Information AboutNatural Number |
| CATEGORIES ABOUT NATURAL NUMBER | |
| cardinal numbers | |
| elementary mathematics | |
| integers | |
| number theory | |
| numbers | |
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Natural numbers have two main purposes: they can be used for Counting ("there are 3 apples on the table"), and they can be used for Ordering ("this is the 3rd largest city in the country"). Properties of the natural numbers related to Divisibility , such as the distribution of Prime Number s, are studied in Number Theory . Problems concerning counting, such as Ramsey Theory , are studied in Combinatorics . HISTORY OF NATURAL NUMBERS AND THE STATUS OF ZERO The natural numbers presumably had their origins in the words used to count things, beginning with the number one. The first major advance in abstraction was the use of Numerals to represent numbers. This allowed systems to be developed for recording large numbers. For example, the Babylonia ns developed a powerful Place-value system based essentially on the numerals for 1 and 10. The ancient Egyptians had a system of numerals with distinct Hieroglyph s for 1, 10, and all the powers of 10 up to one 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, and 6 ones; and similarly for the number 4,622. A much later advance in abstraction was the development of the idea of and Maya Civilization used zero as a separate number as early as 1st Century BC , apparently developed independently, but this usage did not spread beyond Mesoamerica . The concept as used in modern times originated with the India n mathematician Brahmagupta in 628 . 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. Instead, the Latin word for "nothing," ''nullus'', was employed. The first systematic study of numbers as Abstraction s (that is, as abstract Entities ) is usually credited to the Greek philosophers Pythagoras and Archimedes . However, independent studies also occurred at around the same time in India , China , and Mesoamerica . In the nineteenth century, a Set-theoretical Definition of natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the Empty Set ) as a natural number. This convention is followed by Set Theorists , Logicians , and Computer Scientists . Other mathematicians, primarily Number Theorists , often prefer to follow the older tradition and consider zero not to be a natural number. NOTATION Mathematicians use N or (an N in but Countable by definition. This is also expressed by saying that the Cardinal Number of the set is (). To be unambiguous about whether zero is included or not,
Some authors who exclude zero from the naturals use the term ''whole numbers'', denoted , for the set of nonnegative integers. Others use the notation for the positive integers. Set theorists often denote the set of all natural numbers by a lower-case Greek letter Omega : ω. When this notation is used, zero is explicitly included as a natural number. FORMAL DEFINITIONS See Also: Set-theoretic definition of natural numbers Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano Postulates state conditions that any successful definition must satisfy. Certain constructions show that, given Set Theory , Models of the Peano postulates must exist. Peano axioms
It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. All systems that satisfy these axioms are isomorphic, the name "0" is used here for the first element, which is the only element that is not a successor. For example, the natural numbers starting with one also satisfy the axioms. Constructions based on set theory A standard construction A standard construction in Set Theory , a special case of the Von Neumann Ordinal construction, is to define the natural numbers as follows: :We set 0 := { }, the Empty Set , :and define ''S''(''a'') = ''a'' ∪ {''a''} for every set ''a''. ''S''(''a'') is the successor of ''a'', and ''S'' is called the successor function. :If the Axiom Of Infinity holds, then the set of all natural numbers exists and is the intersection of all sets containing 0 which are closed under this successor function. :If the set of all natural numbers exists, then it satisfies the Peano Axioms . :Each natural number is then equal to the set of natural numbers less than it, so that
:and so on. When you see a natural number used as a set, this is typically what is meant. Under this definition, there are exactly ''n'' elements (in the naïve sense) in the set ''n'' and ''n'' ≤ ''m'' (in the naïve sense) If And Only If ''n'' is a Subset of ''m''. :Also, with this definition, different possible interpretations of notations like R''n'' (''n-''tuples versus mappings of ''n'' into R) coincide. :Even if the axiom of infinity fails and the set of all natural numbers does not exist, it is possible to define what it means to be one of these sets. A set ''n'' is a natural number means that it is either 0 (empty) or a successor, and each of its elements is either 0 or the successor of another of its elements. Other constructions Although the standard construction is useful, it is not the only possible construction. For example: :one could define 0 = { } :and ''S''(''a'') = {''a''}, :producing :: 0 = { } :: 1 = {0} = :: 2 = {1} = Or we could even define 0 = :and ''S''(''a'') = ''a'' U {''a''} :producing :: 0 = :: 1 =
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