Talk:Real number - Knowledge

988:

satisfying quadratic constraints (which might be incommensurable, but that was considered fine), while algebra/arithmetic seemed like a substantially separate topic. Irrational numbers were treated with some suspicion, and it was hard to convince mathematicians to accept square roots as numbers per se in solutions to geometric problems. The development of real numbers, mathematical notation, analytic geometry, calculus, the function concept, and mechanics were all intertwined. This idea that continuous curves (either plane curves or smooth functions) must have well-defined intersections is essential to the real number concept.–

1222:– What counts as a definition or an axiom vs. a theorem is to a large extent an arbitrary conventional choice. Mathematicians love to cut out as many properties of objects as they can when making definitions (for a variety of reasons including e.g. avoiding contradictions, making the definitions easier to remember, joy at proving even basic properties as theorems, rhetorical flourish, and personal vanity) so among alternative definitions the one with the fewest essential features is usually preferred. But we shouldn’t confuse 95: 85: 64: 31: 517:). This is important historically in the development of the function concept, and underlies many other kinds of mathematical objects used as domains/codomains that are defined in terms of real numbers. This is not really mentioned here at all. This article does state that real numbers are used in calculus and mathematical analysis, but does not explain how or why, except to say that real numbers were historically used without being entirely well defined. 176: 312:. But not only is "continuous" not wiki-linked, it is nowhere properly defined in this article, and not discussed in an accessible way. The non-continuity of the rational numbers is not described or discussed, nor is it explained why this non-continuity matters, what alternatives we might have to continuity, nor what the real numbers do to address this. (Aside: I think the basic definition here is a bit sloppy, since 1230:. If we choose a different (logically equivalent) definition of a structure or a different (logically equivalent) set of axioms for a mathematical discipline it doesn’t change the scope or nature of the subject, but only re-orders some of the subsequent statements, switches a few labels between 'axiom' vs. 'theorem', and forces some rewriting of some of the proofs. But those are all surface-level changes. – 782:, such a section must be written for most readers, even those who are not interested in geometry. As limits, continuity and derivatives require to know of real numbers, I do not imagine how to write such a section before describing the main properties of real numbers. So, your "Motivation" section would be better as an "Application" section, placed after the "properties" section. 22: 1597:

is "Every infinite decimal expansion represents a real number." This is (a) correct and (b) the basis for the diagonal proof of uncountability, so it captures a deep property of the reals as distinguished from other numbers. Yes, it needs to be made more precise for the proof (that they almost always represent

638:). That too is, as you say, pitched too high for the general introduction. But we might be able to get across the informal idea that you can't pull it apart into two separate pieces without breaking something, unlike, for example, the rationals, which you can pull apart by dividing them at an irrational. -- 430:

for real numbers. This article should at least point out that a "square root" gets its name from being the side length of a square of a given area, and that the Pythagorean theorem gives a way of finding triangle side lengths in terms of square roots, so that square roots come up all the time as lengths in

1202:

as a set of equivalence classes of Cauchy sequences. Given this, I'd probably start with the classical definition in terms of those equivalence classes (easier for most readers to understand), and then mention the universal property later on (more elegant, and simplifies the work in constructing the

405:

is too technical to be relied on for covering this . There should be, early in the article, some kind of explanation about different geometrical interpretations of these operations (e.g. subtraction as a measure of distance; addition as a kind of translation; multiplication/division as scaling). Some

246:

The line "In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more." should be edited or removed, as this is demonstrably false. No measuring instrument can access the infinite precision of real numbers (if such a thing even

1596:

The other is "Every real number can be almost uniquely represented by an infinite decimal expansion." This is certainly true, but not very interesting. As I said above, that statement is also true of every rational number, every algebraic number, heck even of the empty set. The interesting statement

1434:

I reverted this change: the removal of the explanation of “continuous” introduces a confusion with the technical meaning of “continuous”. Also the change of the last sentence of the first paragraph amounts to replace a characterization of the reals with a property of decimal expansions, which is out

760:

I agree with your concern about "continuity" of rational numbers. However, the lack of holes in the real line is not sufficient, for most readers, to explain the difference between real and rational numbers. This is because of this that the first line does not talk of "continuity" of the reals, but

296:

The first section in the article, about history, is (a) somewhat incomplete, and (b) should be deferred until at least after basic discussion of what real numbers are and why they matter. The history section as it is currently does not sufficiently define or motivate the ideas mentioned that someone

1449:

I agree that "continuous" is problematic. But the explanation, as I said above, is meaningless both informally and technically. How can a value have a "variation"? As for characterizing the reals as the numbers represented by decimal expansions, that seems to be by far the best proposed explanation

1327:

is not bad; I think it strikes a reasonable balance. The last sentence of the first paragraph might be tweaked to say a little more explicitly something along the lines of "If you don't want to get into it too deeply, you can think of real numbers as being the values of infinite decimal fractions"

822:

I am not suggesting adding leading questions or problem sets, nor is the goal here to “teach” the subject (for that, someone can consult an analysis textbook). The goal would be to introduce/explain the idea first for a wider audience curious to know what a real number is, before diving into a wall

796:

Yes, the purpose is to make a number system where the solution to calculus problems are “numbers” (instead of just, say, sequences of approximations good to any desired precision). (I think we can address the issue of connectedness/completeness of the real numbers better in a dedicated "motivation"

737:

Do you want to take a shot at rewriting that part of the lead to establish that point? I think the first section after the lead should be something like "Motivation", which could (among other topics) discuss the use of numbers to represent geometric distances, lengths of curves, areas, etc., among

801:

doesn’t really make sense except in the context of calculus. But (a) I think it should go before a “properties” section, and (b) I think the properties section should be broken up into a few sections. I can try to come up with some draft idea by pulling images from other wikipedia articles, but am

657:

Which to an interested layperson is a more or less incomprehensible wall of jargon. I'm not suggesting we should turn this into a whole mini-course on point-set or metric topology. But I think we should spend a section or three at the start of the article trying to explain what the issue is in the

429:

That positive real numbers have real square roots is considered an "advanced property" in this article, and what this means is not described. But the history section mentions this several times (though without really going into much detail), implying it is important to understanding the motivation

1297:

It's not clear that the first paragraph needs to include a rigorous definition. It should give the general, non-technical, reader a sense of what the topic is. Although I suppose "every number defined by a decimal fraction (possibly infinite)" is correct, though certainly not what a mathematician

987:

This sounds like a fine plan. My understanding is that historically, the development of analytic geometry was delayed (until Descartes or later) by the use of rational numbers in arithmetic/algebra. Geometry dominated mathematics of the time and could "exactly" intersect circles, forming segments

458:

In physics" section, but the immediate invocation of "physical constants" and "physical variables" seems less clear than just saying "measured quantities", and there's not too deep a discussion. This seems to me much more important than the other parts of this applications section. In particular,

358:, which also does not really discuss continuity, except via inaccessible jargon.) The idea that the real number line is infinite in extent is mentioned a few times, but this is not defined, nor are there any links to other articles where the concept of infinite extent might be found. (The article 709:

Does anyone have ideas for how to organize 2–3 new sections at the top of this article elaborating on the motivation and some of the uses of real numbers for a general audience before we get to technical definitions? (I think the 'history' section can probably then be moved much further down the

654:

The reals form a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable

1368:

I agree with all of that except this part of the current version: "Here, continuous means that values can have arbitrarily small variations", which makes no sense. How can a value have a variation? I think what it intends to say is that there are numbers that are arbitrarily close to any given

1280:

would be similarly misleading). It should include something about containing its limit points or the like. It’s tricky to come up with a clear/precise enough wording which is also accessible to non-technical readers. It would be good for someone to do some searching through past sources for a

722:

I'm not super comfortable with a description that would treat the rational numbers as "continuous". I think what we want to get across here, intuitively, is that the real line "doesn't have any holes in it". Completeness will imply that, but it's really closer conceptually to connectedness.

396:

The uncountability of real numbers is mentioned, but there is no explanation given for why real numbers are uncountable; this is just taken as an obvious fact, and readers curious to understand what "uncountable" means or why it matters are going to have to do a lot more searching to find out.

1316:

the characterization that's both (1) extensionally accurate and (2) quickly understandable by the lay reader. That would be a pretty good argument for leading with that, except that it doesn't get at the motivation at all, and it leaves the impression that the real numbers are somehow deeply

292:

The lead section of this article is too long and rambling, and spends too much time on advanced material explained using inaccessible jargon. The advanced material is fine to include in the article, but should be pared back if not eliminated from the lead section, which should should aim for

456:

As an abstract model for arbitrary kinds of measured quantities in which basic arithmetic operations are well defined so that people don’t need to worry about too many caveats when doing basic symbolic manipulations. This is discussed in the "Applications and connections to other areas :

1353:

If infinite decimal expansions are mentioned in the first paragraph, this is not to provide a definition; this is because, they are often taught very early to kids (unfortunately, in my opinion). So, mentioning them provides an informal explanation that refers to the background of many.

1197:

Do I understand correctly that the notion of Cauchy completion requires the notion of a complete metric space, which in turn requires the notion of Cauchy sequence? In spelling out the details of Cauchy completion, one would then already be most of the way to the definition of

587:

In mathematics, "continuous" is an adjective most commonly applied to a function. It is not standard to say that a number is continuous, or that a set of numbers is continuous. What definition of continuous are you thinking of when you say that "real numbers are continuous"?

1385:

I think what it's getting at is that it can describe physical quantities that aren't granular. Distance, mass, time, things you measure with no preset limit to precision. This is at an intuitive level; it's not really about whether those things are in fact quantized.

761:

talk only of continuity of measured quantities. This is because I do not know any non-technical way to explain the "non-continuity" of the reals, that I have added an explanatory footnote. This leaves the first paragraph non-misleading, although mathematically correct.

843:

For a rough outline, I am thinking about something like the following, with everything before 'formal definitions' trying to stay somewhat accessible to a broad audience (say, someone with a high school or at most undergraduate engineering/science background) :

968:

that implies directly that the intersection points of two (continuous) curves have real coordinates, that a continuous function that changes of sign has a zero, and, in particular, that a univariate polynomial of odd degree has a root (an important part of the

1252:

means that values can have arbitrarily small variations". Though this language has been in the article for some time, it's not at all clear what it means. What does it mean for a value to "have variations"? I realize that this sentence is not intended to

1660:

I object. I think the line about decimal expansions should be removed, as it does not precisely characterize the real numbers. I don't think it gives readers a clear idea of what the real numbers are. Decimal expansions are already treated in a separate

1561:

I think something like “variable quantity” would be significantly better than “pairs of values”. But arbitrarily small differences is also not the key point here; the important feature is that the reals don’t have “gaps” the way the rational numbers do.

400:

The basic property of real numbers (that they share with rational numbers, etc.) is that they can be added, subtracted, multiplied, and divided, but there is no explanation in this article of what that means or how it works in practice. The article

1574:

Macrakis's objection seems to be to applying the word "vary" to a "value". I do think it's better to say that a "quantity" can vary. In some sense that means that the value of that quantity varies, and that's the sense in which the value can

802:

also glad to hear what other folks think the scope here can be. My goal would be that someone who does not know what the “real numbers” are could read a couple sections into the article and end up with a reasonable not-too-technical summary. –

1837:

However, I agree that it is too technical. In fact, technical accuracy, is much easier to obtain this way. So, this section must be rewritten with examples and less proofs in view of a better compromise better accuracy and readibility.

797:

section than in a sentence or two in the lead section.) But calculus was developed to solve problems arising from analytic geometry and mechanics. Anyway, I agree that a "motivation" section should talk about calculus! Something like

384:

is invoked right off the bat in the lead (which describes real numbers as consisting of both rational and irrational numbers) and then in the history section, but is not properly defined or discussed, and if someone clicks through

1038:, and that the explicit constructions are alternative definitions. This is wrong, since without the constructions one would not know that such a field existed. Instead it is only the constructions that give definitions. That 1454:

of the numbers representable by an infinite decimal expansion (after all, that statement is also true of the rationals or the algebraics); it doesn't say that the infinite decimal expansions characterize all the reals.

1578:

Some of the mathematical objections are a bit beside the point when applied to this language, which is meant to appeal to physical or quasi-physical intuition (geometric intuitions count as quasi-physical).

820:

The purpose of Knowledge is to summarize accepted knowledge, not to teach subject matter. Articles should not read like textbooks, with leading questions and systematic problem solutions as examples.

1406:

sections with rigorous definitions. If anyone has a problem with the sloppy descriptions in the lead, perhaps we could precede them with something like "Loosely, . . ." or "Loosely speaking, . . ."—

1765: 151: 389:

they get a circular definition that irrational numbers are real numbers that aren't rational, and are going to have to work hard to figure out what irrational really means. The concept of

1931: 1100: 1805: 1450:

of the reals for the non-technical reader. The fact that "Every real number can be almost uniquely represented by an infinite decimal expansion" tells us that the reals are a

1129: 1173: 1151: 1880:

As for the bijection: Since the decimal fraction part is a sum of only a finite number of digits, how do we distinguish between, for example, 1, 1.0, 1.00, 1.000, . . . ?

1921: 973:). These properties are fairly intuitive, and, as they are wrong when restricted to rational numbers, show clearly that one need more numbers than the rational ones. 1936: 324: 1319:

I note in passing that so-called "terminating" decimal fractions are still infinitely long; it's just that all but finitely many of the places are occupied by 0.

1342:"certainly not what a mathematician would use as a definition”: I disagree. There are many textbooks written by mathematicians that use this definition. Also, 1811:

appears to be incorrect, but I am not sure because it is so difficult to understand. Finally, what is the point of this section? What is it trying to say?—

445:, but that page does not describe what a sequence is, and is too technical for much of the presumed audience of this article. There are links from here to 247:

exists); they usually use decimal numbers which are like rational numbers, i.e. have a finite representation, can be represented as a ratio of integers.

1946: 610:, as this article states, but just stating that is too formal/technical and not in itself all that useful to non-specialists; it is important to explain 141: 1916: 1826:

This section is required, and must be near to the top, since, most people identify real numbers and their decimal representations. Also, the section

679:

Before reading this thread, I rewrote the lead (IMO, the two last paragraph do not belong to it, and must still be removed). I hope that this solves

289:

This article does not do an adequate job of explaining what the real numbers are, because it presumes too much background and skips basic material.

1926: 614:

this is the definition used, what intuitive notion it is getting at. There is discussion of this under the section "completeness" and the article

1261:

means that between any two real numbers there is another real number" -- which of course is necessary but not sufficient to define the reals. --

437:

That real numbers contain the limits of converging sequences of rational numbers is mentioned several times and is clearly very important, but

117: 1941: 606:. But what do you think this article means when it defines real number as "the value of a continuous quantity"? (This is formalized as the 1546:

means that pairs of values can have arbitrarily small differences." This is technically correct and should be understandable by readers.—

562:

Much of the rest of this article seems more or less okay. It is not very accessible to non-experts, but doesn’t necessarily have to be. –

1153:

is defined, nobody uses these constructions). Then, it is a theorem that this completion is an ordered topological field. The fact that

254: 1346:

are very often defined by infinite p-adic expansions, which are similar to decimal expansions. I am pretty sure that Dedekind invented

630:

I think the closest you can get to capturing the intuitive concept we're trying to get at by a precise topological notion is actually

1874:

article is clearer. We should be careful how we define an infinite set of numbers, each with a (possibly) infinite number of digits.

1175:

is the unique Dedekind-complete ordered field is an interesting theorem, but, IMO, not a fundamental one, as rarely used in practice.

1487:“How can a value have a "variation"?”: I understand that the speed of your car cannot vary or that its variation is not continuous. 343:

which is absurdly terse and technical, the kind of definition you’d find in a journal paper for an audience of mathematicians, etc.

