space and thus carries a topological structure, induced by the semi-norm. Of special interest are complete normed spaces called Banach spaces; every normed vector space V sits as a dense subspace inside a Banach space.
All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology. And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space V is locally compact if and only if the unit ball B = is compact, the case if and only if V is finite-dimensional; the topology of a seminormed vector space has many nice properties. Given a neighbourhood system N around 0 we can construct all other neighbourhood systems
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format