From Wikipedia, the free encyclopedia
.^ NaturalNumber ONE The natural number one.- NaturalNumber (JML and MultiJava documentation) 1 February 2010 3:28 UTC www.eecs.ucf.edu [Source type: Academic]
^ One apple, two apples, three apples.- Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]
^ I said, "The cardinal numbers are used for counting.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
) from top to bottom.
.^ Hence there exist two natural numbered u and v such that a + u = b and c + v = d.- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
^ According to this convention, zero is not a natural number.- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ What are the natural numbers or positive integers?- Goalfinder - Integers, Natural and Whole Numbers ( Math Software ) - Animated Easy Science, Technology Software, Online Education, medical, K12 animation, & e-Learning 1 February 2010 3:28 UTC www.goalfinder.com [Source type: Academic]
} according to the traditional definition or the set of
.^ Hence the word “count.” FACT: Whole Numbers are non-negative integers that are uncut, undivided, and not in pieces (0, 1, 2, 3…).- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ Stepan Wolfram recommends using the terms "positive integers" and "non-negative integers" to avoid confusion.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ This is not tautological, unlike expressions like ‘Whole Numbers are non-negative integers that are uncut, undivided, and not in pieces (0, 1, 2, 3…).- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
} according to a definition first appearing in the nineteenth century.
.^ Hence there exist two natural numbered u and v such that a + u = b and c + v = d.- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
^ Hence, the natural numbers are totally ordered.- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
^ Consider ordered pairs of natural numbers (a, b).- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
.^ We are explaining how the ordinal numbers -- third, fourth, fifth, and so on -- name the parts of the cardinal numbers.- Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]
^ The natural numbers have two forms, cardinal and ordinal .- Ratio of natural numbers. Evolution of the real numbers. 1 February 2010 3:28 UTC www.themathpage.com [Source type: Original source]
^ Some of these tests are not perfect: there may be some composite numbers, called pseudoprimes for the respective test, that will be declared "probably prime" no matter what witness is chosen.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
(See
English numerals.)
.^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.- Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]
^ Note that in the above-quoted treatment, natural numbers were used not only for illustration as data items but also in the construct ‘a n ’ as exponents.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
^ If the assertion “There is only one zero” is taken as something else (something more general) than “In any system of X-numbers, the X-number zero is unique”, then it is false .- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
.^ The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ The distribution of the prime numbers .- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ Whatever we do with natural numbers ultimately relates to properties of exponentiation of operations.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
Problems concerning counting and ordering, such as
partition enumeration, are studied in
combinatorics.
History of natural numbers and the status of zero
The natural numbers had their origins in the words used to count things, beginning with the number 1.^{[1]}
.^ The first major advance in abstraction was the use of numerals to represent numbers.- Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
- WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]
^ The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes .- WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]
^ Nevertheless, zero was used as a number by all medieval computists (calculators of Easter) beginning with Dionysius Exiguus in 525, but in general no Roman numeral was used to write it.- WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]
.^ This allowed systems to be developed for recording large numbers.- Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
- WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]
^ In Spain, a large number of photovoltaic systems were damaged.- The FINANCIAL - Few major natural catastrophe losses in 2009 General trend confirmed by large number of weather extr 1 February 2010 3:28 UTC finchannel.com [Source type: News]
- Munich Re - Few major natural catastrophe losses in 2009 General trend confirmed by large number of weather extremes 1 February 2010 3:28 UTC www.munichre.com [Source type: FILTERED WITH BAYES]
^ Over time, several systems for counting things were developed; the first of which was the natural numbers .- Arithmetic/Introduction to Natural Numbers - Wikibooks, collection of open-content textbooks 1 February 2010 3:28 UTC en.wikibooks.org [Source type: FILTERED WITH BAYES]
.^ MW - Megawatt - A megawatt is a unit of electrical power equal to one million watts.- Natural Gas Marketwatch Glossary 1 February 2010 3:28 UTC www.spragueenergy.com [Source type: FILTERED WITH BAYES]
^ The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ Note also the difference between the even powers and the odd powers: the numerators are negated for the even powers as compared to the Harmonic Sum which is an odd power of one .- Sums of Powers of the Natural Numbers 1 February 2010 3:28 UTC members.bex.net [Source type: Reference]
The
Babylonians had a
place-value system based essentially on the numerals for 1 and 10. A stone carving from
Karnak, dating from around 1500 BC and now at the
Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622.
