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The article currently claims:
However, I checked the Oxford English Dictionary, and it's not clear that this is quite true.
In particular, the OED has an entry for the mathematical usage of "real" as an adjective to describe "quantities (Opposed to IMAGINARY, or IMPOSSIBLE)" (emphasis added). In particular, the earliest usages they cite actually seem to be using the term "real" to describe only positive roots, and in another case it seems to be restricted to only rational roots:
In short, the term "real" was originally a broader reflection of the prejudices of pre-modern mathematicians against certain classes of numbers as "impossible" or "unreal". As time went on, eventually negative and irrational solutions were considered "real", and only the imaginary solutions were excluded...which apparently became fixed terminology by the end of the 19th century (when most of these prejudices had been cast aside in favor of a more axiomatic rigor).
—Steven G. Johnson ( talk) 00:45, 17 August 2008 (UTC)
I doubt this, and it is unsourced. The OED quotes Descartes: "Les..racines..ne sont pas tousiours reeles; mais quelquefois seulement imaginaires; " which suggests that the terms are twins. (This refers to real and imaginary roots, of course; but the earliest uses will be in that context.) Septentrionalis PMAnderson 22:11, 13 December 2006 (UTC)
The retronym question is something like a hoax, in my opinion. The term "real number", in various languages, appears well before 1872: Hamilton used it in English, Liouville in French, Leibniz and Gauss in Latin (realis quantitas). It comes straight from Descartes, the quotation from OED is correct: he divided the solutions of equations in "réelles" and "imaginaires", and the former into "vraies" (positive) and "fausses" (negative or zero). He also used "quantité réelle" and "quantité imaginaire". Later on, but already in the 18th century, mathematician started to say "number" instead of "quantity". But the contraposition between "real" and "imaginary" is there from the start. Bballmath ( talk) 10:32, 3 July 2009 (UTC)
Whilst a science article can't always subscribe to the axiom if you can't explain it to a layman, then you don't know your subject, I'd go as far as to say that this article is obfuscatory in the extreme. I'm not a stupid person (my mother once told me), but I still fail to understand what the difference between a real and unreal (or is it imaginary) number is from the first paragraph - and that is something I'm afraid I always complain about! We have a duty to non-specialists to inform, as well as provide the specialists with hard reference material. Blitterbug ( talk) 07:29, 9 July 2009 (UTC)
Iam so lost —Preceding
unsigned comment added by
67.248.211.132 (
talk)
21:34, 20 October 2009 (UTC)
rotational expression —Preceding unsigned comment added by 124.104.66.200 ( talk) 00:23, 15 July 2010 (UTC)
What is the meaning of "strictly more". Isn't it just the same thing as "more"? —Preceding unsigned comment added by 216.188.231.253 ( talk) 05:41, 29 September 2010 (UTC)
In the "Uses" section, we have the following statement:
"Because there are only countably many algorithms, but an uncountable number of reals, "most" real numbers fail to be computable."
First of all, I kinda feel the truthines of this statement, but have never seen it proved. So, proof needed. Second, if this or these proofs exist, there should definitely be an inline citation, as this would be material enough for an entire article on its own (IMO).
Cheers! Trolle3000 [talk] 06:51, 23 October 2010 (UTC)
4X[2+3]-4+3 —Preceding unsigned comment added by 109.236.36.8 ( talk) 13:11, 4 December 2010 (UTC)
After two reverts to restore a previous controversial section on Nelson's theory of the reals, tkuvho( talk) send me a private mail saying that I am wrong in my edits and that I have to read the IST page (There is no IST page, but I guess he meant internal set theory).
Here is my answer, which may be interesting to everybody:
"May be your are right in your assertion about Nelson. But
Comments are needed to avoid an edit war. D.Lazard ( talk) 14:43, 22 December 2010 (UTC)
My comments:
— Carl ( CBM · talk) 16:15, 22 December 2010 (UTC)
Also, the article on constructivism links Nelson with constructivism for his study of predicative arithmetic. That's correct, but IST is not a predicative system, nor one that people in constructive mathematics seem to study very much. — Carl ( CBM · talk) 16:29, 22 December 2010 (UTC)
If you don't specify the background set theory then you have considerable latitude as to what your reals could look like. They could for example look like Nelson's reals, assorted with the "standard" predicate, in such a way that some real numbers already behave as infinitesimals. This is of course fabulous news for analysis; you can define a derivative by actually forming the infinitesimal quotient a la ancienne; you can write down a Dirac delta function that's actually a function rather than a distribution. So all in all I am sympathetic to your point of view. Though I am not entirely convinced this is what you meant :) Tkuvho ( talk) 21:56, 22 December 2010 (UTC)
You seem to want to eat the cake and have it, too. The background theory is there, and presumably it is ZF, but it's not axiomatic ZF. It's real life, platonic, sweaty under the armpits kind of ZF, that certainly does not tolerate any competition from funny standard predicates. Tkuvho ( talk) 22:06, 22 December 2010 (UTC)
The following entry seems highly questionable:
If you measure a long distance by counting how many times you can fit a 1 m^2 square plate diagonally on the way, your measurement will yield an irrational number of meters. Lapasotka ( talk) 09:21, 5 January 2011 (UTC)
As I see it, the valid point being made by the existing text is the following: Real numbers are infinite-precision objects. It is impossible to make a measurement to infinite precision, nor can computers store infinite-precision representations.
