Concept
Real number
Overview
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
The real numbers are fundamental in calculus and in many other branches of mathematics, in particular by their role in the classical definitions of limits, continuity and derivatives.
The set of real numbers, sometimes called "the reals", is usually notated as a bold R or the blackboard bold
R
{\displaystyle \mathbb {R} }
.
The adjective real, used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of negative numbers.
The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. Real numbers that are not rational are irrational. Those real numbers that are roots of polynomials with rational coefficients are algebraic numbers, which include all the rational numbers and also irrational numbers such as √2 = 1.414.... Other real numbers, such as π = 3.1415..., are not roots of polynomials; these are the transcendental numbers.
The real numbers can be thought of as the points on a line, called the number line or real line, on which the points corresponding to integers (..., −2, −1, 0, 1, 2, ...) are equally spaced.
The informal descriptions above of the real numbers are not sufficient for rigorous reasoning about real numbers. The development of a suitable formal definition was a major achievement of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. One modern axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.
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 Real number is discussed.
Our ideas will find expression more conveniently if, instead of points upon a line, we speak of real numbers. Every real number is, in one sense, the limit of a series, for it can be indefinitely approximated to. Whether every real number is a limit of a _regular_ series may perhaps be open to doubt. But the series referred to in the definition of Aristotelicity must be understood as including all series whether…
Charles S. Peirce, Chance, Love, and Logic
states what it is; names the idea; opens on it
Can it be thought that the pain and discomfort of sentient creatures, and above all the happiness and unhappiness of intelligent substances, are a matter of indifference to God? And what shall be said of his justice? Is it also something arbitrary, and would he have acted wisely and justly if he had resolved to condemn the innocent? I know that there have been writers so ill-advised as to maintain an opinion so…
Gottfried Wilhelm Leibniz, Theodicy Essays on the Goodness of God, the Freedom of Man and the Origin of Evil
states what it is; author linked in the graph
It is thus manifest that to say we perceive colours in objects is in reality equivalent to saying we perceive something in objects and are yet ignorant of what it is, except as that which determines in us a certain highly vivid and clear sensation, which we call the sensation of colours. There is, however, very great diversity in the manner of judging: for so long as we simply judge that there is an unknown…
René Descartes, Selections from the Principles of Philosophy
states what it is; author linked in the graph
_Ten_ must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of anything. Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects, all existences of every kind, known to our experience. All things possess quantity; consist of parts which can be…
John Stuart Mill, A System of Logic
states what it is
By cottages are understood those who neither pay to church or poor, and are, by act of parliament in 1747, in consideration of the poverty of the people, declared to be exempted both from the tax and the two-shillings duty, and they only remain not accurately known, to ascertain the whole number of houses. However, they are so far known, that from all the accounts that are hitherto given in, they do not appear to be…
Various, Philosophical transactions, Vol. L. Part I. For the year 1757.
states what it is; names the idea
Philosophers who discussed Real number
Related ideas
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Place in the tradition
Part of Fuzzy logic.