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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 Zero as a number with its own numeral. A zero Digit had been used in place-value notation as early as 700 BC by the Babylonians, but it was never used as a final element. 1 The Olmec 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 AD . 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," ''nullae'', 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 exclude zero from the natural numbers. The term Whole Number is used informally by some authors for an element of the set of integers, the set of non-negative integers, or the set of positive integers. NOTATION Mathematicians use N or (an N in Blackboard Bold ) to refer to the Set of all natural numbers. This set is Infinite but Countable by definition. To be unambiguous about whether zero is included or not,
Less frequently, W or is used for the set of "whole numbers", which are sometimes identified with the natural numbers as defined here, sometimes with the Integer s. Set theorists often denote the set of all natural numbers by ω. When this notation is used, zero is explicitly included as a natural number. FORMAL DEFINITIONS 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. There are many systems that satisfy these axioms, including the natural numbers (either starting from zero or one). Constructions based on set theory The standard construction A standard construction in Set Theory is to define the natural numbers as follows: :We set 0 := { } :and define ''S''(''a'') = ''a'' U {''a''} for all ''a''. :The set of natural numbers is then ''defined'' to be the intersection of all sets containing 0 which are closed under the successor function. :Assuming the Axiom Of Infinity , this definition can be shown to satisfy 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) Iff ''n'' is a Subset of ''m''. :Also, with this definition, different possible interpretations of notations like R''n'' (''n''-tuples vs. mappings of ''n'' into R) coincide. 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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