That is true and worth saying. What's fishy is the idea that, given that the objects represented are not infinite-precision real numbers, they must be rational numbers. I don't think they are. To me they seem more like "fuzzy real numbers", real numbers represented within a tolerance.
Unfortunately, going into too much detail along those lines is likely to run afoul of WP:OR. So I think it would be better to say that you can't make measurements to infinite precision nor represent them in computers, and leave it at that. -- Trovatore ( talk) 22:06, 5 January 2011 (UTC)
How about dividing the material in the article as follows:
I could write an elementary geometric decription with pictures into the first section. That is also a good place to explain the decimal expansions and perhaps uncountability of real numbers, which can be done in relatively simple terms. Lapasotka ( talk) 12:33, 14 January 2011 (UTC)
I assume that reordering or merging stubbish sections is an incremental change if their content is not changed. The real proposition here is to divide the definitions/descriptions (1 and 3 in this list) into two parts. The "Elementary description" should be accessible to a wide audience and a good reading for precalculus students in US system. It should also follow the historical development of real numbers, explaining the topics in the order
The "Formal definitions", I think, is an appropriate place to go Bourbaki, and to include the current Properties section in it. Also the wordings could be a little bit more precise. For example, real numbers "is not a complete metric space", but rather it "has the standard metric d(x,y)=|x-y|, which is complete." Lapasotka ( talk) 20:53, 15 January 2011 (UTC)
I think the introduction should start with a conscise non-jargon description of the concept of real numbers. For instance "continuum" is jargon and does not describe much to anyone who does not understand the concept of real numbers already. Also the dichotomy of reals into rationals and irrationals should be postponed a little bit, because it doesn't describe anything either (unless you already understand what real number AND rational numbers are).
Proposal for introduction:
Real numbers are mathematical objects that are used to measure various quantities in mathematics, physics and other sciences. Real numbers include natural numbers, which are used in counting objects (two apples, four cars, three Cadillacs etc.) as well as ordering them (first, second and third place in a contest etc.), but real numbers are also useful in measuring quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.
The question "What real numbers really are?" is largely philosophical. Instead, in mathematics the real numbers are defined rigorously by how they behave. The real number system consists of the set of all real numbers, the operations of addition and multiplication, and the order relation between two real numbers. The axioms of real numbers describe precisely how this system of real numbers behaves, and they are tailored to encapsulate the geometric idea of an infinitely long line with two special points 0 (the origin) and 1 on it (this particular line is frequently referred to as the number line or the real line). Real numbers correspond to the points on the number line and the order relation x<y means that the points x and y appear on the number line in the same order as 0 and 1. Addition of real numbers x and y corresponds to parallel translation of the corresponding line segments starting from 0, and their product is determined by the proportions of these line segments and the line segment from 0 to 1.
The set of natural numbers contains those real numbers 1, 2=1+1, 3=1+1+1, ... that can be reached by adding the number 1 arbitrarily many times to itself, the set of integers contains 0, all natural numbers and their negatives, and the set of rational numbers contains all real numbers , where m is an integer and n is a natural number. Rational numbers are dense in the set of real numbers in the sense that for any real number there is a rational number arbitrarily close to it. However, there are points on the real line that do not correspond to rational numbers, a fact of which the ancient Greek and Indian mathematicians were already aware of. The most common example is the diagonal of the unit square, which is of irrational length .
The completeness property of the real number system encapsulates the idea that there are no holes (or gaps) on the real line, and also that there are no infinitely small or infinitely large real numbers. Informally it states that if one cuts the real line into two halves, then exactly one of the halves will contain an end point. The set of rational numbers does not satisfy this property, since one can cut it into two halves at . There are extensions of the real number system where both infinitely small and infinitely large real numbers exist, and they are studied in non-standard analysis.
A common way to represent real numbers is by decimal expansions such as 523, -4.25, and 4.91648367... where "..." signifies that the decimal expansion continues indefinitely. Integers correspond to decimal expressions without a fractional part (the sequence of numerals after the decimal point) and rational numbers to those decimal expressions for which the tail of the rational part is periodic, such as =2.08333... Irrational numbers always have infinitely long decimal representations without any periodicity, for example, π=3.1415926535...
