1002:), they motivate the real numbers by examples such as the square root of two. Then they say (informally) "The real numbers can be thought of as the set of all numbers with a finite or infinite decimal expansion". Would that be a reasonable way for us to proceed here? I also found at least one calculus book that gives the same characterization. This seems like a different way to avoid any awkward "continuous quantity" phrasing, and it has the benefit of being sourced to a high quality publication. â Carl
717:
to enter the philosophical realm discussing whether numbers are a realization or an abstraction. I do admit that a certain axiomatic system describing/generating(?) the reals is one "realization" or "instantiation" of them, but I think discussing a meta-equivalence of axiomatic systems for the reals-in-themselves is way beyond the scope of this article. The category of "is an idealization" is similar to "is beautiful" and not in my focus here. Maybe I could get some consent against your categorical
125:, in the first sentence, as kind of "introduction" for the notion of reals, is a logical flaw of circularity. I am convinced that there must be abundant citations about reals forming "the" or "a" continuum, a term I would prefer to talking about "continuous quantity". I am averse to using the term "measurement" in connection with reals, even when I disagree that physicist make measurements of "complex numbers". All these mathematical constructs are just models in the physicist's worlds.
1332:"Intuitively well-specified" means that, if you've understood the intuition correctly, then your mental picture must correspond to the true Platonic reals. A line has zero width, infinite length in both directions, and can't be pulled apart into two pieces without breaking it. That's enough to specify the reals up to local topology, anyway. If you have understood the intuition, you can use it to distinguish between correct and incorrect formalizations.
31:
246:@Purgy: as Trovatore says, people measured distances along lines for a long time, so there is no circularity in saying that this is one thing that real numbers can measure. Your "very bold" edit included the phrase "quantity along a line" which has, as far as I can tell, no meaning at all. The thing that real numbers measure along a line is distance, not some unnameable "quantity along".
477:"Line" also means all sorts of things, but I think most readers have come across the Euclidean concept â a line is infinitely thin (that is, zero-width), infinitely long in both directions, consists of points of zero size, and has no holes in it. That description is enough to make the reals well-specified, with a couple of quibbles. That's the concept we should be starting with. --
1603:
Others turn to the second half. By the way: is the (rational!) number 1+10 a possible result of a physical measurement? This naive question can be a kind of a bridge from the first part of the truth to the second part. Possible values versus possible results of measurement (of, say, a distance, or a volume, or a mass); the same or not the same?
1939:"Real numbers are, in fact, pretty much any number that you can think of. This can include whole numbers or integers, fractions, rational numbers and irrational numbers. Real numbers can be positive or negative, and include the number zero. They are called real numbers because they are not imaginary, which is a different system of numbers."
1704:, very worthwhile because---the need is: make the discussion accessible tothe lay reader). Citing this particular geometric form (a straight line) does not offend the value of other forms that contain real numbers---it merely takes best advantage of what's already out there and provides an efficient, intuitive connection to the lay public.
1175:
decimals as their way to explain the set of real numbers. (As for privileging base 10 numbers, I think that may be a lost cause for a general audience article, unless there is still any society that uses non-base-10 in their everyday affairs.) I do think that an article like this should be particularly careful to be accessible. â Carl
4263:
mneasurement every real lenght can be given by an integer. a square which side is planck length ha diaginal that has to be either plasnch length or twice that but there is no squareroot of 2 in reality as it is irrational and thus it's actual experssion would use all the energy in the universe and you still haven't got it.
1795:
have an intuitive grasp of any mathematical concept of "continuous quantity"---nor is the concept explained very well in Wp. :::: IMO, the (proposed) phrase "distance along a straight, continuous line" provides the most intuitive feel (for grasping the concept of "real number") to the largest number
1790:
to inform the reader re the actual subject, ie, a "real number" as a "continuous quantity"; 2) it fails to make any meaningful connection between the two concepts; 3) and thus it is unhelpful to the lay person reading Wp and attempting to learn the basic parameters of real numbers; 4) the lay public
1290:
The line is the underlying notion and is intuitively well-specified. The ancients understood what it was; they just didn't know how to work with it. You don't need the reals to specify what the line is. When you have the concept of the line, and elaborate it in the only possible correct way, you will
716:
Again I oppose: I hold dear that mathematicians, talking about reals, do talk about formal objects from the formalization of an idea, and I am convinced that a large majority of potential readers consider this very idea (take any construction of reals out there) a "highly abstract one". I do not want
96:
As we all know, there are some quantities in physics that are measured with complex numbers, rather than real numbers. Are these not "continuous quantities" as well? It seems to me that the identification of real numbers with continuous quantities only works when continuous quantities themselves are
3412:
in places where it could be using LaTeX for consistency with the image, sometimes does in fact use LaTeX where either style would be OK, and is also using the
Unicode character â, which is not allowed at all (except when discussing the character itself, which it looks like this passage isn't). Given
1904:
I'm just going to pick on one little point, without commenting yet on the rest of the proposal. That is that I see no particular reason to mention a "straight" line. A Euclidean line, of course, is straight, but that seems to have almost nothing to do with its connection to the reals; a curvy line
1699:
between "real number" and"continuous quantity" (Pls, don't presume it's just fine---read for yourself). Neither is "real number" actually discussed in the linked article---it is mentioned only 3 times in 3 separate sections. (All this reading for no good explanation of the subject is simply harmful
1602:
True. Abstractions are a half of the truth. The second half, I'd say. And here is the first half of the truth: real numbers are all (integer or not) positive and negative numbers (and zero), widely used as possible results of measurements (geometric, physical, etc). For some readers, this is enough.
1565:
As said, I am a fan of using an informal notion of a line as guiding paradigm to the notion of the reals, but am rather skeptic about all the epitheta like continuous, straight, geometric, ... and especially "real". While I mostly agree to the above arguments, I find not much of an improving move in
1285:
is very wrong on this issue, which has occupied the last few talk sections, and in which he has been an active disputant. But he's not alone in this error -- he's just expressing an orthodoxy that has been drummed into several generations of high school and college students, at least in the US. I'm
1077:
define the reals in terms of decimal representations, modding out by the appropriate equivalence relation. And it's true, you can. In that approach, you make 0.999... equal to 1 by definition. But it's not a very perspicuous approach, and I think not one of the more popular ones, for good reason.
1575:
Wholesale, I think that it is helpful to tell the readership that the real numbers are not very real (in the sense of 'physically manifest') in the real (in the sense of 'as encountered') world. Real numbers exist in the way they do exactly to satisfy (most) mathematicians' needs for rigor, so that
1224:'s point raised in the previous thread. Perhaps it's a French thing?) It's not clear that this gives us any good insights into how the first sentence of the article should be written, but I think the rest of the lead should focus at least somewhat on measurement and less on the real line per se.
