field of reals

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 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.

The set of real numbers is denoted or $\mathbb$ and is sometimes called "the reals".
The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality.

The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers, and include algebraic numbers (such as the square root ) and transcendental numbers (such as ).

Real numbers can be thought of as all points on an infinitely long line called the number line or real line, where the points corresponding to integers () are equally spaced.

Conversely, analytic geometry is the association of points on lines (especially axis lines) to real numbers such that geometric displacements are proportional to differences between corresponding numbers.

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.

Properties

Basic properties

* The real numbers include ''zero'' (), the ''additive identity'': adding to any real number leaves that number unchanged: .
* Every real number has an ''additive inverse'' satisfying .
* The real numbers include a ''unit'' (), the ''multiplicative identity'': multiplying by any real number leaves that number unchanged: .
* Every nonzero real number has a ''multiplicative inverse'' satisfying .
* Given any two real numbers and , the results of addition (), subtraction (), and multiplication () are also real numbers, as is the result of division () if is not zero. Thus the real numbers are closed under elementary arithmetic operations.
* The real numbers form a '' field''.
* The real numbers are ''linearly ordered''. For any distinct real numbers and , either or . If and then . (See also inequality (mathematics).)
* Any nonzero real number is either ''negative'' () or ''positive'' ().
* The real numbers are an ''ordered field'' because the order is compatible with addition and multiplication: if then ; if and then . Because the square of any real number is non-negative, and the sum and product of non-negative real numbers is itself non-negative, non-negative real numbers are a ''positive cone'' of .
* The real numbers make up an infinite set of numbers that cannot be injectively mapped to the infinite set of natural numbers, i.e., there are uncountably infinitely many real numbers, whereas the natural numbers are called ''countably infinite''. This establishes that in some sense, there are ''more'' real numbers than there are elements in any countable set.
* Any nonempty bounded ''open interval'' (the set of all real numbers between two specified endpoints) can be mapped bijectively by an affine function ( scaling and translation of the number line) to any other such interval. Every nonempty open interval contains uncountably infinitely many real numbers.
* The real numbers are ''unbounded''. There is no greatest or least real number; the real numbers extend infinitely in both positive and negative directions.
* There is a hierarchy of countably infinite subsets of the real numbers, e.g., the integers, the rational numbers, the algebraic numbers and the computable numbers, each set being a proper subset of the next in the sequence. The complements of each of these sets in the reals (irrational, transcendental, and non-computable real numbers) is uncountably infinite.
* Real numbers can be used to express measurements of continuous quantities. They may be expressed by decimal representations, most of them having an infinite sequence of digits to the right of the decimal point; these are often represented like 324.823122147..., where the ellipsis indicates that infinitely many digits have been omitted.

More formally, the real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that, if a nonempty set of real numbers has an upper bound, then it has a real least upper bound. The second condition distinguishes the real numbers from the rational numbers: for example, the set of rational numbers whose square is less than 2 is a set with an upper bound (e.g. 1.5) but no (rational) least upper bound: hence the rational numbers do not satisfy the least upper bound property.

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 (''x''_''n'') of real numbers is called a ''Cauchy sequence'' if for any there exists an integer ''N'' (possibly depending on ε) such that the distance is less than ε for all ''n'' and ''m'' that are both greater than ''N''. This definition, originally provided by Cauchy, formalizes the fact that the ''x''_''n'' eventually come and remain arbitrarily close to each other.

A sequence (''x''_''n'') ''converges to the limit'' ''x'' if its elements eventually come and remain arbitrarily close to ''x'', that is, if for any there exists an integer ''N'' (possibly depending on ε) such that the distance is less than ε for ''n'' greater than ''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
:$e^x = \sum_^ \frac$

converges to a real number for every ''x'', because the sums
:$\sum_^ \frac$

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

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element ''z'', is larger).

Additionally, an order can be Dedekind-complete, see . The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness; the description in § Completeness is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterization of the real numbers.) It is not true that $\mathbb$ is the ''only'' uniformly complete ordered field, but it is the only uniformly complete ''Archimedean field'', and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the ''largest'' Archimedean field in the sense that every other Archimedean field is a subfield of $\mathbb$. Thus $\mathbb$ is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

Cardinality

The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers. The cardinality of the set of all real numbers is denoted by $\mathfrak c.$ and called the cardinality of the continuum. It is strictly greater than the cardinality of the set of all natural numbers (denoted $\aleph_0$ and called 'aleph-naught'), and equals the cardinality of the power set of the set of the natural numbers.

The statement that there is no subset of the reals with cardinality strictly greater than $\aleph_0$ and strictly smaller than $\mathfrak c$ is known as the continuum hypothesis (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.

Advanced 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 ''x'' and ''y'' is defined as the absolute value . 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 $\mathbb$, although no negative number does. This shows that the order on $\mathbb$ is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make $\mathbb$ 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 ;1has 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 $\mathbb$. Ordered fields that satisfy the same first-order sentences as $\mathbb$ are called nonstandard models of $\mathbb$. 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 $\mathbb$), we know that the same statement must also be true of $\mathbb$.

