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"edit": 15, "editSummary": "", "prevEdit": 14, "currentEdit": 15, "wasPublished": true, "type": "wiki", "title": "Real number", "clickbait": "", "textLength": 3588, "alias": "real_number", "externalUrl": "", "sortChildrenBy": "likes", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "4tg", "editCreatedAt": "2016-08-16 22:23:25", "pageCreatorId": "47c", "pageCreatedAt": "2016-06-14 21:50:45", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 2, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 191, "text": "A **real number** is any number that can be used to represent a physical quantity.\n\nIntuitively, real numbers are any number that can be found between two [48l integers], such as $0,$ $1,$ $-1,$ $\\frac{3}{2},$ $\\frac{-7}{2},$ [49r $\\pi,$] [e $e$], $100 \\cdot \\sqrt{2},$ and so on. The set of real numbers is written $\\mathbb R.$ You can think of $\\mathbb R$ as [4zq $\\mathbb Q$] extended to include the [54z irrational numbers] like $\\pi$ and $e$ which can be found between rational numbers but which cannot be completely written out in [-4sl].\n\n## Definitions of the real numbers\n\nThe most commonly used definitions of the real numbers are constructions as extensions of the [4zq rational numbers], which involve either [53b Cauchy sequences] or [dedekind_cut Dedekind cuts].\n\n### Cauchy sequences\n\nBroadly speaking, a Cauchy sequence is a sequence where as the sequence goes on, all the elements past that point get closer and closer together. In the real numbers, every Cauchy sequence [convergence_analysis converges] to a real number. However, in the set of rational numbers, not all Cauchy sequences converge to a rational number.\nIn the set of rationals, a Cauchy sequence which does not converge to a rational number cannot really be said to \"converge\" at all: the set of rationals is \"missing some of the points\" that would be required to make every Cauchy sequence converge.\n\nFor example, the sequence of fractions of consecutive Fibonacci numbers $1/1, 2/1, 3/2, 5/3, 8/5, \\ldots$ gets closer and closer to $\\frac{1 + \\sqrt{5}}{2}$, but cannot be said to converge to that number because it is not in the set of rational numbers.\n\nFor each of these non-convergent Cauchy sequences, we define a new irrational number to \"fill in the gap\", and for the Cauchy sequences that do converge, we define a real number equal to that rational number.\n\n### Dedekind cuts\n\nA Dedekind cut of a [-540] is a [-partition] of that set into two sets so that every element in the first set is [ less than] every element in the second set, and the second set has no smallest element. 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Obviously every element of $A$ is less than every element of $B$, but $A$ has no greatest element either, because we can create a sequence of numbers in $A$ that gets bigger and bigger (as it approaches $\\sqrt{2}$) but never stops at a maximum value.\n\nFor each of these \"strict cuts\" where neither set has a \"boundary element\", we define a new irrational number to \"fill in the gap\", just like with the Cauchy sequences. For the Dedekind cuts where one of the sets does have a least or greatest element, we define a real number equal to that rational number.\n\nThis definition has the advantage that each real number is represented by a unique Dedekind cut, unlike the Cauchy sequences where multiple sequences can converge to the same number.", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 2, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": true, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, "creatorIds": [ "47c", "4tg", "1yq", "299", "5", "32", "1", "202", "267" ], "childIds": [ "50d", "50g", "6fk" ], "parentIds": [ "1lw" ], "commentIds": [ "4bp", "4bv", "4h5", "503", "5dd" ], "questionIds": [], "tagIds": [ "3rk", "433" ], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [ { "id": "61", "pageId": "4bc", "lensId": "50d", "lensIndex": 0, "lensName": "Cauchy sequence definition", "lensSubtitle": "", "createdBy": "267", "createdAt": "2016-07-02 14:09:58", "updatedBy": "267", "updatedAt": "2016-07-02 14:10:11" }, { "id": "65", "pageId": "4bc", "lensId": "50g", "lensIndex": 1, "lensName": "DEDEKIND CUT DEFINITION", "lensSubtitle": "", "createdBy": "4tg", "createdAt": "2016-07-05 19:34:06", "updatedBy": "4tg", "updatedAt": "2016-07-05 19:34:20" } ], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [ { "likeableId": "0", "likeableType": "changeLog", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "id": "20211", "pageId": "4bc", "userId": "1yq", "edit": 0, "type": "newChild", "createdAt": "2016-10-21 11:05:26", "auxPageId": "6fk", "oldSettingsValue": "", "newSettingsValue": "" }, { "likeableId": "0", "likeableType": "changeLog", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "id": "18768", "pageId": "4bc", "userId": "4tg", "edit": 15, "type": "newEdit", "createdAt": "2016-08-16 