108: 69: 1885:

I think this article should simply link to the decimal representation article and we should make any necessary improvements there.—

1501:

A value is something like 4096 or π/7. It does not have "variations". You apparently want to introduce not just the concept of a

615: 446: 35: 452:

In my view there are a few primary uses of real numbers all over science and mathematics which should be discussed prominently:

1911: 449:

but except for one sentence that article does not describe concretely what completeness means, except in very technical terms.

339:

which just describes having an uncountable set of values (not quite technically complete/precise, or helpful as a basic idea),

327:

which does not provide any basic conceptual description of what continuity means but just links to more advanced articles like

778:. This is the reason for which I have added a "motivation" paragraph in the lead about that. About a section "Motivation": As 970: 1106:, and, as such, if it exists, it is unique up to an isomorphism. So, it is natural to define the reals as the completion of 543:). There should be some discussion of why real numbers are used for this and how it works, since there are surely links to 1852:

I rewritten the section, making the hypothesis that readers well know decimal numbers. I hope that the result is clearer.

558:. This article should define complex numbers and discuss the relationship between real and complex numbers in more detail. 1690: 44: 1831: 487: 231: 1402:

I tend to agree that the first paragraph does not need to include a rigorous definition. The article already has a

1180:

So I suggest to rewrite along this line the parts of the article that are devoted to the definitions of the reals.

965: 607: 201: 197: 193: 189: 618:, but they are pitched at too high a level for the widest audience that might be expected to read this article.) – 738:

which some are clearly not rational and thus "missing" from a number line which only includes rational numbers. –

850:

Motivation (not too long, just explaining why the rationals are insufficient and the rough concept of the reals)

210: 320: 498:

doesn't do a very good job of) that it can be used for purposes other than Euclidean geometry, e.g. for other

1877:

This section only defines nonnegative real numbers. I think it should address negative real numbers and zero.

1220:

That R is the unique complete ordered field up to isomorphism is a property of R, after it has been defined.

1208: 1059: 593: 503: 258: 1871: 423: 328: 815: 779: 579: 536: 495: 491: 475: 50: 94: 393:

numbers is likewise invoked (as presumed background knowledge) but not really described or discussed.

1076: 532: 340: 285:

severe inaccessibility to a nonspecialist audience; entirely missing most of the fundamental concepts

250: 1770: 21: 1890: 1816: 1666: 1584: 1566: 1551: 1411: 1391: 1333: 1285: 1234: 1013: 992: 956: 827: 806: 742: 728: 714: 662: 643: 622: 566: 555: 528: 402: 336: 274: 1109: 1069:

I agree that the alternative "definitions" are only constructions, but I think that starting form

116:

on Knowledge. If you would like to participate, please visit the project page, where you can join

1857: 1843: 1767:

is not clear because it uses negative indexing, which I think is unnecessary. The statement that

1639: 1606: 1514: 1492: 1478: 1460: 1440: 1425: 1374: 1359: 1303: 1266: 1204: 1185: 1156: 1134: 1103: 1055: 1006: 978: 787: 688: 589: 540: 216: 100: 84: 63: 1369:

number, but as was pointed out above, that doesn't distinguish the reals from the rationals. --

490:. This article should summarize more of this material directly, perhaps including a definition 1070: 774:

I do not think that the main motivation of introducing real numbers is geometry. IMO, this is

703: 499: 483: 471: 467: 431: 386: 300:

The important feature of the real numbers (per se) is right there in our current definition,

1470: 1203:

field operations I suppose, but perhaps not enough is gained over the direct construction).

603: 576: 514: 460: 332: 212: 175: 1420:

Based on the above discussion, I've reworded the lead. Obviously open to improvement... --

1257:

the reals (which is more subtle) but as it stands, it doesn't even make sense. How about "

1194:

I agree with your assessment of the fact about the unique Dedekind-complete ordered field.

635: 442: 1005:

There is a nice discussion of some of the history of these concepts in Hartshorne (2000)

346:

The notion of a line is repeatedly invoked, but again not really defined; at the article

1886: 1812: 1662: 1625: 1621: 1617: 1580: 1563: 1547: 1407: 1387: 1329: 1282: 1231: 1010: 989: 953: 824: 803: 769: 755: 739: 724: 711: 680: 659: 639: 619: 563: 551: 419: 270: 369:

is mentioned, but not described in an accessible way. Someone who follows the link on

1905: 1853: 1839: 1635: 1629: 1602: 1510: 1488: 1474: 1456: 1436: 1421: 1370: 1355: 1343: 1299: 1262: 1181: 974: 783: 699: 684: 631: 1347: 1131:

Cauchy's and Dedekind's constructions are proofs that this completion exists (once

1073:

would be clearer and closer to the common use of the reals (it is fundamental that

521: 510: 370: 1870:

This is better, but still difficult to follow. The description in the lead of the

1593:

There are two issues here. One is the "vary" language; let's leave that to later.

1827: 1403: 544: 415: 407: 374: 359: 355: 113: 1601:

real numbers), but it is an excellent first cut at an intuitive definition. --

1634:

If there are no objections, I will use the wording in the previous comment. --

306:

is a value of a continuous quantity that can represent a distance along a line

90: 479: 441:

is not linked and the notion of a sequence is not defined. The term used is

351: 1687:

I find this section extremely difficult to follow. The summation defining

775: 438: 411: 297:

who does not already know what real numbers are will get much out of it.