1219:
Another interesting discussion of the reals is to be found in the historical note to
Bourbaki's General Topology, Volume 1, Section IV, which begins thus: "Every measurement of quantities implies a vague notion of real numbers." The line is scarcely mentioned in their historical discussion. (With
908:
is really a topic for a philosophy-of-math article more than for this one, though the reals are an interesting enough dividing line (there are people who are realist about the natural numbers but formalist about the reals, for example) that some discussion might not be out of place. I don't really
4262:
Should it be addressed that real numbers are not actually real. One can zoom
Mandelbrot's set as far as one can and find the self similiarity but if you do that for eaxmaple to englisg coastline you come up with atomic level where the self similiarity ends. In fact if one uses planck lenght as the
3745:
I'm not sure there's a graceful way to avoid singular "real" in the context of forcing (or why you would want to avoid it). Add a Cohen real; add a random real. It would sound really odd to say "add a Cohen real number". I suppose that's because you really aren't interested in it as a "number";
2540:
Nevertheless, I have had a quick look on this article, and it seems not to be extraordinary at all: It is well known that the binary expansion of a real number may be identified to a subset of the integers (the set of the places where there is a one). So, the real numbers may be identified with a
1965:
My opinion that the proposals by Jbeans are to my measures no improvement to the status quo is not affected by the recent statements by Jbeans. To me, the list by Boris
Tsirelson indicates that it is not easy to find something better, and my preference of defaming the reals as something abstractly
1174:
I am perfectly fine with referring to the real number line as the model for the real numbers; I was suggesting the decimal option as a possible alternative, but I don't really have a preference. I spent a few minutes looking at elementary algebra books, and they seem to either use the real line or
899:
As I'm sure you know, D., and I'm less sure about Purgy but probably, it is standard in mathematics to talk like a
Platonist whether you are one or not. This is the convention we should follow here. In the lead, or at least the early part of the lead, we should describe the reals uncritically as
532:
May I point to the effort of avoiding introducing "reals" as "numbers", but instead as a "formalization" of something in "everyday use", similar to the notion of "line" in this here context. I share the opinion of
Trovatore that mentioning the line is helpful for most readers, and is no obtrusion,
297:
The mention of a line in the first sentence is a reference to a mental representation (intuitive representation). Even if this mental representation is commonly accepted, there is no reason to impose it to everybody in the first sentence of an article. On the other hand, it should be useful to say
4081:
Hmm, now that my attention as been drawn to it specifically, I am not convinced we need to say "or real" in the first sentence, particularly in boldface. I'm against removing the usage where it comes up naturally, but I don't really see a need for it there, in spite of the link from the disambig
2573:
The main article claim is that the basic operations on reals can be realized by set-theoretic operations on subsets. This is not a surprise, and has essentially been used by computer scientists for hardware implementation of arithmetic. As I have read only the beginning of the article, I can have
1256:
I was not serious in my speculation of the French connection. While I share
Trovatore's view that the Princeton Companion is not ideal, and that the primary conceptual motivation for the reals should be the line, I think something about the practical importance for measurement, calculus, and the
491:
IMO, "of everyday use" excludes the complexes for everybody. By the way, a reader who knows of the complexes knows enough mathematics for not needing any explanation of what are the reals. Thus we do not have to take care of the complexes in this sentence. Here, understandability by the layman is
1579:
I.e., I like to start saying that reals can be (vaguely) found along a line or in other measurements, and that their overall usefulness abounds, in spite of the next, that their accessibility results from one of several "unwieldy realizations" of highly abstract and partly unintuitive ideas (the
1546:
More important point: we understand why we emphasize distances along a line, but some readers do not (I guess). They are puzzled: just distances on a line? Not distances on the plane, or space? Not areas and volumes? Not angles? Not masses, voltages etc? I'd say, the first phrase cannot just say
965:
numbers with line segments. So where we would say "x is a multiple of y", he would say "the line segment x is measured by the line segment y". It was the (shocking!) discovery that not every pair of line segments had a common measure (i.e. that their ratio was not always a rational number), that
3795:
A comment: "the reals" and "the real numbers" are not exactly synonymous. The latter refers implicitly to the set, while the former refers to the whole structure. So "the multiplicative group of the reals" sounds better than "the multiplicative group of the real numbers". Moreover the latter is
3252:
In
Unicode, the character â is labelled âreal number symbolâ. As far as I can tell it was present, along with this label, in the very first version of Unicode. Does anyone know what is meant by this? Is there an area in mathematics that uses lower case g as a symbol for (a/the) real number(s)?
3726:
Yes, the OED agrees that it is "usually" used in the plural. The only case in the current version of the article that uses it as a singular noun is in the lead sentence. I am not sure how to rewrite that to both introduce the use of "real" as a noun, and not to add too much detail too early. I
2536:
asserts that one can include in
Knowledge only results that are mentioned in secondary sources (this is necessary, but not sufficient, as notability is also required). So a full review is not a work for Knowledge, and, for considering this result, one must wait that some reliable sources cite
960:
I donât think mentioning the idea of âformalizationâ in the first sentence, is the right way to go here. For me, the most basic property of real numbers is that they measure distance along a line. This is born out, it seems to me, by the historical development of what a number is. Euclid, for
337:
That's a good idea to think about. I would replace "integer" with "rational", since that is the bigger leap (cf. the history behind the irrationality of the square root of 2). The other thing is to get a phrasing that does include the integer and rational numbers among the real numbers. One
3523:
I wanted to record here that "the reals" is not proper
English, because "real" is not a noun. Also, there is a difference between "the real numbers" and "the set of real numbers"; the latter (or "the space of real numbers" or "the field of real numbers", as appropriate to context) should be
2368:
seems much more in need of support, and not really subject to "proof" in a single paper. It's something that the mathematical community would need to come to a consensus about, and that would take years at a minimum, decades more likely. To uncritically cite the claims of a paper published
1966:
engineered, at least a little bit, has not changed. Negatives yes, complexes no, straight, geometrical, continuous, "number lines", ... are all constructs way above the accessible math for the targeted readership, and partly of no obvious help at all. I still like something along Trovatore's
1865:
No, this change would make the opening sentence significantly worse. "straight, continuous line, or geometric line" doesn't really make much sense. How can a line be not straight? or not geometric? What's a continuous line? Trying to cram in stuff about positive and negative, rational or
857:
first sentence would be "The concept of real number is a formalization ...". However, such a formulation would be a little pedantic for the first sentence of the lead, and "The real numbers are a formalization ..." seems acceptable, as, for users of mathematics, mathematics is essentially a
1241:
I does not think that this difference of point of view is a question of geographic origin. It is rather a different between a pedagogic approach, which emphasizes on intuitive motivation for young people, and a working-mathematician approach, which emphasizes on applications and historical
1096:
popular approaches to defining the reals in settings below the level of real analysis, along with references to the real line. Dedekind cuts and Cauchy sequences are not going to appear in grade school texts. The point of the first sentence is to have something at that lower level. â Carl
2162:
It seems to me that Trovatore agrees with Deacon Vorbis and IP that the pipe should not exist, based on the comment for the edit, but undid the removal of the pipe. This confuses me greatly. Is there a clear consensus to linking just Quantity or is there something still to be discussed?
1328:
This is really not the place to discuss it at length, but having been called out directly, I will give brief answers. I'm happy to discuss it further on my talk page, or yours, but any further discussion here ought to be justified by some direct connection to what should appear in the
2314:
This sentence could easily placed after the first paragraph, either as a paragraph for itself or as a first sentence of the second paragraph. In the latter case, the transition with the second sentence could be assured by adding at the beginning of the second sentence something like
2291:
It certainly doesn't absolutely have to be in the second sentence, but I still think it should be in the lead. It's kind of an important note about the naming, that other numbers aren't any more or less real, but that it's just the name that got attached and stuck. (The article at
1994:. The proposal moves further away from a satisfactory beginning. The first sentence emphasizes the straightness of the line or its other (unspecified) geometric properties, which are irrelevant. The second sentence is completely redundant with the rest of the first paragraph.
3578:
I grant that it is sometimes used colloquially, and yes, now that you mention it, I can even find it in published dictionaries! Still, it is the kind of sloppy verbiage that I would not expect to find in a respectable encyclopedia (though someone will surely prove me wrong...)
3032:
inclusion in Knowledge, which strongly favours material supported by secondary and tertiary sources. I recommend drafting the material you want included in your userspace and seek feedback on it, here and elsewhere, which might lead to discovery of related previous work. â
1884:
Add: on second thought, some sort of rewording to make it clear that negative numbers are included might be good (a distance is often taken to be positive only, as the current version states). But the whole rigamarole that's proposed doesn't directly address that anyway.
2240:
is essential for reals, and I am along the lines of Trovatore in not being sure, if at this point a link is appropriate at all. A "broader founding" of vagueness for "quantity" might not be really optimal. I apologize for having caused these troubles by my inadvertency.
1547:"what is a real number?" (this is desirable, of course; but maybe impossible); rather, it should introduce the reader into the distinction between "practical" idea of numbers (approximate, subtleties aside) and "theoretical" idea of numbers (continuum, irrationals etc).
3319:
2. AMS went through several font transitions. At one point, they switched to Times Roman, but at first they opted to use a single-storey g in italic text like their previous font did, rather than the double-storey g of italic Times Roman. Neither looked script-like
3408:), today I was making the rest of the article consistently use LaTeX instead of regular bold. Except, obviously, where the article is talking about the difference between the two. After this revert, it looks to me like this "Vocabulary and notation" section uses
1020:
I think that is not a good approach. It starts with the representation rather than with an intuitive description of what is being represented. And it does it specifically in base 10, whereas there is nothing special about base 10 when talking about the reals.
1197:-base-10 is not really the point. The point is that base 10 is a representation, not the the thing being represented. The importance of other radices is not so much that anyone uses them, as that they show that a particular radix cannot be of the essence.