End of proposal, last edited by Lapasotka ( talk) 11:17, 14 January 2011 (UTC), D.Lazard ( talk) 16:07, 14 January 2011 (UTC)
Please comment. I will wait for few days before tampering with anything. Maybe some pictures will be helpful in visualizing the meaning of multiplication and division of real numbers. Lapasotka ( talk) 15:35, 9 January 2011 (UTC)
I added "gaps" by your suggestion, and also something about irrational numbers. The geometric constructions for the multiplication and addition are of course the same for rational numbers, but the crux is that they work equally well for all real numbers, and that they can be explained to people without knowledge of higher mathematics or axiomatic systems.
Of course one can measure the temperature of a refrigerator with quite a few different number systems, of even with some suitable refrigeratory Performance Index taking values from blue to red, for that matter, but this article is about real numbers, which is the standard system for measuring physical quantities "taking values in the continuum". I was being careful not to state that one needs real numbers to do physical measurements, but only that they are useful in things like that. Lapasotka ( talk) 22:47, 9 January 2011 (UTC)
You cannot do all Euclid and not notice that there are gaps on the rational line (which I suppose you meant). Algebraic numbers will do, however. But back to the point:
We need more opinions on the relevance of the discussion of what real numbers are really needed in physical measurements within the article. I see it as an unneccessary complification, since one can also turn the question upside down and ask what numbers need to be discarded. I think we agree that there should be a section on the subsets of real numbers that can be constructed by certain finite processes, such as floating point numbers, algebraic numbers and geometrically constructible numbers, as well as some others subsets such as definable real numbers. However, my impression is that mainstream physicists (or other scientists) do not care too much about these finesses and perhaphs they should not be mentioned in the introduction even though they are of mathematical and (perhaps) philosophical importance.
If there are no objections, corrections or improvements I will paste the text above into the introduction within a few days. Lapasotka ( talk) 07:12, 10 January 2011 (UTC)
To Tkuvho: Please re-read 'but real numbers are also useful in measuring quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.' I agree on the questionability of the statement 'real numbers are needed to measure the temperature in the refrigerator', but this is a different claim. What do you think of the first (actual) one? Lapasotka ( talk) 13:27, 10 January 2011 (UTC)
Alright. Maybe one needs to be more precise: 'but real numbers are also useful in modeling quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.' Either in Real_number#Uses subsection or in another subsection covering decimal representations and floating point numbers one could underline your point. BTW, I don't think I have confused the meaning of the word "real" as the part of the combined word "real number" into its other possible meanings. Whether any numbers are " real" in the everyday sense of the word is a philosophical question that I will not try to answer. Lapasotka ( talk) 14:37, 10 January 2011 (UTC)
To Trovatore: Thank you for the comment. I feel myself that the geometric explanation of the real number system is too lengthy, so it is probably better to cut it down to a few words and move the rest together with some pictures into a short subsection. I somehow feel the same about N,Z,Q-part, but from the previous comments it seems that constrasting real numbers with rational numbers cannot be left out of the introduction. Most importantly, saying only that "real numbers can be defined axiomatically" is not a satisfactory solution. Can you please point out the things you do not agree with. Also some ideas how to switch the bias would be useful, but let us keep in mind that this article is about real numbers (as defined in mathematics), not about "real world" or practices of storing measured data in science laboratories. Lapasotka ( talk) 19:58, 10 January 2011 (UTC)
I did a little streamlining on my original proposal, deleted the mentions on proportionality and re-explained multiplication by comparison of areas. I really think there should be a section where the relationship between real numbers and plane geometry is explained more throughoutly in elementary (pun intented) terms. The fact that real numbers represent abstract proportions should be presented as well, but I don't have a very good idea how. Lapasotka ( talk) 16:43, 11 January 2011 (UTC)
I copied here my major change on the introduction that Trovatore just reverted. I am happy to discuss more, but nothing has happened in the talk page recently. Please comment so we can reach some resolution. The current vesion of the introduction is worse than B-class. Lapasotka ( talk) 09:23, 14 January 2011 (UTC)
Structuralism vs. constructivism is a good point, and writing down a short description taking into account both points is a challenging task. My understanding is that nowadays most mathematicians tend to be somewhere between platonists and structuralists, and constructivists are in the minority, so I didn't consider this as a major fault. I compared the length of the introduction to those of the featured math articles and it seemed to be about 50% longer on average. I added a comment on structuralism and shortened the proposition a little bit. Actually it is not much longer than the introductions of some featured math articles any more. To shorten it further I think the discussion of N,Z and Q could be almost entirely discarded. Lapasotka ( talk) 10:27, 14 January 2011 (UTC)
To Trovatore: I think I misunderstood your earlier comment. Indeed, geometric intuition should be emphasized either in the intro (which is hard if we want to keep it short) or in the first section about basic properties. Lapasotka ( talk) 09:56, 16 January 2011 (UTC)
