{
localUrl: '../page/pi_is_irrational.html',
arbitalUrl: 'https://arbital.com/p/pi_is_irrational',
rawJsonUrl: '../raw/513.json',
likeableId: '2925',
likeableType: 'page',
myLikeValue: '0',
likeCount: '2',
dislikeCount: '0',
likeScore: '2',
individualLikes: [
'EricBruylant',
'JaimeSevillaMolina'
],
pageId: 'pi_is_irrational',
edit: '4',
editSummary: '',
prevEdit: '3',
currentEdit: '4',
wasPublished: 'true',
type: 'wiki',
title: 'Pi is irrational',
clickbait: 'The number pi is famously not rational, in spite of joking attempts at legislation to fix its value at 3 or 22/7.',
textLength: '3345',
alias: 'pi_is_irrational',
externalUrl: '',
sortChildrenBy: 'likes',
hasVote: 'false',
voteType: '',
votesAnonymous: 'false',
editCreatorId: 'PatrickStevens',
editCreatedAt: '2016-07-21 19:36:18',
pageCreatorId: 'PatrickStevens',
pageCreatedAt: '2016-07-03 10:25:33',
seeDomainId: '0',
editDomainId: 'AlexeiAndreev',
submitToDomainId: '0',
isAutosave: 'false',
isSnapshot: 'false',
isLiveEdit: 'true',
isMinorEdit: 'false',
indirectTeacher: 'false',
todoCount: '1',
isEditorComment: 'false',
isApprovedComment: 'true',
isResolved: 'false',
snapshotText: '',
anchorContext: '',
anchorText: '',
anchorOffset: '0',
mergedInto: '',
isDeleted: 'false',
viewCount: '137',
text: 'The number [49r $\\pi$] is not [4zq rational].\n\n# Proof\n\nFor any fixed real number $q$, and any [-45h] $n$, let $$A_n = \\frac{q^n}{n!} \\int_0^{\\pi} [x (\\pi - x)]^n \\sin(x) dx$$\nwhere $n!$ is the [-5bv] of $n$, $\\int$ is the [-definite_integral], and $\\sin$ is the [-sin_function].\n\n## Preparatory work\n\nExercise: $A_n = (4n-2) q A_{n-1} - (q \\pi)^2 A_{n-2}$.\n%%hidden(Show solution):\nWe use [-integration_by_parts].\n\n[todo: show this]\n%%\n\nNow, $$A_0 = \\int_0^{\\pi} \\sin(x) dx = 2$$\nso $A_0$ is an integer.\n\nAlso $$A_1 = q \\int_0^{\\pi} x (\\pi-x) \\sin(x) dx$$ which by a simple calculation is $4q$.\n%%hidden(Show calculation):\nExpand the integrand and then integrate by parts repeatedly:\n$$\\frac{A_1}{q} = \\int_0^{\\pi} x (\\pi-x) \\sin(x) dx = \\pi \\int_0^{\\pi} x \\sin(x) dx - \\int_0^{\\pi} x^2 \\sin(x) dx$$\n\nThe first integral term is $$[-x \\cos(x)]_0^{\\pi} + \\int_0^{\\pi} \\cos(x) dx = \\pi$$\n\nThe second integral term is $$[-x^2 \\cos(x)]_{0}^{\\pi} + \\int_0^{\\pi} 2x \\cos(x) dx$$\nwhich is $$\\pi^2 + 2 \\left( [x \\sin(x)]_0^{\\pi} - \\int_0^{\\pi} \\sin(x) dx \\right)$$\nwhich is $$\\pi^2 -4$$\n\nTherefore $$\\frac{A_1}{q} = \\pi^2 - (\\pi^2 - 4) = 4$$\n%%\n\nTherefore, if $q$ and $q \\pi$ are integers, then so is $A_n$ [5fz inductively], because $(4n-2) q A_{n-1}$ is an integer and $(q \\pi)^2 A_{n-2}$ is an integer.\n\nBut also $A_n \\to 0$ as $n \\to \\infty$, because $\\int_0^{\\pi} [x (\\pi-x)]^n \\sin(x) dx$ is in modulus at most $$\\pi \\times \\max_{0 \\leq x \\leq \\pi} [x (\\pi-x)]^n \\sin(x) \\leq \\pi \\times \\max_{0 \\leq x \\leq \\pi} [x (\\pi-x)]^n = \\pi \\times \\left[\\frac{\\pi^2}{4}\\right]^n$$\nand hence $$|A_n| \\leq \\frac{1}{n!} \\left[\\frac{\\pi^2 q}{4}\\right]^n$$\n\nFor $n$ larger than $\\frac{\\pi^2 q}{4}$, this expression is getting smaller with $n$, and moreover it gets smaller faster and faster as $n$ increases; so its limit is $0$.\n%%hidden(Formal treatment):\nWe claim that $\\frac{r^n}{n!} \\to 0$ as $n \\to \\infty$, for any $r > 0$.\n\nIndeed, we have $$\\frac{r^{n+1}/(n+1)!}{r^n/n!} = \\frac{r}{n+1}$$\nwhich, for $n > 2r-1$, is less than $\\frac{1}{2}$.\nTherefore the ratio between successive terms is less than $\\frac{1}{2}$ for sufficiently large $n$, and so the sequence must shrink at least geometrically to $0$.\n%%\n\n## Conclusion\n\nSuppose (for [46z contradiction]) that $\\pi$ is rational; then it is $\\frac{p}{q}$ for some integers $p, q$.\n\nNow $q \\pi$ is an integer (indeed, it is $p$), and $q$ is certainly an integer, so by what we showed above, $A_n$ is an integer for all $n$.\n\nBut $A_n \\to 0$ as $n \\to \\infty$, so there is some $N$ for which $|A_n| < \\frac{1}{2}$ for all $n > N$; hence for all sufficiently large $n$, $A_n$ is $0$.