{ localUrl: '../page/transcendental_number.html', arbitalUrl: 'https://arbital.com/p/transcendental_number', rawJsonUrl: '../raw/5wx.json', likeableId: '3405', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'PatrickStaples' ], pageId: 'transcendental_number', edit: '4', editSummary: '', prevEdit: '3', currentEdit: '4', wasPublished: 'true', type: 'wiki', title: 'Transcendental number', clickbait: 'A transcendental number is one which is not the root of any integer-coefficient polynomial.', textLength: '2649', alias: 'transcendental_number', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-08-20 21:59:36', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-08-17 17:58:55', 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: '48', text: '[summary(Technical): A [4bc real] or [4zw complex] number $z$ is said to be *transcendental* if there is no (nonzero) [-48l]-coefficient [-polynomial] which has $z$ as a [root_of_polynomial root].]\n\n[summary: A *transcendental* number $z$ is one such that there is no (nonzero) [-polynomial] function which outputs $0$ when given $z$ as input. $\\frac{1}{2}$, $\\sqrt{6}$, $i$ and $e^{i \\pi/2}$ are not transcendental; $\\pi$ and $e$ are both transcendental.]\n\nA [4bc real] or [4zw complex] number is said to be *transcendental* if it is not the root of any (nonzero) [-48l]-coefficient [-polynomial].\n("Transcendental" means "not [algebraic_number algebraic]".)\n\n# Examples and non-examples\n\nMany of the most interesting numbers are *not* transcendental.\n\n- Every integer is *not* transcendental (i.e. is algebraic): the integer $n$ is the root of the integer-coefficient polynomial $x-n$.\n- Every [4zq rational] is algebraic: the rational $\\frac{p}{q}$ is the root of the integer-coefficient polynomial $qx - p$.\n- $\\sqrt{2}$ is algebraic: it is a root of $x^2-2$.\n- $i$ is algebraic: it is a root of $x^2+1$.\n- $e^{i \\pi/2}$ (or $\\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{2}}{2}i$) is algebraic: it is a root of $x^4+1$.\n\nHowever, $\\pi$ and $e$ are both transcendental. (Both of these are difficult to prove.)\n\n# Proof that there is a transcendental number\n\nThere is a very sneaky proof that there is some transcendental real number, though this proof doesn't give us an example.\nIn fact, the proof will tell us that "[almost_every almost all]" real numbers are transcendental.\n(The same proof can be used to demonstrate the existence of [54z irrational numbers].)\n\nIt is a fairly easy fact that the *non*-transcendental numbers (that is, the algebraic numbers) form a [-countable] subset of the real numbers.\nIndeed, the [fundamental_theorem_of_algebra Fundamental Theorem of Algebra] states that every polynomial of degree $n$ has exactly $n$ complex roots (if we count them with [multiplicity multiplicity], so that $x^2+2x+1$ has the "two" roots $x=-1$ and $x=-1$).\nThere are only countably many integer-coefficient polynomials [todo: spell out why], and each has only finitely many complex roots (and therefore only finitely many—possibly $0$—*real* roots), so there can only be countably many numbers which are roots of *any* integer-coefficient polynomial.\n\nBut there are uncountably many reals ([reals_are_uncountable proof]), so there must be some real (indeed, uncountably many!) which is not algebraic.\nThat is, there are uncountably many transcendental numbers.\n\n# Explicit construction of a transcendental number\n[todo: Liouville's constant]', 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: [ 'PatrickStevens', 'ChrisBarnett', 'JoeZeng' ], childIds: [], parentIds: [ 'number' ], 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: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19052', pageId: 'transcendental_number', userId: 'PatrickStevens', edit: '4', type: 'newEdit', createdAt: '2016-08-20 21:59:36', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19043', pageId: 'transcendental_number', userId: 'JoeZeng', edit: '3', type: 'newEdit', createdAt: '2016-08-20 18:38:49', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '3407', likeableType: 'changeLog', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [], id: '18820', pageId: 'transcendental_number', userId: 'ChrisBarnett', edit: '2', type: 'newEdit', createdAt: '2016-08-18 17:26:12', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18793', pageId: 'transcendental_number', userId: 'EricBruylant', edit: '0', type: 'newParent', createdAt: '2016-08-17 22:00:25', auxPageId: 'number', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18791', pageId: 'transcendental_number', userId: 'EricBruylant', edit: '0', type: 'deleteParent', createdAt: '2016-08-17 22:00:05', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18784', pageId: 'transcendental_number', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-08-17 17:58:57', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18782', pageId: 'transcendental_number', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-08-17 17:58:55', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }