{ localUrl: '../page/rationals_are_a_field.html', arbitalUrl: 'https://arbital.com/p/rationals_are_a_field', rawJsonUrl: '../raw/4zr.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'rationals_are_a_field', edit: '4', editSummary: '', prevEdit: '3', currentEdit: '4', wasPublished: 'true', type: 'wiki', title: 'The rationals form a field', clickbait: '', textLength: '2611', alias: 'rationals_are_a_field', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JoeZeng', editCreatedAt: '2016-07-06 18:28:22', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-07-01 16:15:57', 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: '23', text: 'The set $\\mathbb{Q}$ of [4zq rational numbers] is a [481 field].\n\n# Proof\n\n$\\mathbb{Q}$ is a ([3jb commutative]) [3gq ring] with additive identity $\\frac{0}{1}$ (which we will write as $0$ for short) and multiplicative identity $\\frac{1}{1}$ (which we will write as $1$ for short): we check the axioms individually.\n\n- $+$ is commutative: $\\frac{a}{b} + \\frac{c}{d} = \\frac{ad+bc}{bd}$, which by commutativity of addition and multiplication in $\\mathbb{Z}$ is $\\frac{cb+da}{db} = \\frac{c}{d} + \\frac{a}{b}$\n- $0$ is an identity for $+$: have $\\frac{a}{b}+0 = \\frac{a}{b} + \\frac{0}{1} = \\frac{a \\times 1 + 0 \\times b}{b \\times 1}$, which is $\\frac{a}{b}$ because $1$ is a multiplicative identity in $\\mathbb{Z}$ and $0 \\times n = 0$ for every integer $n$.\n- Every rational has an additive inverse: $\\frac{a}{b}$ has additive inverse $\\frac{-a}{b}$.\n- $+$ is [3h4 associative]: $$\\left(\\frac{a_1}{b_1}+\\frac{a_2}{b_2}\\right)+\\frac{a_3}{b_3} = \\frac{a_1 b_2 + b_1 a_2}{b_1 b_2} + \\frac{a_3}{b_3} = \\frac{a_1 b_2 b_3 + b_1 a_2 b_3 + a_3 b_1 b_2}{b_1 b_2 b_3}$$\nwhich we can easily check is equal to $\\frac{a_1}{b_1}+\\left(\\frac{a_2}{b_2}+\\frac{a_3}{b_3}\\right)$. [todo: actually do this]\n- $\\times$ is associative, trivially: $$\\left(\\frac{a_1}{b_1} \\frac{a_2}{b_2}\\right) \\frac{a_3}{b_3} = \\frac{a_1 a_2}{b_1 b_2} \\frac{a_3}{b_3} = \\frac{a_1 a_2 a_3}{b_1 b_2 b_3} = \\frac{a_1}{b_1} \\left(\\frac{a_2 a_3}{b_2 b_3}\\right) = \\frac{a_1}{b_1} \\left(\\frac{a_2}{b_2} \\frac{a_3}{b_3}\\right)$$\n- $\\times$ is commutative, again trivially: $$\\frac{a}{b} \\frac{c}{d} = \\frac{ac}{bd} = \\frac{ca}{db} = \\frac{c}{d} \\frac{a}{b}$$\n- $1$ is an identity for $\\times$: $$\\frac{a}{b} \\times 1 = \\frac{a}{b} \\times \\frac{1}{1} = \\frac{a \\times 1}{b \\times 1} = \\frac{a}{b}$$ by the fact that $1$ is an identity for $\\times$ in $\\mathbb{Z}$.\n- $+$ distributes over $\\times$: $$\\frac{a}{b} \\left(\\frac{x_1}{y_1}+\\frac{x_2}{y_2}\\right) = \\frac{a}{b} \\frac{x_1 y_2 + x_2 y_1}{y_1 y_2} = \\frac{a \\left(x_1 y_2 + x_2 y_1\\right)}{b y_1 y_2}$$\nwhile $$\\frac{a}{b} \\frac{x_1}{y_1} + \\frac{a}{b} \\frac{x_2}{y_2} = \\frac{a x_1}{b y_1} + \\frac{a x_2}{b y_2} = \\frac{a x_1 b y_2 + b y_1 a x_2}{b^2 y_1 y_2} = \\frac{a x_1 y_2 + a y_1 x_2}{b y_1 y_2}$$\nso we are done by distributivity of $+$ over $\\times$ in $\\mathbb{Z}$.\n\nSo far we have shown that $\\mathbb{Q}$ is a ring; to show that it is a field, we need all nonzero fractions to have inverses under multiplication.\nBut if $\\frac{a}{b}$ is not $0$ (equivalently, $a \\not = 0$), then $\\frac{a}{b}$ has inverse $\\frac{b}{a}$, which does indeed exist since $a \\not = 0$.\n\nThis completes the proof.', 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', 'JoeZeng' ], childIds: [], parentIds: [ 'rational_number' ], commentIds: [], questionIds: [], tagIds: [ 'proof_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [ { id: '4664', parentId: 'algebraic_field', childId: 'rationals_are_a_field', type: 'requirement', creatorId: 'PatrickStevens', createdAt: '2016-07-01 15:57:21', 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: '2965', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '15731', pageId: 'rationals_are_a_field', userId: 'JoeZeng', edit: '4', type: 'newEdit', createdAt: '2016-07-06 18:28:22', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15137', pageId: 'rationals_are_a_field', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-07-02 23:33:15', auxPageId: 'proof_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15065', pageId: 'rationals_are_a_field', userId: 'PatrickStevens', edit: '3', type: 'newEdit', createdAt: '2016-07-01 16:17:12', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15064', pageId: 'rationals_are_a_field', userId: 'PatrickStevens', edit: '2', type: 'newEdit', createdAt: '2016-07-01 16:16:19', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15063', pageId: 'rationals_are_a_field', userId: 'PatrickStevens', edit: '0', type: 'newRequirement', createdAt: '2016-07-01 16:15:59', auxPageId: 'algebraic_field', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15062', pageId: 'rationals_are_a_field', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-07-01 16:15:58', auxPageId: 'rational_number', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15060', pageId: 'rationals_are_a_field', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-07-01 16:15:57', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }