Real numbers

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385 nodes use this node, directly or through other nodes.

• Closure is the set of limits of sequences 003QP7A8HG
• Variance 007QED3E0Y
• Finite intersection property 00BWV5CKCF
• Modulus of the conjugate 00KRCVE7CK
• Compact set 00R0X1WGPX
• Absolute value 00S7C68D51
• Chernoff bound 014PEBGDYF
• Residue at a simple pole 01991NCP7H
• Derivative of the complex exponential 019H1Z6419
• Chapman–Kolmogorov equation 01APZ8GDQH
• Bias of an estimator 01BD2TFFP4
• Totally disconnected space 01CYW8S04D
• Urysohn's lemma 01EY3VR7ZD
• Triangle inequality for a metric 01M6CW9VM6
• Cauchy's integral theorem 01MPZXYVYF
• Isolated point 01PKQYTTMT
• Cantor set is closed 01VT24J1SD
• Arguments add under multiplication 01W6RKJP01
• Identically distributed random variables 01X8YSDTBW
• Open set 01ZQQDCERP
• Law of the unconscious statistician 01ZWX875AT
• Density of the normal distribution 027JF1FBB0
• Exponential distribution 028Z8WC5Y9
• Holomorphic functions on a disc have primitives 02AC7JR6G6
• Covariance matrix 02CBHV4ZZQ
• Open balls form a basis of the metric topology 02EEX96DFS
• Casorati–Weierstrass theorem 02HHNPC8ED
• Chebyshev's inequality 02HZ1BK3MW
• Isometries are injective 02J4G4J4TJ
• Characteristic function of a sum of independent random variables 02JGYG0T7M
• Stationary distribution 02KYGSCTCN
• Closure is closed 02TZDKG50Q
• Riemann mapping theorem 02WHMEG5ZR
• Bernoulli process 02XPXR1FBK
• The real part of a holomorphic function is harmonic 031BG625Y3
• Residue theorem 032HKYS329
• Markov chain 033TJBZDG0
• Formula for the modulus 034FGCEYX4
• Statistic 036756NFD6
• Arzelà–Ascoli theorem 0381S8Z5P0
• Law of total variance 03BY5XTNA4
• Variance of the normal distribution 03EESZDB9A
• Triangle inequality 03HF5PMY5T
• Equivalence of norms in finite dimension 03MC70ZRNR
• Torus 03NM7DZXXC
• Cauchy–Schwarz inequality for expectations 03PGHKAJG3
• Open mapping theorem 03YEM7JEDN
• Lebesgue number lemma 040CFNJEAP
• Entire function 0419794R0T
• Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
• Uniform limits of holomorphic functions are holomorphic 0431RN0R1N
• Product metric 044D4AQV8V
• Every real number has a decimal expansion 047KEVEZ9Q
• Maximum modulus principle 048F2P90WR
• Correlation coefficient 04CTY6XYWN
• Variance under translation 04CYN5G6RY
• Density of the irrationals 04J6G4H78J
• Almost sure convergence 04P7D63W8P
• Law of total expectation 04RRB6HM00
• Probability density function 04YG690N5V
• Compact iff complete and totally bounded 054Q5PW5GX
• The nth roots of unity 057BH3JC9S
• Expectation of a product of independent random variables 05AW1G0WE1
• Closed ball 05DSV40EB9
• Likelihood function 05FSTXM92W
• T1 space 05GQJXBP3F
• Homeomorphism 05GZ11PWRY
• Composition of continuous maps is continuous 05NK26G0DR
• Markov's inequality 05PWWHYBX5
• Standard score 05RZ4YZG4E
• Poisson point process 05TG0SE4HT
• Gamma distribution 06395GBNWD
• Metric space 063W5JMY85
• Order of a zero 064N002TX4
• Inclusion of a subspace is continuous 065J822CKN
