Real number

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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.^[a] Every real number can be almost uniquely represented by an infinite decimal expansion.^[b]

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.^[c]

The set of real numbers, sometimes called "the reals", is usually notated as a bold R or the blackboard bold ${\displaystyle \mathbb {R} }$.

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.^[1]

The real numbers include the rational numbers, such as the integer ${\displaystyle -5}$ and the fraction ${\displaystyle 4/3}$. 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.^[1]

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.^[d] 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.

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 a and b produces a real number denoted ${\displaystyle a+b,}$ which is the sum of a and b.
• The multiplication of two real numbers a and b produces a real number denoted ${\displaystyle ab,}$ ${\displaystyle a\cdot b}$ or ${\displaystyle a\times b,}$ which is the product of a and b.
• Addition and multiplication are both commutative, which means that ${\displaystyle a+b=b+a}$ and ${\displaystyle ab=ba}$ for all real numbers a and b.
• Addition and multiplication are both associative, which means that ${\displaystyle (a+b)+c=a+(b+c)}$ and ${\displaystyle (ab)c=a(bc)}$ for all real numbers a, b and c.
• Multiplication is distributive over addition, which means that ${\displaystyle a(b+c)=ab+ac}$ and ${\displaystyle (a+b)c=ac+bc}$ for all real numbers a, b and c.
• There is a real number called zero and denoted 0 which is an additive identity, which means that ${\displaystyle a+0=0+a=a}$ for every real number a.
• There is a real number denoted 1 which is a multiplicative identity, which means that ${\displaystyle a\times 1=1\times a=a}$ for every real number a.
• Every real number a has an additive inverse denoted ${\displaystyle -a.}$ This means that ${\displaystyle a+(-a)=(-a)+a=0}$ for every real number a.
• Every nonzero real number a has a multiplicative inverse denoted ${\displaystyle a^{-1}}$ or ${\displaystyle {\tfrac {1}{a}}.}$ This means that ${\displaystyle aa^{-1}=a^{-1}a=1}$ for every nonzero real number a.
• The total order is denoted ${\displaystyle a<b.}$ Being a total order means that it has the following two properties:

1. For any two real numbers a and b, exactly one of ${\displaystyle a<b}$, ${\displaystyle \;a=b}$, ${\displaystyle \;b<a}$ is true.
2. If ${\displaystyle a<b}$ and ${\displaystyle b<c,}$ then ${\displaystyle a<c}$.

• The order is compatible with addition and multiplication, which means that ${\displaystyle a<b}$ implies ${\displaystyle a+c<b+c}$ for every real number c, and ${\displaystyle 0<ab}$ is implied by ${\displaystyle 0<a}$ and ${\displaystyle 0<b.}$

Many other properties can be deduced from the above properties. In particular:

• ${\displaystyle 0\cdot a=0}$ for every real number a
• ${\displaystyle 0<1}$
• ${\displaystyle 0<a^{2}}$ for every nonzero real number a

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 ${\displaystyle S}$ is bounded above if there is a real number ${\displaystyle u}$ such that ${\displaystyle s\leq u}$ for all ${\displaystyle s\in S}$; such a ${\displaystyle u}$ is called an upper bound of ${\displaystyle S.}$ So, Dedekind completeness means that, if S 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 (see below), but also has some important consequences.

• Archimedean property: for every real number x, there is an integer n such that ${\displaystyle x<n}$ (take ${\displaystyle n=u+1,}$ where ${\displaystyle u}$ is the least upper bound of the integers less than x).
• Equivalently, if x is a positive real number, there is a positive integer n such that ${\displaystyle 0<{\tfrac {1}{n}}<x}$.
• Every positive real number x has a positive square root, that is, there exists a positive real number ${\displaystyle r}$ such that ${\displaystyle r^{2}=x.}$
• Every univariate polynomial of odd degree with real coefficients has at least one real root (if the leading coefficient is positive, take the least upper bound of real numbers for which the value of the polynomial is negative).

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 x whose decimal representation extends k places to the left, the standard notation is the juxtaposition of the digits ${\displaystyle b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots ,}$ in descending order by power of ten, with non-negative and negative powers of ten separated by a decimal point, representing the infinite series

${\displaystyle x=b_{k}10^{k}+b_{k-1}10^{k-1}+\cdots +b_{0}+{\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2}}}+\cdots .}$

For example, for the circle constant ${\displaystyle \pi =3.14159\cdots ,}$ k is zero and ${\displaystyle b_{0}=3,}$ ${\displaystyle a_{1}=1,}$ ${\displaystyle a_{2}=4,}$ etc.

More formally, a decimal representation for a nonnegative real number x consists of a nonnegative integer k and integers between zero and nine in the infinite sequence

${\displaystyle b_{k},b_{k-1},\ldots ,b_{0},a_{1},a_{2},\ldots .}$

(If ${\displaystyle k>0,}$ then by convention ${\displaystyle b_{k}\neq 0.}$)

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 n, the truncation of the sequence at the place n is the finite partial sum

${\displaystyle {\begin{aligned}D_{n}&=b_{k}10^{k}+b_{k-1}10^{k-1}+\cdots +b_{0}+{\frac {a_{1}}{10}}+\cdots +{\frac {a_{n}}{10^{n}}}\\&=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{j=1}^{n}a_{j}10^{-j}\end{aligned}}}$

The real number x defined by the sequence is the least upper bound of the ${\displaystyle D_{n},}$ which exists by Dedekind completeness.

Conversely, given a nonnegative real number x, one can define a decimal representation of x by induction, as follows:

Define ${\displaystyle b_{k}\cdots b_{0}}$ as decimal representation of the largest integer ${\displaystyle D_{0}}$ such that ${\displaystyle D_{0}\leq x}$ (this integer exists because of the Archimedean property). Then, supposing by induction that the decimal fraction ${\displaystyle D_{i}}$ has been defined for ${\displaystyle i<n,}$ one defines ${\displaystyle a_{n}}$ as the largest digit such that ${\displaystyle D_{n-1}+{\tfrac {a_{n}}{10^{n}}}\leq a,}$ and one sets ${\displaystyle D_{n}=D_{n-1}+{\tfrac {a_{n}}{10^{n}}}.}$

One can use the defining properties of the real numbers to show that x is the least upper bound of the ${\displaystyle D_{n}.}$ So, the resulting sequence of digits is called a decimal representation of x.

Another decimal representation can be obtained by replacing ${\displaystyle \leq x}$ with ${\displaystyle <x}$ in the preceding construction. These two representations are identical, unless x is a decimal fraction of the form ${\textstyle {\tfrac {m}{10^{h}}}.}$ In this case, in the first decimal representation, all ${\displaystyle a_{n}}$ are zero for ${\displaystyle n>h,}$ and, in the second representation, all ${\displaystyle a_{n}}$ 9. (see 0.999... for details).

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 ${\displaystyle B\geq 2,}$ simply by replacing 10 with ${\displaystyle B}$ and 9 with ${\displaystyle B-1.}$

Topological completeness

A main reason for using real numbers is so that many sequences have limits. More formally, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section):

A sequence ${\displaystyle (x_{n})}$ of real numbers is called a Cauchy sequence if for any ${\displaystyle \varepsilon >0}$ there exists an integer ${\displaystyle N}$ (possibly depending on ${\displaystyle \varepsilon }$) such that the distance ${\displaystyle |x_{n}-x_{m}|}$ is less than ${\displaystyle \varepsilon }$ for all ${\displaystyle n}$ and ${\displaystyle m}$ that are both greater than ${\displaystyle N}$. This definition, originally provided by Cauchy, formalizes the fact that the ${\displaystyle x_{n}}$ eventually come and remain arbitrarily close to each other.

A sequence ${\displaystyle (x_{n})}$ converges to the limit ${\displaystyle x}$ if its elements eventually come and remain arbitrarily close to ${\displaystyle x}$, that is, if for any ${\displaystyle \varepsilon >0}$ there exists an integer ${\displaystyle N}$ (possibly depending on ${\displaystyle \varepsilon }$) such that the distance ${\displaystyle |x_{n}-x|}$ is less than ${\displaystyle \varepsilon }$ for ${\displaystyle n}$ greater than ${\displaystyle N}$.

Every convergent sequence is a Cauchy sequence, and the converse is true for real numbers, and this means that the topological space of the real numbers is complete.

The set of rational numbers is not complete. For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).