\nWe already know that $A_0 = 2$ and $A_1 = 4q$, neither of which is $0$; so let $N$ be the first integer such that $A_n = 0$ for all $n \\geq N$, and we can already note that $N > 1$.\n\nThen $$0 = A_{N+1} = (4N-2) q A_N - (q \\pi)^2 A_{N-1} = - (q \\pi)^2 A_{N-1}$$\nwhence $q=0$ or $\\pi = 0$ or $A_{N-1} = 0$.\n\nCertainly $q \\not = 0$ because $q$ is the denominator of a fraction; and $\\pi \\not = 0$ by whatever definition of $\\pi$ we care to use.\nBut also $A_{N-1}$ is not $0$ because then $N-1$ would be an integer $m$ such that $A_n = 0$ for all $n \\geq m$, and that contradicts the definition of $N$ as the *least* such integer.\n\nWe have obtained the required contradiction; so it must be the case that $\\pi$ is irrational.',
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: [
'PatrickStevens',
'EricRogstad'
],
childIds: [],
parentIds: [
'pi',
'irrational_number'
],
commentIds: [],
questionIds: [],
tagIds: [
'proof_meta_tag'
],
relatedIds: [],
markIds: [],
explanations: [],
learnMore: [],
requirements: [
{
id: '4716',
parentId: 'rational_number',
childId: 'pi_is_irrational',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-07-03 09:52:22',
level: '1',
isStrong: 'false',
everPublished: 'true'
},
{
id: '4718',
parentId: 'math3',
childId: 'pi_is_irrational',
type: 'requirement',
creatorId: 'PatrickStevens',
createdAt: '2016-07-03 10:27:15',
level: '1',
isStrong: 'false',
everPublished: 'true'
}
],
subjects: [],
lenses: [],
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: '17248',
pageId: 'pi_is_irrational',
userId: 'PatrickStevens',
edit: '4',
type: 'newEdit',
createdAt: '2016-07-21 19:36:18',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '17240',
pageId: 'pi_is_irrational',
userId: 'EricRogstad',
edit: '3',
type: 'newEdit',
createdAt: '2016-07-21 17:57:09',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: '[pi_real_number $\\pi$] -> [pi $\\pi$]'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15619',
pageId: 'pi_is_irrational',
userId: 'EricBruylant',
edit: '0',
type: 'newParent',
createdAt: '2016-07-06 06:49:42',
auxPageId: 'irrational_number',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15617',
pageId: 'pi_is_irrational',
userId: 'EricBruylant',
edit: '0',
type: 'deleteParent',
createdAt: '2016-07-06 06:49:36',
auxPageId: 'math',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15615',
pageId: 'pi_is_irrational',
userId: 'EricBruylant',
edit: '0',
type: 'newParent',
createdAt: '2016-07-06 06:49:35',
auxPageId: 'pi',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15386',
pageId: 'pi_is_irrational',
userId: 'PatrickStevens',
edit: '2',
type: 'newEdit',
createdAt: '2016-07-05 08:37:32',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: 'Minor insertion of a word'
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15265',
pageId: 'pi_is_irrational',
userId: 'EricBruylant',
edit: '0',
type: 'newParent',
createdAt: '2016-07-04 18:04:26',
auxPageId: 'math',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15172',
pageId: 'pi_is_irrational',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-07-03 10:27:15',
auxPageId: 'math3',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15170',
pageId: 'pi_is_irrational',
userId: 'PatrickStevens',
edit: '0',
type: 'newRequirement',
createdAt: '2016-07-03 10:25:35',
auxPageId: 'rational_number',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15171',
pageId: 'pi_is_irrational',
userId: 'PatrickStevens',
edit: '0',
type: 'newTag',
createdAt: '2016-07-03 10:25:35',
auxPageId: 'proof_meta_tag',
oldSettingsValue: '',
newSettingsValue: ''
},
{
likeableId: '0',
likeableType: 'changeLog',
myLikeValue: '0',
likeCount: '0',
dislikeCount: '0',
likeScore: '0',
individualLikes: [],
id: '15169',
pageId: 'pi_is_irrational',
userId: 'PatrickStevens',
edit: '1',
type: 'newEdit',
createdAt: '2016-07-03 10:25:33',
auxPageId: '',
oldSettingsValue: '',
newSettingsValue: ''
}
],
feedSubmissions: [],
searchStrings: {},
hasChildren: 'false',
hasParents: 'true',
redAliases: {},
improvementTagIds: [],
nonMetaTagIds: [],
todos: [],
slowDownMap: 'null',
speedUpMap: 'null',
arcPageIds: 'null',
contentRequests: {}
}