335:

which circularly describe a continuum as being "like the real numbers",

214: 1054:

that there is at most one complete ordered field up to isomorphism.

1894: 1861: 1847: 1820: 1670: 1643: 1610: 1588: 1569: 1555: 1518: 1496: 1482: 1464: 1444: 1429: 1415: 1395: 1378: 1363: 1337: 1307: 1288: 1270: 1237: 1212: 1189: 1063: 1016: 995: 982: 959: 830: 809: 791: 745: 732: 717: 692: 665: 647: 625: 597: 582: 569: 278: 262: 652:

Yes, that’s right. What the article currently says about this is

823:

of more advanced jargon useful to the “mathematically mature”. –

347: 1469:

A possible solution for explaining "continuous" is to link to

1312:"Number represented by an infinite decimal fraction" probably 217: 169: 15: 1276:

Agreed, the current one is insufficient and misleading (and

1042:

is the unique complete ordered field up to isomorphism is a

1026:

The article suggests that the axiomatic characterization of

1278:"between any two real numbers there is another real number" 655:

order topologies are necessarily homeomorphic to the reals.

520:

In particular as the codomain of the distance function any

1830:

use implicitly the decimal (or binary) representation for

1473:, but unfortunately that's a rather technical article. -- 1102:

is complete): The Cauchy completion is the solution of a

1328:(obviously cleaned up into a more encyclopedic tone). -- 575:

On a first reading, I generally agree with all of that.

486:(frankly unnecessary technical jargon in this context), 362:

mentions but does not describe or discuss this either.)

1324: 698:

That is definitely an improvement. I added mentions of

509:

As the domain or codomain of functions. (Also cf. e.g.

377:

finds an incomprehensible wall of jargon and symbols.