22:23:25", "auxPageId": "", "oldSettingsValue": "", "newSettingsValue": "" }, { "likeableId": "3352", "likeableType": "changeLog", "myLikeValue": 0, "likeCount": 1, "dislikeCount": 0, "likeScore": 1, "individualLikes": [], "id": "18494", "pageId": "4bc", "userId": "267", "edit": 14, "type": "newEdit", "createdAt": "2016-08-06 12:40:03", "auxPageId": "", "oldSettingsValue": "", "newSettingsValue": "" }, { "likeableId": "0", "likeableType": "changeLog", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "id": "18476", "pageId": "4bc", "userId": "202", "edit": 13, "type": "newEdit", "createdAt": "2016-08-06 03:09:51", "auxPageId": "", "oldSettingsValue": "", "newSettingsValue": "Before, these sentences looked contradictory until I read the next paragraph(which indicated that they do converge, but not to a rational). That is not good." }, { "likeableId": "0", "likeableType": "changeLog", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "id": "17729", "pageId": "4bc", "userId": "1", "edit": 12, "type": "newEdit", "createdAt": "2016-07-29 18:15:02", "auxPageId": "", "oldSettingsValue": "", "newSettingsValue": "" }, { "likeableId": "3090", "likeableType": "changeLog", "myLikeValue": 0, "likeCount": 2, "dislikeCount": 0, "likeScore": 2, "individualLikes": [], "id": "16922", "pageId": "4bc", "userId": "299", "edit": 11, "type": "newEdit", "createdAt": "2016-07-16 20:17:17", "auxPageId": "", "oldSettingsValue": "", "newSettingsValue": "Previous edit explanation: The definition was circular because infinite summations are defined in terms of limits, and limits of rationals are defined in terms of real numbers. 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"edit": 1, "editSummary": "", "prevEdit": 0, "currentEdit": 1, "wasPublished": true, "type": "comment", "title": "\"Approved, but the summary c...\"", "clickbait": "", "textLength": 190, "alias": "4bp", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "1yq", "editCreatedAt": "2016-06-14 22:44:15", "pageCreatorId": "1yq", "pageCreatedAt": "2016-06-14 22:44:15", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 190, "text": "Approved, but the summary could do with a bit of improvement, make it something that a non-mathematician will get something out of. Give examples of things that are and are not real numbers.", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 1, "maintainerCount": 1, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, "creatorIds": [], "childIds": [], "parentIds": [ "4bc" ], "commentIds": [], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": true, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "4bv": { "likeableId": "0", "likeableType": "page", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "pageId": "4bv", "edit": 2, "editSummary": "", "prevEdit": 1, "currentEdit": 2, "wasPublished": true, "type": "comment", "title": "\"I'm pretty tired right now,...\"", "clickbait": "", "textLength": 350, "alias": "4bv", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "299", "editCreatedAt": "2016-06-14 23:49:35", "pageCreatorId": "299", "pageCreatedAt": "2016-06-14 23:49:07", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 190, "text": "I'm pretty tired right now, but this definition seems kind of circular to me. It involves an infinite sum, and infinite sums are defined in terms of limits. But a limit of rational numbers is defined in terms of the set of real numbers. Maybe it would be better to present the definition of real numbers that one would find in a real analysis text.\n ", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 1, "maintainerCount": 1, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, "creatorIds": [], "childIds": [], "parentIds": [ "4bc" ], "commentIds": [], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": true, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "4h5": { "likeableId": "0", "likeableType": "page", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "pageId": "4h5", "edit": 1, "editSummary": "", "prevEdit": 0, "currentEdit": 1, "wasPublished": true, "type": "comment", "title": "\"Hey Kevin. I think I accide...\"", "clickbait": "", "textLength": 1041, "alias": "4h5", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "47c", "editCreatedAt": "2016-06-17 08:21:27", "pageCreatorId": "47c", "pageCreatedAt": "2016-06-17 08:21:27", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 189, "text": "Hey Kevin. I think I accidentally clicked ignore on your query. I'll look into that definition. I think potentially this definition isn't circular, but you might be right. I don't know quite how to put it, but it seems like this criticism could be directed at any algorithm that attempts to point towards all real numbers. The algorithm, it seems, would have to keep gesturing closer and closer to the real number it is trying to indicate. It's not as if the algorithm can take the set of real numbers for granted, and then say that we just need to find the number in the set of real numbers that satisfies