Everything about Natural Number totally explained
In
mathematics, a
natural number (also called
counting number) can mean either an element of the
set,,, ...} (the
positive integers) or an element of the set, 1, 2, 3, ...} (the
non-negative integers). The former is generally used in
number theory, while the latter is preferred in
mathematical logic,
set theory, and
computer science. A more formal definition will follow.
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 3
rd largest city in the country").
Properties of the natural numbers related to
divisibility, such as the distribution of
prime numbers, 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 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
Babylonians 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
hieroglyphs 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, they omitted it when it would have been the last symbol in the number. The
Olmec and
Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but this usage didn't spread beyond
Mesoamerica. The concept as used in modern times originated with the
Indian mathematician
Brahmagupta in 628. Nevertheless, medieval
computists (calculators of
Easter), beginning with
Dionysius Exiguus in 525, used zero as a number without using a
Roman numeral to write it. Instead
nullus, the Latin word for "nothing", was employed.
The first systematic study of numbers as
abstractions (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 convenient to include zero (corresponding to the
empty set) as a natural number. Including zero in the natural numbers is now the common convention among
set theorists,
logicians and
computer scientists. Other mathematicians, such as
number theorists, have kept the older tradition and take 1 to be the first natural number.
Notation
Mathematicians use
N or
;
is the lowest possible value (the
initial ordinal).
For
finite well-ordered sets there's one-to-one correspondence between ordinal and cardinal number; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite,
sequence.
Other generalizations are discussed in the article on
numbers.
Further Information
Get more info on 'Natural Number'.
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