The page is a pain to elaborate. Almost everything is reverted instead of modified and no-one seems to be able to get anything done with it. It is a shame, since real numbers is one of the most important topics in mathematics, and the page as it is right now is a mess. Before I started on it I took a look on featured articles in mathematics and noticed that an elementary description, accessible at least to high school students, is an essential part of them. Here are some comments:
And a kind request:
Lapasotka ( talk) 08:09, 20 January 2011 (UTC)
A recent change by a user made subtle modifications to a sentence that drastically altered the meaning of the statement. Here is the link to the change in question. No sources were added. Is the new meaning or the old one correct? Cliff ( talk) 18:12, 18 February 2011 (UTC)
My mistake, I was not clear enough. I don't challenge the information about Dedekind, that's easy enough to verify. My question about the modifications were actually to the second paragraph in the History section. The change implies that the the acceptance of these types of numbers was local to Europe, but that Chinese and Arab scholars had accepted these ideas earlier. The previous incarnation of the paragraph indicated that these three regions accepted these notions at about the same time. Cliff ( talk) 20:20, 18 February 2011 (UTC)
The changes have now been reverted, but which is correct? since there is no source for either contribution, how do we know which is correct? reverted article. --- Cliff ( talk) 20:24, 18 February 2011 (UTC)
I am not sure if this is encoded in some formal wiki regulations somewhere, but it seems to me that the appropriate way to proceed with a lede is incremental: one generally avoids making sweeping changes, and adds items one at a time. This is surely the best way of avoiding reverts. Tkuvho ( talk) 09:33, 14 January 2011 (UTC)
I copied the proposition on the talk page for the reason that it can be incrementally changed before replacing the original one. I felt that the original introduction needed so many structural changes that it was better to do them at once to avoid unreadable versions of the page. Here is a list of (seemingly) unmentioned things in the original intro:
Now the description of fundamental operations (multiplication and addition) is mentioned but not explained in the proposal. Lapasotka ( talk) 10:53, 14 January 2011 (UTC)
Note - Just a thought as a reminder, in order to avoid going through this (or a similar) process every time a new contributor arrives, wouldn't it be a good idea to provide a solid standard textbook source with each change? - DVdm ( talk) 11:39, 14 January 2011 (UTC)
Why is the following metaphysical "deduction" still there?
And what about the other even more questionable statement, which is certainly not widely agreed upon and only backed up by a non-peer-reviewed citation?
I skimmed through the citation and even if it was reliable and peer-reviewed (which ArXiv is not), it did not imply such a strong and controversial conclusion. Besides, the existence of any sort of numbers in the physical universe is a philosophical question, and should be left out of mathematics articles. At most, there could be a section called "Real numbers and metaphysics" where one can include statements like these. In the way how they are stated now they might give a false impression on the importance and definiteness of real number system at the heart of modern day mathematics and its applications. Lapasotka ( talk) 09:11, 27 May 2011 (UTC)
I elaborated the "physical sciences" part into a form which most working mathematicians and physicists can probably agree with. I am still waiting for more comments on the "infinite divisibility". As far as I understand, the space-time is actually modeled by a Minkowski-space (effectively R4) in relativistic quantum mechanics, or by some higher dimesional fibre bundle in super string theories. I don't see how in such framework "infinite precision real numbers" do not exist in the "physical universe". After all, statements as above can only be made with respect to some physical model of the reality. Lapasotka ( talk) 13:23, 27 May 2011 (UTC)
I edited the part on the existence of infinite precision real numbers in the physical universe in accordance to the Wikipedia links. I am not an expert on these matters, but I tried to be true to the referred articles. Somekind of elaboration was needed, because the previous statement was simply too radical to be stated only "as matter of fact". Next I would like to draw attention to the P=NP -part. There are no citations. What might that piece of text actually try to achieve? Lapasotka ( talk) 13:57, 27 May 2011 (UTC)
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?
The lead says that the reals can be defined axiomatically as the "complete Archimedean ordered field." The main body of the article, however, gives an axiomatization that simply defines them as a "complete ordered field." The second version is correct, and the word "Archimedean" in the lead should be deleted. The Archimedean property is not independent of completeness. If you make a conservative extension of the reals to a nonarchimedean system, you get the hyperreals, which lack completeness. The Archimedean property is important, but it shouldn't be given in the lead as if it were independent of completeness. You only have to specify whether it's Archimedean if it's not complete. The rationals are Archimedean and not complete; the hyperreals are non-Archimedean and not complete.-- 76.167.77.165 ( talk) 15:16, 22 March 2009 (UTC)
In the Advanced Properties section the article claims that the real numbers are everywhere dense. I am familiar with the concept "dense" and "nowhere dense", but not "everywhere dense". Surely when claiming that a set is dense one has to specify the set in which it is dense. The complex numbers are an example of a set in which the real numbers are nowhere dense. Can it still be possible that they are everywhere dense? 70.72.220.88 ( talk) 03:21, 21 December 2011 (UTC)
This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.
Please help by viewing the entry for this article shown at the page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed.
I searched the page history, and found 3 edits by Jagged 85. Tobby72 ( talk) 22:38, 20 January 2012 (UTC)
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The article currently claims:
However, I checked the Oxford English Dictionary, and it's not clear that this is quite true.
