{ localUrl: '../page/sqrt_2_is_irrational.html', arbitalUrl: 'https://arbital.com/p/sqrt_2_is_irrational', rawJsonUrl: '../raw/548.json', likeableId: '2950', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'sqrt_2_is_irrational', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'The square root of 2 is irrational', clickbait: 'The number whose square is 2 can't be written is a quotient of natural numbers', textLength: '839', alias: 'sqrt_2_is_irrational', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'NateSoares', editCreatedAt: '2016-07-06 20:40:58', pageCreatorId: 'DylanHendrickson', pageCreatedAt: '2016-07-05 20:34:33', seeDomainId: '0', editDomainId: 'AlexeiAndreev', 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: '24', text: '$\\sqrt 2$, the unique [-positive] [-4bc] whose square is 2, is not a [-4zq].\n\n#Proof\n\nSuppose $\\sqrt 2$ is rational. Then $\\sqrt 2=\\frac{a}{b}$ for some integers $a$ and $b$; [-without_loss_of_generality] let $\\frac{a}{b}$ be in [-lowest_terms], i.e. $\\gcd(a,b)=1$. We have\n\n$$\\sqrt 2=\\frac{a}{b}$$\n\nFrom the definition of $\\sqrt 2$,\n\n$$2=\\frac{a^2}{b^2}$$\n$$2b^2=a^2$$\n\nSo $a^2$ is a multiple of $2$. Since $2$ is [4mf prime], $a$ must be a multiple of 2; let $a=2k$. Then\n\n$$2b^2=(2k)^2=4k^2$$\n$$b^2=2k^2$$\n\nSo $b^2$ is a multiple of $2$, and so is $b$. But then $2|\\gcd(a,b)$, which contradicts the assumption that $\\frac{a}{b}$ is in lowest terms! So there isn't any way to express $\\sqrt 2$ as a fraction in lowest terms, and thus there isn't a way to express $\\sqrt 2$ as a ratio of integers at all. That is, $\\sqrt 2$ is irrational.', 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: [ 'DylanHendrickson', 'NateSoares' ], childIds: [], parentIds: [ 'irrational_number' ], commentIds: [], questionIds: [], tagIds: [ 'proof_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], 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: '15773', pageId: 'sqrt_2_is_irrational', userId: 'NateSoares', edit: '2', type: 'newEdit', createdAt: '2016-07-06 20:40:58', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15612', pageId: 'sqrt_2_is_irrational', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-06 06:48:47', auxPageId: 'irrational_number', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15610', pageId: 'sqrt_2_is_irrational', userId: 'EricBruylant', edit: '0', type: 'deleteParent', createdAt: '2016-07-06 06:48:40', auxPageId: 'rational_number', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15460', pageId: 'sqrt_2_is_irrational', userId: 'EricBruylant', edit: '0', type: 'deleteParent', createdAt: '2016-07-05 21:54:43', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15458', pageId: 'sqrt_2_is_irrational', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-05 21:53:49', auxPageId: 'rational_number', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15449', pageId: 'sqrt_2_is_irrational', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-07-05 21:46:47', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15447', pageId: 'sqrt_2_is_irrational', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-05 21:46:37', auxPageId: 'proof_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15430', pageId: 'sqrt_2_is_irrational', userId: 'DylanHendrickson', edit: '1', type: 'newEdit', createdAt: '2016-07-05 20:34:33', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }