55-410: 1000 or one thousand is the natural number following 999 and preceding 1001 . In most English-speaking countries , it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000 . A group of one thousand things is sometimes known, from Ancient Greek , as a chiliad . A period of one thousand years may be known as a chiliad or, more often from Latin , as
110-680: A and b with b ≠ 0 there are natural numbers q and r such that The number q is called the quotient and r is called the remainder of the division of a by b . The numbers q and r are uniquely determined by a and b . This Euclidean division is key to the several other properties ( divisibility ), algorithms (such as the Euclidean algorithm ), and ideas in number theory. The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: Two important generalizations of natural numbers arise from
165-425: A + c = b . This order is compatible with the arithmetical operations in the following sense: if a , b and c are natural numbers and a ≤ b , then a + c ≤ b + c and ac ≤ bc . An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number ; for
220-466: A + 1 = S ( a ) and a × 1 = a . Furthermore, ( N ∗ , + ) {\displaystyle (\mathbb {N^{*}} ,+)} has no identity element. In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where
275-508: A millennium . The number 1000 is also sometimes described as a short thousand in medieval contexts where it is necessary to distinguish the Germanic concept of 1200 as a long thousand . It is the first 4-digit integer . 1000 is the 10th icositetragonal number, or 24-gonal number. It is also the 16th generalized 30-gonal number. 1000 is the Wiener index of cycle length 20 , also
330-427: A totient value of 1000 (1111, 1255, ..., 3750). One thousand is also equal to the sum of Euler's totient summatory function Φ ( n ) {\displaystyle \Phi (n)} over the first 57 integers. In decimal , multiples of one thousand are totient values of four-digit repdigits : In the list of composite numbers , 7777 is very nearly the composite index of 8888: 8886
385-401: A × ( b + c ) = ( a × b ) + ( a × c ) . These properties of addition and multiplication make the natural numbers an instance of a commutative semiring . Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that N {\displaystyle \mathbb {N} }
440-404: A × 0 = 0 and a × S( b ) = ( a × b ) + a . This turns ( N ∗ , × ) {\displaystyle (\mathbb {N} ^{*},\times )} into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers . Addition and multiplication are compatible, which is expressed in the distribution law :
495-474: 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 BCE , 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 CE. However, 0 had been used as a number in the medieval computus (the calculation of
550-606: A matter of definition. In 1727, Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889, Giuseppe Peano used N for the positive integers and started at 1, but he later changed to using N 0 and N 1 . Historically, most definitions have excluded 0, but many mathematicians such as George A. Wentworth , Bertrand Russell , Nicolas Bourbaki , Paul Halmos , Stephen Cole Kleene , and John Horton Conway have preferred to include 0. Mathematicians have noted tendencies in which definition
605-460: A natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell's paradox . To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements. In 1881, Charles Sanders Peirce provided
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#1732765762773660-461: A natural number is to use one's fingers, as in finger counting . Putting down a tally mark for each object is another primitive method. Later, 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
715-526: A need to improve upon 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 really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications. Set-theoretical definitions of natural numbers were initiated by Frege . He initially defined
770-482: A number like any other. Independent studies on numbers also occurred at around the same time in India , China, and Mesoamerica . Nicolas Chuquet used the term progression naturelle (natural progression) in 1484. The earliest known use of "natural number" as a complete English phrase is in 1763. The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article. Starting at 0 or 1 has long been
825-507: A powerful system of numerals with distinct hieroglyphs for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak , dating back from around 1500 BCE and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on
880-509: A set (because of Russell's paradox ). The standard solution is to define a particular set with n elements that will be called the natural number n . The following definition was first published by John von Neumann , although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set as a definition of ordinal number , the sets considered below are sometimes called von Neumann ordinals . The definition proceeds as follows: It follows that
935-441: A sports team, where they serve as nominal numbers and do not have mathematical properties. The natural numbers form a set , commonly symbolized as a bold N or blackboard bold N {\displaystyle \mathbb {N} } . Many other number sets are built from the natural numbers. For example, the integers are made by adding 0 and negative numbers. The rational numbers add fractions, and
990-574: A subscript (or superscript) "0" is added in the latter case: This section uses the convention N = N 0 = N ∗ ∪ { 0 } {\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}} . Given the set N {\displaystyle \mathbb {N} } of natural numbers and the successor function S : N → N {\displaystyle S\colon \mathbb {N} \to \mathbb {N} } sending each natural number to
1045-513: Is consistent (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong. The five Peano axioms are the following: These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic,
1100-427: Is 10000, where 6000 is the totient of 9999, one less than 10. The sum of the first nine prime numbers up to 23 is 100, with φ ( p ( 23 ) ) = 1000 {\displaystyle \varphi (p(23))=1000} , where p ( 23 ) = 1255 {\displaystyle p(23)=1255} is the number of integer partitions of 23. Using decimal representation as well, On
1155-505: Is a free monoid on one generator. This commutative monoid satisfies the cancellation property , so it can be embedded in a group . The smallest group containing the natural numbers is the integers . If 1 is defined as S (0) , then b + 1 = b + S (0) = S ( b + 0) = S ( b ) . That is, b + 1 is simply the successor of b . Analogously, given that addition has been defined, a multiplication operator × {\displaystyle \times } can be defined via
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#17327657627731210-438: Is a subset of m . In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order . 1600 (number) 1000 or one thousand is the natural number following 999 and preceding 1001 . In most English-speaking countries , it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000 . A group of one thousand things
1265-408: Is a 1000-sided polygon , of order 2000 in its regular form . 1000 has a reduced totient value λ ( n ) {\displaystyle \lambda (n)} of 100 , and Euler totient φ ( n ) {\displaystyle \varphi (n)} of 400 . 11 integers have a totient value of 1000 (1111, 1255, ..., 3750). One thousand is also equal to
1320-412: Is also the 16th generalized 30-gonal number. 1000 is the Wiener index of cycle length 20 , also the sum of labeled boxes arranged as a pyramid with base 1 – 20. 1000 is the element of multiplicity in a 24 × 24 {\displaystyle 24\times 24} toroidal board in the n -Queens problem , with respective indicator of 25 and count of 51 . 1000
1375-552: Is based on set theory . It defines the natural numbers as specific sets . More precisely, each natural number n is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set S has n elements" means that there exists a one to one correspondence between the two sets n and S . The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem that can be stated and proved in Peano arithmetic can also be proved in set theory. However,
1430-584: Is based on an axiomatization of the properties of ordinal numbers : each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory . One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem . The set of all natural numbers
1485-410: Is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that N {\displaystyle \mathbb {N} } is not a ring ; instead it is a semiring (also known as a rig ). If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with
1540-446: Is sometimes known, from Ancient Greek , as a chiliad . A period of one thousand years may be known as a chiliad or, more often from Latin , as a millennium . The number 1000 is also sometimes described as a short thousand in medieval contexts where it is necessary to distinguish the Germanic concept of 1200 as a long thousand . It is the first 4-digit integer . 1000 is the 10th icositetragonal number, or 24-gonal number. It
1595-429: Is standardly denoted N or N . {\displaystyle \mathbb {N} .} Older texts have occasionally employed J as the symbol for this set. Since natural numbers may contain 0 or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as: Alternatively, since
1650-457: Is the 7779th composite number. Also, 1600 = 40 is the totient value of 4000, as well as 6000, whose collective sum is 10000, where 6000 is the totient of 9999, one less than 10 . The sum of the first nine prime numbers up to 23 is 100, with φ ( p ( 23 ) ) = 1000 {\displaystyle \varphi (p(23))=1000} , where p ( 23 ) = 1255 {\displaystyle p(23)=1255}
1705-473: Is the number of integer partitions of 23. Using decimal representation as well, On the other hand, the largest prime number less than 10000 is the 1229th prime number, 9973 . 1000 is also the smallest number in base-ten that generates three primes in the fastest way possible by concatenation with decremented numbers: all represent prime numbers. Adding the prime 853 with its prime index of 147 yields 1000. The one-thousandth prime number
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1760-443: Is the number of strict partitions of 50 containing the sum of no subset of the parts . A chiliagon is a 1000-sided polygon , of order 2000 in its regular form . 1000 has a reduced totient value λ ( n ) {\displaystyle \lambda (n)} of 100 , and Euler totient φ ( n ) {\displaystyle \varphi (n)} of 400 . 11 integers have
1815-422: Is used, such as algebra texts including 0, number theory and analysis texts excluding 0, logic and set theory texts including 0, dictionaries excluding 0, school books (through high-school level) excluding 0, and upper-division college-level books including 0. There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include division by zero and
1870-406: The non-negative integers 0, 1, 2, 3, ... , while others start with 1, defining them as the positive integers 1, 2, 3, ... . Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers plus zero. In other cases, the whole numbers refer to all of the integers , including negative integers. The counting numbers are another term for
1925-410: The order of the smallest sporadic group : | M 11 | = 7920 {\displaystyle |\mathrm {M} _{11}|=7920} . There are 135 prime numbers between 1000 and 2000: Natural number In mathematics , the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as
1980-515: The real numbers add infinite decimals. Complex numbers add the square root of −1 . This chain of extensions canonically embeds the natural numbers in the other number systems. Natural numbers are studied in different areas of math. Number theory looks at things like how numbers divide evenly ( divisibility ), or how prime numbers are spread out. Combinatorics studies counting and arranging numbered objects, such as partitions and enumerations . The most primitive method of representing
2035-574: The date of Easter), beginning with Dionysius Exiguus in 525 CE, without being denoted by a numeral. Standard Roman numerals do not have a symbol for 0; instead, nulla (or the genitive form nullae ) from nullus , the Latin word for "none", was employed to denote a 0 value. The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes . Some Greek mathematicians treated
2090-409: The first axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of Dedekind's axioms in his book The principles of arithmetic presented by a new method ( Latin : Arithmetices principia, nova methodo exposita ). This approach is now called Peano arithmetic . It
2145-450: The natural numbers are defined iteratively as follows: It can be checked that the natural numbers satisfy the Peano axioms . With this definition, given a natural number n , the sentence "a set S has n elements" can be formally defined as "there exists a bijection from n to S ." This formalizes the operation of counting the elements of S . Also, n ≤ m if and only if n
2200-403: The natural numbers naturally form a subset of the integers (often denoted Z {\displaystyle \mathbb {Z} } ), they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a superscript " ∗ {\displaystyle *} " or "+" is added in the former case, and
2255-470: The natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are six coins on the table", in which case they are called cardinal numbers . They are also used to put things in order, like "this is the third largest city in the country", which are called ordinal numbers . Natural numbers are also used as labels, like jersey numbers on
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2310-435: The natural numbers, this is denoted as ω (omega). In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder or Euclidean division is available as a substitute: for any two natural numbers
2365-439: The next one, one can define addition of natural numbers recursively by setting a + 0 = a and a + S ( b ) = S ( a + b ) for all a , b . Thus, a + 1 = a + S(0) = S( a +0) = S( a ) , a + 2 = a + S(1) = S( a +1) = S(S( a )) , and so on. The algebraic structure ( N , + ) {\displaystyle (\mathbb {N} ,+)} is a commutative monoid with identity element 0. It
2420-413: The number 1 differently than larger numbers, sometimes even not as a number at all. Euclid , for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2). However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as
2475-490: 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 later 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 (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such
2530-599: The ordinary natural numbers via the ultrapower construction . Other generalizations are discussed in Number § Extensions of the concept . Georges Reeb used to claim provocatively that "The naïve integers don't fill up N {\displaystyle \mathbb {N} } ". There are two standard methods for formally defining natural numbers. The first one, named for Giuseppe Peano , consists of an autonomous axiomatic theory called Peano arithmetic , based on few axioms called Peano axioms . The second definition
2585-419: The other hand, the largest prime number less than 10000 is the 1229th prime number, 9973 . 1000 is also the smallest number in base-ten that generates three primes in the fastest way possible by concatenation with decremented numbers: all represent prime numbers. Adding the prime 853 with its prime index of 147 yields 1000. The one-thousandth prime number is 7919 . It is a difference of 1 from
2640-479: The same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence . A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from
2695-399: The size of the empty set . Computer languages often start from zero when enumerating items like loop counters and string- or array-elements . Including 0 began to rise in popularity in the 1960s. The ISO 31-11 standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as ISO 80000-2 . In 19th century Europe, there
2750-433: The successor of x {\displaystyle x} is x + 1 {\displaystyle x+1} . Intuitively, the natural number n is the common property of all sets that have n elements. So, it seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence ". This does not work in all set theories , as such an equivalence class would not be
2805-463: The sum of Euler's totient summatory function Φ ( n ) {\displaystyle \Phi (n)} over the first 57 integers. In decimal , multiples of one thousand are totient values of four-digit repdigits : In the list of composite numbers , 7777 is very nearly the composite index of 8888: 8886 is the 7779th composite number. Also, 1600 = 40 is the totient value of 4000, as well as 6000, whose collective sum
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#17327657627732860-410: The sum of labeled boxes arranged as a pyramid with base 1 – 20. 1000 is the element of multiplicity in a 24 × 24 {\displaystyle 24\times 24} toroidal board in the n -Queens problem , with respective indicator of 25 and count of 51 . 1000 is the number of strict partitions of 50 containing the sum of no subset of the parts . A chiliagon
2915-402: The two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not provable inside Peano arithmetic. A probable example is Fermat's Last Theorem . The definition of the integers as sets satisfying Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set theory
2970-423: The two uses of counting and ordering: cardinal numbers and ordinal numbers . The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0 ) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω . For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by
3025-430: Was mathematical and philosophical discussion about the exact nature of the natural numbers. Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act. Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man". The constructivists saw
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