The completeness property of the reals is the basis on which calculus, and more generally mathematical analysis, are built. In particular, the test that a sequence is a Cauchy sequence allows proving that a sequence has a limit, without computing it, and even without knowing it.

For example, the standard series of the exponential function

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$

converges to a real number for every ${\displaystyle x}$, because the sums

${\displaystyle \sum _{n=N}^{M}{\frac {x^{n}}{n!}}}$

can be made arbitrarily small (independently of ${\displaystyle M}$) by choosing ${\displaystyle N}$ sufficiently large. This proves that the sequence is Cauchy, and thus converges, showing that ${\displaystyle e^{x}}$ is well defined for every ${\displaystyle x}$.

Cardinality

The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers {1, 2, 3, 4, ...} and the set of all real numbers are infinite sets, there exists no one-to-one function from the real numbers to the natural numbers. The cardinality of the set of all real numbers is called the cardinality of the continuum and commonly denoted by ${\displaystyle {\mathfrak {c}}.}$ It is strictly greater than the cardinality of the set of all natural numbers, denoted ${\displaystyle \aleph _{0}}$ and called aleph-zero or aleph-nought. The cardinality of the continuum equals the cardinality of the power set of the natural numbers, that is, the set of all subsets of the natural numbers.

The statement that there is no cardinality strictly greater than ${\displaystyle \aleph _{0}}$ and strictly smaller than ${\displaystyle {\mathfrak {c}}}$ is known as the continuum hypothesis (CH). The axiom system most commonly used in mathematics, Zermelo-Fraenkel set theory with the axiom of choice (ZFC), is insufficient to decide whether CH holds: assuming that ZFC is consistent, CH can be neither proved nor disproved within ZFC, since some models of ZFC satisfy CH, while others violate it.^[2]

Other properties

As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

The real numbers form a metric space: the distance between ${\displaystyle x}$ and ${\displaystyle y}$ is defined as the absolute value ${\displaystyle |x-y|}$. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.

Every nonnegative real number has a square root in ${\displaystyle \mathbb {R} }$, although no negative number does. This shows that the order on ${\displaystyle \mathbb {R} }$ is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make ${\displaystyle \mathbb {R} }$ the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.

The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalized such that the unit interval ${\displaystyle [0;1]}$ has measure 1. There exist sets of real numbers that are not Lebesgue measurable, e.g. Vitali sets.

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as ${\displaystyle \mathbb {R} }$. Ordered fields that satisfy the same first-order sentences as ${\displaystyle \mathbb {R} }$ are called nonstandard models of ${\displaystyle \mathbb {R} }$. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in ${\displaystyle \mathbb {R} }$), we know that the same statement must also be true of ${\displaystyle \mathbb {R} }$.

The field ${\displaystyle \mathbb {R} }$ of real numbers is an extension field of the field ${\displaystyle \mathbb {Q} }$ of rational numbers, and ${\displaystyle \mathbb {R} }$ can therefore be seen as a vector space over ${\displaystyle \mathbb {Q} }$. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis of this vector space: there exists a set B of real numbers such that every real number can be written uniquely as a finite linear combination of elements of this set, using rational coefficients only, and such that no element of B is a rational linear combination of the others. However, this existence theorem is purely theoretical, as such a base has never been explicitly described.

The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on ${\displaystyle \mathbb {R} }$ with the property that every nonempty subset of ${\displaystyle \mathbb {R} }$ has a least element in this ordering. (The standard ordering ${\displaystyle \leq }$ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described. If the axiom of constructibility is assumed in addition to the axioms of ZF, a well ordering of the real numbers can be shown to be explicitly definable by a formula.^[3]

A real number may be either computable or uncomputable; either algorithmically random or not; and either arithmetically random or not.

History

Simple fractions were used by the Egyptians around 1000 BC; the Vedic "Shulba Sutras" ("The rules of chords") in c. 600 BC include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava (c. 750–690 BC), who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined.^[4]

Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational.

For Greek mathematicians, numbers were only the natural numbers. Real numbers were called "proportions", being the ratios of two lengths, or equivalently being measures of a length in terms of another length, called unit length. Two lengths are "commensurable", if there is a unit in which they are both measured by integers, that is, in modern terminology, if their ratio is a rational number. Eudoxus of Cnidus (c. 390−340 BC) provided a definition of the equality of two irrational proportions in a way that is similar to Dedekind cuts (introduced more than 2,000 years later), except that he did not use any arithmetic operation other than multiplication of a length by a natural number (see Eudoxus of Cnidus). This may be viewed as the first definition of the real numbers.

