
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the point on an infinite line (mathematics)—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational number or irrational number; algebraic number or transcendental number; and positive number, negative number, or 0 (number). Real numbers measure continuum quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247... (where the three dots express that there would still be more digits to come, no matter how many more might be added at the end). Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be ''computable number'' if there exists an algorithm that yields its digits. Because there are only countably infinite many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivism (math) accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point arithmetic numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. Mathematicians use the symbol R (or alternatively, , the letter "R" in blackboard bold) to represent the set of all real numbers. The mathematical notation R''n'' refers to an ''n''-dimension space of real numbers; for example, a value from R3 consists three real numbers and specifies a location in 3-dimensional space (also known as 3-D). In mathematics, the term "real XXX" means that the underlying field is the field of real numbers. For example matrix (mathematics), polynomial and Lie algebra. == History == Vulgar fractions had been used by the History of Egypt around 1000 BC; around 500 BC, the History of Greece mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers were invented by Indian mathematicians around 600 Anno Domini, and then possibly reinvented in China shortly after. They were not used in Europe until the 17th century, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. == Definition == === Construction from the rational numbers === The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers. === Axiomatic approach === Let R denote the set of all real numbers. Then: * The set R is a field (mathematics), i.e., addition, subtraction, multiplication and division (mathematics) are defined and have the usual properties. * The field R is ordered field, i.e., 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 completion, i.e., every empty set subset ''S'' of R with an upper bound in R has a supremum (also called supremum) in R. The last property is what differentiates the reals from the rational number. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object. == Properties == === Completeness === The main reason for introducing the reals is that the reals contain all limit (mathematics). More technically, the reals are completeness (topology) (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). This means the following: A sequence (''x''''n'') of real numbers is called a ''Cauchy sequence'' if for any ε > 0 there exists an integer ''N'' (possibly depending on ε) such that the distance |''x''''n'' - ''x''''m''| is less than ε provided that ''n'' and ''m'' are both greater than ''N''. In other words, a sequence is a Cauchy sequence if its elements ''x''''n'' eventually come and remain arbitrarily close to each other. A sequence (''x''''n'') ''converges to the limit'' ''x'' if for any ε > 0 there exists an integer ''N'' (possibly depending on ε) such that the distance |''x''''n'' - ''x''| is less than ε provided that ''n'' is greater than ''N''. In other words, a sequence has limit ''x'' if its elements eventually come and remain arbitrarily close to ''x''. It is easy to see that every convergent sequence is a Cauchy sequence. Now the important fact about the real numbers is that the converse is true: :Every Cauchy sequence of real numbers is convergent. That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example the standard series of the exponential function : converges to a real number because for every ''x'' the sums : can be made arbitrarily small by choosing ''N'' sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we do not know ahead of time what the limit is. === "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 complete lattice. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element ''z'', ''z'' + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind completion, as defined in the section Axioms. 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 (mathematics) (and a field is a group under the operations of addition and subtraction) defines a uniform space structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above 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 characterisation of the real numbers.) It is not true that R 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". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies 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 R. Thus R 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. === Advanced properties === The reals are uncountable, that is, there are strictly more real numbers than natural numbers (even though both sets are infinity). This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2ω (see cardinality of the continuum), i.e., the cardinality of the set of subsets of the natural numbers. Since only a countable set of real numbers can be algebraic number, almost all real numbers are transcendental number. The nonexistence of a subset of the reals with cardinality strictly in between that of the integers and the reals is known as the continuum hypothesis. This can neither be proved nor be disproved, because it is independent from the axioms of set theory. The real numbers form a metric space: the distance between ''x'' and ''y'' is defined to be the absolute value |''x'' - ''y''|. By virtue of being a total order set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected space and simply connected), local compactness separable metric space, of dimension 1, and are first category. The real numbers are not compact space. There are various properties that uniquely specify them; for instance, all unbounded, continuous, and separable total order are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make 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 normalised such that the unit interval [0,1] has measure 1. 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 is much bigger than R but also satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of 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 R), we know that the same statement must also be true of R. == Generalizations and extensions == The real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean group. Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers. Hermitians on a Hilbert space (for example, self-adjoint square complex matrix (math)) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvector 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. Elementary mathematics Real numbers Set theory
I've just (mistakenly) did an update to this entry. ''No changes have been made at all''. The update I meant to do was that of the Portuguese version, which I have just forked off the English one. --Doshell ---- "Real numbers" has just been redirected to "Real number". Fine with me, but presently "Rational number" redirects to "Rational number". What do we want to do, Wikipedians? :Thanks for noticing that "rational number" has the same problem--it's fixed now. The singular form is preferred for simple nouns like this to make linking easier. For example, ''"...probability can be expressed as a :real number in the interval [0,1]..."'' --User:Lee Daniel Crocker ---- ''A real number is one that can be expressed in the form 'DDD.ddd'. DDD is zero or more decimal digits ddd is zero or more decimal digits Of course, DDD must be finite in length. This restriction does not apply to ddd.'' Why must DDD be finite in length? If a sequence of real numbers goes to infinity, then there must be an (countably) infinite number of digits in ...DDD. What am I missing? :I don't really understand the question. The sequence 101,102,103,104,... goes to infinity, but none of the numbers have infinite digits. :In effect, a sequence of numbers may go to infinity, but a single number can't. (Consider the problem of comparing two such "infinite integers". How could you decide which was bigger without calculating all the (infinite) digits ..DDDD for both numbers. ---- ''If we have a space where Cauchy sequences are meaningful (a metric space, i.e. a space where distance is defined), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completing). When applied to the rational numbers, it gives the following useful construction of the real numbers:'' It should point out that this only works with a Euclidean metric or one equivalent to a Euclidean metric; using other metrics gives you the p-adic numbers instead. ---- The following comment was moved from the main page: ''RB: The dimension is actually difficult to define: the reals have dimension 1 for pretty much any sensible definition, but the best definition I know is that cohomology with compact support is non trivial in dimension 1 and vanishes above it.'' :That's the cohomology dimension; I don't see why it's better than any other. ---- Tarquin, do you plan to add to the symbols for subsets of the real line? Otherwise, I'll add to it, including the uses of the symbols. Right now, it's not very clear. — User:Toby Bartels 07:16 Aug 3, 2002 (PDT) ---- As I understand it, ''local field'' means a field complete with respect to a discrete valuation. Are the reals and complexes really local fields? --User:Alodyne I moved the statement here for the moment, until we sort this out: :The reals are one of the two local fields of field (mathematics) 0 (the other one being the complex numbers). See also Talk:Local field. User:AxelBoldt 19:38 Nov 9, 2002 (UTC) Oh and by the way: at least one change certainly must be made in that statement since the ''p''-adic numbers are local fields of characteristic 0 by all the various definitions I've seen. Any dispute? --User:Alodyne :No. The Encyclopedic Dictionary of Mathematics defines a local field as a field that's complete with respect to a discrete valuation and such that it's residue field is finite. They mention that the the reals and complex numbers are also sometimes considered as local fields, but they explicitly exclude them. User:AxelBoldt 19:14 Nov 12, 2002 (UTC) ---- To tie up this article with the entries on Model Theory, can someone tell me whether or not there is a first-order theory model for the real numbers? A maths teacher friend of mine told me that there is, but this would imply, by Lowenhein-Skolem theorems that the reals are denumerable, or have a countable model, which which seem to be inconsistent with Cantor's diagonal proof of uncountability of the Reals. --B. Smith. No, it would not imply that. It would imply only that there is a countable field satisfying all of the same first-order sentences that the real numbers satisfy. The Loewenheim-Skolem theorem does imply the existence of such a model. If the model and the language are elaborate enough, one could write a first-order