937:
Actually, anyone who wants to say that the reals "are" a formalization is the one who should show sources. And even then it might be "undue weight". I think very few high-quality sources in English are going to describe the reals as a "formalization".
652:
Purgy, for question 2, I think the idiomatic phrase you want is "rounding out," rather than "rounding off." Then there's no danger of some dumbhead thinking you're talking about removing precision. Or, of course, you can just cay "completion." Cheers,
3760:
Well even if the singular usage occurs, it is apparently not common, (I don't think "Cohen real" necessarily counts as a use, since one could think of "Cohen real" as the noun), so I don't think it needs mentioning in the lead (frankly it sounds
266:
Related to being bold - bold edits are good when they work towards consensus. On this page (previous section), several people have indicated they are in favor of talking about the real line as a way of understanding the real numbers, so simply
3623:
It is not "colloquial". It is perfectly standard usage in research mathematics, including in journal papers. Admittedly encyclopedic writing is a tad more formal even than research publications, but still I see no need to change it.
4286:, roughly speaking, consider statements about them to be shorthand for claims about what can be proved. Intuitonists and constructivists have many varied approaches that we need not belabor here, even if I had the expertise to do so.
3765:
odd to me there.) If it must stay, then explanatory details could (and should) be provided in a note, which should also include a source. I'd also be okay, instead, with including somewhere "The set of real numbers is also called the
2108:. I see nothing wrong here with referring to a real number as a continuous quantity in the text, as this is clear enough in natural language without the need to supply precise technical details. It does not even require a link to
3796:
incorrect, as it should be written "the multiplicative group of the nonzero real numbers". So using "the reals" allows often a simpler wording, and not using it may lead to some pedantry if one want to be mathematically correct.
3435:
Now, it is true that BBB R often does appear in other places than blackboards, and it often does mean the reals. I could live with using it that way in the article, as long as we explain at first reference that it can also be
1700:
in terms of reaching (and informing) the lay reader.) ... {Two} New specific text is added, saying a real number can be visualized as: "distance on a straight, continuous line" (which is both a phrase and a geometric form
511:
I actually don't agree that readers who know about the complex numbers don't need any explanation about what the reals are. The reals are a subtle and non-obvious concept. Their most salient distinguishing feature is their
338:
challenge is finding a sentence that does not apply to the complex numbers just as well (e.g. measurable quantities, continuous quantities). How to do that without referring to a line is something to think about. â Carl
2276:(Speaking of history, this article is still missing any mention of Zeno, which as I have said before is a hard-to-understand omission. I should probably quit whinging about it and just try to do something about it.) --
1150:
Yes, but not in the first sentence. One sticking point for me is that it privileges base 10. We could make it not privilege base 10, but only at the expense of lots more verbiage that's not really going to fit there.
4289:
It would be possible to bring in some of this discussion, attributing various views to various schools, but the question is, why this article specifically? The same questions arise for lots of sorts of mathematical
2232:. Mislead by the IP's edit summary I wrongly assumed, after only sloppily looking, that the word "continous" had been removed completely (cf. my summary). If it had not been for the possibility of misinterpretion as
675:
I thought maybe you wanted to say that they're an "idealization" â that is, you start with a messy group of empirical observations, and you abstract out an underlying idea, one that perhaps none of the observations
3282:
mentioned that, at some time, in the publications of the AMS, g did not have the same shape in superscript and on the normal line. One of the shapes was â. The explanation, if any, could probably be found there.
1427:
Even I, long of a technical/engineering career, cannot relate to a singular "number" itself being described as a "continuous quantity"---algebraic variables excluded. That is, here the phrase seems to imply: the
3709:
I'm happy with the "reals" in the plural (since I'm familiar with that usage), but I'm not familiar (so not happy) with it's use in the singular e.g. "x is a real". I don't see any such usage in my copy of the
355:
I think we should hold the line on "line". That's how the reals are typically introduced to children, and for once it's an excellent choice. The ancients weren't disturbed about the irrationality of â2 for
1394:
Hi, sorry, I seem to have saved this section inadvertently while it was still a very rough work in progress, and while it still didn't say any of the main things I wanted to say. Please ignore. Sorry again.
3101:
I can see it is very wise to wait for second opinions in most everything. In mathematics, there may be an exception to this because if you prove something, its proven. All you have to do is read the proof.
621:
Well, ... I think the phrase "everyday use" excludes some of the incriminated deviances, and integers are a nice object to continue a "rounding off"/completioning process, which might have started at the
516:, which is what we should be trying to get across (informally) right up front. Just because you're comfortable with â1 having a square root doesn't mean you've thought about the topology of the reals. --
2809:
Well, it's not really our opinion that counts, either, at least not in theory. Naturally people's individual views are likely to color how they see the literature, and I don't think that can really be
923:"The reals are not a formalization, period." Which sources allow you to making such an authoritative assertion. Authoritative assertions are of no value here. You are confusing your opinion with truth.
683:
But then I noticed you say they're a "formalization" of a "very abstract idea" (what idea?) so maybe you actually want the other direction from idealization â say, "realization" or "instantiation".
187:
I am fond of the "line", especially of the blank-faced Euclidean version (curtly behind "continuum", but immensely before "continuous line/quantity"), possibly even when linked (ridiculously?) to "
4054:
The cite is included in my comment above. The text of the entry is quite short. I've added a ref to the article, with the full text of the entry, which should be short enough to avoid copyvio. --
3028:
I don't think reaching the bar for some kind of representation of content requires all that widespread a conversation in the literature, but that proposed material is very new actively counts
1522:
Both alternatives have the advantage that the reader can ignore linking to "geometric line" and still have an intuitive feel for: ".. a distance along a straight, continuous line". Thanks,
470:
As for defining "real number" in terms of "number", no, I don't like that at all. "Number" is a very fuzzy word; means all sorts of things. Defining "real number" in terms of "number" is a
3345:
3260:
1412:
I agree with keeping the geometric line in the lede description for "real number"---it has an intuitive feel for much of the reading lay public. But my guess is, the typical lay reader will
3329:
In Latex using the default font, regular text gets a two-storey g and italic text gets a single-storey g, regardless of whether it appears in body text, a formula, subscript or superscript.
2464:
On second glance, maybe the author is actually making these sorts of claims, but to echo the points made above, this is more philosophical and probably not shared by the wider community. â
1929:"The short, simple answer used in calculus courses is that a real number is a point on the number line. That's not the whole truth, but it is adequate for the needs of freshman calculus."
2345:
Changes to this article are proposed, given new results provided in . The proposal is to include a construction of real numbers as an extension of natural numbers in the universe of sets.
249:
But my question still stands: is there any modern source that defines continuous quantity? If not, I think we should avoid it in the first sentence as an overly specialized term. â Carl
2755:
Thats clear, but youll have to see the paper to see what kind of "new results" we have. In my opinion they are well worth including, but its not just my opinion that counts so ill wait.
452:
circularity I think, but it definitely doesn't sound good. I hadn't realized that was the sticking point for you. I agree that "line" is better than "real line". We can point to the
1416:
have any (good) intuitive feeling for the phrase "continuous quantity"; in fact, he/she will likely suffer a 'lost cause' or 'eyes-glaze-over' feeling upon encountering it. The linked
618:
I humbly ask for linguistic help in finding the right wording for the meaning of "completion", which I had in mind, and wanted to circumscribe in a most accessible way by "round off".
885:
No, the reals are not a formalization, period. The concept of the reals is not a formalization either. There is no reason at all to mention formalizations in the first sentence. --
1747:
Fellow editors, please review my two recent posts---they are in the section immediately above---regarding my two recommendations that follow. Please note the current lede sentence:
3404:) or the standard boldface must be used. As with all such choices, each article should be consistent with itself". Given that this article has an image that uses blackboard bold (
492:
much more important than logical and mathematical correctness. As "fuzzy words" are not really a problem for the layman, if he think understand them, they are not a problem here.
2571:
3942:
3920:
3867:
3831:
3544:
3482:
3402:
4143:
It's not a "rare" usage in the area of academic math I was in, but I don't see any need to call it out specially. When it comes up, it's pretty obvious what it means. --
2178:
It's not clear to me there should be a link there at all, but what is clear to me is that piped links of the form ] are almost never a good idea. The link just points to
2080:
Re your final question, I tend to the opinion (please, refer also Boris Tsirelson's punctuation) that for the highly sophisticated real numbers there might be no "clearly
571:
2. Saying that the real numbers "round off the numbers" does not make sense to me. "Round off" means to round a number to an approximation, i.e. we can round off Ï to 3.1.