the condition that we keep getting closer and closer to it, and then finally we add that to the set of real numbers. And yet, any algorithm that doesn't have a set in mind that it can pull the answer from, I think it will involve something along the lines of pointing towards a number that doesn't yet come from a well-defined spot. Otherwise, the algorithm can't create the well-defined spot. That might all be utter nonsense though.", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 2, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, "creatorIds": [], "childIds": [], "parentIds": [ "4bc" ], "commentIds": [ "4hr", "4j0" ], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": true, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "4hr": { "likeableId": "0", "likeableType": "page", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "pageId": "4hr", "edit": 1, "editSummary": "", "prevEdit": 0, "currentEdit": 1, "wasPublished": true, "type": "comment", "title": "\"I understand what you're sa...\"", "clickbait": "", "textLength": 827, "alias": "4hr", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "299", "editCreatedAt": "2016-06-17 15:16:54", "pageCreatorId": "299", "pageCreatedAt": "2016-06-17 15:16:54", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 189, "text": "I understand what you're saying and I think it's a good point. The problem is that you're developing an algorithm (a non-terminating one) that finds real numbers rather than providing a definition of them. It turns out that providing a definition of real numbers is not a simple as it may at first seem. This presentation is somewhat similar constructive analysis, in which a real number is defined as regularly converging sequence of rational numbers; importantly, constructive analysis does not define real numbers as *infinite sums* of these sequences, because as I've said, that would be a circular definition.\n\nIf you want to learn more about rigorous foundations for real numbers and related topics, I think that the book Calculus by Michael Spivak is a very approachable and well respected introduction to the topic. ", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 0, "maintainerCount": 0, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, "creatorIds": [], "childIds": [], "parentIds": [ "4bc", "4h5" ], "commentIds": [], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": true, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "4j0": { "likeableId": "0", "likeableType": "page", "myLikeValue": 0, "likeCount": 0, "dislikeCount": 0, "likeScore": 0, "individualLikes": [], "pageId": "4j0", "edit": 1, "editSummary": "", "prevEdit": 0, "currentEdit": 1, "wasPublished": true, "type": "comment", "title": "\"Is an infinite sum of ratio...\"", "clickbait": "", "textLength": 470, "alias": "4j0", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "47c", "editCreatedAt": "2016-06-17 17:12:09", "pageCreatorId": "47c", "pageCreatedAt": "2016-06-17 17:12:09", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 190, "text": "Is an infinite sum of rationals isomorphic with a regularly converging sequence of rationals (something along the lines of 1/2, 5/8, 11/16, etc.), where each rational in the sequence is the sum of all the addends up until then? I agree it is probably worth putting up a different definition anyway. I'm not sure I'll be able to do that for a little bit, since I haven't studied real analysis yet, but if you want to do that sooner, go for it. This is a fun conversation!", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 0, "maintainerCount": 0, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, 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"2016-07-05 16:21:22", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 3, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 52, "text": "[summary: The real numbers can be defined using [dedekind_cut Dedekind cuts], which are [set_partition partitions] of the rational numbers into two sets where every element in one set is less than every element in the other set. Each real number is uniquely defined by a one-sided Dedekind cut.]\n\n%%comment: Mnemonics for defined macros: \\Ql = Q left, \\Qr = Q right, \\Qls = Q left strict, \\Qrs = Q right strict.%%\n\nThe rational numbers have a problem that makes them unsuitable for use in calculus — they have \"gaps\" in them. This may not be obvious or even make sense at first, because between any two rational numbers you can always find infinitely many other rational numbers. How could there be *gaps* in a set like that? $\\newcommand{\\rats}{\\mathbb{Q}} \\newcommand{\\Ql}{\\rats^\\le} \\newcommand{\\Qr}{\\rats^\\ge} \\newcommand{\\Qls}{\\rats^<} \\newcommand{\\Qrs}{\\rats^>}$\n$\\newcommand{\\set}[1]{\\left\\{#1\\right\\}} \\newcommand{\\sothat}{\\ |\\ }$ \n\nBut using the construction of [dedekind_cut Dedekind cuts], we can suss out these gaps into plain view. A Dedekind cut of a [-540] $S$ is a pair of sets $(A, B)$ such that:\n\n1. Every element of $S$ is in exactly one of $A$ or $B$. (That is, $(A, B)$ is a [set_partition partition] of $S$.)