sentence saying the reals are not countable. It would be true within the countable model. That means that although there would be a sequence containing every member of the model, such a sequence could not itself belong to the model; there could be no enumeration within the model. User:Michael Hardy 00:12 Apr 11, 2003 (UTC) ---- "The term 'real number' is a retronym coined in response to 'imaginary number'." That hardly seems likely, since extension to the reals should always have seemed more urgent than extension to complex, and the reals need a name as soon as you consider them. IMO, real is intended to contrast with rational, and the motivation for "real" is that the rationals are an unrealistically limited set. That follows from the mind necessarily regarding all integer-legged right triangles as precise models of something from reality, if anything in math beyond counting is such a model; from irrational hypotenuses, it follows that rationals aren't quite the full set of numbers needed to correspond to reality, so the choice of the term "real number" reflects the hypothesis that there are no problems in the real world that the reals don't suffice to express the answer to. (In contrast, wanting imaginaries and quaternions takes some sophisticated mathematical abstraction.) In fact, the best evidence that "real" led to "imaginary" (not the other way around) is that "real" is a better fit to the reals than "imaginary" is to the imaginaries: IIRC how one of my physics profs put it, it's not so much a matter of complex numbers having an imaginary part as that they have *two* parts, both real numbers representing something in the physical world, and it's just that there are systems like electromagnetic fields whose behavior looks simpler if you do the bookkeeping by labeling the electric field as the first component of a complex number and the magnetic as the second, and hiding the fact that the "multiplication" you do is, according to the rules of complex arithmetic, more complicated than the multiplication table you learned in grade school. So where's this "retronym" thing coming from; is it more than someone's conjecture? --User:Jerzy 06:41, 30 Sep 2003 (UTC) I don't think there was a monolithic nomenclature that all mathematicians subscribed to at the time when negative numbers, imaginary numbers, and real numbers were being investigated and systematized (15th century through 19 century-ish). History is messy like that. It is well known that negative numbers (first) and imaginary numbers (later) inspired censure (and sometimes even disgust) in the European mathematicians of the day, so the retronym thing is at least plausible, e.g. "The square root of -1? That's not a ''real'' number, it's some kind of imaginary thing!" I'd like to see a source too though. -- User:Cyan 07:06, 30 Sep 2003 (UTC) :Just for information, the first reference to ''imaginary'' numbers in the Oxford English Dictionary is to Descartes (in French) in 1637, and the first reference in English is in 1706: ''W. JONES Syn. Palmar. Matheseos 127 The Original Components or Roots of all Equations, may be either Affirmative, Negative, Mix'd, or Imaginary.'' In contrast, the first reference to a real number that they give is in a 1727 encyclopedia: ''CHAMBERS Cycl. s.v. Root, If the value of x be positive, i.e. if x be a positive quantity,..the root [of an equation] is called a real or true root''. (These are actually citations for the use of ''real'' or ''imaginary'' as adjectives with the modern meaning, so they weren't looking specifically for the phrase ''real number'', which they don't cite until the 1910 Encyc. Brit.) Although I'm not sure the OED is as good about backdating mathematical terms as they are in general, this gives some indication at least. Anyway, since a rigorous definition and theory of real numbers wasn't really developed until the 19th century, it's hard to call it a ''retronym'' (what people had looked at in the past ''were'' real numbers but didn't ''define'' them; e.g. if you look at just the roots of polynomials, those are the ''algebraic'' numbers which are a countable subset of the reals). User:Stevenj 02:02, 12 Oct 2003 (UTC) ---- The article begins with: :''The real numbers are practically any numbers that can be expressed.'' I have to say I find this unsatisfying, especially since ''most'' (i.e. all but a countable subset of) real numbers arguably ''cannot'' be specifically expressed. That is, if you take the meaning of ''expressed'' in a natural way: i.e. to be uniquely defined by a finite-length description (such as 4.73, √2, sin(1), ...). And only a strict subset of these ''expressible'' numbers are computable (i.e. to an arbitrary precision in a finite time). User:Stevenj 01:34, 12 Oct 2003 (UTC) ---- :''It's impossible to explicitly specify a non-recursive number;'' That depends on what "explicitly specify" means. Consider, for example, Chaitin's constant. This is non-recursive, but in some sense can be specified. User:Josh Cherry 23:31, 19 Oct 2003 (UTC) And what is this Russian school that assumes all numbers are recursive? This sounds like a provably false statement to me, unless something nonobvious is meant by "recursive algorithm". It's provably impossible to specify every real numbers uniquely with a recursive algorithm in the computer-science sense, i.e. a finite-size finite-state machine, or equivalently a finite-length program in any Turing-complete computer language. (There are only a countable number of finite-length computer programs.) (One can even explicitly