Concept

Number

Overview

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. 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. 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), negative numbers such as negative one (−1), rational numbers such as one half

(

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{\displaystyle \left({\tfrac {1}{2}}\right)}

, real numbers such as the square root of 2

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{\displaystyle \left({\sqrt {2}}\right)}

, and pi (π), and complex numbers which extend the real numbers with a square root of −1 (i), and its combinations with real numbers by adding or subtracting its multiples. 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.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

In the original texts

Passages from public-domain books where Number is discussed.

It seems, then, that there are two kinds of units, the one ultimate, out of which a number is formed by a process of addition, and the other provisional, the number so formed, which is multiple in itself, and owes its unity to the simplicity of the act by which the mind perceives it.
Henri Bergson, Time and Free Will
names the idea; states what it is; opens on it
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The mode in which our existing numerical nomenclature is contrived possesses this advantage, with the additional one, that it happily conveys to the mind two of the modes of formation of every number. Each number is considered as formed by the addition of a unit to the number next below it in magnitude, and this mode of formation is conveyed by the place which it occupies in the series. And each is also considered…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
names the idea; states what it is; opens on it
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The houses and the land will be divided in the same way, so that every man may correspond to a lot. Let the whole number be first divided into two parts, and then into three; and the number is further capable of being divided into four or five parts, or any number of parts up to ten. Every legislator ought to know so much arithmetic as to be able to tell what number is most likely to be useful to all cities; and we…
Plato, Laws
names the idea; states what it is; author linked in the graph
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The fact that in so many instances, remote, and seemingly unrelated nations, have adopted _ten_ as their basic number; together with the fact that in the remaining instances the basic number is either _five_ (the fingers of one hand) or _twenty_ (the fingers and toes); almost of themselves show that the fingers were the original units of numeration. The still surviving use of the word _digit_, as the general name…
Herbert Spencer, Illustrations of Universal Progress
names the idea; states what it is; opens on it
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In union with itself, it does not by its nature produce anything imperfect or foreign. All its changes are defined; it either produces itself or the Decad, either the homogeneous or the perfect.
Plutarch, Selected Essays of Plutarch, Vol. II.
names the idea; states what it is; opens on it
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Number

Overview

In the original texts

Philosophers who discussed Number

Related ideas

Opposing perspectives

Place in the tradition