\n2. Every element of $A$ is less than every element of $B$.\n3. Neither $A$ nor $B$ is [empty_set empty]. (We'll see why this restriction matters in a moment.)\n\nOne example of such a cut might be the set where $A$ is the negative rational numbers and $B$ is the nonnegative rational numbers (positive or zero). We see that it satisfies the three properties of a Dedekind cut:\n\n1. Every rational number is either negative or nonnegative, but not both.\n2. Every rational number which is negative is less than a rational number that is nonnegative.\n3. There exists at least one negative rational number (e.g. $-1$) and one nonnegative rational number (e.g. $1$).\n\nIn fact, Dedekind cuts are intended to represent sets of rational numbers that are less than or greater than a specific real number (once we've defined them). To represent this, let's call them $\\Ql$ and $\\Qr$.\n\nKnowing this, why does it matter that neither set in a Dedekind cut is empty?\n\n%%hidden(Show solution): \nIf $\\Ql$ were empty, then we'd have a real number less than _all_ the rational numbers, which is $-\\infty$, which we don't want to define as a real number. Similarly, if $\\Qr$ were empty, then we'd get $+\\infty$.\n%%\n\n## Completion of a space\n\nIf a space is [ complete] (doesn't have any gaps in it), then in any Dedekind cut $(\\Ql, \\Qr)$, either $\\Ql$ will have a greatest element or $\\Qr$ will have a least element. (We can't have both at the same time — why?)\n\n%%hidden(Show solution):\nSuppose $\\Ql$ had a greatest element $q_u$ and $\\Qr$ had a least element $q_v$. We can't have $q_u = q_v$, because the same number would be in both sets. So then because the rational numbers are a dense space, there must exist a rational number $r$ so that $q_u < r < q_v$. Then $r$ is not in either $\\Ql$ or $\\Qr$, contradicting property 1 of a Dedekind cut.\n%%\n\nBut in the rational numbers, we can find a Dedekind cut where neither $\\Ql$ nor $\\Qr$ have a greatest or least element respectively.\n\nConsider the pair of sets $(\\Ql, \\Qr)$ where $\\Ql = \\set{x \\in \\rats \\mid x^3 \\le 2}$ and $\\Qr = \\set{x \\in \\rats \\mid x^3 \\ge 2}$.\n\n1. Every rational number has a cube either greater than 2 or less than 2, \n2. Because $f(x) = x^3$ is a [ monotonically increasing] function, we have that $p < q \\iff p^3 < q^3$, which means that every element in $\\Ql$ is less than every element in $\\Qr$.\n\nSo $(\\Ql, \\Qr)$ is a Dedekind cut. However, there is no rational number whose cube is *equal to* $2$, so $\\Ql$ has no greatest element and $\\Qr$ has no least element.\n\nThis represents a gap in the numbers, because we can invent a new number to place in that gap (in this case $\\sqrt[3]{2}$), which is \"between\" any two numbers in $\\Ql$ and $\\Qr$.\n\n## Definition of real numbers\n\nBefore we move on, we will define one more structure that makes the construction more elegant. Define a *one-sided Dedekind cut* as any Dedekind cut $(\\Ql, \\Qr)$ with the additional property that the set $\\Ql$ has no greatest element (in which case we now call it $\\Qls$). The case where $\\Ql$ has a greatest element $q_g$ can be trivially transformed into the equivalent case on the other side by moving $q_g$ to $\\Qr$ where it is automatically the least element due to being less than any other element in $\\Qr$.\n\nThen we define the real numbers as the set of one-sided Dedekind cuts of the rational numbers.\n\n* A rational number $r$ is mapped to itself by the Dedekind cut where $r$ itself is the least element of $\\Qr$. (If the cuts weren't one-sided, $r$ would also be mapped to the set where $r$ was the greatest element of $\\Ql$, which would make the mapping non-unique.)\n* An irrational number $q$ is newly defined by the Dedekind cut where all the elements of $\\Qls$ are less than $q$ and all the elements of $\\Qr$ are (strictly) greater than $q$.\n\nNow we can define the [549 total order] $\\le$ for two real numbers $a = (\\Qls_a, \\Qr_a)$ and $b = (\\Qls_b, \\Qr_b)$ as follows: $a \\le b$ when $\\Qls_a \\subseteq \\Qls_b$. \n\nUsing this, we can show that unlike in the [50d Cauchy sequence definition], we don't need to define any equivalence classes — every real number is uniquely defined by a one-sided Dedekind cut.\n\n%%hidden(Proof):\nIf $a = b$, then $a \\le b$ and $b \\le a$. By the definition of the order, we have that $\\Qls_a \\subseteq \\Qls_b$ and $\\Qls_b \\subseteq \\Qls_a$, which means that $\\Qls_a = \\Qls_b$, which means that the Dedekind cuts corresponding to $a$ and $b$ are also equal.\n%%\n\n[todo: Proof of the field structure of Dedekind cuts.]", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 1, "maintainerCount": 1, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": true, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": 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