.^ In any case, Wittgenstein did not explicitly develop further his idea of the system of natural numbers, although this system played an important rôle in his metamathematical investigations.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
^ For any natural number n other than zero, n, The exception in the wording reminds me; why isn't 0 defined to be its own additive inverse and 1 its own multiplicative inverse, for simplicity?- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
.^ We will use B n to notate the n'th Bernoulli Number .- Sums of Powers of the Natural Numbers 1 February 2010 3:28 UTC members.bex.net [Source type: Reference]
^ It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ When creating a field that would be used for small numbers, you can set its data type to either DB_BYTE or dbByte .- Lesson 16: Data Entry and Natural Numbers 1 February 2010 3:28 UTC www.functionx.com [Source type: Reference]
^{[2]} .^ Whenever we needed 0, we would use the Notation of N with a small zero to signify that we assumed 0 to be present in the natural numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ In any case, Wittgenstein did not explicitly develop further his idea of the system of natural numbers, although this system played an important rôle in his metamathematical investigations.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
^ At the start of the 19th century, Legendre and Gauss independently conjectured that as x tends to infinity, the number of primes up to x is asymptotic to x/log(x), where log(x) is the natural logarithm of x.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
.^ We frequently use concepts like “the n th derivative of a function”, “the n th iterated kernel of an integral equation”, “recurrence relations”, “applying a formula n times”, etc.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
people who calculated the date of
.^ Whenever we needed 0, we would use the Notation of N with a small zero to signify that we assumed 0 to be present in the natural numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ They are numbered with Roman numerals, beginning in the upper right quadrant and proceeding in a counter-clockwise rotation.- Math Glossary N-Z 1 February 2010 3:28 UTC www.blc.edu [Source type: Reference]
^ You write: "The integers (written Z) are all of the numbers, both larger and smaller than zero, that can be written without fractions."- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
Instead
nullus, the Latin word for "nothing", was employed.
.^ After the Greeks, little happened with the study of prime numbers until the 17th century.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ ObsessiveMathsFreak wrote: "For this reason, the Greeks never really studied true 'numbers' and their properties.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
.^ Many mathematicians have worked on primality tests for large numbers, often restricted to specific number forms.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20).- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ If you consider as they did, that a real number means a length of a line segment, then you can do things with numbers that don't make much sense with "lengths".- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^{[3]}
.^ The same prime factor may occur multiple times.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
.^ Take the set of natural numbers N .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ We usually write N for the set of natural numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
.^ Take the set of natural numbers N .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ These labels are what we recognise as Natural Numbers.- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ We usually write N for the set of natural numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
.^ So the set theorists and the computer scientists should just conform… .- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ A computer science definition or a set theorist one?- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
.^ Take the set of natural numbers N .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Many numbers occur in nature, and inevitably some of these are prime.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ There are a number of different kinds of natural law legal theories, differing from each other with respect to the role that morality plays in determining the authority of legal norms.- Natural Law [Internet Encyclopedia of Philosophy] 1 February 2010 3:28 UTC www.utm.edu [Source type: Original source]
.^ Location: Natural History -- Call Number: 581.529 R313 .
^ Besides the Byte and the integer, another natural number supported in the libraries is called Long or Long Integer .- Lesson 16: Data Entry and Natural Numbers 1 February 2010 3:28 UTC www.functionx.com [Source type: Reference]
^ Location: Natural History -- Call Number: 639.92 G375h .
Notation
.^ I’d be surprised to see natural log taken on a straight number (total assets or sales) — usually it’d be used for a growth factor or a series.- Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]
^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.- Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]
^ Here the attribute “X-number” only refers to the fact that distinct sets of objects have distinct zeros.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
This set is countably infinite: it is
infinite but
countable by definition.
.^ As you say, while it is “possible” to have a negative base, it’s likely not very useful because it only works for a small set of numbers.- Demystifying the Natural Logarithm (ln) | BetterExplained 1 February 2010 3:28 UTC betterexplained.com [Source type: General]
^ Cardinal numbers are the ones that are used to count the elements of a set.- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ But induction is essential: the natural numbers are an infinite set - so if we want to be able to say anything about the entire set, then we need to be able to use that kind of reasoning to extend from the finite to the infinite.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "*" or subscript "1" is added in the latter case:
(Sometimes, an index or
superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as
R^{+} = [0,∞) and
Z^{+} = { 0, 1, 2,... }, at least in European literature. The notation "*", however, is standard for nonzero, or rather,
invertible elements.)
.^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Cardinal numbers are the ones that are used to count the elements of a set.- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ Take the set of natural numbers N .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
.^ There are infinitely many prime numbers .- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ I was taught that natural numbers are the positive integers (i.e.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Legendre's conjecture: There is a prime number between n2 and (n + 1)2 for every positive integer n.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
.^ Take the set of natural numbers N .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ According to this convention, zero is not a natural number.- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ We usually write N for the set of natural numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
This stems from the identification of an
ordinal number with the set of ordinals that are smaller.
Algebraic properties
The addition and multiplication operations on natural numbers have several algebraic properties:
- Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.
- Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
- Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.
- Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
- Distributivity for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c)
- No zero divisors: if a and b are natural numbers such that a × b = 0 then a = 0 or b = 0
Properties
.^ All of these theories subscribe to one or more basic tenets of natural law legal theory and are important to its development and influence.- Natural Law [Internet Encyclopedia of Philosophy] 1 February 2010 3:28 UTC www.utm.edu [Source type: Original source]
^ I, too, had been under the impression that zero was not a natural, or counting, number but, rather, the first in the set of whole numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Addition is a function "+" from a pair of natural numbers to another natural number called their sum .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
.^ Predecessor rule: 0 is not the successor of any natural number.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Successor rule : For every natural number n there is exactly one other natural number called its successor s(n) .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Location: Natural History -- Call Number: 581.529 R313 .
This monoid satisfies the
cancellation property and can be embedded in a
group.
.^ The smallest group containing the natural numbers is the integers .- Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
- WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]
^ I was taught that natural numbers are the positive integers (i.e.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ What are the natural numbers or positive integers?- Goalfinder - Integers, Natural and Whole Numbers ( Math Software ) - Animated Easy Science, Technology Software, Online Education, medical, K12 animation, & e-Learning 1 February 2010 3:28 UTC www.goalfinder.com [Source type: Academic]
If we define 1 := S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.
.^ Lemma : Matrices constitute an Abelian group under addition. We define subtraction in an analogous manner to that employed for the integers.- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
^ Definition : The reciprocal operation is defined as an indication of the multiplicative-inverse of a given number (a/b); that is 1/(a/b) = b/a.- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
^ For example, addition and multiplication can be defined as compositions at two levels: (σ+τ)(f) = σ(f)∘τ(t), (σ×τ)(f) = (σ∘τ)(t).- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
.^ Consider how we establish whether two collections of physical object (two sets in the everyday sense) have the same number of elements.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
^ The notion of prime number has been generalized in many different branches of mathematics.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ Prime ideals are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
Addition and multiplication are compatible, which is expressed in the
distribution law:
a × (b + c) =
(a × b) + (a × c).
.^ Addition is a function "+" from a pair of natural numbers to another natural number called their sum .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Many numbers occur in nature, and inevitably some of these are prime.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ In general, the properties of ℕ X depend on X, and only by setting restrictions on X can we interpret X-numbers as natural numbers.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
.^ If the natural number a is divisible by the natural number b, then a is a multiple of b.