The Middle Ages brought about the acceptance of zero, negative numbers, integers, and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra).^[5] Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers.^[6] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations, or as coefficients in an equation (often in the form of square roots, cube roots, and fourth roots).^[7] In Europe, such numbers, not commensurable with the numerical unit, were called irrational or surd ("deaf").

In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard.

In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" numbers.

In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Lambert (1761) gave a flawed proof that π cannot be rational; Legendre (1794) completed the proof^[8] and showed that π is not the square root of a rational number.^[9] Liouville (1840) showed that neither e nor e² can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Cantor (1873) extended and greatly simplified this proof.^[10] Hermite (1873) proved that e is transcendental, and Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), Hilbert (1893), Hurwitz,^[11] and Gordan.^[12]

The concept that many points existed between rational numbers, such as the square root of 2, was well known to the ancient Greeks. The existence of a continuous number line was considered self-evident, but the nature of this continuity, presently called completeness, was not understood. The rigor developed for geometry did not cross over to the concept of numbers until the 1800s.^[13]

Formal definitions

The real number system ${\displaystyle (\mathbb {R}$ ;{}+{};{}\cdot {};{}<{})} can be defined axiomatically up to an isomorphism, which is described hereinafter. There are also many ways to construct "the" real number system, and a popular approach involves starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers.^[16] Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or of Tarski), and then define the real number system geometrically. All these constructions of the real numbers have been shown to be equivalent, in the sense that the resulting number systems are isomorphic.

Applications and connections

Vocabulary and notation

The set of all real numbers is denoted as ${\displaystyle \mathbb {R} }$ (blackboard bold) or R (upright bold).^[22]

As it is naturally endowed with the structure of a field, the expression field of real numbers is frequently used when its algebraic properties are under consideration.

The sets of positive real numbers and negative real numbers are often noted ${\displaystyle \mathbb {R} ^{+}}$ and ${\displaystyle \mathbb {R} ^{-}}$,^[23] respectively; ${\displaystyle \mathbb {R} _{+}}$ and ${\displaystyle \mathbb {R} _{-}}$ are also used.^[24] The non-negative real numbers can be noted ${\displaystyle \mathbb {R} _{\geq 0}}$ but one often sees this set noted ${\displaystyle \mathbb {R} ^{+}\cup \{0\}.}$^[23]

The notation ${\displaystyle \mathbb {R} ^{n}}$ refers to the set of the n-tuples of elements of ${\displaystyle \mathbb {R} }$ (real coordinate space), which can be identified to the Cartesian product of n copies of ${\displaystyle \mathbb {R} .}$ It is an n-dimensional vector space over the field of the real numbers, often called the coordinate space of dimension n; this space may be identified to the n-dimensional Euclidean space as soon as a Cartesian coordinate system has been chosen in the latter. In this identification, a point of the Euclidean space is identified with the tuple of its Cartesian coordinates.

In mathematics real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").

Generalizations and extensions

The real numbers can be generalized and extended in several different directions:

• The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field. The complex numbers contain the real numbers. Sometimes a complex number which turns out to be real is said to have the property of "reality", as in "the reality of the eigenvalues".^[25]
• The affinely extended real number system adds two elements +∞ and −∞. It is a compact space. It is no longer a field, or even an additive group, but it still has a total order; moreover, it is a complete lattice.
• The real projective line adds only one value ∞. It is also a compact space. Again, it is no longer a field, or even an additive group. However, it allows division of a nonzero element by zero. It has cyclic order with topology described by point-pair separation.
• The long real line pastes together ${\displaystyle \aleph _{1}^{*}+\aleph _{1}}$ copies of the real line plus a single point (here ${\displaystyle \aleph _{1}^{*}}$ denotes the reversed ordering of ${\displaystyle \aleph _{1}}$) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ${\displaystyle \aleph _{1}}$ in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group.
• Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and are therefore non-Archimedean ordered fields.
• Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.

See also

• Completeness of the real numbers
• Continued fraction
• Definable real numbers
• Positive real numbers
• Real analysis

Notes

References

External links

• "Real number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]