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An abstract measure of quantity. In everyday terms, numbers are usually used with units: e.g., three meters (or 3m); 6.5893 kilograms (or 6.5893kg).

The most familiar numbers are the natural numbers, 0, 1, 2, ..., used for counting. If negative numbers are included, the result is the integers. Ratios of integers are called rational numbers, which can be expressed as terminating or repeating decimals. If all infinite and non-repeating decimal expansions are thrown in as well, the scope of numbers extends to all real numbers, which can be extended to the complex numbers in order to include all possible solutions to algebraic equations. More recent developments are the hyperreal numbers and the surreal numbers, which extend the real numbers by adding infinitesimal and infinitely large numbers (see infinity). For measuring the size of infinite sets, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers.

All ancient cultures devised their own number systems for the practical purposes of counting and measuring. The development of the idea of place-value notation and the introduction of zero into the base-10 number system that we use today. This system seems to have reached the West from India via 12th-century translations of Arabic mathematical writings. From the basic process of counting we get the natural numbers.

See also numeral and number system.

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