define a real number, whose digits are e.g. based on the halting problem, that is uniquely specified by a finite-length description but which is not computable in finite time by any program.) User:Stevenj 20:00, 21 Oct 2003 (UTC) Certainly one can explicitly specify particular non-computable numbers. I haven't read everything above, so I'm not certain which \"Russian school\" is referred to, but it's probably something about constructivism, which is a philosophy that holds that an existence proof is not valid unless it "constructs" the object whose existence is to be proved. For example, if you were to deduce a contradiction from the proposition that every even number greater than 2 is a sum of two primes, that would not be taken by constructivists to be a proof of the existence of a counterexample. User:Michael Hardy 00:03, 22 Oct 2003 (UTC) :However, ''most'' (all but a countable subset of) real numbers cannot be uniquely specified by a finite description of any sort (whether by a computer algorithm or otherwise). The current Wikipedia statement about the Russian school needs clarification (or deletion), because on its face it seems to imply the contrary. User:Stevenj 04:11, 22 Oct 2003 (UTC) == Quantity Box == This has been removed several times by one person. I am among those who sees it as an enhancement of the article and the "explanations" for its removal wrong -- unless the word is just "silly". It's time to stop deleting and start defending the preceding deletions on this page: convince someone if you're so sure. --User:JerzyUser talk:Jerzy 01:56, 2004 Mar 28 (UTC) :The box is not silly but unnecessary and distracting. The box is already too long (you have to scroll to see the whole list and it has a potential to get longer certainly. This kind of the box is only useful and necessary in cases like 1) the article is separated into several pages because the article would be too long if those were in one page. 2) The article is about one entity among several ones. There is no much relationship between real and natural numbers except they are numbers. You want me to convince you then can I also ask you to convince me first? To my knowledge, there was no consensus about having such a box. In other words, I am just reverting the article to what it used to be and see if we can have concensus or something. -- User:TakuyaMurata 03:04, Mar 28, 2004 (UTC) :I've just found this page. Wikipedia_talk:Article_series. It would help us to see whether to have a box or not. -- User:TakuyaMurata 16:08, Mar 28, 2004 (UTC) == Negatives are that recent? == From the History section: "Negative numbers began to be generally accepted in the 1600s and were invented by Muslim mathematicians." I find it hard to believe that negative numbers were just being accepted in Isaac Newton's time. I'm no history geek, but I'd like to get that confirmed. :Same here. Newton obviously used negative numbers, and what about Napier? I had thought that negative numbers were introduced by medieval Italian accountants. User:Michael Hardy 20:59, 6 Dec 2004 (UTC) :A serious mistake here. Chinese mathematicians (indeed they maybe more "accountants" than "mathematicians") developed similar concept long time ago. I believe that negative numbers have been reinvented many times, as surplus and deficiency are fundamental concepts in commerce. -User:Wshun 11:45, 10 Dec 2004 (UTC) ==Packing real numbers together== You can "pack" any finite number of integers or rational numbers into one (e.g. by alternating digits), but is this legal for real numbers? User:Fredrik | User talk:Fredrik 02:47, 20 Feb 2005 (UTC) : Yes you can do it with real numbers too, at least when they are all in the interval [0, 1). : Start by writing each of those numbers in the form math>0.a_1a_2a_3\dots, and for uniqueness, suppose that none of your numbers after a while contain only 9's. :Then, write the first digit of the first number, then the first digit of the second number, all the way to the first digit of the last number. Then work on the second digit of each number and so on. :Now, presumably you could do same thing without the assumption that the numbers are in [0, 1), but then you need to worry in addition about the exponent. User:Oleg Alexandrov 19:12, 20 Feb 2005 (UTC) == "Almost all" means...? == : ''Since only a countable set of real numbers can be algebraic number, almost all real numbers are transcendental number.'' This is somewhat puzzling, because the link to the "almost all" article says : ''"Almost all" is sometimes used synonymously with "all but finitely many"; see almost.'' and the other meanings given don't apply. Surely there are not just finitely many algebraic numbers? -- User:KittySaturn 08:47, 2005 May 23 (UTC) :The meaning is actually more in line with the description at ''almost everywhere''. That is, the set of algebraic numbers have measure zero. -- User:Fropuff 15:26, 2005 May 23 (UTC) ::I've taken no number theory classes, so I may very well be wrong about this, but if i'm reading http://mathworld.wolfram.com/AlmostAll.html correctly then "almost all" can also mean "all of an uncountable set but countable number". —User:MilesKUser_talk:MilesK 19:50, May 23, 2005 (UTC)
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Words begining with Real_number:
Real_number
Real_number
Real_Numbers
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Real number
