Natural Numbers Examples
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In mathematics, the Natural Numbers are the ordinary whole numbers used for counting ("there
are 6 coins on the table") and ordering ("this is the 3rd largest city in the country"). These
purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see
English numerals). A later notion is that of a nominal number, which is used only for naming.
Properties of the natural numbers related to divisibility, such as the distribution of prime numbers,
are studied in number theory. Problems concerning counting and ordering, such as partition
enumeration, are studied in combinatorics.
There is no universal agreement about whether to include zero in the set of natural numbers:
some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term
designates the nonnegative integers {0, 1, 2, 3, ...}.
The former definition is the traditional one, with the latter definition first appearing in the 19th
century. Some authors use the term "natural number" to exclude zero and "whole number" to
include it; others use "whole number" in a way that excludes zero, or in a way that includes both
zero and the negative integers.
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.
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
a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to
over 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. The Babylonians had
a placevalue system based essentially on the numerals for 1 and 10.
A much later advance was the development of the idea that zero can be considered as a number,
with its own numeral. The use of a zero digit in placevalue notation (within other numbers) dates
back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have
been the last symbol in the number. The Olmec and Maya civilizations used zero as a separate
number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.
The use of a numeral zero in modern times originated with the Indian mathematician
Brahmagupta in 628. However, zero had been used as a number in the medieval computus (the
calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted
by a numeral (standard Roman numerals do not have a symbol for zero); instead nulla or nullae,
genitive of nullus, the Latin word for "none", was employed to denote a zero value.
The first systematic study of numbers as abst ractions (that is, as abstract entities) is usually
credited to the Greek philosophers Pythagoras and Archimedes. Note that many Greek
mathematicians did not consider 1 to be "a number", so to them 2 was the smallest number.
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.
By the axiom of infinity, the set of all natural numbers exists and is the intersection of all sets
containing 0 which are closed under this successor function. This then satisfies the Peano
axioms.
Each natural number is then equal to the set of all natural numbers less than it, so that
and so on. When a natural number is used as a set, this is typically what is meant. Under this
definition, there are exactly n elements (in the naive sense) in the set n and n m (in the naive
sense) if and only if n is a subset of m.
Als o, with this definition, different possible inte rpretations of notations like Rn (ntuples versus
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
Each natural number is then equal to the set of the natural number preceding it.
Or we could even define 0 = {{ }}
and S(a) = a
{a}
Producing
The oldest and most "classical" settheoretic definition of the natural numbers is the definition
commonly ascribed to Frege and Russell under which each concrete natural number n is defined
as the set of all sets with n elements.
This may appear circular, but can be made rigorous with care. Define 0 as {{ }} (clearly the set of
all sets with 0 elements) and define S(A) (for any set A) as {x
{y}  x
A y
x }
(see set
bui lder notation).
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