Real Numbers
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", was traditionally denoted by a bold R, or the letters I and R close together to create a double-struck appearance of the leftmost part of the letter R. More recently, as of the mid 1990s, a double-struck blackboard bold,
R
{\displaystyle \mathbb {R} }
is used. When the more traditional (with only the left side of the R double-struck) form is desired, LaTeX \mathds font or Asana Math font may now be used.
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.
Some real numbers are roots of polynomials with rational coefficients; these are called algebraic numbers, which include all rational numbers but also some irrational numbers such as √2 = 1.414.... Other real numbers, such as π = 3.1415..., are not; these are called 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 ensuring the correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. A current 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.
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