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the point on an infinite line (mathematics)—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational number or irrational number; algebraic number or transcendental number; and positive number, negative number, or 0 (number). Real numbers measure continuum quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247... (where the three dots express that there would still be more digits to come, no matter how many more might be added at the end). Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be ''computable number'' if there exists an algorithm that yields its digits. Because there are only countably infinite many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivism (math) accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point arithmetic numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. Mathematicians use the symbol R (or alternatively, , the letter "R" in blackboard bold) to represent the set of all real numbers. The mathematical notation R''n'' refers to an ''n''-dimension space of real numbers; for example, a value from R3 consists three real numbers and specifies a location in 3-dimensional space (also known as 3-D). In mathematics, the term "real XXX" means that the underlying field is the field of real numbers. For example matrix (mathematics), polynomial and Lie algebra. == History == Vulgar fractions had been used by the History of Egypt around 1000 BC; around 500 BC, the History of Greece mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers were invented by Indian mathematicians around 600 Anno Domini, and then possibly reinvented in China shortly after. They were not used in Europe until the 17th century, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. == Definition == === Construction from the rational numbers === The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers. === Axiomatic approach === Let R denote the set of all real numbers. Then: * The set R is a field (mathematics), i.e., addition, subtraction, multiplication and division (mathematics) are defined and have the usual properties. * The field R is ordered field, i.e., 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 completion, i.e., every empty set subset ''S'' of R with an upper bound in R has a supremum (also called supremum) in R. The last property is what differentiates the reals from the rational number. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R1 and R2, there exists a unique field isomorphism from R1 to R2, allowing us to think of them as essentially the same mathematical object. == Properties == === Completeness === The main reason for introducing the reals is that the reals contain all limit (mathematics). More technically, the reals are completeness (topology) (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). This means the following: A sequence (''x''''n'') of real numbers is called a ''Cauchy sequence'' if for any ε > 0 there exists an integer ''N'' (possibly depending on ε) such that the distance |''x''''n'' - ''x''''m''| is less than ε provided that ''n'' and ''m'' are both greater than ''N''. In other words, a sequence is a Cauchy sequence if its elements ''x''''n'' eventually come and remain arbitrarily close to each other. A sequence (''x''''n'') ''converges to the limit'' ''x'' if for any ε > 0 there exists an integer ''N'' (possibly depending on ε) such that the distance |''x''''n'' - ''x''| is less than ε provided that ''n'' is greater than ''N''. In other words, a sequence has limit ''x'' if its elements eventually come and remain arbitrarily close to ''x''. It is easy to see that every convergent sequence is a Cauchy sequence. Now the important fact about the real numbers is that the converse is true: :Every Cauchy sequence of real numbers is convergent. That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example the standard series of the exponential function : converges to a real number because for every ''x'' the sums : can be made arbitrarily small by choosing ''N'' sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we do not know ahead of time what the limit is. === "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 complete lattice. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element ''z'', ''z'' + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind completion, as defined in the section Axioms. 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 (mathematics) (and a field is a group under the operations of addition and subtraction) defines a uniform space structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above 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 characterisation of the real numbers.) It is not true that R 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". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies 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 R. Thus R 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. === Advanced properties === The reals are uncountable, that is, there are strictly more real numbers than natural numbers (even though both sets are infinity). This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2ω (see cardinality of the continuum), i.e., the cardinality of the set of subsets of the natural numbers. Since only a countable set of real numbers can be algebraic number, almost all real numbers are transcendental number. The nonexistence of a subset of the reals with cardinality strictly in between that of the integers and the reals is known as the continuum hypothesis. This can neither be proved nor be disproved, because it is independent from the axioms of set theory. The real numbers form a metric space: the distance between ''x'' and ''y'' is defined to be the absolute value |''x'' - ''y''|. By virtue of being a total order set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected space and simply connected), local compactness separable metric space, of dimension 1, and are first category. The real numbers are not compact space. There are various properties that uniquely specify them; for instance, all unbounded, continuous, and separable total order are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make 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 normalised such that the unit interval [0,1] has measure 1. 