1934:"All numbers on the number line. This includes (but is not limited to) positives and negatives, integers and rational numbers, square roots, cube roots , Ï (pi), etc."
136:
Real numbers are mathematical constructs, defined in a way that satisfy most needs in arbitrary continua. They may/can be envisioned as distances or points on a line.
3688:
4113:
doesn't need to be in the first sentence. I just put it there because it seemed like the smallest reasonable incremental change. I do think it needs to be mentioned
3811:@D.Lazard: Yes, those are good points, though I would also say that it is quite easy to avoid the pedantry and to save even more space: "the multiplicative group of
2424:"Some modern constructions propose real numbers built directly from the structure of integers. However, it has been recently proven that real numbers are subsets of
568:
but the real numbers themselves are not a formalization, unless every mathematical object is somehow a formalization (in which case there's no reason to mention it).
2863:
it's the canonical one. I haven't checked the nice properties, but even assuming that part of the argument is correct, I'm just not really seeing the "therefore".
3349:
3264:
1831:; and note, the phrase "geometric line", immediately above, is presented in apposition to the phrase "a straight, continuous line", and they are not redundant. }
615:" Roughly: I do not believe in real numbers in any philosophical reality, besides as reals-in-themselves, they are no measurements, neither real nor complex, ...
1518:
Among its other properties, a real number takes either positive or negative values; it will be either a rational or irrational number, and not a complex number.
360:
reasons â if they had been, they would have just said, OK, 2 doesn't have a square root, maybe surprising, but that's how it is. But they could see that there
2182:, so the "frumious" has nothing to do with the link and should not appear in blue. The user who clicks on the link has no warning that the article is merely
89:
It seems to me that "continuous quantity" is not a common phrase, at least not in the mathematics books I see that talk about real numbers. I was wondering:
1669:
Among its other properties, a real number is either a positive or negative value; it can be either a rational or irrational number, but not a complex number.
833:. "Formal" and "formalization" have been used in mathematics a long time before the introduction of the first formal systems. This being said, I agree that,
3847:
Generally bad to use symbols in place of prose, though, provided the prose is easily understood. Particularly given that not everyone notates the reals as
97:
identified as signed magnitudes on the real line, at which point the first sentence here might as well cut out the middleman. But this takes us back to #1.
93:
Is there is any good reference in a contemporary book for the term "continuous quantity"? In particular, in a book that is not about history or statistics?
1362:"The real numbers are a number system that includes the integers and the rational numbers" â rather, one of such number systems; recall complex numbers.
1866:
irrational, and complex numbers so quickly isn't a very good approach either. The current opening paragraph already eases into the rest more gently. â
1295:
This is gibberish. In the first place, what does he mean by "intuitively well-specified?" Whose intuition his he referring to here? His? Mine? Yours?
1721:
Either one of the proposed alternatives plus the 'follow-on' sentence would serve the lay reader much better than the current lede sentence. Regards,
2084:" text available at all, but just some motivation leading to various axiomatizations, probably not perceived as "clear" by an uninitiated audience.
1055:
In addition I bother that decimal expansion taken as the fundamental idea could be interpreted by some readers as a green light to the negation of
574:
3. The word "number" on its own does not have much meaning. It could mean an ordinal number, surreal number, natural number, or many other things.
4161:
page points to this page. But it is pointing to "Real numbers", and it also links to dozens of other phrases that include the word real, such as
2077:
for any of your two suggestions, compared to this status. To me it is obvious that for the time being there is no agreed upon improvement visible.
686:
I'm not sure exactly what word you want because I haven't quite understood what you're getting at. But "formalization" is definitely not it. --
4278:
This gets at very fundamental disputes in the philosophy of mathematics, on which we have no business taking a position in Knowledge's voice.
2434:, and this paper just seems to give yet another, not make any sort of grand claim about the true nature of what the real numbers Really Are. â
1092:
To be fair, I believe they are also defined that way in many elementary algebra and calculus books. I would suspect it is probably one of the
172:
with it. You don't need the reals to specify what the line is. When you have the concept of the line, and elaborate it in the only possible
3602:
2756:
4220:
Is 0 a rational number? As per my classification every rational number has got some value .if I talk about (0/1) it doesn't have any value.
1286:
picking on him because he has defined his position so succinctly above, i.e., because he writes well. In particular, he has said above that
1438:(one) real number. It's doubtful the phrase "continuous quantity" has any knowledge recognition among the lay public, and the linked page
271:
references to the line is not likely to work towards consensus. It's also worth remembering that the first sentence of the article is not
164:
No, I disagree that it is circular. The line is the underlying notion and is intuitively well-specified. The ancients understood what it
279:
the real numbers, so the reader knows what we are talking about. That kind of identification may well have some circularity to it. â Carl
3674:
3428:
My concern is that we not give the impression that BBB R is "the" symbol for the reals. At least in origin, the symbol for the reals is
800:
Look, I'm not saying you can't be a formalist. If you're a formalist, then you may well take the position that the objects of discourse
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missed something, but, again, if there is more in the paper, we must wait a secondary source that explains the true new result, if any.
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they can be used without bothering about exceptions and quirks (irrational, constructible, algebraic, computable, transcendental, ???).
4168:
So let's remove "real" from the first sentence, and move the OED reference to "the reals" at the end of the first paragraph. (P.S. to
2859:
for canonicity. It kind of looks like it might go something like, this construction has this and this and the other nice properties,
2431:
2273:
important. It strikes me that it distracts from the flow of thought, and should be deferred to a "History" or "Etymology" section.
2011:
It seems that no one is in favor that "straight, continuous line" be put (directly) before the reader. Regardless, please remember,
1823:. A real number is either a positive or negative value; it can be either a rational or irrational number, but not a complex number.
2855:
That said, though it's off-topic, I'll indulge myself to say that I took a quick scan through the paper, and I don't see any real
302:
the concept has been introduced, and why real numbers are not simply called numbers. Therefore I suggest for the first sentence:
3310:
1. For years AMS used a slightly different font in subscripts and superscripts than on the baseline; he wasn't happy about this.
3171:
Yes; but "canonicity" cannot be proven, just because it is not defined. Or did I miss its (mathematical!) definition somewhere?
456:
article somewhere later, if we want a link to that article (and if indeed that article shouldn't just be merged here anyway). --
1359:"he's just expressing an orthodoxy" â wow... I'd say, yes, Knowledge is generally expressing an orthodoxy in the first place.
1313:
874:
607:
Reals ARE a formalization of an indeed very abstract idea, and the reason for mentioning this here is not that math is about
2908:
This is not the place to discuss it, but if you want to leave a note on my talk page clarifying that key step, feel free. --
2119:
2001:
1264:
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It seems, no one found a good one-line "definition" for real numbers. I guess, this is not possible. Here are some quotes:
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intuition. There is a right and wrong about this, though it is not subject to proof in the ordinary mathematical sense. --
4165:. Also, although the OED reference supports the use of "the reals", it indicates that the singular form is not typical.
323:
This proposal is a first draft, that may and should be improved. In particular, I do not like the use of "non-integer".
672:
in the first place. I'm not sure what you're trying to say, exactly, but "formalization" is definitely the wrong word.
3601:
I agree, and I think "reals" should be changed to "real numbers" throughout this article. This article also links to
2323:". After all, the concept of number line consists of associating each point of a line with its Cartesian coordinate.
3650:
Not only is it standard today, it has a long history with respectable authors. For example, Hamilton uses it in his
364:
to be a square root of 2, because it was the length of the hypotenuse of an isosceles right triangle with unit side.
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My main point, though, was that we should spell out "the reals" in running text, rather than resort to a symbol. --
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are a different topic from real numbers, and the first sentence of the article should not reinforce any ambiguity.
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Sections 6 and 7 justify this. Reading those, and the following essay on Benacerraf's paper will help seeing this:
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motivations (Dedekind and Cauchy were not concerned by geometry when they developed the modern concept of reals),
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The extraordinary claims made are fully supported in the paper. Maybe a few full review of the paper is in order.
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1995:
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suppose we could remove "or real" and add a second sentence saying "The set of real numbers is also called the
3714:(1971), and I don't have a subscription to the online version linked to above. Can anyone point to such usage?
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2187:
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I don't see why. It is a rare usage at best. If it needs mentioning at all it can be mentioned in a footnote.
3458:, the intro does say at the end of the first paragraph: "The set of real numbers is denoted using the symbol
448:
There's a linguistic or rhetorical infelicity in defining the reals in terms of the "real line", not really a
4228:
1537:
A real number takes values? No, it is not a variable nor function, it does not take values; each real number
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the real numbers -- it is not talking about Dedekind cuts or Cauchy sequences, etc. Instead it is trying to
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The paper makes reference to Paul Benacerraf's "What Numbers Cannot Be" as validation we have new results.
2236:, I would have even thanked Deacon Vorbis for his revert. I think that a hint to an intuitive notion of a
1424:---for what it's worth; and there is no lay explanation in the article of any connection to real numbers.
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real than, say, imaginary numbers (though it was likely thought at some point that it did mean that). --
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In any case, I am convinced by the OED references you all gave (going back to Hamilton!) that it is not
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in the lead paragraph, though, and not just "the reals" as a synonym for "the set of real numbers". --
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1924:"A rational number or the limit of a sequence of rational numbers, as opposed to a complex number."
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This is the current version of my suggestion, based on D.Lazard's draft and critique a perceived.
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To date, no comments, pro or con, have been posted re the 2nd point---which raises the question:
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2035:", some specific text that helps the lay reader to better understand real numbers--- ... please
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R2---re improvements, there are two chiefly---proposed by both alternatives: {One} The link to
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No, sorry, the reals are not a formalization of anything. There are formal theories that talk
191:", but I ferociously oppose to "real line", linked or unlinked, the use of which is, I insist,
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Dedekind cuts and Cauchy sequences do not appear in grade-school texts, but lines absolutely
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consider real numbers to be real, albeit, as you observe, almost definitely non-physical.
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explained above, that the results are "new" is not a sufficient reason for inclusion here.
2054:" clearly narrates (to the lay reader) a meaningful connection to "real number"? Regards,
4172:: Out of curiosity, are you referring to descriptive set theory, or something like that?)
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The reason given for including "real" as singular noun in the first sentence was that the
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2031:" Editors, please reply to both points, pro or con. Particularly---if you have found in "
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The real numbers are a number system that includes the integers and the rational numbers
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the reals. But you must not confuse the formal theories with their objects of discourse.
768:, which are in everyday use. They may/can be envisioned as distances or points on a line.
429:, which are of everyday use. They may/can be envisioned as distances or points on a line.
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Editors, please reply to both points, pro or con. Particularly---if you have found in "
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I've consulted it and your memory is a blend of two independent observations of Knuth.
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I agree, "proper English" is (in the appropriate context) whatever is commonly used.
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that it's a point worth mentioning. But in the second sentence? I don't think it's
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and "real" mathematics is, if it depends on the radix, it's probably recreational. --
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doesn't seem to be supported by the reference you give. There are many, equivalent
1838:", some specific text that helps the lay reader to better understand real numbers---
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entry for this usage, I could do the cite. I'm interested to know what exactly the
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Zero is a rational number. Zero is the ratio of zero to one; that is, 0/1. Zero
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I agree with Trovatore. Decimal represtations of real numbers are important, but
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If you wish to start a new discussion or revive an old one, please do so on the
1925:
744:
as a first draft. I just consider this a good approach, and tried to work upon:
367:
It would be a serious mistake not to mention the line in the first sentence. --
1935:
1136:
Maybe we should mention both approaches, with a note about their equivalence.
2100:
If the second part of the question concerns a lack of clarity in the article
1819:
is a value that can represent distance along a straight, continuous line, or
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is a value that can represent distance along a straight, continuous line, or
1905:
would do just as well. So at the very least I would leave that word out. --
1443:
453:
122:
2752:"that the results are "new" is not a sufficient reason for inclusion here."
2069:@Jbeans, I do not think that the above discussion was targeted to express
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does something similar, and I think that's good, for the same reason). â
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1786:"---as (nominally) discussed in that page---is faulty and unhelpful: 1)
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R1---Agree; the text of the follow-on sentence can be adjusted, such as:
564:
1. The real numbers are not a "formalization". There are formalizations
3379:
966:
pushed the idea of number beyond rational numbers, in the first place.
4004:
to write "the reals" for the set of real numbers. I can't say that I
3687:
has multiple quotes from math journals, math books, and popular books.
1799:
Therefore, for a better description of "real number", I move that the
1420:
page 'Quantity' doesn't read in 'lay terms' until the last section---
853:, I assume them even I have no formal definition of them :-). Thus a
765:
739:
426:
314:
3413:
that, does my edit make sense or am I missing something? Thanks! --
2204:
Oh, now I see that I actually did the opposite of what I thought.
804:. But not existing is not the same thing as being a formalization.
508:
Well, if "fuzzy words" are not a problem, then "line" shouldn't be.
1629:
Response to: Boris Tsirelson, Purgy, (rather than require chasing
4008:
that terminology, but this is a matter of taste. Thank you all,
1503:
is the value of a distance along a straight, continuous line, or
121:
I still think that referring to the "real line", with or without
2373:
for extraordinary assertions about the importance of results in
1282:
1073:
I think this is one of Gowers's hobby-horses, the fact that you
994:
I thought it might be good to look at a couple sources. In the
680:
match, but which simplifies discussing what they have in common.
668:
The reals are not a "formalization" of anything. They aren't
227:
And incidentally, the correct word here is "let," not "left."
25:
2073:
for some detail or linking of the status quo, but only shows
4297:
be worth discussing, if there are good sources, is that the
4223:
Eg.1/1=1 , 5/3=1.6...etc. But, 0/1= doesn't have any value.
4034:
here for this usage. If someone would provide a copy of the
1697:
that the discussion there does not describe any relationship
1200:
A fairly reliable rule of thumb for the distinction between
3873:
on a blackboard, and is not necessarily distinguished from
1220:
some irony, I note that this was very closely connected to
3432:, and it's rendered on a blackboard using blackboard bold.
3338:
I think that makes this explanation unlikely, for now.
4186:
Agree, and yes, my research was largely in the area of
3731:", but that seems a bit clumsy. What do you propose? --
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3375:
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533:
regardless if it helps against complex numbers or not.
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the reals, whether or not they exist. But the reals
3869:; that symbol is in origin just a way of drawing an
2377:â well, no, I don't think we're going to do that. --
1842:
identify the text you see. (Please do not offer the
3972:By the way, what other things are commonly denoted
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you're interested in the information it encodes. --
1507:, from the zero point to a given point of the line.
1450:---does not speak to the phrase orto real numbers.
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2360:The construction given in the paper may very well
1702:that a maximal number of the lay public relates to
1378:Shall we return to discussing the article please?
434:Please, just don't make it the "real line" again.
130:Here is my effort to contribute to the discussion:
3880:Long story short, we should keep "the reals". --
1848:re ratios of magnitudes as real numbers, as here
845:by the introduction by Dedekind and others of a
474:circularity than defining it in terms of "line".
3238:https://hilton.org.uk/what_numbers_are_not.html
1580:history, leading to the reals should be told).
176:way, you will necessarily recover the reals. --
2422:If nothing else, the statement you added that
1773:concluding arguments for editing this sentence
1453:I offer the following for your consideration:
3877:, which can mean other things than the reals.
2039:." (See my previous post, closing paragraph).
755:
396:
8:
2612:Yes. It is good to wait for a second source.
829:You must not confuse "formal theories" with
147:rephrased for below 08:20, 10 May 2018 (UTC)
145:Should Eleuther have let sleeping dogs lie?
143:Eleuther should have left sleeping dogs lie.
1840:I will be most grateful if you would please
1512:A follow-on descriptive sentence might read
3339:
3254:
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1257:sciences should be drafted for the lead.
841:, this is the concept of reals, which was
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2130:Link Continuous Quantity vs just Quantity
719:But "formalization" is definitely not it.
4301:"real" does not mean that the reals are
4242:have value. The value of zero is zero.â
1692:(with its phrase "continuous quantity")
3382:says "the LaTeX rendering (for example
3219:
909:think it belongs in the lead though. --
3922:. But the trend seems to be favoring
2423:
2316:
1763:that can represent a distance along a
1469:that can represent a distance along a
730:
718:
422:more correct? 07:10, 11 May 2018 (UTC)
409:more correct? 07:10, 11 May 2018 (UTC)
393:I think this excludes complex numbers.
305:
44:Do not edit the contents of this page.
3227:https://www.mdpi.com/2075-1680/8/1/31
189:line (geometry)#In Euclidean geometry
7:
3603:Tarski's axiomatization of the reals
3346:2A02:A457:9497:1:F938:499E:52BC:CB4C
3261:2A02:A457:9497:1:F938:499E:52BC:CB4C
3605:, which I think should be renamed.â
3562:https://en.wiktionary.org/real#Noun
2017:points were proposed; the 2nd was:
837:, this is not the reals that are a
996:Princeton Companion to Mathematics
24:
2566:{\displaystyle 2^{\mathbb {Z} }.}
2432:constructions of the real numbers
2186:, and this is a violation of the
764:, that rounds out the notion of
3492:) 01:29, 17 January 2021 (UTC) (
2257:Etymology in the second sentence
1850:. Thanks & grins). Regards,
858:formalization of the methods of
611:, but here it is exactly about "
29:
3900:; I think it looks better than
3275:I remembre that in his article
1829:Here the current text continues
1803:lede sentence be replaced with
1782:" and particularly its phrase "
168:; they just didn't know how to
1291:necessarily recover the reals.
213:Sleeping dogs are still dogs.
1:
4200:19:43, 13 February 2021 (UTC)
4182:21:52, 12 February 2021 (UTC)
4153:21:42, 11 February 2021 (UTC)
4139:21:00, 11 February 2021 (UTC)
4127:18:22, 11 February 2021 (UTC)
4105:00:05, 11 February 2021 (UTC)
4092:23:23, 10 February 2021 (UTC)
4064:20:42, 10 February 2021 (UTC)
2112:, if that's your main beef.
1805:new lede and second sentences
4050:13:17, 9 February 2021 (UTC)
4018:03:01, 9 February 2021 (UTC)
3988:02:29, 9 February 2021 (UTC)
3968:01:36, 9 February 2021 (UTC)
3954:01:33, 9 February 2021 (UTC)
3937:{\displaystyle \mathbb {R} }
3915:{\displaystyle \mathbb {R} }
3890:01:16, 9 February 2021 (UTC)
3862:{\displaystyle \mathbb {R} }
3843:01:10, 9 February 2021 (UTC)
3826:{\displaystyle \mathbb {R} }
3806:20:53, 8 February 2021 (UTC)
3778:20:09, 8 February 2021 (UTC)
3756:19:36, 8 February 2021 (UTC)
3741:19:29, 8 February 2021 (UTC)
3722:19:07, 8 February 2021 (UTC)
3705:16:16, 8 February 2021 (UTC)
3646:14:10, 8 February 2021 (UTC)
3634:06:36, 8 February 2021 (UTC)
3615:04:29, 8 February 2021 (UTC)
3589:04:23, 8 February 2021 (UTC)
3574:00:54, 8 February 2021 (UTC)
3556:00:31, 8 February 2021 (UTC)
3539:{\displaystyle \mathbb {R} }
3524:preferred when referring to
3510:19:45, 17 January 2021 (UTC)
3477:{\displaystyle \mathbb {R} }
3450:03:31, 11 January 2021 (UTC)
3423:03:26, 11 January 2021 (UTC)
3397:{\displaystyle \mathbb {Z} }
3181:09:33, 1 December 2019 (UTC)
2341:Construction of Real Numbers
2333:09:25, 19 January 2019 (UTC)
2310:23:04, 18 January 2019 (UTC)
2286:22:59, 18 January 2019 (UTC)
2033:Quantity/continuous quantity
2023:Quantity/continuous quantity
1836:Quantity/continuous quantity
1743:In conclusion, re "the line"
742:, which are of everyday use.
317:, which are of everyday use.
4188:Borel equivalence relations
3661:are, therefore, simply the
3456:the version with my changes
3363:Blackboard vs. regular bold
1694:should be dropped for cause
1277:Real numbers and "the line"
815:are not a formalization. --
4331:
4252:19:50, 17 April 2022 (UTC)
4233:19:32, 17 April 2022 (UTC)
3354:19:15, 28 April 2020 (UTC)
3293:10:43, 25 April 2020 (UTC)
3269:10:22, 25 April 2020 (UTC)
2224:@GiovanniSidwell, already
990:In the Princeton Companion
762:mathematical formalization
736:mathematical formalization
613:the formalization that ...
413:mathematical formalization
311:mathematical formalization
4315:21:33, 24 July 2022 (UTC)
4273:20:27, 24 July 2022 (UTC)
3684:Oxford English Dictionary
3111:17:50, 9 April 2019 (UTC)
3045:07:02, 9 April 2019 (UTC)
2976:19:51, 8 April 2019 (UTC)
2918:00:28, 8 April 2019 (UTC)
2765:23:07, 7 April 2019 (UTC)
2712:13:25, 4 April 2019 (UTC)
2664:11:16, 4 April 2019 (UTC)
2622:11:12, 4 April 2019 (UTC)
2584:10:45, 4 April 2019 (UTC)
2508:09:35, 4 April 2019 (UTC)
2478:21:19, 3 April 2019 (UTC)
2448:21:14, 3 April 2019 (UTC)
2403:21:08, 3 April 2019 (UTC)
2387:21:06, 3 April 2019 (UTC)
2355:20:24, 3 April 2019 (UTC)
2251:07:03, 14 June 2018 (UTC)
2218:17:33, 13 June 2018 (UTC)
2200:17:31, 13 June 2018 (UTC)
2173:15:22, 13 June 2018 (UTC)
2045:Is there anyone who will
2037:identify the text you see
1788:in that the article fails
1422:quantity#Further examples
561:Here are a few thoughts:
4030:We should also cite the
2188:least surprise principle
2124:11:02, 5 June 2018 (UTC)
2104:, it should be asked at
2094:08:30, 5 June 2018 (UTC)
2064:02:38, 5 June 2018 (UTC)
2006:11:13, 23 May 2018 (UTC)
1987:09:41, 23 May 2018 (UTC)
1956:04:48, 23 May 2018 (UTC)
1915:04:30, 23 May 2018 (UTC)
1899:03:46, 23 May 2018 (UTC)
1880:03:37, 23 May 2018 (UTC)
1860:02:29, 23 May 2018 (UTC)
1731:13:48, 19 May 2018 (UTC)
1613:10:32, 17 May 2018 (UTC)
1590:09:14, 17 May 2018 (UTC)
1557:05:58, 17 May 2018 (UTC)
1532:03:07, 17 May 2018 (UTC)
1405:22:25, 12 May 2018 (UTC)
1386:10:26, 12 May 2018 (UTC)
1372:10:21, 12 May 2018 (UTC)
1349:09:56, 12 May 2018 (UTC)
1318:09:35, 12 May 2018 (UTC)
1269:15:24, 12 May 2018 (UTC)
1252:14:07, 12 May 2018 (UTC)
1236:10:10, 12 May 2018 (UTC)
1214:21:45, 10 May 2018 (UTC)
1202:recreational mathematics
1188:21:41, 10 May 2018 (UTC)
1161:21:37, 10 May 2018 (UTC)
1146:21:33, 10 May 2018 (UTC)
1130:21:37, 10 May 2018 (UTC)
1110:20:39, 10 May 2018 (UTC)
1088:19:19, 10 May 2018 (UTC)
1069:19:15, 10 May 2018 (UTC)
1051:18:07, 10 May 2018 (UTC)
1031:17:17, 10 May 2018 (UTC)
1015:13:58, 10 May 2018 (UTC)
974:10:08, 12 May 2018 (UTC)
948:10:03, 12 May 2018 (UTC)
933:09:59, 12 May 2018 (UTC)
919:09:36, 12 May 2018 (UTC)
895:09:20, 12 May 2018 (UTC)
879:09:08, 12 May 2018 (UTC)
825:07:54, 12 May 2018 (UTC)
789:07:40, 12 May 2018 (UTC)
696:06:40, 12 May 2018 (UTC)
663:06:09, 12 May 2018 (UTC)
636:07:10, 11 May 2018 (UTC)
592:20:37, 10 May 2018 (UTC)
543:18:07, 10 May 2018 (UTC)
526:18:20, 10 May 2018 (UTC)
502:18:02, 10 May 2018 (UTC)
487:17:50, 10 May 2018 (UTC)
466:17:43, 10 May 2018 (UTC)
444:17:34, 10 May 2018 (UTC)
377:17:24, 10 May 2018 (UTC)
351:13:49, 10 May 2018 (UTC)
333:12:59, 10 May 2018 (UTC)
292:12:34, 10 May 2018 (UTC)
262:12:25, 10 May 2018 (UTC)
237:07:32, 10 May 2018 (UTC)
223:07:01, 10 May 2018 (UTC)
205:10:36, 10 May 2018 (UTC)
158:06:43, 10 May 2018 (UTC)
4284:Mathematical formalists
3652:Elements of Quaternions
3406:File:Number-systems.svg
3277:Mathematical typography
1430:set of all real numbers
114:22:28, 9 May 2018 (UTC)
4258:Are real numbers real?