1773: 1693: 1159: 1137: 1112: 1079: 1050:, after it has been defined. Alternatively, it is a 112:, a collaborative effort to improve the coverage of 1760:{\displaystyle S_{n}=\sum _{i=-k}^{n}a_{-i}10^{-i}} 964:IMO, the "Motivation" section must be based on the 1799: 1759: 1167: 1145: 1123: 1094: 323:would explain the basic idea, but it redirects to 1350:for having a definition that is base independent. 474:. There is a brief mention of this with links to 1932:Knowledge level-3 vital articles in Mathematics 531:of scalars, e.g. for use in vector spaces (cf. 350:there is no discussion of continuity. (Aside: 325:List of continuity-related mathematical topics 225:This page has archives. Sections older than 8: 248: 58: 1791: 1778: 1772: 1748: 1735: 1725: 1711: 1698: 1692: 1161: 1160: 1158: 1139: 1138: 1136: 1114: 1113: 1111: 1086: 1082: 1081: 1078: 547:from many other articles in this context. 1922:Knowledge vital articles in Mathematics 1007:"Teaching Geometry According to Euclid" 60: 19: 1277: 1219: 819: 653: 301: 235:when more than 5 sections are present. 1937:C-Class vital articles in Mathematics 1030:as "the complete ordered field" is a 933:Projectively extended complex numbers 7: 106:This article is within the scope of 856:Fundamental operations (+, –, ×, /) 706:; hopefully those are clear enough. 49:It is of interest to the following 1542:I changed this sentence to "Here, 930:Projectively extended real numbers 459:scalar quantities in the sense of 410:. Some could maybe be pulled from 14: 1947:Top-priority mathematics articles 1509:, which is not the same thing. -- 634:(our more specific article is at 293:legibility for a wider audience. 229:may be automatically archived by 126:Knowledge:WikiProject Mathematics 1917:Knowledge level-3 vital articles 1095:{\displaystyle \mathbb {R} ^{n}} 616:completeness of the real numbers 494:, and should clarify (which the 447:completeness of the real numbers 174: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 1800:{\displaystyle S_{n}<10^{k}} 406:of this material is covered at 365:The idea that real numbers are 146:This article has been rated as 1927:C-Class level-3 vital articles 1834:. So the section is important. 971:fundamental theorem of algebra 927:Affinely extended real numbers 316:is usually an unsigned idea.) 1: 1298:would use as a definition. -- 1124:{\displaystyle \mathbb {Q} .} 683:'s concerns about the lead.) 658:most accessible way we can. – 120:and see a list of open tasks. 1942:C-Class mathematics articles 1213:05:45, 28 October 2022 (UTC) 1190:11:36, 24 October 2022 (UTC) 1168:{\displaystyle \mathbb {R} } 1146:{\displaystyle \mathbb {R} } 1064:23:05, 23 October 2022 (UTC) 1017:11:24, 20 January 2023 (UTC) 996:22:24, 24 October 2022 (UTC) 983:17:17, 24 October 2022 (UTC) 960:16:05, 24 October 2022 (UTC) 831:18:25, 22 October 2022 (UTC) 810:18:11, 22 October 2022 (UTC) 792:13:08, 22 October 2022 (UTC) 746:00:06, 22 October 2022 (UTC) 733:23:40, 21 October 2022 (UTC) 718:22:02, 21 October 2022 (UTC) 693:10:19, 21 October 2022 (UTC) 666:07:24, 21 October 2022 (UTC) 648:06:39, 21 October 2022 (UTC) 626:06:04, 21 October 2022 (UTC) 602:Here we are talking about a 598:03:56, 21 October 2022 (UTC) 583:21:29, 20 October 2022 (UTC) 570:20:40, 20 October 2022 (UTC) 488:Cartesian coordinate system 466:As the coordinates of some 308:. Namely, real numbers are 279:21:43, 24 August 2022 (UTC) 263:20:37, 24 August 2022 (UTC) 1963: 1832:Cantor's diagonal argument 1671:17:01, 20 April 2023 (UTC) 1644:16:03, 20 April 2023 (UTC) 1611:03:04, 18 April 2023 (UTC) 966:Intermediate value theorem 608:least-upper-bound property 1589:20:48, 6 April 2023 (UTC) 1570:18:32, 6 April 2023 (UTC) 1556:15:47, 6 April 2023 (UTC) 1519:15:41, 6 April 2023 (UTC) 1497:15:31, 6 April 2023 (UTC) 1483:14:50, 6 April 2023 (UTC) 1465:14:35, 6 April 2023 (UTC) 1445:08:23, 6 April 2023 (UTC) 1430:21:12, 5 April 2023 (UTC) 1416:18:41, 5 April 2023 (UTC) 1396:19:44, 5 April 2023 (UTC) 1379:18:33, 5 April 2023 (UTC) 1364:17:40, 5 April 2023 (UTC) 1338:17:37, 5 April 2023 (UTC) 1308:17:02, 5 April 2023 (UTC) 1289:16:57, 4 April 2023 (UTC) 1271:20:34, 3 April 2023 (UTC) 1238:17:09, 4 April 2023 (UTC) 145: 78: 57: 1895:23:13, 23 May 2023 (UTC) 1862:17:20, 20 May 2023 (UTC) 1848:17:25, 19 May 2023 (UTC) 1821:16:25, 19 May 2023 (UTC) 1317:connected with base 10. 1224:how something is defined 321:continuity (mathematics) 319:I would have hoped that 152:project's priority scale 1323:The first paragraph of 504:generalized coordinates 242:Useful for measurement? 109:WikiProject Mathematics 1912:C-Class vital articles 1872:decimal representation 1801: 1761: 1730: 1683:Decimal representation 1169: 1147: 1125: 1096: 1022:Complete ordered field 550:As the "real part" of 424:division (mathematics) 329:continuum (set theory) 232:Lowercase sigmabot III 1802: 1762: 1707: 1281:legible definition. – 1244:Lead characterization 1170: 1148: 1126: 1097: 952:What do folks think? 