The field $\mathbb$ of real numbers is an extension field of the field $\mathbb$ of rational numbers, and $\mathbb$ can therefore be seen as a vector space over $\mathbb$. Zermelo–Fraenkel set theory with the axiom of choice guarantees the existence of a basis (linear algebra), 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 $\mathbb$ with the property that every nonempty subset of $\mathbb$ has a least element in this ordering. (The standard ordering ≤ 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 V=L 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.

A real number may be either computable number, computable or uncomputable; either Algorithmically random sequence, algorithmically random or not; and either Algorithmically random sequence#Stronger than Martin-Löf randomness, arithmetically random or not.

History

Fraction (mathematics)#Simple, common, or vulgar fractions, Simple fractions were used by the History of Egypt, Egyptians around 1000 BC; the Vedic civilization, Vedic "Shulba Sutras" ("The rules of chords") in include what may be the first "use" of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematics, Indian mathematicians such as Manava , who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined. Around 500 BC, the Greek mathematics, Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational.

The Middle Ages brought about the acceptance of zero, negative numbers, integers, and Fraction (mathematics), fractional numbers, first by Indian mathematics, Indian and Chinese mathematics, Chinese mathematicians, and then by Mathematics in medieval Islam, Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects (the latter being made possible by the development of algebra). Arabic mathematicians merged the concepts of "number" and "Magnitude (mathematics), magnitude" into a more general idea of real numbers. The Egyptian mathematician Abū Kāmil Shujā ibn Aslam 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 Nth root, fourth roots). In Europe, such numbers, not commensurable with the numerical unit, were called ''irrational'' or nth_root, ''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" ones.

In the 18th and 19th centuries, there was much work on irrational and transcendental numbers. Johann Heinrich Lambert, Lambert (1761) gave a flawed proof that cannot be rational; Adrien-Marie Legendre, Legendre (1794) completed the proof and showed that is not the square root of a rational number. Joseph Liouville, Liouville (1840) showed that neither nor can be a root of an integer quadratic equation, and then established the existence of transcendental numbers; Georg Cantor, Cantor (1873) extended and greatly simplified this proof. Charles Hermite, Hermite (1873) proved that e (mathematical constant), is transcendental, and Ferdinand von Lindemann, Lindemann (1882), showed that is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), David Hilbert, Hilbert (1893), Adolf Hurwitz, Hurwitz, and Paul Gordan, Gordan.

The developers of calculus used real numbers without having defined them rigorously. The first rigorous definition was published by Cantor in 1871. In 1874, he showed that the set of all real numbers is uncountable, uncountably infinite, but the set of all algebraic numbers is countable, countably infinite. Cantor's first uncountability proof was different from his famous Cantor's diagonal argument, diagonal argument published in 1891.

Formal definitions

The real number system $(\mathbb ; + ; \cdot ; <)$ can be defined Axiomatic system, axiomatically up to an isomorphism, which is described hereafter. 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. Another approach is to start from some rigorous axiomatization of Euclidean geometry (say of Hilbert or of Alfred Tarski, 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.

Axiomatic approach

Let $\mathbb$ denote the Set (mathematics), set of all real numbers, then:

* The set $\mathbb$ is a field, meaning that addition and multiplication are defined and have the usual properties.
* The field $\mathbb$ is ordered, meaning that there is a total order ≥ such that for all real numbers ''x'', ''y'' and ''z'':
** if ''x'' ≥ ''y'', then ''x'' + ''z'' ≥ ''y'' + ''z'';
** if ''x'' ≥ 0 and ''y'' ≥ 0, then ''xy'' ≥ 0.
* The order is Dedekind-complete, meaning that every nonempty subset ''S'' of $\mathbb$ with an upper bound in $\mathbb$ has a least upper bound (a.k.a., supremum) in $\mathbb$.

The last property is what differentiates the real numbers from the rational numbers (and from Ordered field#Examples of ordered fields, other more exotic ordered fields). For example, $\$ has a rational upper bound (e.g., 1.42), but no ''least'' rational upper bound, because square root, $\sqrt$ is not rational.

These properties imply the Archimedean property (which is not implied by other definitions of completeness), which states that the set of integers has no upper bound in the reals. In fact, if this were false, then the integers would have a least upper bound ''N''; then, ''N'' – 1 would not be an upper bound, and there would be an integer ''n'' such that , and thus , which is a contradiction with the upper-bound property of ''N''.

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields $\mathbb_1$ and $\mathbb_2$, there exists a unique field isomorphism from $\mathbb_1$ to $\mathbb$. This uniqueness allows us to think of them as essentially the same mathematical object.

For another axiomatization of $\mathbb$, see Tarski's axiomatization of the reals.