3938:
3916:
3863:
3827:
3540:
3478:
3398:
2567:
2317:Following his idea of
770:
431:
4280:Mathematical realists
4163:Real Colorado Cougars
4159:Real (disambiguation)
3939:
3917:
3864:
3828:
3541:
3479:
3399:
3374:Greetings! Regarding
2568:
1775:---as the following:
904:. What exactly that
849:(many occurrences of
42:of past discussions.
3926:
3904:
3851:
3815:
3528:
3466:
3386:
2545:
1633:all over the place):
1335:"Whose intuition"?
1193:Whether anyone uses
2052:continuous quantity
2025:" be deleted unless
1796:of the lay public.
1784:continuous quantity
1761:continuous quantity
1467:continuous quantity
738:of the non-integer
602:And here some more:
566:of the real numbers
313:of the non-integer
85:Continuous quantity
3934:
3912:
3859:
3823:
3536:
3474:
3394:
2563:
2208:. I'll fix it. --
807:The formalization
3356:
3344:comment added by
3271:
3259:comment added by
3043:
2532:Knowledge policy
2421:
2050:where: the link "
2047:identify the text
1320:
1304:comment added by
1186:
1108:
1013:
882:
847:formal definition
729:, who introduced
590:
424:
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54:
53:
48:current talk page
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2364:. Its claim to
2294:Imaginary number
2265:: I agree with
2161:
2154:
2151:Purgy Purgatorio
2147:
2140:
2116:
1998:
1759:is a value of a
1566:the suggestions.
1465:is a value of a
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1228:
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855:formally correct
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18:Talk:Real number
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3667:real quantities
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3173:Boris Tsirelson
3036:Charles Stewart
2757:187.189.208.183
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2165:GiovanniSidwell
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2021:that the link "
1996:
1948:Boris Tsirelson
1745:
1605:Boris Tsirelson
1549:Boris Tsirelson
1364:Boris Tsirelson
1279:
1259:
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1138:Boris Tsirelson
1061:Boris Tsirelson
1041:
1037:decimal numbers
992:
864:
425:rounds off the
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2390:
2389:
2375:the same paper
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2255:
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2228:revert was an
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2115:SÅawomir BiaÅy
2097:
2096:
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2009:
2008:
1997:SÅawomir BiaÅy
1989:
1959:
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1821:geometric line
1807:, as follows:
1778:Page-article "
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4174:Ebony Jackson
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3692:
3689:s.v. 'real',
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3581:Ebony Jackson
3577:
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3548:Ebony Jackson
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2661:
2657:
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2652:
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2650:
2649:
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2647:
2646:
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2640:
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2585:
2581:
2577:
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2549:
2539:
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2509:
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2497:
2496:
2495:
2494:
2493:
2492:
2491:
2490:
2489:
2488:
2479:
2475:
2471:
2467:
2466:Deacon Vorbis
2463:
2462:
2461:
2460:
2459:
2458:
2457:
2456:
2449:
2445:
2441:
2437:
2436:Deacon Vorbis
2433:
2429:
2427:
2419:
2418:edit conflict
2414:
2413:
2412:
2411:
2410:
2409:
2404:
2401:
2398:
2394:
2393:
2392:
2391:
2388:
2384:
2380:
2376:
2372:
2367:
2363:
2359:
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2357:
2356:
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2334:
2330:
2326:
2322:
2320:
2313:
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2307:
2303:
2299:
2298:Deacon Vorbis
2295:
2290:
2289:
2288:
2287:
2283:
2279:
2274:
2272:
2268:
2267:Deacon Vorbis
2264:
2256:
2252:
2248:
2244:
2239:
2235:
2231:
2227:
2223:
2222:
2219:
2215:
2211:
2207:
2203:
2201:
2197:
2193:
2189:
2185:
2181:
2177:
2176:
2175:
2174:
2170:
2166:
2159:
2152:
2145:
2144:Deacon Vorbis
2138:
2129:
2125:
2121:
2117:
2111:
2107:
2106:Talk:Quantity
2103:
2099:
2098:
2095:
2091:
2087:
2083:
2079:
2076:
2072:
2068:
2067:
2066:
2065:
2061:
2057:
2053:
2049:
2048:
2038:
2034:
2030:
2029:
2028:
2026:
2024:
2018:
2016:
2015:
2007:
2003:
1999:
1993:
1990:
1988:
1984:
1980:
1976:
1975:formalization
1972:
1969:
1964:
1961:
1960:
1957:
1953:
1949:
1946:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1922:
1920:
1919:
1916:
1912:
1908:
1902:
1900:
1896:
1892:
1888:
1887:Deacon Vorbis
1883:
1881:
1877:
1873:
1869:
1868:Deacon Vorbis
1864:
1863:
1862:
1861:
1857:
1853:
1849:
1845:
1841:
1837:
1832:
1830:
1822:
1818:
1814:
1810:
1809:
1808:
1806:
1802:
1797:
1794:
1789:
1785:
1781:
1776:
1774:
1766:
1762:
1758:
1754:
1750:
1749:
1748:
1742:
1732:
1728:
1724:
1720:
1719:
1718:
1717:
1716:
1715:
1714:
1713:
1703:
1698:
1695:
1691:
1687:
1686:
1685:
1684:
1683:
1682:
1681:
1680:
1668:
1667:
1666:
1665:
1664:
1663:
1662:
1661:
1660:
1659:
1649:
1648:
1647:
1646:
1645:
1644:
1643:
1642:
1632:
1628:
1627:
1626:
1625:
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1622:
1621:
1614:
1610:
1606:
1601:
1600:
1599:
1598:
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1596:
1591:
1587:
1583:
1578:
1574:
1573:
1572:
1571:
1564:
1563:
1562:
1561:
1558:
1554:
1550:
1545:
1544:
1540:
1536:
1535:
1534:
1533:
1529:
1525:
1517:
1516:
1515:
1513:
1506:
1502:
1498:
1494:
1493:
1489:
1485:
1481:
1477:
1476:
1472:
1468:
1464:
1460:
1457:(Current) In
1456:
1455:
1454:
1451:
1449:
1448:discontinuity
1445:
1441:
1437:
1436:
1431:
1425:
1423:
1419:
1415:
1406:
1402:
1398:
1393:
1392:
1391:
1390:
1387:
1384:
1381:
1377:
1376:
1373:
1369:
1365:
1361:
1358:
1357:
1350:
1346:
1342:
1338:
1334:
1331:
1327:
1326:
1325:
1324:
1323:
1322:
1321:
1319:
1315:
1311:
1307:
1303:
1296:
1289:
1288:
1287:
1284:
1281:I think that
1276:
1270:
1266:
1262:
1255:
1254:
1253:
1249:
1245:
1240:
1239:
1238:
1237:
1233:
1229:
1223:
1222:User:D.Lazard
1215:
1211:
1207:
1203:
1199:
1196:
1192:
1191:
1190:
1189:
1184:
1180:
1162:
1158:
1154:
1149:
1148:
1147:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1114:
1113:
1112:
1111:
1106:
1102:
1095:
1091:
1090:
1089:
1085:
1081:
1076:
1072:
1071:
1070:
1066:
1062:
1058:
1054:
1052:
1048:
1044:
1038:
1034:
1032:
1028:
1024:
1019:
1018:
1017:
1016:
1011:
1007:
1001:
997:
989:
975:
972:
969:
964:
959:
949:
945:
941:
936:
935:
934:
930:
926:
922:
921:
920:
916:
912:
907:
903:
898:
896:
892:
888:
884:
883:
880:
876:
872:
868:
861:
856:
852:
848:
844:
840:
839:formalization
836:
832:
828:
827:
826:
822:
818:
814:
810:
806:
803:
799:
796:
792:
791:
790:
786:
782:
778:
777:
776:
775:
774:
773:
772:
771:
769:
767:
763:
759:
754:
743:
741:
737:
733:
728:
724:
720:
715:
714:
713:
712:
711:
710:
709:
708:
707:
706:
697:
693:
689:
685:
682:
679:
674:
671:
666:
664:
660:
656:
651:
650:
649:
648:
647:
646:
645:
644:
637:
633:
629:
626:
625:
620:
617:
614:
610:
609:formalization
606:
605:
601:
600:
599:
598:
597:
596:
593:
588:
584:
578:
573:
570:
567:
563:
562:
560:
559:
544:
540:
536:
531:
527:
523:
519:
515:
510:
507:
506:
505:
504:
503:
499:
495:
490:
489:
488:
484:
480:
476:
473:
469:
467:
463:
459:
455:
451:
447:
446:
445:
441:
437:
433:
430:
428:
418:
414:
405:
401:
395:
392:
391:
390:
389:
388:
387:
386:
385:
378:
374:
370:
366:
363:
359:
354:
353:
352:
347:
343:
336:
335:
334:
330:
326:
322:
318:
316:
312:
308:
304:
303:
301:
296:
295:
294:
293:
288:
284:
278:
274:
270:
264:
263:
258:
254:
247:
238:
234:
230:
226:
224:
220:
216:
212:
206:
202:
198:
194:
190:
186:
185:
183:
179:
175:
171:
167:
163:
162:
161:
160:
159:
155:
151:
144:
141:
140:
139:
138:
129:
128:
124:
120:
119:
118:
117:
116:
115:
110:
106:
95:
92:
91:
90:
84:
77:
73:
71:
68:
66:
63:
61:
58:
57:
49:
45:
41:
40:
35:
28:
27:
19:
4302:
4298:
4294:
4261:
4239:
4222:
4219:
4114:
4110:
4109:Agreed that
4080:
4039:
4035:
4031:
4005:
4001:
3974:
3896:
3874:
3870:
3794:
3767:
3762:
3728:
3711:
3690:
3682:
3670:
3666:
3658:
3651:
3522:
3459:
3437:
3429:
3409:
3366:
3340:â Preceding
3280:Donald Knuth
3276:
3255:â Preceding
3251:
3233:
3222:
3214:
3035:
3029:
2860:
2856:
2425:
2374:
2371:this January
2370:
2365:
2361:
2344:
2275:
2270:
2260:
2237:
2233:
2229:
2225:
2205:
2184:bandersnatch
2180:bandersnatch
2133:
2081:
2074:
2070:
2051:
2046:
2044:
2042:
2036:
2020:
2019:
2013:
2012:
2010:
1991:
1974:
1970:
1967:
1962:
1839:
1833:
1828:
1826:
1816:
1804:
1800:
1798:
1792:
1787:
1777:
1772:
1770:
1756:
1746:
1701:
1696:
1693:
1630:
1538:
1521:
1511:
1510:
1500:
1483:
1462:
1452:
1434:
1433:
1429:
1426:
1417:
1413:
1411:
1336:
1300:â Preceding
1297:
1294:
1280:
1218:
1194:
1173:
1117:
1093:
1074:
995:
993:
962:
905:
901:
860:modelization
854:
850:
846:
842:
838:
834:
812:
808:
801:
794:
758:Real numbers
757:
756:
732:Real numbers
731:
726:
677:
669:
612:
608:
565:
513:
471:
449:
416:
403:
400:Real numbers
399:
397:
361:
357:
307:Real numbers
306:
299:
276:
273:constructing
272:
268:
265:
248:
245:
192:
173:
169:
165:
142:
100:
88:
75:
43:
37:
4133:Paul August
4099:Paul August
4044:Paul August
4042:says here.
3772:Paul August
3716:Paul August
3640:Paul August
3376:this revert
2706:Paul August
2397:Paul August
2319:coordinates
2158:37.71.17.36
1973:D.Lazard's
1846:blokequote
1844:John Wallis
1817:real number
1813:mathematics
1771:I offer my
1757:real number
1753:mathematics
1501:real number
1497:mathematics
1495:(alt-B) In
1484:real number
1480:mathematics
1478:(alt-A) In
1463:real number
1459:mathematics
1380:Paul August
998:(edited by
968:Paul August
809:talks about
802:don't exist
36:This is an
3215:References
2541:subset of
2366:canonicity
963:identified
843:formalized
813:themselves
472:much worse
4307:Trovatore
4244:Anita5192
4192:Trovatore
4170:Trovatore
4145:Trovatore
4115:somewhere
4084:Trovatore
3960:Trovatore
3882:Trovatore
3748:Trovatore
3626:Trovatore
3607:Anita5192
3519:The reals
3496:Trovatore
3442:Trovatore
3370:Trovatore
2910:Trovatore
2861:therefore
2379:Trovatore
2278:Trovatore
2263:this edit
2238:continuum
2210:Trovatore
2206:Mea culpa
2192:Trovatore
2137:Trovatore
2082:narrating
1907:Trovatore
1444:continuum
1418:(C-class)
1341:Trovatore
1283:Trovatore
1206:Trovatore
1153:Trovatore
1122:Trovatore
1080:Trovatore
1023:Trovatore
961:example,
940:Trovatore
911:Trovatore
887:Trovatore
817:Trovatore
725:, it was
688:Trovatore
622:naturals.
518:Trovatore
479:Trovatore
458:Trovatore
454:real line
369:Trovatore
358:algebraic
178:Trovatore
76:Archive 4
70:Archive 3
65:Archive 2
60:Archive 1
4290:objects.
4265:Linkato1
4119:Macrakis
4082:page. --
4056:Macrakis
3798:D.Lazard
3733:Macrakis
3697:Macrakis
3654:(1866):
3342:unsigned
3285:D.Lazard
3257:unsigned
2857:argument
2810:avoided.
2702:D.Lazard
2576:D.Lazard
2325:D.Lazard
2110:quantity
2102:quantity
2075:no favor
1793:does not
1780:Quantity
1690:Quantity
1541:a value.
1440:Quantity
1397:Eleuther
1329:article.
1314:contribs
1306:Eleuther
1302:unsigned
1244:D.Lazard
1057:0.999...
925:D.Lazard
875:contribs
867:D.Lazard
835:formally
760:are the
734:are the
723:D.Lazard
655:Eleuther
514:topology
494:D.Lazard
325:D.Lazard
309:are the
277:identify
269:removing
229:Eleuther
215:Eleuther
193:circular
3671:Algebra
3659:scalars
3380:MOS:BBB
3320:though.
3103:JPMural
3030:against
2968:JPMural
2656:JPMural
2614:JPMural
2500:JPMural
2395:Agree.
2347:JPMural
2234:sarcasm
1963:Oppose.
1801:current
1337:Correct
902:objects
766:numbers
740:numbers
678:exactly
579:â Carl
450:logical
427:numbers
315:numbers
174:correct
101:â Carl
39:archive
4006:prefer
3566:Modocc
3502:Beland
3486:Beland
3484:." --
3415:Beland
3041:(talk)
2474:videos
2470:carbon
2444:videos
2440:carbon
2306:videos
2302:carbon
2056:Jbeans
1992:Oppose
1895:videos
1891:carbon
1876:videos
1872:carbon
1852:Jbeans
1723:Jbeans
1524:Jbeans
851:formal
727:not me
670:formal
4295:might
4293:What
4097:Yes.
4002:wrong
3768:reals
3729:reals
3693:, B.4
3675:p. 10
3669:) of
3663:Reals
3657:Such
2534:WP:OR
2321:, ...
2243:Purgy
2086:Purgy
2071:favor
1979:Purgy
1582:Purgy
1514:: -->
1120:. --
1059:=1.
906:means
795:about
781:Purgy
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