818:... what it says is, 537:scalar multiplication 496:real coordinate space 492:real coordinate space 476:real coordinate space 36:level-3 vital article 1771: 1691: 1157: 1135: 1110: 1077: 556:hypercomplex numbers 554:or various kinds of 533:scalar (mathematics) 341:continuum (topology) 132:mathematics articles 1325:the current version 907:Advanced properties 884:Basic applications 403:field (mathematics) 373:which redirects to 337:continuous variable 1797: 1757: 1404:Formal definitions 1165: 1143: 1121: 1092: 941:Other applications 904:Formal definitions 541:scaling (geometry) 500:coordinate systems 101:Mathematics portal 45:content assessment 1320: 1228:what something is 1104:universal problem 1071:Cauchy completion 890:Analytic geometry 767: 753: 704:analytic geometry 484:Cartesian product 472:analytic geometry 468:coordinate system 432:analytic geometry 387:irrational number 265: 253:comment added by 239: 238: 166: 165: 162: 161: 158: 157: 1954: 1810: 1806: 1804: 1803: 1798: 1796: 1795: 1783: 1782: 1766: 1764: 1763: 1758: 1756: 1755: 1743: 1742: 1729: 1724: 1703: 1702: 1633: 1505:, but that of a 1471:linear continuum 1318: 1248:The lead reads " 1174: 1172: 1171: 1166: 1164: 1152: 1150: 1149: 1144: 1142: 1130: 1128: 1127: 1122: 1117: 1101: 1099: 1098: 1093: 1091: 1090: 1085: 910:Use in analysis 773: 765: 759: 751: 604:linear continuum 515:parametric curve 461:scalar (physics) 333:linear continuum 234: 218: 178: 170: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 1962: 1961: 1957: 1956: 1955: 1953: 1952: 1951: 1902: 1901: 1808: 1787: 1774: 1769: 1768: 1744: 1731: 1694: 1689: 1688: 1685: 1615: 1246: 1155: 1154: 1133: 1132: 1108: 1107: 1080: 1075: 1074: 1024: 938:Generalizations 896:Complex numbers 764: 750: 636:connected space 552:complex numbers 443:Cauchy sequence 380:The concept of 287: 244: 230: 219: 213: 183: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 1960: 1958: 1950: 1949: 1944: 1939: 1934: 1929: 1924: 1919: 1914: 1904: 1903: 1900: 1899: 1898: 1897: 1883: 1882: 1881: 1878: 1865: 1864: 1850: 1835: 1794: 1790: 1786: 1781: 1777: 1754: 1751: 1747: 1741: 1738: 1734: 1728: 1723: 1720: 1717: 1714: 1710: 1706: 1701: 1697: 1684: 1681: 1680: 1679: 1678: 1677: 1676: 1675: 1674: 1673: 1651: 1650: 1649: 1648: 1647: 1646: 1613: 1594: 1576: 1540: 1539: 1538: 1537: 1536: 1535: 1534: 1533: 1532: 1531: 1530: 1529: 1528: 1527: 1526: 1525: 1524: 1523: 1522: 1521: 1467: 1400: 1399: 1398: 1351: 1344:p-adic numbers 1340: 1321: 1292: 1291: 1245: 1242: 1241: 1240: 1217: 1216: 1215: 1195: 1177: 1176: 1163: 1141: 1120: 1116: 1089: 1084: 1023: 1020: 1003: 1002: 1001: 1000: 999: 998: 950: 949: 948: 945: 942: 939: 936: 935: 934: 931: 928: 922: 921: 920: 919:Measure theory 917: 914: 908: 905: 902: 901: 900: 899:Linear algebra 897: 894: 891: 888: 882: 881: 880: 869: 866: 863: 860: 857: 851: 848: 841: 840: 839: 838: 837: 836: 835: 834: 833: 816:WP:NOTTEXTBOOK 812: 780:WP:NOTTEXTBOOK 762: 707: 677: 676: 675: 674: 673: 672: 671: 670: 669: 668: 560: 559: 548: 525: 518: 507: 464: 420:multiplication 286: 283: 282: 281: 243: 240: 237: 236: 224: 221: 220: 215: 211: 209: 206: 205: 185: 184: 179: 173: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 1959: 1948: 1945: 1943: 1940: 1938: 1935: 1933: 1930: 1928: 1925: 1923: 1920: 1918: 1915: 1913: 1910: 1909: 1907: 1896: 1892: 1888: 1884: 1879: 1876: 1875: 1873: 1869: 1868: 1867: 1866: 1863: 1859: 1855: 1851: 1849: 1845: 1841: 1836: 1833: 1829: 1828:§ Cardinality 1825: 1824: 1823: 1822: 1818: 1814: 1792: 1788: 1784: 1779: 1775: 1752: 1749: 1745: 1739: 1736: 1732: 1726: 1721: 1718: 1715: 1712: 1708: 1704: 1699: 1695: 1682: 1672: 1668: 1664: 1659: 1658: 1657: 1656: 1655: 1654: 1653: 1652: 1645: 1641: 1637: 1631: 1627: 1623: 1619: 1614: 1612: 1608: 1604: 1600: 1595: 1592: 1591: 1590: 1586: 1582: 1577: 1573: 1572: 1571: 1568: 1565: 1560: 1559: 1558: 1557: 1553: 1549: 1545: 1520: 1516: 1512: 1508: 1507:real variable 1504: 1500: 1499: 1498: 1494: 1490: 1486: 1485: 1484: 1480: 1476: 1472: 1468: 1466: 1462: 1458: 1453: 1448: 1447: 1446: 1442: 1438: 1433: 1432: 1431: 1427: 1423: 1419: 1418: 1417: 1413: 1409: 1405: 1401: 1397: 1393: 1389: 1384: 1383: 1382: 1381: 1380: 1376: 1372: 1367: 1366: 1365: 1361: 1357: 1352: 1349: 1348:Dedekind cuts 1345: 1341: 1339: 1335: 1331: 1326: 1322: 1315: 1311: 1310: 1309: 1305: 1301: 1296: 1295: 1294: 1293: 1290: 1287: 1284: 1279: 1275: 1274: 1273: 1272: 1268: 1264: 1260: 1256: 1251: 1243: 1239: 1236: 1233: 1229: 1225: 1221: 1218: 1214: 1210: 1206: 1205:Ebony Jackson 1201: 1196: 1193: 1192: 1191: 1187: 1183: 1179: 1178: 1118: 1105: 1087: 1072: 1068: 1067: 1066: 1065: 1061: 1057: 1056:Ebony Jackson 1053: 1049: 1045: 1041: 1037: 1033: 1029: 1021: 1019: 1018: 1015: 1012: 1008: 997: 994: 991: 986: 985: 984: 980: 976: 972: 