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# Number

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.

## 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 groupss, ringss and fieldss.

## 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 simple 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

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