Natural number

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

Species

In biology, a species is the basic unit of classification and a taxonomic rank of an organism, as well as a unit of biodiversity. A species is defined as the largest group of organisms in which any two individuals of the appropriate sexes or mating types can produce fertile offspring by sexual reproduction. Other ways of defining species include their karyotype, DNA sequence, behaviour or ecological niche. In addition, paleontologists use the concept of the chronospecies since fossil reproduction cannot be examined. While these definitions may seem adequate, when looked at more they represent problematic species concepts. For example, the boundaries between related species become unclear with hybridisation, in a species complex of hundreds of similar microspecies, in a ring species. Among organisms that reproduce only asexually, the concept of a reproductive species breaks down, each clone is a microspecies. All species are given a two-part name, a "binomial"; the first part of a binomial is the genus.

The second part is called the specific epithet. For example, Boa constrictor is one of four species of the genus Boa. None of these is satisfactory definitions, but scientists and conservationists need a species definition which allows them to work, regardless of the theoretical difficulties. If species were fixed and distinct from one another, there would be no problem, but evolutionary processes cause species to change continually, to grade into one another. Species were seen from the time of Aristotle until the 18th century as fixed kinds that could be arranged in a hierarchy, the great chain of being. In the 19th century, biologists grasped. Charles Darwin's 1859 book The Origin of Species explained how species could arise by natural selection; that understanding was extended in the 20th century through genetics and population ecology. Genetic variability arises from mutations and recombination, while organisms themselves are mobile, leading to geographical isolation and genetic drift with varying selection pressures.

Genes can sometimes be exchanged between species by horizontal gene transfer. Viruses are a special case, driven by a balance of mutation and selection, can be treated as quasispecies. Biologists and taxonomists have made many attempts to define species, beginning from morphology and moving towards genetics. Early taxonomists such as Linnaeus had no option but to describe what they saw: this was formalised as the typological or morphological species concept. Ernst Mayr emphasised reproductive isolation, but this, like other species concepts, is hard or impossible to test. Biologists have tried to refine Mayr's definition with the recognition and cohesion concepts, among others. Many of the concepts are quite similar or overlap, so they are not easy to count: the biologist R. L. Mayden recorded about 24 concepts, the philosopher of science John Wilkins counted 26. Wilkins further grouped the species concepts into seven basic kinds of concepts: agamospecies for asexual organisms biospecies for reproductively isolated sexual organisms ecospecies based on ecological niches evolutionary species based on lineage genetic species based on gene pool morphospecies based on form or phenotype and taxonomic species, a species as determined by a taxonomist.

A typological species is a group of organisms in which individuals conform to certain fixed properties, so that pre-literate people recognise the same taxon as do modern taxonomists. The clusters of variations or phenotypes within specimens would differentiate the species; this method was used as a "classical" method of determining species, such as with Linnaeus early in evolutionary theory. However, different phenotypes are not different species. Species named in this manner are called morphospecies. In the 1970s, Robert R. Sokal, Theodore J. Crovello and Peter Sneath proposed a variation on this, a phenetic species, defined as a set of organisms with a similar phenotype to each other, but a different phenotype from other sets of organisms, it differs from the morphological species concept in including a numerical measure of distance or similarity to cluster entities based on multivariate comparisons of a reasonably large number of phenotypic traits. A mate-recognition species is a group of sexually reproducing organisms that recognize one another as potential mates.

Expanding on this to allow for post-mating isolation, a cohesion species is the most inclusive population of individuals having the potential for phenotypic cohesion through intrinsic cohesion mechanisms. A further development of the recognition concept is provided by the biosemiotic concept of species. In microbiology, genes can move even between distantly related bacteria extending to the whole bacterial domain; as a rule of thumb, microbiologists have assumed that kinds of Bacteria or Archaea with 16S ribosomal RNA gene sequences more similar than 97% to each other need to be checked by DNA-DNA hybridisation to decide if they belong to the same species or not. This concept was narrowed in 2006 to a similarity of 98.7%. DNA-DNA hybri

Divisor

In mathematics, a divisor of an integer n called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one says that n is a multiple of m. An integer n is divisible by another integer m. If m and n are nonzero integers, more nonzero elements of an integral domain, it is said that m divides n, m is a divisor of n, or n is a multiple of m, this is written as m ∣ n, if there exists an integer k, or an element k of the integral domain, such that m k = n; this definition is sometimes extended to include zero. This does not add much to the theory, as 0 does not divide any other number, every number divides 0. On the other hand, excluding zero from the definition simplifies many statements. In ring theory, an element a is called a "zero divisor" only if it is nonzero and ab = 0 for a nonzero element b. Thus, there are no zero divisors among the integers. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors.

For example, there are six divisors of 4. 1 and −1 divide every integer. Every integer is a divisor of itself. Integers divisible by 2 are called and integers not divisible by 2 are called odd. 1, −1, n and −n are known as the trivial divisors of n. A divisor of n, not a trivial divisor is known as a non-trivial divisor. A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits. 7 is a divisor of 42 because 7 × 6 = 42, so we can say 7 ∣ 42. It can be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42; the non-trivial divisors of 6 are 2, −2, 3, −3. The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42; the set of all positive divisors of 60, A = ordered by divisibility, has the Hasse diagram: There are some elementary rules: If a ∣ b and b ∣ c a ∣ c, i.e. divisibility is a transitive relation.

If a ∣ b and b ∣ a a = b or a = − b. If a ∣ b and a ∣ c a ∣ holds, as does a ∣. However, if a ∣ b and c ∣ b ∣ b does not always hold. If a ∣ b c, gcd = 1 a ∣ c; this is called Euclid's lemma. If p is a prime number and p ∣ a b p ∣ a or p ∣ b. A positive divisor of n, different from n is called a proper divisor or an aliquot part of n. A number that does not evenly leaves a remainder is called an aliquant part of n. An integer n > 1 {\displ

Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example, are subsets of A. Sets can themselves be elements. For example, consider the set B =; the elements of B are not 1, 2, 3, 4. Rather, there are only three elements of B, namely the numbers 1 and 2, the set; the elements of a set can be anything. For example, C =, is the set whose elements are the colors red and blue; the relation "is an element of" called set membership, is denoted by the symbol " ∈ ". Writing x ∈ A means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A"; the expressions "A includes x" and "A contains x" are used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos urged that "contains" be used for membership only and "includes" for the subset relation only.

For the relation ∈, the converse relation ∈T may be written A ∋ x, meaning "A contains x". The negation of set membership is denoted by the symbol "∉". Writing x ∉ A means that "x is not an element of A"; the symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X: Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b. So a ∈ b is read; every relation R: U → V is subject to two involutions: complementation R → R ¯ and conversion RT: V → U. The relation ∈ has for its domain a universal set U, has the power set P for its codomain or range; the complementary relation ∈ ¯ = ∉ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ A, the complement of A in U; the converse relation ∈ T = ∋ swaps the domain and range with ∈. For any A in P, A ∋ x is true when x ∈ A; the number of elements in a particular set is a property known as cardinality. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3.

An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers. Using the sets defined above, namely A =, B = and C =: 2 ∈ A ∈ B 3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite and equal to 5; the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not axiomatized, not that it is silly or easy. Jech, Thomas, "Set Theory", Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY: Dover Publications, Inc. ISBN 0-486-61630-4 - Both the notion of set, membership or element-hood, the axiom of extension, the axiom of separation, the union axiom are needed for a more thorough understanding of "set element". Weisstein, Eric W. "Element". MathWorld

Mathematics

Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.

The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.

The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to