967: 963: 962: 961: 958: 955: 951: 946: 943: 940: 937: 932: 929: 926: 925: 923: 918: 916:Metric spaces 915: 912: 911: 909: 906: 903: 898: 895: 892: 889: 886: 885: 883: 878: 874: 870: 867: 864: 861: 858: 855: 854: 852: 849: 846: 845: 842: 832: 829: 826: 821: 817: 813: 811: 808: 805: 800: 795: 794: 793: 789: 785: 781: 777: 771: 763: 757: 749: 748: 747: 744: 741: 736: 735: 734: 730: 726: 721: 720: 719: 716: 713: 708: 705: 701: 700:real analysis 697: 696: 695: 694: 690: 686: 682: 667: 664: 661: 656: 651: 650: 649: 645: 641: 637: 633: 632:connectedness 629: 628: 627: 624: 621: 617: 613: 609: 605: 601: 600: 599: 595: 591: 590:Ebony Jackson 586: 585: 584: 581: 578: 574: 573: 572: 571: 568: 565: 557: 553: 549: 546: 542: 538: 534: 530: 526: 523: 519: 516: 512: 508: 505: 501: 497: 493: 489: 485: 481: 477: 473: 469: 465: 462: 455: 454: 453: 450: 448: 444: 440: 435: 433: 427: 425: 421: 417: 413: 409: 404: 398: 394: 392: 388: 383: 378: 376: 372: 368: 363: 361: 357: 354:redirects to 353: 349: 344: 342: 338: 334: 330: 326: 322: 317: 315: 311: 307: 305: 298: 294: 290: 284: 280: 276: 272: 268: 267: 266: 264: 260: 256: 255:199.46.11.200 252: 241: 233: 228: 223: 222: 208: 207: 204: 203: 199: 195: 191: 187: 186: 182: 177: 172: 171: 168: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 1686: 1598: 1543: 1541: 1506: 1502: 1451: 1313: 1258: 1254: 1249: 1247: 1227: 1223: 1199: 1051: 1047: 1043: 1039: 1035: 1031: 1027: 1025: 1004: 876: 872: 862:Completeness 859:Order (<) 847:Lead section 798: 678: 611: 561: 522:metric space 511:scalar field 451: 436: 428: 399: 395: 390: 381: 379: 371:linear order 366: 364: 345: 318: 313: 309: 303: 299: 295: 291: 288: 249:— Preceding 245: 226: 188: 180: 167: 148:Top-priority 147: 107: 73:Top‑priority 51:WikiProjects 34: 1503:real number 924:Extensions 865:Cardinality 577:Paul August 545:real number 470:, e.g. for 416:subtraction 408:number line 375:total order 360:number line 356:number line 304:real number 123:Mathematics 114:mathematics 70:Mathematics 1906:Categories 1807:for every 1544:continuous 1435:of scope. 1259:continuous 1250:continuous 1032:definition 887:Arithmetic 853:Structure 382:irrational 310:continuous 1887:Anita5192 1813:Anita5192 1663:Anita5192 1661:section.— 1626:Anita5192 1622:Jacobolus 1618:Trovatore 1581:Trovatore 1564:jacobolus 1548:Anita5192 1408:Anita5192 1388:Trovatore 1330:Trovatore 1283:jacobolus 1232:jacobolus 1011:jacobolus 990:jacobolus 954:jacobolus 871:Subsets ( 868:Intervals 825:jacobolus 804:jacobolus 770:Jacobolus 756:Trovatore 740:jacobolus 725:Trovatore 712:jacobolus 681:jacobolus 660:jacobolus 640:Trovatore 620:jacobolus 564:jacobolus 480:dimension 352:real line 271:Anita5192 39:is rated 1854:D.Lazard 1840:D.Lazard 1636:Macrakis 1630:D.Lazard 1603:Macrakis 1599:distinct 1511:Macrakis 1489:D.Lazard 1475:Macrakis 1457:Macrakis 1437:D.Lazard 1422:Macrakis 1371:Macrakis 1356:D.Lazard 1300:Macrakis 1263:Macrakis 1182:D.Lazard 1044:property 975:D.Lazard 893:Calculus 784:D.Lazard 776:calculus 710:page.) – 685:D.Lazard 439:sequence 412:addition 391:rational 314:distance 251:unsigned 227:365 days 181:Archives 1628:, and 1052:theorem 944:History 879:, etc.) 814:As for 768:editor 754:editor 367:ordered 150:on the 41:C-class 1452:subset 1255:define 506:, etc. 502:, for 269:Done.— 47:scale. 1575:vary. 1226:with 529:field 527:As a 28:This 1891:talk 1858:talk 1844:talk 1817:talk 1785:< 1667:talk 1640:talk 1607:talk 1585:talk 1552:talk 1515:talk 1493:talk 1479:talk 1461:talk 1441:talk 1426:talk 1412:talk 1392:talk 1375:talk 1360:talk 1334:talk 1304:talk 1267:talk 1209:talk 1186:talk 1060:talk 979:talk 947:...? 913:...? 788:talk 729:talk 702:and 689:talk 644:talk 594:talk 348:line 331:and 275:talk 259:talk 1567:(t) 1286:(t) 1235:(t) 1046:of 1034:of 1014:(t) 1009:. – 993:(t) 957:(t) 828:(t) 807:(t) 743:(t) 715:(t) 663:(t) 623:(t) 612:why 567:(t) 457:--> 142:Top 1908:: 1893:) 1860:) 1846:) 1819:) 1789:10 1750:− 1746:10 1737:− 1719:− 1709:∑ 1669:) 1642:) 1624:, 1620:, 1609:) 1587:) 1579:-- 1554:) 1517:) 1495:) 1481:) 1463:) 1455:-- 1443:) 1428:) 1414:) 1394:) 1386:-- 1377:) 1362:) 1336:) 1314:is 1306:) 1269:) 1211:) 1188:) 1062:) 981:) 875:, 790:) 766:To 752:To 731:) 723:-- 691:) 646:) 596:) 539:, 535:, 513:, 482:, 478:, 434:. 426:. 422:, 418:, 414:, 302:a 277:) 261:) 200:, 196:, 192:, 1889:( 1856:( 1842:( 1815:( 1809:n 1793:k 1780:n 1776:S 1753:i 1740:i 1733:a 1727:n 1722:k 1716:= 1713:i 1705:= 1700:n 1696:S 1665:( 1638:( 1632:: 1616:@ 1605:( 1583:( 1562:– 1550:( 1513:( 1491:( 1477:( 1459:( 1439:( 1424:( 1410:( 1390:( 1373:( 1358:( 1332:( 1302:( 1265:( 1207:( 1200:R 1184:( 1162:R 1140:R 1119:. 1115:Q 1088:n 1083:R 1058:( 1048:R 1040:R 1036:R 1028:R 977:( 877:Q 873:N 799:π 786:( 772:: 758:: 727:( 687:( 642:( 592:( 580:☎ 524:. 463:. 273:( 257:( 202:4 198:3 194:2 190:1 154:. 53::

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index