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Number ○꠹｜Definition｜1st｜20251119205401-00-⌔
Number
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth.1 Individual numbers can be represented in spoken or written language with number words, or with dedicated symbols called numerals; for example, “eleven” is a number word and “11” is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, a decimal system which can display any non-negative integer using a combination of ten Arabic numeral symbols called digits.23 Numerals can be used for counting (as with cardinal number of a collection or set), for labelling (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, however, a numeral is not clearly distinguished from the number that it represents.
In mathematics, the notion of a number has been extended over the centuries to include zero (0),4 negative numbers such as negative one (−1),5 rational numbers such as one half , real numbers such as the square root of 2 , and pi (π),6 and complex numbers7 which extend the real numbers with a square root of −1 (i), and its combinations with real numbers by adding or subtracting its multiples.5 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.
Viewing the concept of zero as a number required a fundamental shift in philosophy, identifying nothingness with a value. During the 19th century, mathematicians began to develop the various systems now called algebraic structures, which share certain properties of numbers, and may be seen as extending the concept. Some algebraic structures are explicitly referred to as numbers (such as the p -adic numbers and hypercomplex numbers) while others are not, but this is more a matter of convention than a mathematical distinction.8
Printed 2026-06-28.
Link to original Footnotes
“number, n.” OED Online. Oxford University Press. Archived from the original on 4 October 2018. Retrieved 16 May 2017. ↩
“numeral, adj. and n.” OED Online. Oxford University Press. Archived from the original on 30 July 2022. Retrieved 16 May 2017. ↩
In linguistics, a numeral can refer to a symbol like 5, but also to a word or a phrase that names a number, like “five hundred”; numerals include also other words representing numbers, like “dozen”. ↩
Matson, John. “The Origin of Zero”. Scientific American. Archived from the original on 26 August 2017. Retrieved 16 May 2017. ↩
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. ↩ ↩2
Puttaswamy, T. K. (2012). “The Mathematical Accomplishments of Ancient Indian Mathematics”. In Selin, Helaine (ed.). Mathematics across cultures: the history of non-western mathematics. Dordrecht: Springer Science & Business Media. pp. 409–422. ISBN 978-94-011-4301-1. ↩
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. ↩
Gouvêa, Fernando Q. (28 September 2008). “II.1, The Origins of Modern Mathematics”. The Princeton Companion to Mathematics. Princeton University Press. p. 82. 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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