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# Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). 

132 relations: Abstraction, Addition, Additive inverse, Ancient Greece, Archimedes, Arithmetic, Associative property, Axiom of infinity, Axiomatic system, Babylonia, Bijection, Blackboard bold, Brahmagupta, Cancellation property, Cardinal number, Cardinal number (linguistics), Cardinality, Charles Sanders Peirce, China, Closure (mathematics), Combinatorics, Commutative property, Complex number, Computer scientist, Computus, Constructivism (mathematics), Countable set, Counting, Dionysius Exiguus, Discrete mathematics, Distributive property, Divisibility rule, Division (mathematics), Easter, Egyptian hieroglyphs, Embedding, Empty set, Enumerative combinatorics, Equiconsistency, Equinumerosity, Ernst Zermelo, Euclid's Elements, Euclidean algorithm, Euclidean division, Europe, European Mathematical Society, Finite set, Finitism, Foundations of mathematics, Free object, ... Expand index (82 more) »

## Abstraction

Abstraction in its main sense is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods.

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

## Ancient Greece

Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 13th–9th centuries BC to the end of antiquity (AD 600).

## Archimedes

Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

## Arithmetic

Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## Axiom of infinity

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory.

## Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.

## Babylonia

Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia (present-day Iraq).

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Blackboard bold

Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled.

## Brahmagupta

Brahmagupta (born, died) was an Indian mathematician and astronomer.

## Cancellation property

In mathematics, the notion of cancellative is a generalization of the notion of invertible.

## Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

## Cardinal number (linguistics)

In linguistics, more precisely in traditional grammar, a cardinal number or cardinal numeral (or just cardinal) is a part of speech used to count, such as the English words one, two, three, but also compounds, e.g. three hundred and forty-two (Commonwealth English) or three hundred forty-two (American English).

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Charles Sanders Peirce

Charles Sanders Peirce ("purse"; 10 September 1839 – 19 April 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism".

## China

China, officially the People's Republic of China (PRC), is a unitary one-party sovereign state in East Asia and the world's most populous country, with a population of around /1e9 round 3 billion.

## Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

## Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Computer scientist

A computer scientist is a person who has acquired the knowledge of computer science, the study of the theoretical foundations of information and computation and their application.

## Computus

Computus (Latin for "computation") is a calculation that determines the calendar date of Easter.

## Constructivism (mathematics)

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Counting

Counting is the action of finding the number of elements of a finite set of objects.

## Dionysius Exiguus

Dionysius Exiguus (Latin for "Dionysius the Humble"; –) was a 6th-century monk born in Scythia Minor (probably modern Dobruja, in Romania and Bulgaria).

## Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

## Distributive property

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.

## Divisibility rule

A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.

## Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

## Easter

Easter,Traditional names for the feast in English are "Easter Day", as in the Book of Common Prayer, "Easter Sunday", used by James Ussher and Samuel Pepys and plain "Easter", as in books printed in,, also called Pascha (Greek, Latin) or Resurrection Sunday, is a festival and holiday celebrating the resurrection of Jesus from the dead, described in the New Testament as having occurred on the third day of his burial after his crucifixion by the Romans at Calvary 30 AD.

## Egyptian hieroglyphs

Egyptian hieroglyphs were the formal writing system used in Ancient Egypt.

## Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

## Empty set

In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Enumerative combinatorics

Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.

## Equiconsistency

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa.

## Equinumerosity

In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (a bijection) between them, i.e. if there exists a function from A to B such that for every element y of B there is exactly one element x of A with f(x).

## Ernst Zermelo

Ernst Friedrich Ferdinand Zermelo (27 July 1871 – 21 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics.

## Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

## Euclidean algorithm

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## Euclidean division

In arithmetic, Euclidean division is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor.

## Europe

Europe is a continent located entirely in the Northern Hemisphere and mostly in the Eastern Hemisphere.

## European Mathematical Society

The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe.

## Finite set

In mathematics, a finite set is a set that has a finite number of elements.

## Finitism

Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects.

## Foundations of mathematics

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.

## Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra.

## Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

## Georges Reeb

Georges Henri Reeb (12 November 1920 – 6 November 1993) was a French mathematician.

## Giuseppe Peano

Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist.

## Goodstein's theorem

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0.

## Gottlob Frege

Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

## Hermann Grassmann

Hermann Günther Grassmann (Graßmann; April 15, 1809 – September 26, 1877) was a German polymath, known in his day as a linguist and now also as a mathematician.

## History of ancient Egypt

The history of ancient Egypt spans the period from the early prehistoric settlements of the northern Nile valley to the Roman conquest, in 30 BC.

## Hyperinteger

In non-standard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part.

## Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.

## Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

## India

India (IAST), also called the Republic of India (IAST), is a country in South Asia.

## Infinite set

In set theory, an infinite set is a set that is not a finite set.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## International Organization for Standardization

The International Organization for Standardization (ISO) is an international standard-setting body composed of representatives from various national standards organizations.

## ISO 80000-2

ISO 80000-2:2009 is a standard describing mathematical signs and symbols developed by the International Organization for Standardization (ISO), superseding ISO 31-11.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Karnak

The Karnak Temple Complex, commonly known as Karnak (from Arabic Ka-Ranak meaning "fortified village"), comprises a vast mix of decayed temples, chapels, pylons, and other buildings in Egypt.

## Leopold Kronecker

Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic.

## Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

## Linguistics

Linguistics is the scientific study of language, and involves an analysis of language form, language meaning, and language in context.

## List of continuity-related mathematical topics

In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways.

## Logic

Logic (from the logikḗ), originally meaning "the word" or "what is spoken", but coming to mean "thought" or "reason", is a subject concerned with the most general laws of truth, and is now generally held to consist of the systematic study of the form of valid inference.

## Louvre

The Louvre, or the Louvre Museum, is the world's largest art museum and a historic monument in Paris, France.

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Maya civilization

The Maya civilization was a Mesoamerican civilization developed by the Maya peoples, and noted for its hieroglyphic script—the only known fully developed writing system of the pre-Columbian Americas—as well as for its art, architecture, mathematics, calendar, and astronomical system.

## McGraw-Hill Education

McGraw-Hill Education (MHE) is a learning science company and one of the "big three" educational publishers that provides customized educational content, software, and services for pre-K through postgraduate education.

## Measurement

Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events.

## Merriam-Webster

Merriam–Webster, Incorporated is an American company that publishes reference books which is especially known for its dictionaries.

## Mesoamerica

Mesoamerica is an important historical region and cultural area in the Americas, extending from approximately central Mexico through Belize, Guatemala, El Salvador, Honduras, Nicaragua, and northern Costa Rica, and within which pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and 16th centuries.

## Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x&minus;1, is a number which when multiplied by x yields the multiplicative identity, 1.

## Naturalism (philosophy)

In philosophy, naturalism is the "idea or belief that only natural (as opposed to supernatural or spiritual) laws and forces operate in the world." Adherents of naturalism (i.e., naturalists) assert that natural laws are the rules that govern the structure and behavior of the natural universe, that the changing universe at every stage is a product of these laws.

## Nominal number

Nominal numbers or categorical numbers are numeric codes, meaning numerals used for labelling or identification only.

## Non-standard model of arithmetic

In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers.

## Number

A number is a mathematical object used to count, measure and also label.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Numeral system

A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

## Numerical digit

A numerical digit is a single symbol (such as "2" or "5") used alone, or in combinations (such as "25"), to represent numbers (such as the number 25) according to some positional numeral systems.

## Olmecs

The Olmecs were the earliest known major civilization in Mexico following a progressive development in Soconusco.

## Operation (mathematics)

In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value.

## Order isomorphism

In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets).

## Order of operations

In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.

## Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

## Ordinal number (linguistics)

In linguistics, ordinal numbers (or ordinal numerals) are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on (e.g., "third", "tertiary").

## Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers.

## Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

## Penguin Group

The Penguin Group is a trade book publisher and part of Penguin Random House.

## Positional notation

Positional notation or place-value notation is a method of representing or encoding numbers.

## Primary school

A primary school (or elementary school in American English and often in Canadian English) is a school in which children receive primary or elementary education from the age of about seven to twelve, coming after preschool, infant school and before secondary school.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

## Project Gutenberg

Project Gutenberg (PG) is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks".

## Pythagoras

Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.

## Quotient

In arithmetic, a quotient (from quotiens "how many times", pronounced) is the quantity produced by the division of two numbers.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Real analysis

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Remainder

In mathematics, the remainder is the amount "left over" after performing some computation.

## Richard Dedekind

Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Roman numerals

The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages.

In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.

## Semiring

In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Set-theoretic definition of natural numbers

Several ways have been proposed to construct the natural numbers using set theory.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Successor function

In mathematics, the successor function or successor operation is a primitive recursive function S such that S(n).

## Thoralf Skolem

Thoralf Albert Skolem (23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory.

## Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

## Ultraproduct

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic.

## Von Neumann cardinal assignment

The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers.

## Well-order

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

## Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

## Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.

## Zero-based numbering

Zero-based numbering or index origin.

## 19th century

The 19th century was a century that began on January 1, 1801, and ended on December 31, 1900.

## References

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