^ These properties of addition and multiplication make the natural numbers an instance of a commutative semiring .- Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
^ Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative.- Natural number - Psychology Wiki 1 February 2010 3:28 UTC psychology.wikia.com [Source type: Reference]
- WikiSlice 1 February 2010 3:28 UTC dev.laptop.org [Source type: Reference]
.^ Definition . A subset S of G is said to be a subgroup of G if S is a group under the same operation as that of G. Theorem . A subset S of a group G is a subgroup; if for any elements a and b of S the sum of a and the additive inverse of b is an element of S, namely, a + (-b) is in S. Theorem : Transitivity of the subgroup property .- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
^ If we don't define the symbol "-0," you can write it to mean "the result of applying the operation of negation to 0," but you can't write it to mean "the symbol which represents the additive inverse of 0" unless we specifically define such a symbol.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ The symbol "-" serves at least 3 distinct purposes: to denote subtraction of one number from another, to denote the function of "taking the negative," and as part of a set of symbols which we use for the additive inverses of the positive integers (where it has no independent meaning).- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
.^ John Foster says that Natural numbers “are the ones we use to count things that are there.” But that’s his definition…his opinion.- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ At my school, it is taught (in my math class) that the set of natural numbers starts at 1.- Is 0 a Natural Number? :: squareCircleZ 1 February 2010 3:28 UTC www.squarecirclez.com [Source type: FILTERED WITH BAYES]
^ Natural numbers as exponents of operations - a formalization of Wittgenstein’s definition Natural numbers as exponents of operations A formalization of Wittgenstein ’s definition .- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
For the remainder of the article, we write
ab to indicate the product
a ×
b, and we also assume the standard
order of operations.
.^ We usually write N for the set of natural numbers.- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ In this case there is only one X-number, and consequently PA2 is true.- Natural numbers as exponents of operations -a formalization of Wittgenstein’s definition 1 February 2010 3:28 UTC www.cs.tut.fi [Source type: Reference]
^ The Bernoulli Numbers seem almost random in nature even though there are precise regular patterns that can be constructed to obtain their exact values.- Sums of Powers of the Natural Numbers 1 February 2010 3:28 UTC members.bex.net [Source type: Reference]
.^ Author: Iverson, Ben & Elkins, Dr. Carl Natural arithmetic, number theory and geometry are the basis of all natural phenomena.- SVP Catalog - Number and Number Theory 1 February 2010 3:28 UTC www.svpvril.com [Source type: FILTERED WITH BAYES]
^ Natural arithmetic, number theory and geometry is the basis of all natural phenomena.- SVP Catalog - Number and Number Theory 1 February 2010 3:28 UTC www.svpvril.com [Source type: FILTERED WITH BAYES]
^ Author: Iverson, Ben Natural arithmetic, number theory and geometry is the basis of all natural phenomena.- SVP Catalog - Number and Number Theory 1 February 2010 3:28 UTC www.svpvril.com [Source type: FILTERED WITH BAYES]
.^ Take the set of natural numbers N .- Basics: Natural Numbers and Integers : Good Math, Bad Math 1 February 2010 3:28 UTC scienceblogs.com [Source type: FILTERED WITH BAYES]
^ Hence, the natural numbers are totally ordered.- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
^ Consider ordered pairs of natural numbers (a, b).- Natural to Complex Numbers 1 February 2010 3:28 UTC www.rism.com [Source type: Academic]
The rank among well-ordered sets is expressed by an
ordinal number; for the natural numbers this is expressed as "ω".
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 is available as a substitute: for any two natural numbers
a and
b with
b ≠ 0 we can find natural numbers
q and
r such that
- a = bq + r and r < b.
.^ The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of one.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
The numbers
q and
r are uniquely determined by
a and
b.
.^ Main article: public key cryptography Several public-key cryptography algorithms, such as RSA, are based on large prime numbers (for example with 512 bits).- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ All prime numbers above 3 are of form 6n − 1 or 6n + 1, because all other numbers are divisible by 2 or 3.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
^ However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms.- The Universe of Prime Numbers 1 February 2010 3:28 UTC www.4987.com [Source type: Reference]
Generalizations
Two generalizations of natural numbers arise from the two uses:
.