Informally, a definable real number is a
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 values can have arbitrarily small variations. Every ...
that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of sym ...
. 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 number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
s of geometry, the
algebraic numbers, and the
computable numbers. Because formal languages can have only
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
formulas, every notion of definable numbers has at most countably many definable real numbers. However, by
Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...
, there are uncountably many real numbers, so
almost every
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
real number is undefinable.
Constructible numbers
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
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related probl ...
.
Real algebraic numbers
A real number
is called a real
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
if there is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
, 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. This means that 0 and 1 are algebraic numbers and, moreover, if
and
are algebraic numbers, then so are
,
,
and, if
is nonzero,
.
The real algebraic numbers also have the property, which goes beyond being a subfield of the reals, that for each positive integer
and each real algebraic number
, all of the
th roots of
that are real numbers are also algebraic.
There are only
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
algebraic numbers, but there are uncountably many real numbers, so in the sense of
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
most real numbers are not algebraic. This
nonconstructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existenc ...
that not all real numbers are algebraic was first published by
Georg Cantor in his 1874 paper "
On a Property of the Collection of All Real Algebraic Numbers".
Non-algebraic numbers are called
transcendental numbers. The best known transcendental numbers are
and .
Computable real numbers
A real number is a
computable number if there is an algorithm that, given a natural number
, produces a decimal expansion for the number accurate to
decimal places. This notion was introduced by
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical ...
in 1936.
The computable numbers include the algebraic numbers along with many transcendental numbers including
Like the algebraic numbers, the computable numbers also form a subfield of the real numbers, and the positive computable numbers are closed under taking
th roots for each
Not all real numbers are computable. Specific examples of noncomputable real numbers include the limits of
Specker sequences, and
algorithmically random real numbers such as
Chaitin's Ω numbers.
Definability in arithmetic
Another notion of definability comes from the formal theories of arithmetic, such as
Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...
. The
language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. Because no variables of this language range over the
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 values can have arbitrarily small variations. Every ...
s, a different sort of definability is needed to refer to real numbers. A real number
is ''definable in the language of arithmetic'' (or ''
arithmetical'') if its
Dedekind cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
can be defined as a
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
in that language; that is, if there is a first-order formula
in the language of arithmetic, with three free variables, such that