In particular, the OED has an entry for the mathematical usage of "real" as an adjective to describe "quantities (Opposed to IMAGINARY, or IMPOSSIBLE)" (emphasis added). In particular, the earliest usages they cite actually seem to be using the term "real" to describe only positive roots, and in another case it seems to be restricted to only rational roots:
In short, the term "real" was originally a broader reflection of the prejudices of pre-modern mathematicians against certain classes of numbers as "impossible" or "unreal". As time went on, eventually negative and irrational solutions were considered "real", and only the imaginary solutions were excluded...which apparently became fixed terminology by the end of the 19th century (when most of these prejudices had been cast aside in favor of a more axiomatic rigor).
—Steven G. Johnson ( talk) 00:45, 17 August 2008 (UTC)
I doubt this, and it is unsourced. The OED quotes Descartes: "Les..racines..ne sont pas tousiours reeles; mais quelquefois seulement imaginaires; " which suggests that the terms are twins. (This refers to real and imaginary roots, of course; but the earliest uses will be in that context.) Septentrionalis PMAnderson 22:11, 13 December 2006 (UTC)
The retronym question is something like a hoax, in my opinion. The term "real number", in various languages, appears well before 1872: Hamilton used it in English, Liouville in French, Leibniz and Gauss in Latin (realis quantitas). It comes straight from Descartes, the quotation from OED is correct: he divided the solutions of equations in "réelles" and "imaginaires", and the former into "vraies" (positive) and "fausses" (negative or zero). He also used "quantité réelle" and "quantité imaginaire". Later on, but already in the 18th century, mathematician started to say "number" instead of "quantity". But the contraposition between "real" and "imaginary" is there from the start. Bballmath ( talk) 10:32, 3 July 2009 (UTC)
Whilst a science article can't always subscribe to the axiom if you can't explain it to a layman, then you don't know your subject, I'd go as far as to say that this article is obfuscatory in the extreme. I'm not a stupid person (my mother once told me), but I still fail to understand what the difference between a real and unreal (or is it imaginary) number is from the first paragraph - and that is something I'm afraid I always complain about! We have a duty to non-specialists to inform, as well as provide the specialists with hard reference material. Blitterbug ( talk) 07:29, 9 July 2009 (UTC)
Iam so lost —Preceding
unsigned comment added by
67.248.211.132 (
talk)
21:34, 20 October 2009 (UTC)
rotational expression —Preceding unsigned comment added by 124.104.66.200 ( talk) 00:23, 15 July 2010 (UTC)
What is the meaning of "strictly more". Isn't it just the same thing as "more"? —Preceding unsigned comment added by 216.188.231.253 ( talk) 05:41, 29 September 2010 (UTC)
In the "Uses" section, we have the following statement:
"Because there are only countably many algorithms, but an uncountable number of reals, "most" real numbers fail to be computable."
First of all, I kinda feel the truthines of this statement, but have never seen it proved. So, proof needed. Second, if this or these proofs exist, there should definitely be an inline citation, as this would be material enough for an entire article on its own (IMO).
Cheers! Trolle3000 [talk] 06:51, 23 October 2010 (UTC)
4X[2+3]-4+3 —Preceding unsigned comment added by 109.236.36.8 ( talk) 13:11, 4 December 2010 (UTC)
After two reverts to restore a previous controversial section on Nelson's theory of the reals, tkuvho( talk) send me a private mail saying that I am wrong in my edits and that I have to read the IST page (There is no IST page, but I guess he meant internal set theory).
Here is my answer, which may be interesting to everybody:
"May be your are right in your assertion about Nelson. But
Comments are needed to avoid an edit war. D.Lazard ( talk) 14:43, 22 December 2010 (UTC)
My comments:
— Carl ( CBM · talk) 16:15, 22 December 2010 (UTC)
Also, the article on constructivism links Nelson with constructivism for his study of predicative arithmetic. That's correct, but IST is not a predicative system, nor one that people in constructive mathematics seem to study very much. — Carl ( CBM · talk) 16:29, 22 December 2010 (UTC)
If you don't specify the background set theory then you have considerable latitude as to what your reals could look like. They could for example look like Nelson's reals, assorted with the "standard" predicate, in such a way that some real numbers already behave as infinitesimals. This is of course fabulous news for analysis; you can define a derivative by actually forming the infinitesimal quotient a la ancienne; you can write down a Dirac delta function that's actually a function rather than a distribution. So all in all I am sympathetic to your point of view. Though I am not entirely convinced this is what you meant :) Tkuvho ( talk) 21:56, 22 December 2010 (UTC)
You seem to want to eat the cake and have it, too. The background theory is there, and presumably it is ZF, but it's not axiomatic ZF. It's real life, platonic, sweaty under the armpits kind of ZF, that certainly does not tolerate any competition from funny standard predicates. Tkuvho ( talk) 22:06, 22 December 2010 (UTC)
The following entry seems highly questionable:
If you measure a long distance by counting how many times you can fit a 1 m^2 square plate diagonally on the way, your measurement will yield an irrational number of meters. Lapasotka ( talk) 09:21, 5 January 2011 (UTC)
As I see it, the valid point being made by the existing text is the following: Real numbers are infinite-precision objects. It is impossible to make a measurement to infinite precision, nor can computers store infinite-precision representations.
