Real number


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 or the blackboard bold.
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 and the fraction. 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. Other real numbers, such as, 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 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 Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences, Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.

Characterizing properties

Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete. Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal definitions and the proof of their equivalence.

Arithmetic

The real numbers form an ordered field. Intuitively, this means that methods and rules of elementary arithmetic apply to them. More precisely, there are two binary operations, addition and multiplication, and a total order that have the following properties.
  • The addition of two real numbers and produce a real number denoted which is the sum of and.
  • The multiplication of two real numbers and produce a real number denoted or which is the product of and.
  • Addition and multiplication are both commutative, which means that and for every real numbers and.
  • Addition and multiplication are both associative, which means that and for every real numbers, and, and that parentheses may be omitted in both cases.
  • Multiplication is distributive over addition, which means that for every real numbers, and.
  • There is a real number called zero and denoted which is an additive identity, which means that for every real number.
  • There is a real number denoted which is a multiplicative identity, which means that for every real number.
  • Every real number has an additive inverse denoted This means that for every real number.
  • Every nonzero real number has a multiplicative inverse denoted or This means that for every nonzero real number.
  • The total order is denoted Being a total order means that it has the following two properties:
  1. For any two real numbers and, exactly one of,, is true.
  2. If and then.
  • The order is compatible with addition and multiplication, which means that implies for every real number, and is implied by and
Many other properties can be deduced from the above ones. In particular:
  • for every real number
  • for every nonzero real number

    Auxiliary operations

Several other operations are commonly used, which can be deduced from the above ones.
  • Subtraction: the subtraction of two real numbers and results in the sum of and the additive inverse of ; that is,
  • Division: the division of a real number by a nonzero real number is denoted or and defined as the multiplication of with the multiplicative inverse of ; that is,
  • Absolute value: the absolute value of a real number, denoted measures its distance from zero, and is defined as

    Auxiliary order relations

The total order that is considered above is denoted and read as " is less than ". Three other order relations are also commonly used:
  • Greater than: read as " is greater than ", is defined as if and only if
  • Less than or equal to: read as " is less than or equal to " or " is not greater than ", is defined as or equivalently as
  • Greater than or equal to: read as " is greater than or equal to " or " is not less than ", is defined as or equivalently as

    Integers and fractions as real numbers

The real numbers and are commonly identified with the natural numbers and. This allows identifying any natural number with the sum of real numbers equal to.
This identification can be pursued by identifying a negative integer with the additive inverse of the real number identified with Similarly a rational number is identified with the division of the real numbers identified with and.
These identifications make the set of the rational numbers an ordered subfield of the real numbers The Dedekind completeness described below implies that some real numbers, such as are not rational numbers; they are called irrational numbers.
The above identifications make sense, since natural numbers, integers and real numbers are generally not defined by their individual nature, but by defining properties. So, the identification of natural numbers with some real numbers is justified by the fact that Peano axioms are satisfied by these real numbers, with the addition with taken as the successor function.
Formally, one has an injective homomorphism of ordered monoids from the natural numbers to the integers an injective homomorphism of ordered rings from to the rational numbers and an injective homomorphism of ordered fields from to the real numbers The identifications consist of not distinguishing the source and the image of each injective homomorphism, and thus to write
These identifications are formally abuses of notation, and are generally harmless. It is only in very specific situations, that one must avoid them and replace them by using explicitly the above homomorphisms. This is the case in constructive mathematics and computer programming. In the latter case, these homomorphisms are interpreted as type conversions that can often be done automatically by the compiler.

Dedekind completeness

Previous properties do not distinguish real numbers from rational numbers. This distinction is provided by Dedekind completeness, which states that every non-empty set of real numbers with an upper bound admits a least upper bound. This means the following. A set of real numbers is bounded above if there is a real number such that for all ; such a is called an upper bound of So, Dedekind completeness means that, if is non-empty and bounded above, it has an upper bound that is less than any other upper bound.
Dedekind completeness implies other sorts of completeness, but also has some important consequences.
The last two properties are summarized by saying that the real numbers form a real closed field. This implies the real version of the fundamental theorem of algebra, namely that every polynomial with real coefficients can be factored into polynomials with real coefficients of degree at most two.

Decimal representation

The most common way of describing a real number is via its decimal representation, a sequence of decimal digits each representing the product of an integer between zero and nine times a power of ten, extending to finitely many positive powers of ten to the left and infinitely many negative powers of ten to the right. For a number whose decimal representation extends places to the left, the standard notation is the juxtaposition of the digits in descending order by power of ten, with non-negative and negative powers of ten separated by a decimal point, representing the infinite series
For example, for the circle constant is zero and etc.
More formally, a decimal representation for a nonnegative real number consists of a nonnegative integer and integers between zero and nine in the infinite sequence
Such a decimal representation specifies the real number as the least upper bound of the decimal fractions that are obtained by truncating the sequence: given a positive integer, the truncation of the sequence at the place is the finite partial sum
The real number defined by the sequence is the least upper bound of the which exists by Dedekind completeness.
Conversely, given a nonnegative real number, one can define a decimal representation of by induction, as follows. Define as decimal representation of the largest integer such that . Then, supposing by induction that the decimal fraction has been defined for one defines as the largest digit such that and one sets
One can use the defining properties of the real numbers to show that is the least upper bound of the So, the resulting sequence of digits is called a decimal representation of.
Another decimal representation can be obtained by replacing with in the preceding construction. These two representations are identical, unless is a decimal fraction of the form In this case, in the first decimal representation, all are zero for and, in the second representation, all 9..
In summary, there is a bijection between the real numbers and the decimal representations that do not end with infinitely many trailing 9.
The preceding considerations apply directly for every numeral base simply by replacing 10 with and 9 with