Set inclusions between the natural numbers (ℕ), the integers (ℤ), the rational numbers (ℚ), the real numbers (ℝ), and the complex numbers (ℂ)

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth.[1] Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits.[2][a] In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0),[3] negative numbers,[4] rational numbers such as one half , real numbers such as the square root of 2 and π,[5] and complex numbers[6] which extend the real numbers with a square root of −1 (and its combinations with real numbers by adding or subtracting its multiples).[4] Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

Besides their practical uses, numbers have cultural significance throughout the world.[7][8] For example, in Western society, the number 13 is often regarded as unlucky, and "a million" may signify "a lot" rather than an exact quantity.[7] Though it is now regarded as pseudoscience, belief in a mystical significance of numbers, known as numerology, permeated ancient and medieval thought.[9] Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.[9]

During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.[10]

  1. ^ "number, n." OED Online. Oxford University Press. Archived from the original on 4 October 2018. Retrieved 16 May 2017.
  2. ^ "numeral, adj. and n." OED Online. Oxford University Press. Archived from the original on 30 July 2022. Retrieved 16 May 2017.
  3. ^ Matson, John. "The Origin of Zero". Scientific American. Archived from the original on 26 August 2017. Retrieved 16 May 2017.
  4. ^ a b Hodgkin, Luke (2 June 2005). A History of Mathematics: From Mesopotamia to Modernity. OUP Oxford. pp. 85–88. ISBN 978-0-19-152383-0. Archived from the original on 4 February 2019. Retrieved 16 May 2017.
  5. ^ Mathematics across cultures : the history of non-western mathematics. Dordrecht: Kluwer Academic. 2000. pp. 410–411. ISBN 1-4020-0260-2.
  6. ^ Descartes, René (1954) [1637]. La Géométrie | The Geometry of René Descartes with a facsimile of the first edition. Dover Publications. ISBN 0-486-60068-8. Retrieved 20 April 2011.
  7. ^ a b Gilsdorf, Thomas E. (2012). Introduction to cultural mathematics : with case studies in the Otomies and the Incas. Hoboken, N.J.: Wiley. ISBN 978-1-118-19416-4. OCLC 793103475.
  8. ^ Restivo, Sal P. (1992). Mathematics in society and history : sociological inquiries. Dordrecht. ISBN 978-94-011-2944-2. OCLC 883391697.{{cite book}}: CS1 maint: location missing publisher (link)
  9. ^ a b Ore, Øystein (1988). Number theory and its history. New York: Dover. ISBN 0-486-65620-9. OCLC 17413345.
  10. ^ Gouvêa, Fernando Q. The Princeton Companion to Mathematics, Chapter II.1, "The Origins of Modern Mathematics", p. 82. Princeton University Press, September 28, 2008. ISBN 978-0-691-11880-2. "Today, it is no longer that easy to decide what counts as a 'number.' The objects from the original sequence of 'integer, rational, real, and complex' are certainly numbers, but so are the p-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions."

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