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A number is an abstract entity used to describe quantity. There
are different types of numbers. The most familiar numbers are the natural
numbers {0, 1, 2, ...} used for counting and denoted by N. If
the negative whole numbers are included, one obtains the integers Z. Ratios of integers are called rational numbers or fractions; the set of all
rational numbers is denoted by Q. If all infinite and non-repeating decimal expansions are included, one obtains
the real numbers R. Those real numbers which are not
rational are called irrational numbers. The real numbers are in
turn extended to the complex numbers C in order to be
able to solve all algebraic equations. The above symbols are often written in blackboard bold, thus:
-
Complex numbers can, in turn, be extended to quaternions, but multiplication
of quaternions is not commutative. Octonions, in turn, extend the quaternions, but this time, associativity is lost. In fact, the only finite-dimensional associative division algebras over R are the reals, the complex numbers, and the
quaternions.
Numbers should be distinguished from numerals, which are (combinations
of) symbols used to represent numbers. The notation of numbers as a series of digits is
discussed in numeral systems.
People like to assign numbers to objects in order to have unique names. There are various numbering schemes for doing so.
Many languages have the concept of grammatical number, an attribute of certain words and phrases that affects their syntactic usage and
meaning. um
Extensions
Newer developments are the hyperreal numbers and the surreal numbers, which extend the real numbers by adding infinitesimal and
infinitely large numbers. While (most) real numbers have infinitely long expansions to the right of the decimal point, one can
also try to allow for infinitely long expansions to the left, leading to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the
ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite
case, the ordinal and cardinal numbers are equivalent; they diverge in the infinite case.)
The arithmetical operations of numbers, such as addition, subtraction, multiplication
and division, are generalized in the branch of mathematics called abstract algebra; one obtains
the groups, rings and fields.
Biological basis of culture-free similarities
In many cultures, the notation for the numbers one, two, and three is very similar. In Roman
numerals, the corresponding numerals are I, II, and III, and in Chinese, the same notation is used but with the tally marks
written horizontally.
However, neither the Roman nor the Chinese systems use simply tally marks for four. The Roman numeral for
four is IV, meaning one less than V, which stands for five. Evidently, five has significance because of the
number of digits on each human hand. However, there is more here than mere human anatomy. Psychologists explain that the reason
for the shift from a simple tally notation to one involving more symbols is the difficulty humans have in visually separating
similar patterns with more than three identical elements. For example, it's hard to tell at a glance which is greater: IIIIIIII,
or IIIIIII, but it is easy to tell X from XI.
The Arabic numeral system uses modified tally marks for 1, 2, and 3: 1 has undergone only very minor modification,
and 2 and 3 are evidently based on horizontal lines written without lifting the pen. And again, the simple
tally is abandoned with the numeral for 4.
Particular numbers
See: List of numbers, mathematical constants, even and
odd numbers, negative and
non-negative numbers, small numbers, large numbers, orders of magnitude (numbers), prime
numbers; umpteen
See also
External links
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