Definable real number

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers. Because formal languages can have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument, there are uncountably many real numbers, so almost every real number is undefinable. Constructible number One way of specifying a real number uses geometric techniques. A real number is a constructible number if there is a method to construct a line segment of length using a compass and straightedge, beginning with a fixed line segment of length 1. Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube. A real number is called a real algebraic number if there is a polynomial , with only integer coefficients, so that is a root of , that is, . Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial has 5 real roots, the third one can be defined as the unique such that and such that there are two distinct numbers less than at which is zero. All rational numbers are algebraic, and all constructible numbers are algebraic. There are numbers such as the cube root of 2 which are algebraic but not constructible. The real algebraic numbers form a subfield of the real numbers.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, 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 more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Mathematical constant

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and pi occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (pi).

Definable real number

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