{
localUrl: '../page/real_number.html',
arbitalUrl: 'https://arbital.com/p/real_number',
rawJsonUrl: '../raw/4bc.json',
likeableId: '2685',
likeableType: 'page',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
pageId: 'real_number',
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: 'JoeZeng',
editCreatedAt: '2016-08-16 22:23:25',
pageCreatorId: 'MichaelCohen',
pageCreatedAt: '2016-06-14 21:50:45',
seeDomainId: '0',
editDomainId: 'AlexeiAndreev',
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: '367',
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. The latter restriction requires that the set also be a [ perfect set] (have no [isolated_point isolated points]), in the sense used in topology.\n\nIn the real numbers, such a partition will always have the first set having a greatest element, which is known as the least-upper-bound property. However, in the rational numbers, we might come across a partition where the first set does not have such an element.\n\nFor example, define a Dedekind cut $(A, B)$ of the rational numbers such that $B = \\{x \\in \\mathbb{Q} \\ | \\ x > 0 \\wedge x^2 > 2\\}$ and $A$ is the complement of $B$. In plainer language, $B$ consists of all the numbers greater than $\\sqrt{2}$, but because $\\sqrt{2}$ doesn't exist in the space of rational numbers, we can't use that to formulate our definition. 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: [
'MichaelCohen',
'JoeZeng',
'EricBruylant',
'KevinClancy',
'EricRogstad',
'NateSoares',
'AlexeiAndreev',
'MYass',
'PatrickStevens'
],
childIds: [
'real_number_as_cauchy_sequence',
'real_number_as_dedekind_cut',
'real_numbers_uncountable'
],
parentIds: [
'math'
],
commentIds: [
'4bp',
'4bv',
'4h5',
'503',
'5dd'
],
questionIds: [],
tagIds: [
'start_meta_tag',
'needs_summary_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [],
subjects: [],
lenses: [
{
id: '61',
pageId: 'real_number',
lensId: 'real_number_as_cauchy_sequence',
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: 'real_number',
lensId: 'real_number_as_dedekind_cut',
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: 'real_number',
userId: 'EricBruylant',
edit: '0',
type: 'newChild',
createdAt: '2016-10-21 11:05:26',
auxPageId: 'real_numbers_uncountable',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '18768',
pageId: 'real_number',
userId: 'JoeZeng',
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: 'real_number',
userId: 'PatrickStevens',
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: 'real_number',
userId: 'MYass',
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: 'real_number',
userId: 'AlexeiAndreev',
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: 'real_number',
userId: 'KevinClancy',
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. '
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '16883',
pageId: 'real_number',
userId: 'KevinClancy',
edit: '10',
type: 'newEdit',
createdAt: '2016-07-16 17:41:44',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '16733',
pageId: 'real_number',
userId: 'AlexeiAndreev',
edit: '0',
type: 'newTag',
createdAt: '2016-07-14 23:59:52',
auxPageId: 'needs_summary_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15899',
pageId: 'real_number',
userId: 'JoeZeng',
edit: '9',
type: 'newEdit',
createdAt: '2016-07-07 02:01:22',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15409',
pageId: 'real_number',
userId: 'JoeZeng',
edit: '0',
type: 'newChild',
createdAt: '2016-07-05 19:34:04',
auxPageId: 'real_number_as_dedekind_cut',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15234',
pageId: 'real_number',
userId: 'NateSoares',
edit: '8',
type: 'newEdit',
createdAt: '2016-07-04 07:07:24',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15233',
pageId: 'real_number',
userId: 'NateSoares',
edit: '0',
type: 'newTag',
createdAt: '2016-07-04 07:07:16',
auxPageId: 'start_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15116',
pageId: 'real_number',
userId: 'PatrickStevens',
edit: '0',
type: 'deleteRequirement',
createdAt: '2016-07-02 14:10:31',
auxPageId: 'math3',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15114',
pageId: 'real_number',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-07-02 14:10:18',
auxPageId: 'math3',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15112',
pageId: 'real_number',
userId: 'PatrickStevens',
edit: '0',
type: 'newChild',
createdAt: '2016-07-02 14:09:56',
auxPageId: 'real_number_as_cauchy_sequence',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '2920',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '1',
dislikeCount: '0',
likeScore: '1',
individualLikes: [],
id: '15108',
pageId: 'real_number',
userId: 'JoeZeng',
edit: '7',
type: 'newEdit',
createdAt: '2016-07-02 02:24:55',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '13453',
pageId: 'real_number',
userId: 'MichaelCohen',
edit: '6',
type: 'newEdit',
createdAt: '2016-06-17 07:49:16',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12892',
pageId: 'real_number',
userId: 'EricRogstad',
edit: '5',
type: 'newEdit',
createdAt: '2016-06-15 00:03:48',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12862',
pageId: 'real_number',
userId: 'EricBruylant',
edit: '4',
type: 'newParent',
createdAt: '2016-06-14 22:42:59',
auxPageId: 'math',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12860',
pageId: 'real_number',
userId: 'EricBruylant',
edit: '4',
type: 'newEdit',
createdAt: '2016-06-14 22:42:47',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12810',
pageId: 'real_number',
userId: 'EricBruylant',
edit: '3',
type: 'newEdit',
createdAt: '2016-06-14 22:15:15',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12799',
pageId: 'real_number',
userId: 'MichaelCohen',
edit: '2',
type: 'newEdit',
createdAt: '2016-06-14 21:51:10',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '12797',
pageId: 'real_number',
userId: 'MichaelCohen',
edit: '1',
type: 'newEdit',
createdAt: '2016-06-14 21:50:46',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'true',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}