That is true and worth saying. What's fishy is the idea that, given that the objects represented are not infinite-precision real numbers, they must be rational numbers. I don't think they are. To me they seem more like "fuzzy real numbers", real numbers represented within a tolerance.
Unfortunately, going into too much detail along those lines is likely to run afoul of WP:OR. So I think it would be better to say that you can't make measurements to infinite precision nor represent them in computers, and leave it at that. -- Trovatore ( talk) 22:06, 5 January 2011 (UTC)
How about dividing the material in the article as follows:
I could write an elementary geometric decription with pictures into the first section. That is also a good place to explain the decimal expansions and perhaps uncountability of real numbers, which can be done in relatively simple terms. Lapasotka ( talk) 12:33, 14 January 2011 (UTC)
I assume that reordering or merging stubbish sections is an incremental change if their content is not changed. The real proposition here is to divide the definitions/descriptions (1 and 3 in this list) into two parts. The "Elementary description" should be accessible to a wide audience and a good reading for precalculus students in US system. It should also follow the historical development of real numbers, explaining the topics in the order
The "Formal definitions", I think, is an appropriate place to go Bourbaki, and to include the current Properties section in it. Also the wordings could be a little bit more precise. For example, real numbers "is not a complete metric space", but rather it "has the standard metric d(x,y)=|x-y|, which is complete." Lapasotka ( talk) 20:53, 15 January 2011 (UTC)
I think the introduction should start with a conscise non-jargon description of the concept of real numbers. For instance "continuum" is jargon and does not describe much to anyone who does not understand the concept of real numbers already. Also the dichotomy of reals into rationals and irrationals should be postponed a little bit, because it doesn't describe anything either (unless you already understand what real number AND rational numbers are).
Proposal for introduction:
Real numbers are mathematical objects that are used to measure various quantities in mathematics, physics and other sciences. Real numbers include natural numbers, which are used in counting objects (two apples, four cars, three Cadillacs etc.) as well as ordering them (first, second and third place in a contest etc.), but real numbers are also useful in measuring quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.
The question "What real numbers really are?" is largely philosophical. Instead, in mathematics the real numbers are defined rigorously by how they behave. The real number system consists of the set of all real numbers, the operations of addition and multiplication, and the order relation between two real numbers. The axioms of real numbers describe precisely how this system of real numbers behaves, and they are tailored to encapsulate the geometric idea of an infinitely long line with two special points 0 (the origin) and 1 on it (this particular line is frequently referred to as the number line or the real line). Real numbers correspond to the points on the number line and the order relation x<y means that the points x and y appear on the number line in the same order as 0 and 1. Addition of real numbers x and y corresponds to parallel translation of the corresponding line segments starting from 0, and their product is determined by the proportions of these line segments and the line segment from 0 to 1.
The set of natural numbers contains those real numbers 1, 2=1+1, 3=1+1+1, ... that can be reached by adding the number 1 arbitrarily many times to itself, the set of integers contains 0, all natural numbers and their negatives, and the set of rational numbers contains all real numbers , where m is an integer and n is a natural number. Rational numbers are dense in the set of real numbers in the sense that for any real number there is a rational number arbitrarily close to it. However, there are points on the real line that do not correspond to rational numbers, a fact of which the ancient Greek and Indian mathematicians were already aware of. The most common example is the diagonal of the unit square, which is of irrational length .
The completeness property of the real number system encapsulates the idea that there are no holes (or gaps) on the real line, and also that there are no infinitely small or infinitely large real numbers. Informally it states that if one cuts the real line into two halves, then exactly one of the halves will contain an end point. The set of rational numbers does not satisfy this property, since one can cut it into two halves at . There are extensions of the real number system where both infinitely small and infinitely large real numbers exist, and they are studied in non-standard analysis.
A common way to represent real numbers is by decimal expansions such as 523, -4.25, and 4.91648367... where "..." signifies that the decimal expansion continues indefinitely. Integers correspond to decimal expressions without a fractional part (the sequence of numerals after the decimal point) and rational numbers to those decimal expressions for which the tail of the rational part is periodic, such as =2.08333... Irrational numbers always have infinitely long decimal representations without any periodicity, for example, π=3.1415926535...
