
Review of Short Phrases and Links 
This Review contains major "Natural Number" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 A natural number is a nonnegative integer.
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 A natural number is an isomorphism class of a finite set.
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 A natural number is smaller than another if and only if it is an element of the other.
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 A natural number is any positive nonzero integer.
 The natural number is called the ramification index of over.
 At no point is n larger than any natural number.
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 Please forgive me for reducing the problem to only 10 dimensions in total, this is a natural number made up of nine spatial and one temporal dimension.
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 For every natural number n, the n  sphere is compact.
 Note that in this field of fractions, every number can be uniquely written as p −n u with a natural number n and a p adic integer u.
 Chinese number gestures refers to the Chinese method of using one hand to signify the natural number s one through ten.
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 Suppose S is a set and f n: I  R are realvalued functions for every natural number n.
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 A rational number is a number that can be expressed as a fraction with an integer numerator and a nonzero natural number denominator.
 Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial.
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 For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number.
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 For example, suppose n is a nonstandard natural number, then and, and so on for any actual natural number k,.
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 They are fake natural numbers which are "larger" than any actual natural number.
 In mathematics a pairing function is a process to uniquely encode two natural numbers into a single natural number.
 Namely by showing that there is no digit 1 at any position other than indicated by a natural number, which *by your* definition makes the number indexable.
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 Furthermore, one defines a total order on the natural numbers by writing a ≤ b if and only if there exists another natural number c with a + c = b.
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 F,G dialgebras also enable us to define the natural number object, the object for finite lists and other familiar data types in programming.
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 Call a set finite if it can be put in onetoone correspondence with some natural number n.
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 I'll call this category Set[[x]], because it's really a categorification of the set of formal power series with natural number coefficients, N[[x]].
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 The rational numbers are defined as the natural number i divided by the natural number j.
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 Rational numbers are made up of all numbers that can be expressed as a fraction, with integer numerator and nonzero natural number denominator.
 We are very familiar with this matter in the case of flat Euclidean space, in which the coordinate system is denoted as for, for a natural number.
 This would be the case if D 1 read "the smallest natural number".
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 As an extension of this idea, we call a group G nilpotent if there is some natural number n such that A n is trivial.
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 If n is the smallest natural number such that A n is trivial, then we say that G is nilpotent of class n.
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 The answer is yes, because for every natural number n there is a square number n 2, and likewise the other way around.
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 Mapping each natural number to the corresponding real number gives an example for an order embedding.
 Topos theory with natural number object is insufficient to develop undergraduate real analysis  although many fom postings conceal this fact.
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 This is clearly false; it asserts that there is a single natural number s that is at once the square of every natural number.
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 On the other hand, "For any natural number x, x = 2" is false, because you could put, say, 3 in for x and get the false statement "3 = 2".
 Choose a natural number N greater than all types assigned to variables by this stratification.
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 BoxPierce statistic: Defined on a time series sample for each natural number k by the sum of the squares of the first k sample autocorrelations.
 This is clearly true; it just asserts that every natural number has a square.
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 For example, it might begin by showing that if a statement is true for a natural number n it must also be true for some smaller natural number m (m < n).
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 The natural number system (N, 0, S) constructed above is an object in this category; it is called the natural unary system.
 In category theory, a natural number object (nno) is an object endowed with a recursive structure similar to natural numbers.
 In fact, in Peano's original formulation, the first natural number was 1.
 Letting the first natural number be 1 merely requires replacing 0 with 1 in the above axioms, to no substantive effect.
 Addition is first defined on the natural number s.
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 The idea is simply to define a natural number n as the set of all smaller natural numbers: {0, 1, …, n − 1}.
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 The negative of a natural number is defined as a number that produces zero when it is added to the number.
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 A counter stores a single natural number (initially zero) and can be arbitrarilymany digits long.
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 Ten thousand (10,000) is the natural number following 9999 and preceding 10,001.
 It is the natural number following 1 and preceding 3.
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 Define S = { f(n): n a natural number}, the range of f, which can be seen to be a set from the formal definition of a function.
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 Applying this function simply increments every natural number in x.
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 Mathematicians have proved that the square root of every natural number is either an integer or an irrational number.
 For each (meta) natural number n, type n +1 objects are sets of type n objects; sets of type n have members of type n 1.
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 The Natural Number s, N, form a commutative monoid under addition (identity element Zero), or multiplication (identity element One).
 If n is not a natural number, then the multiplication may still make sense, so that we have a sort of notion of adding a term, say, two and a half times.
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 Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring.
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 One has to add to topoi: [not only a natural number object, but] well pointedness, and choice.
 Other mathematicians, primarily number theorists, often prefer to follow the older tradition and consider zero not to be a natural number.
 Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number[ 4].
Definition
 A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers.
 Zero may or may not be counted as a natural number, depending on the definition of natural numbers.
 Then the ramification index is defined to be the unique natural number such that or if.
 In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural number s used to measure the cardinality (size) of sets.
 In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself.
 In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist.
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 The next limit ordinal above the first is ω + ω = ω2, and then we have ω n for any n a natural number.
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 We illustrate this aspect of set theory by extending the natural number system to include infinite ordinals and cardinals.
 Several ways have been proposed to define the natural number s using set theory.
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 The natural number system (N, 0, S) can be shown to satisfy the Peano axioms.
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 In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n.
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 The most familiar numbers are the natural number s, which to some mean the nonnegative integers and to others mean the positive integers.
Categories
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 Science > Mathematics > Infinity > Cardinality
 Society > Culture > Languages > Natural
 Nature > Form > Part > Number
 Ordinals
Related Keywords
* Arithmetic
* Axiom
* Axioms
* Cardinality
* Element
* Elements
* Empty Set
* Finite
* Finitely
* Finite Set
* Finite Sets
* First Proposed
* Infinity
* Integers
* Largest Element
* Natural
* Naturals
* Number
* Ordinals
* Power Set
* Prime
* Primes
* Reals
* Sequence
* Set
* Successor

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