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 is much bigger than R but also satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of 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 R), we know that the same statement must also be true of R. == Generalizations and extensions == The real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean group. Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers. Hermitians on a Hilbert space (for example, self-adjoint square complex matrix (math)) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvector 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. Elementary mathematics Real numbers Set theory
Real number
I've just (mistakenly) did an update to this entry. ''No changes have been made at all''. The update I meant to do was that of the Portuguese version, which I have just forked off the English one. --Doshell ---- "Real numbers" has just been redirected to "Real number". Fine with me, but presently "Rational number" redirects to "Rational number". What do we want to do, Wikipedians? :Thanks for noticing that "rational number" has the same problem--it's fixed now. The singular form is preferred for simple nouns like this to make linking easier. For example, ''"...probability can be expressed as a :real number in the interval [0,1]..."'' --User:Lee Daniel Crocker ---- ''A real number is one that can be expressed in the form 'DDD.ddd'. DDD is zero or more decimal digits ddd is zero or more decimal digits Of course, DDD must be finite in length. This restriction does not apply to ddd.'' Why must DDD be finite in length? If a sequence of real numbers goes to infinity, then there must be an (countably) infinite number of digits in ...DDD. What am I missing? :I don't really understand the question. The sequence 101,102,103,104,... goes to infinity, but none of the numbers have infinite digits. :In effect, a sequence of numbers may go to infinity, but a single number can't. (Consider the problem of comparing two such "infinite integers". How could you decide which was bigger without calculating all the (infinite) digits ..DDDD for both numbers. ---- ''If we have a space where Cauchy sequences are meaningful (a metric space, i.e. a space where distance is defined), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completing). When applied to the rational numbers, it gives the following useful construction of the real numbers:'' It should point out that this only works with a Euclidean metric or one equivalent to a Euclidean metric; using other metrics gives you the p-adic numbers instead. ---- The following comment was moved from the main page: ''RB: The dimension is actually difficult to define: the reals have dimension 1 for pretty much any sensible definition, but the best definition I know is that cohomology with compact support is non trivial in dimension 1 and vanishes above it.'' :That's the cohomology dimension; I don't see why it's better than any other. ---- Tarquin, do you plan to add to the symbols for subsets of the real line? Otherwise, I'll add to it, including the uses of the symbols. Right now, it's not very clear. — User:Toby Bartels 07:16 Aug 3, 2002 (PDT) ---- As I understand it, ''local field'' means a field complete with respect to a discrete valuation. Are the reals and complexes really local fields? --User:Alodyne I moved the statement here for the moment, until we sort this out: :The reals are one of the two local fields of field (mathematics) 0 (the other one being the complex numbers). See also Talk:Local field. User:AxelBoldt 19:38 Nov 9, 2002 (UTC) Oh and by the way: at least one change certainly must be made in that statement since the ''p''-adic numbers are local fields of characteristic 0 by all the various definitions I've seen. Any dispute? --User:Alodyne :No. The Encyclopedic Dictionary of Mathematics defines a local field as a field that's complete with respect to a discrete valuation and such that it's residue field is finite. They mention that the the reals and complex numbers are also sometimes considered as local fields, but they explicitly exclude them. User:AxelBoldt 19:14 Nov 12, 2002 (UTC) ---- To tie up this article with the entries on Model Theory, can someone tell me whether or not there is a first-order theory model for the real numbers? A maths teacher friend of mine told me that there is, but this would imply, by Lowenhein-Skolem theorems that the reals are denumerable, or have a countable model, which which seem to be inconsistent with Cantor's diagonal proof of uncountability of the Reals. --B. Smith. No, it would not imply that. It would imply only that there is a countable field satisfying all of the same first-order sentences that the real numbers satisfy. The Loewenheim-Skolem theorem does imply the existence of such a model. If the model and the language are elaborate enough, one could write a first-order sentence saying the reals are not countable. It would be true within the countable model. That means that although there would be a sequence containing every member of the model, such a sequence could not itself belong to the model; there could be no enumeration within the model. User:Michael Hardy 00:12 Apr 11, 2003 (UTC) ---- "The term 'real number' is a retronym coined in response to 'imaginary number'." That hardly seems likely, since extension to the reals should always have seemed more urgent than extension to complex, and the reals need a name as soon as you consider them. IMO, real is intended to contrast with rational, and the motivation for "real" is that the rationals are an unrealistically limited set. That follows from the mind necessarily regarding all integer-legged right triangles as precise models of something from reality, if anything in math beyond counting is such a model; from irrational hypotenuses, it follows that rationals aren't quite the full set of numbers needed to correspond to reality, so the choice of the term "real number" reflects the hypothesis that there are no problems in the real world that the reals don't suffice to express the answer to. (In contrast, wanting imaginaries and quaternions takes some sophisticated mathematical abstraction.) In fact, the best evidence that "real" led to "imaginary" (not the other way around) is that "real" is a better fit to the reals than "imaginary" is to the imaginaries: IIRC how one of my physics profs put it, it's not so much a matter of complex numbers having an imaginary part as that they have *two* parts, both real numbers representing something in the physical world, and it's just that there are systems like electromagnetic fields whose behavior looks simpler if you do the bookkeeping by labeling the electric field as the first component of a complex number and the magnetic as the second, and hiding the fact that the "multiplication" you do is, according to the rules of complex arithmetic, more complicated than the multiplication table you learned in grade school. So where's this "retronym" thing coming from; is it more than someone's conjecture? --User:Jerzy 06:41, 30 Sep 2003 (UTC) I don't think there was a monolithic nomenclature that all mathematicians subscribed to at the time when negative numbers, imaginary numbers, and real numbers were being investigated and systematized (15th century through 19 century-ish). History is messy like that. It is well known that negative numbers (first) and imaginary numbers (later) inspired censure (and sometimes even disgust) in the European mathematicians of the day, so the retronym thing is at least plausible, e.g. "The square root of -1? That's not a ''real'' number, it's some kind of imaginary thing!" I'd like to see a source too though. -- User:Cyan 07:06, 30 Sep 2003 (UTC) :Just for information, the first reference to ''imaginary'' numbers in the Oxford English Dictionary is to Descartes (in French) in 1637, and the first reference in English is in 1706: ''W. JONES Syn. Palmar. Matheseos 127 The Original Components or Roots of all Equations, may be either Affirmative, Negative, Mix'd, or Imaginary.'' In contrast, the first reference to a real number that they give is in a 1727 encyclopedia: ''CHAMBERS Cycl. s.v. Root, If the value of x be positive, i.e. if x be a positive quantity,..the root [of an equation] is called a real or true root''. (These are actually citations for the use of ''real'' or ''imaginary'' as adjectives with the modern meaning, so they weren't looking specifically for the phrase ''real number'', which they don't cite until the 1910 Encyc. Brit.) Although I'm not sure the OED is as good about backdating mathematical terms as they are in general, this gives some indication at least. Anyway, since a rigorous definition and theory of real numbers wasn't really developed until the 19th century, it's hard to call it a ''retronym'' (what people had looked at in the past ''were'' real numbers but didn't ''define'' them; e.g. if you look at just the roots of polynomials, those are the ''algebraic'' numbers which are a countable subset of the reals). User:Stevenj 02:02, 12 Oct 2003 (UTC) ---- The article begins with: :''The real numbers are practically any numbers that can be expressed.'' I have to say I find this unsatisfying, especially since ''most'' (i.e. all but a countable subset of) real numbers arguably ''cannot'' be specifically expressed. That is, if you take the meaning of ''expressed'' in a natural way: i.e. to be uniquely defined by a finite-length description (such as 4.73, √2, sin(1), ...). And only a strict subset of these ''expressible'' numbers are computable (i.e. to an arbitrary precision in a finite time). User:Stevenj 01:34, 12 Oct 2003 (UTC) ---- :''It's impossible to explicitly specify a non-recursive number;'' That depends on what "explicitly specify" means. Consider, for example, Chaitin's constant. This is non-recursive, but in some sense can be specified. User:Josh Cherry 23:31, 19 Oct 2003 (UTC) And what is this Russian school that assumes all numbers are recursive? This sounds like a provably false statement to me, unless something nonobvious is meant by "recursive algorithm". It's provably impossible to specify every real numbers uniquely with a recursive algorithm in the computer-science sense, i.e. a finite-size finite-state machine, or equivalently a finite-length program in any Turing-complete computer language. (There are only a countable number of finite-length computer programs.) (One can even explicitly define