End of proposal, last edited by Lapasotka ( talk) 11:17, 14 January 2011 (UTC), D.Lazard ( talk) 16:07, 14 January 2011 (UTC)
Please comment. I will wait for few days before tampering with anything. Maybe some pictures will be helpful in visualizing the meaning of multiplication and division of real numbers. Lapasotka ( talk) 15:35, 9 January 2011 (UTC)
I added "gaps" by your suggestion, and also something about irrational numbers. The geometric constructions for the multiplication and addition are of course the same for rational numbers, but the crux is that they work equally well for all real numbers, and that they can be explained to people without knowledge of higher mathematics or axiomatic systems.
Of course one can measure the temperature of a refrigerator with quite a few different number systems, of even with some suitable refrigeratory Performance Index taking values from blue to red, for that matter, but this article is about real numbers, which is the standard system for measuring physical quantities "taking values in the continuum". I was being careful not to state that one needs real numbers to do physical measurements, but only that they are useful in things like that. Lapasotka ( talk) 22:47, 9 January 2011 (UTC)
You cannot do all Euclid and not notice that there are gaps on the rational line (which I suppose you meant). Algebraic numbers will do, however. But back to the point:
We need more opinions on the relevance of the discussion of what real numbers are really needed in physical measurements within the article. I see it as an unneccessary complification, since one can also turn the question upside down and ask what numbers need to be discarded. I think we agree that there should be a section on the subsets of real numbers that can be constructed by certain finite processes, such as floating point numbers, algebraic numbers and geometrically constructible numbers, as well as some others subsets such as definable real numbers. However, my impression is that mainstream physicists (or other scientists) do not care too much about these finesses and perhaphs they should not be mentioned in the introduction even though they are of mathematical and (perhaps) philosophical importance.
If there are no objections, corrections or improvements I will paste the text above into the introduction within a few days. Lapasotka ( talk) 07:12, 10 January 2011 (UTC)
To Tkuvho: Please re-read 'but real numbers are also useful in measuring quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.' I agree on the questionability of the statement 'real numbers are needed to measure the temperature in the refrigerator', but this is a different claim. What do you think of the first (actual) one? Lapasotka ( talk) 13:27, 10 January 2011 (UTC)
Alright. Maybe one needs to be more precise: 'but real numbers are also useful in modeling quantities which can "change continuously", such as the temperature inside a refrigerator or the length of a fuse.' Either in Real_number#Uses subsection or in another subsection covering decimal representations and floating point numbers one could underline your point. BTW, I don't think I have confused the meaning of the word "real" as the part of the combined word "real number" into its other possible meanings. Whether any numbers are " real" in the everyday sense of the word is a philosophical question that I will not try to answer. Lapasotka ( talk) 14:37, 10 January 2011 (UTC)
To Trovatore: Thank you for the comment. I feel myself that the geometric explanation of the real number system is too lengthy, so it is probably better to cut it down to a few words and move the rest together with some pictures into a short subsection. I somehow feel the same about N,Z,Q-part, but from the previous comments it seems that constrasting real numbers with rational numbers cannot be left out of the introduction. Most importantly, saying only that "real numbers can be defined axiomatically" is not a satisfactory solution. Can you please point out the things you do not agree with. Also some ideas how to switch the bias would be useful, but let us keep in mind that this article is about real numbers (as defined in mathematics), not about "real world" or practices of storing measured data in science laboratories. Lapasotka ( talk) 19:58, 10 January 2011 (UTC)
I did a little streamlining on my original proposal, deleted the mentions on proportionality and re-explained multiplication by comparison of areas. I really think there should be a section where the relationship between real numbers and plane geometry is explained more throughoutly in elementary (pun intented) terms. The fact that real numbers represent abstract proportions should be presented as well, but I don't have a very good idea how. Lapasotka ( talk) 16:43, 11 January 2011 (UTC)
I copied here my major change on the introduction that Trovatore just reverted. I am happy to discuss more, but nothing has happened in the talk page recently. Please comment so we can reach some resolution. The current vesion of the introduction is worse than B-class. Lapasotka ( talk) 09:23, 14 January 2011 (UTC)
Structuralism vs. constructivism is a good point, and writing down a short description taking into account both points is a challenging task. My understanding is that nowadays most mathematicians tend to be somewhere between platonists and structuralists, and constructivists are in the minority, so I didn't consider this as a major fault. I compared the length of the introduction to those of the featured math articles and it seemed to be about 50% longer on average. I added a comment on structuralism and shortened the proposition a little bit. Actually it is not much longer than the introductions of some featured math articles any more. To shorten it further I think the discussion of N,Z and Q could be almost entirely discarded. Lapasotka ( talk) 10:27, 14 January 2011 (UTC)
To Trovatore: I think I misunderstood your earlier comment. Indeed, geometric intuition should be emphasized either in the intro (which is hard if we want to keep it short) or in the first section about basic properties. Lapasotka ( talk) 09:56, 16 January 2011 (UTC)
The page is a pain to elaborate. Almost everything is reverted instead of modified and no-one seems to be able to get anything done with it. It is a shame, since real numbers is one of the most important topics in mathematics, and the page as it is right now is a mess. Before I started on it I took a look on featured articles in mathematics and noticed that an elementary description, accessible at least to high school students, is an essential part of them. Here are some comments:
And a kind request:
Lapasotka ( talk) 08:09, 20 January 2011 (UTC)
A recent change by a user made subtle modifications to a sentence that drastically altered the meaning of the statement. Here is the link to the change in question. No sources were added. Is the new meaning or the old one correct? Cliff ( talk) 18:12, 18 February 2011 (UTC)
My mistake, I was not clear enough. I don't challenge the information about Dedekind, that's easy enough to verify. My question about the modifications were actually to the second paragraph in the History section. The change implies that the the acceptance of these types of numbers was local to Europe, but that Chinese and Arab scholars had accepted these ideas earlier. The previous incarnation of the paragraph indicated that these three regions accepted these notions at about the same time. Cliff ( talk) 20:20, 18 February 2011 (UTC)
The changes have now been reverted, but which is correct? since there is no source for either contribution, how do we know which is correct? reverted article. --- Cliff ( talk) 20:24, 18 February 2011 (UTC)
I am not sure if this is encoded in some formal wiki regulations somewhere, but it seems to me that the appropriate way to proceed with a lede is incremental: one generally avoids making sweeping changes, and adds items one at a time. This is surely the best way of avoiding reverts. Tkuvho ( talk) 09:33, 14 January 2011 (UTC)
I copied the proposition on the talk page for the reason that it can be incrementally changed before replacing the original one. I felt that the original introduction needed so many structural changes that it was better to do them at once to avoid unreadable versions of the page. Here is a list of (seemingly) unmentioned things in the original intro:
Now the description of fundamental operations (multiplication and addition) is mentioned but not explained in the proposal. Lapasotka ( talk) 10:53, 14 January 2011 (UTC)
Note - Just a thought as a reminder, in order to avoid going through this (or a similar) process every time a new contributor arrives, wouldn't it be a good idea to provide a solid standard textbook source with each change? - DVdm ( talk) 11:39, 14 January 2011 (UTC)
Why is the following metaphysical "deduction" still there?
And what about the other even more questionable statement, which is certainly not widely agreed upon and only backed up by a non-peer-reviewed citation?
I skimmed through the citation and even if it was reliable and peer-reviewed (which ArXiv is not), it did not imply such a strong and controversial conclusion. Besides, the existence of any sort of numbers in the physical universe is a philosophical question, and should be left out of mathematics articles. At most, there could be a section called "Real numbers and metaphysics" where one can include statements like these. In the way how they are stated now they might give a false impression on the importance and definiteness of real number system at the heart of modern day mathematics and its applications. Lapasotka ( talk) 09:11, 27 May 2011 (UTC)
I elaborated the "physical sciences" part into a form which most working mathematicians and physicists can probably agree with. I am still waiting for more comments on the "infinite divisibility". As far as I understand, the space-time is actually modeled by a Minkowski-space (effectively R4) in relativistic quantum mechanics, or by some higher dimesional fibre bundle in super string theories. I don't see how in such framework "infinite precision real numbers" do not exist in the "physical universe". After all, statements as above can only be made with respect to some physical model of the reality. Lapasotka ( talk) 13:23, 27 May 2011 (UTC)
I edited the part on the existence of infinite precision real numbers in the physical universe in accordance to the Wikipedia links. I am not an expert on these matters, but I tried to be true to the referred articles. Somekind of elaboration was needed, because the previous statement was simply too radical to be stated only "as matter of fact". Next I would like to draw attention to the P=NP -part. There are no citations. What might that piece of text actually try to achieve? Lapasotka ( talk) 13:57, 27 May 2011 (UTC)
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?
The lead says that the reals can be defined axiomatically as the "complete Archimedean ordered field." The main body of the article, however, gives an axiomatization that simply defines them as a "complete ordered field." The second version is correct, and the word "Archimedean" in the lead should be deleted. The Archimedean property is not independent of completeness. If you make a conservative extension of the reals to a nonarchimedean system, you get the hyperreals, which lack completeness. The Archimedean property is important, but it shouldn't be given in the lead as if it were independent of completeness. You only have to specify whether it's Archimedean if it's not complete. The rationals are Archimedean and not complete; the hyperreals are non-Archimedean and not complete.-- 76.167.77.165 ( talk) 15:16, 22 March 2009 (UTC)
In the Advanced Properties section the article claims that the real numbers are everywhere dense. I am familiar with the concept "dense" and "nowhere dense", but not "everywhere dense". Surely when claiming that a set is dense one has to specify the set in which it is dense. The complex numbers are an example of a set in which the real numbers are nowhere dense. Can it still be possible that they are everywhere dense? 70.72.220.88 ( talk) 03:21, 21 December 2011 (UTC)
This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.
Please help by viewing the entry for this article shown at the page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed.
I searched the page history, and found 3 edits by Jagged 85. Tobby72 ( talk) 22:38, 20 January 2012 (UTC)