Construction from the rational numbers

The real numbers can be constructed as a Complete metric space, completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...) Limit of a sequence, converges to a unique real number—in this case . For details and other constructions of real numbers, see construction of the real numbers.

Applications and connections

Physics

In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers, although actual measurements of physical quantities are of finite accuracy and precision.

Physicists have occasionally suggested that a more fundamental theory would replace the real numbers with quantities that do not form a continuum, but such proposals remain speculative.

Logic

The real numbers are most often formalized using the Zermelo–Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in Constructivism (mathematics), constructive mathematics.

The hyperreal numbers as developed by Edwin Hewitt, Abraham Robinson and others extend the set of the real numbers by introducing infinitesimal and infinite numbers, allowing for building infinitesimal calculus in a way closer to the original intuitions of Leibniz, Euler, Cauchy and others.

Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory).

The continuum hypothesis posits that the cardinality of the set of the real numbers is $\aleph_1$; i.e. the smallest infinite cardinal number after $\aleph_0$, the cardinality of the integers. Paul Cohen (mathematician), Paul Cohen proved in 1963 that it is an axiom independent of the other axioms of set theory; that is: one may choose either the continuum hypothesis or its negation as an axiom of set theory, without contradiction.

Computation

calculator, Electronic calculators and computers cannot operate on arbitrary real numbers, because finite computers cannot directly store infinitely many digits or other infinite representations. Nor do they usually even operate on arbitrary definable real numbers, which are inconvenient to manipulate.

Instead, computers typically work with finite-precision approximations called floating-point numbers, a representation similar to scientific notation. The achievable precision is limited by the computer data storage, data storage space allocated for each number, whether as fixed-point arithmetic, fixed-point, floating-point, or arbitrary-precision arithmetic, arbitrary-precision numbers, or some other representation. Most computational science, scientific computation uses binary number, binary floating-point arithmetic, often a double-precision floating-point format, 64-bit representation with around 16 decimal significant figures, digits of precision. Real numbers satisfy the field (mathematics)#Definition and illustration, usual rules of arithmetic, but floating-point arithmetic#Accuracy problems, floating-point numbers do not. The field of numerical analysis studies the numerical stability, stability and accuracy and precision, accuracy of numerical algorithms implemented with approximate arithmetic.

Alternately, computer algebra systems can operate on irrational quantities exactly by symbolic computation, manipulating symbolic formulas for them (such as $\sqrt,$ $\arctan 5,$ or $\int_0^1 x^x \,dx$) rather than their rational or decimal approximation. But exact and symbolic arithmetic also have limitations: for instance, they are computationally more expensive; it is not in general possible to determine whether two symbolic expressions are equal (the constant problem); and arithmetic operations can cause exponential growth, exponential explosion in the size of representation of a single number (for instance, squaring a rational number roughly doubles the number of digits in its numerator and denominator, and squaring a polynomial roughly doubles its number of terms), overwhelming finite computer storage.

A real number is called ''computable number, computable'' if there exists an algorithm that yields its digits. Because there are only countably infinite, countably many algorithms, but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem. Some constructivism (mathematics), constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

Set theory

In set theory, specifically descriptive set theory, the Baire space (set theory), Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

Vocabulary and notation

Mathematicians use mainly the symbol R to represent the Set (mathematics), set of all real numbers. Alternatively, it may be used $\mathbb$, the R, letter "R" in blackboard bold, which may be encoded in Unicode (and HTML) as . As this set 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 $\mathbb^+$ and $\mathbb^-$, respectively; $\mathbb_+$ and $\mathbb_-$ are also used.École Normale Supérieure of Paris
"" ("Real numbers")
, p. 6 The non-negative real numbers can be noted $\mathbb_$ but one often sees this set noted $\mathbb^+ \cup \.$ In French mathematics, the ''positive real numbers'' and ''negative real numbers'' commonly include zero, and these sets are noted respectively $\mathbb$ and $\mathbb_.$ In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted $\mathbb_^*$ and $\mathbb_^*.$

The notation $\mathbb^n$ refers to the set of the tuple, -tuples of elements of $\R$ (real coordinate space), which can be identified to the Cartesian product of copies of $\mathbb.$ It is an -dimensional vector space over the field of the real numbers, often called the coordinate space of dimension ; this space may be identified to the -dimensional Euclidean space as soon as a Cartesian coordinate system has been chosen in the latter. In this identification, a Point (geometry), 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 (mathematics), 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 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 described by a separation relation.
* The Long line (topology), long real line pastes together copies of the real line plus a single point (here denotes the reversed ordering of ) 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 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 matrix (mathematics), matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvector, 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

Citations

Sources

*
*
*
*
*
*
*
* Vol. 2, 1989. Vol. 3, 1990.
*
* Translated from the Germa
''Grundlagen der Analysis''
1930.
*
*

External links

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Real numbers,
Real algebraic geometry
Elementary mathematics