a real number, whose digits are e.g. based on the halting problem, that is uniquely specified by a finite-length description but which is not computable in finite time by any program.) User:Stevenj 20:00, 21 Oct 2003 (UTC) Certainly one can explicitly specify particular non-computable numbers. I haven't read everything above, so I'm not certain which \"Russian school\" is referred to, but it's probably something about constructivism, which is a philosophy that holds that an existence proof is not valid unless it "constructs" the object whose existence is to be proved. For example, if you were to deduce a contradiction from the proposition that every even number greater than 2 is a sum of two primes, that would not be taken by constructivists to be a proof of the existence of a counterexample. User:Michael Hardy 00:03, 22 Oct 2003 (UTC) :However, ''most'' (all but a countable subset of) real numbers cannot be uniquely specified by a finite description of any sort (whether by a computer algorithm or otherwise). The current Wikipedia statement about the Russian school needs clarification (or deletion), because on its face it seems to imply the contrary. User:Stevenj 04:11, 22 Oct 2003 (UTC) == Quantity Box == This has been removed several times by one person. I am among those who sees it as an enhancement of the article and the "explanations" for its removal wrong -- unless the word is just "silly". It's time to stop deleting and start defending the preceding deletions on this page: convince someone if you're so sure. --User:JerzyUser talk:Jerzy 01:56, 2004 Mar 28 (UTC) :The box is not silly but unnecessary and distracting. The box is already too long (you have to scroll to see the whole list and it has a potential to get longer certainly. This kind of the box is only useful and necessary in cases like 1) the article is separated into several pages because the article would be too long if those were in one page. 2) The article is about one entity among several ones. There is no much relationship between real and natural numbers except they are numbers. You want me to convince you then can I also ask you to convince me first? To my knowledge, there was no consensus about having such a box. In other words, I am just reverting the article to what it used to be and see if we can have concensus or something. -- User:TakuyaMurata 03:04, Mar 28, 2004 (UTC) :I've just found this page. Wikipedia_talk:Article_series. It would help us to see whether to have a box or not. -- User:TakuyaMurata 16:08, Mar 28, 2004 (UTC) == Negatives are that recent? == From the History section: "Negative numbers began to be generally accepted in the 1600s and were invented by Muslim mathematicians." I find it hard to believe that negative numbers were just being accepted in Isaac Newton's time. I'm no history geek, but I'd like to get that confirmed. :Same here. Newton obviously used negative numbers, and what about Napier? I had thought that negative numbers were introduced by medieval Italian accountants. User:Michael Hardy 20:59, 6 Dec 2004 (UTC) :A serious mistake here. Chinese mathematicians (indeed they maybe more "accountants" than "mathematicians") developed similar concept long time ago. I believe that negative numbers have been reinvented many times, as surplus and deficiency are fundamental concepts in commerce. -User:Wshun 11:45, 10 Dec 2004 (UTC) ==Packing real numbers together== You can "pack" any finite number of integers or rational numbers into one (e.g. by alternating digits), but is this legal for real numbers? User:Fredrik | User talk:Fredrik 02:47, 20 Feb 2005 (UTC) : Yes you can do it with real numbers too, at least when they are all in the interval [0, 1). : Start by writing each of those numbers in the form math>0.a_1a_2a_3\dots, and for uniqueness, suppose that none of your numbers after a while contain only 9's. :Then, write the first digit of the first number, then the first digit of the second number, all the way to the first digit of the last number. Then work on the second digit of each number and so on. :Now, presumably you could do same thing without the assumption that the numbers are in [0, 1), but then you need to worry in addition about the exponent. User:Oleg Alexandrov 19:12, 20 Feb 2005 (UTC) == "Almost all" means...? == : ''Since only a countable set of real numbers can be algebraic number, almost all real numbers are transcendental number.'' This is somewhat puzzling, because the link to the "almost all" article says : ''"Almost all" is sometimes used synonymously with "all but finitely many"; see almost.'' and the other meanings given don't apply. Surely there are not just finitely many algebraic numbers? -- User:KittySaturn 08:47, 2005 May 23 (UTC) :The meaning is actually more in line with the description at ''almost everywhere''. That is, the set of algebraic numbers have measure zero. -- User:Fropuff 15:26, 2005 May 23 (UTC) ::I've taken no number theory classes, so I may very well be wrong about this, but if i'm reading http://mathworld.wolfram.com/AlmostAll.html correctly then "almost all" can also mean "all of an uncountable set but countable number". —User:MilesKUser_talk:MilesK 19:50, May 23, 2005 (UTC)
See other meanings of words starting from letter:
R
RA | RB | RC | RD | RE | RF | RG | RH | RI | RJ | RK | RL | RM | RN | RO | RP | RS | RT | RU | RW | RX | RY | RZ |Words begining with Real_number:
Real_number
Real_number
Real_Numbers
Real_numbers
Real_numbers
Real_number_field
Real_number_line
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