• Repeating decimals are rational 065S5B9HF1
• De Moivre–Laplace theorem 065X9S8CM3
• Closed subsets of compact sets are compact 066HW6D9A7
• Existence of square roots of nonnegative reals 068BZBXE0C
• Closure in a metric space 068P0P32DW
• Diameter of a set 0692CGQF00
• Heine–Cantor theorem 06B8BNJCD6
• Branch cut 06JYC06JHY
• Convergence of a sequence in a metric space 06JYPJ4WHH
• Existence of Laurent expansions on an annulus 06WHRARZKY
• Decimal expansion 06XY4X9430
• Kolmogorov's zero–one law 073Q8Z94KG
• Independent random variables are uncorrelated 077GN9HDR6
• Convolution of probability distributions 07CHJ5YX28
• Lipschitz maps are uniformly continuous 07DCKDXRBP
• Continuity at a point of a topological space 07N7W5C29X
• Heine–Borel theorem 07PFYV65MB
• Normal distribution 07RXBVTVZB
• Distance from a point to a set 07TXYYVYZR
• Irrational number 080T2TXVQV
• Holomorphic functions are analytic 0826SGV3R0
• Absolutely convergent series in a Banach space converge 082MGNWNA8
• Mean of the Poisson distribution 086HEVXR9Z
• Isometry 0878DN3WCS
• Statistical hypothesis test 087P03NEND
• Complete metric space 08EWT7J0SC
• Sequence spaces lp 08P6T2WH6R
• Picard's great theorem 08QA8QTM71
• Independence means the joint law is the product of the marginals 08QCA2F1NX
• Topological embedding 08SVRVHJNY
• Median 08W2CW8PKW
• Riesz's lemma 097C7KCAT5
• Domain in the complex plane 097KNJFJ7V
• Intermediate value theorem for connected spaces 098C0KZZAP
• Power series of the complex logarithm 09BG0Y7SZD
• Open sets characterised by neighbourhoods 09CJ00H3NW
• Closed sets contain their limits 09GQ3ZTQJ9
• Completeness of the space of bounded continuous functions 09GRCZVR5E
• Method of moments 09J1H07VAH
• Metrizable space 09KDHB82SG
• Finite intersection of open sets is open 09NDE9V7X1
• Mean of the normal distribution 09PR1V6RWR
• Intersection of closed sets is closed 09YMV8H4X5
• Product of compact sets is compact 09ZKE5XQ74
• Open sets of a metric space 0A26X79RTB
• The reals are Archimedean 0A3GWZ86AY
• Metric topology 0A6YRNZB2R
• Baire category theorem 0A7KGAYHGV
• Finite-dimensional normed spaces are complete 0ABP2F2ZP7
• Metrizable spaces are normal 0AD0BZ49DW
• Annulus 0AJ0YVK743
• Normal space 0AJ646N339
• Independent and identically distributed random variables 0AJN7WHBJZ
• Ceiling function 0ASM4EPMB2
• Marginal distribution 0AXP2Z5N9H
• Completeness axiom 0AXWET80YW
• Moment-generating function of the normal distribution 0B1G4E21G7
• Open balls are open 0B2305GNAZ
• Quantile function 0B3Q3XDM5X
• Continuity via closed sets 0B8GPXFSP7
• Connected subsets of the real line are intervals 0B9ZQFMQNV
• Banach space 0BDTTM24DP
• Weak law of large numbers 0BFDHF2H7N
• Goursat's theorem 0BRG85WXYC
• Polar form 0BRXFNQ5JV
• Riemann's theorem on removable singularities 0BSQDZWF34
• Borel sigma-algebra 0BVHZXWKCD
• Random walk 0C08YMV95H
• Linear maps on finite-dimensional spaces are continuous 0C13E623WD
• Random variable with a density 0C21RZG5V2
• Interior 0C38S0KMJK
• Uncorrelated random variables that are not independent 0C6J8BQXVR
• Reverse triangle inequality 0C9S13WKMF
• Kurtosis 0CBQVVNBMP
• Harmonic conjugate 0CC56MJ4FG
• Martingale 0CDVK9RQBH
• Zero of a holomorphic function 0CEGEASY49
• exp(2 pi i/n) is a primitive root of unity 0CF0XGYHNJ
• Dense set 0D198GPMTJ
• Complex differentiability 0D1C9Y1EXA
• A set is closed iff it equals its closure 0D2W7FS52T
• Rouché's theorem 0D430QR02W
• Baire space 0D5KR33H7X
• Variance of a sum 0D86GJEZ39
• Limits are unique in Hausdorff spaces 0D8T984XEE
• Connected space 0D939CMME7
• Mean of the geometric distribution 0DB0R4N5XX
• Open ball 0DC3MPR7C9
• Expected value 0DCT0QZVQ1
• Strong law of large numbers 0DFWHQ839T
• Holomorphic function 0DG0Z72CQD
• Cauchy's integral formula 0DG10TVYPN
• Covariance 0DJ8W3HP2D
• Random variable 0DKRR22AEQ
• Closure 0DRKT1QJR7
• Irrationality of the square root of 2 0DZGSDHD92
• Number of nth roots of unity 0E021GF2B8
• Independence of two random variables 0E9502NT92
• Existence of the infimum 0EDH403M21
• Neyman–Pearson lemma 0EE19CFEEF
• Convergence in probability 0EFDDXTVD6
• Liouville's theorem 0EHKVRF4J6
• P-value 0EHYZFZ4PK
• Epsilon-delta continuity in metric spaces 0EKGKY5TGH
• Chi-squared test 0EN0V3NY20
• Continuous image of a connected set is connected 0EPD87YVG8
• Convergence in probability implies convergence in distribution 0EPTDZPJ83
• Central limit theorem 0EPZC4Z1FC
• Locally compact space 0EQMWQ4WMJ
• 68–95–99.7 rule 0EYQ03XCR3
• Cantor's intersection theorem 0EZBG8SKJM
• Cauchy's integral formula for derivatives 0F0H5P5DED
• Transition matrix 0F1KBVY161
• Conditional variance 0F1SN93KXE
• Bounded linear maps are continuous 0F3K4C7XYW
• Closed subsets of complete spaces are complete 0FBXJZX3MK
• Universal property of the quotient topology 0FF2FJ8CH2
• The reals form a field 0FGENPMPJX
• Epsilon-delta uniform continuity 0FHSDCQACY
• Lévy's continuity theorem 0FPWF7TTMB
• Convergence of random variables 0FQ968YKA1
• Meagre set 0FQPYAY9PJ
• Inverse of a complex number 0FQRAV5SKH
• Maximum likelihood estimation 0FREPFKNS7
• The complex numbers form a field 0FRSR10DWF
• The complex exponential is entire 0G2YJN0H5S
• Completion of a metric space 0G4J7Z7WPJ
• Multivariate normal distribution 0G558GJQMW
• Closed set 0G5AV9Q4RR
• Separable space 0GKJ78948V
• Absolute value is multiplicative 0GMV7GP8W0
• Skewness 0GQ3H7VW3V
• Pseudometric space 0H3W614GEC
• Galton–Watson process 0H631CZP5K
• Interior is the largest open subset 0HAZEFH137
• Markov property 0HB8SXHPFA
• Limit point 0HBQC2GJXT
• Union of open sets is open 0HEF46RPEK
• First-countable space 0HG11FW8MJ
• Modulus is multiplicative 0HYNZE940T
• Student's t-test 0HYP606RYF
• Mean value property 0HYPR64JM2
• Equivalent norms 0J3578562S
• Discrete random variable 0J6Z6H5VQJ
• Bernoulli distribution 0J97P5KZY0
• Convergence in distribution 0JDAQHWA9N
• The cumulative distribution function determines the distribution 0JEVZAFDY6
• Regular space 0JF9KW2TXB
• Residue 0JFGKKWWFK
• Boundary 0JJ9J1XZ8G
• Cauchy's estimates 0JJRENWMP6
• Sample variance 0JN9P4A944
• Fundamental theorem of algebra 0JP0W07Y9A
• Tychonoff's theorem 0JTRSJKCFB
• Bayesian inference 0JWRJX2RMP
• Path-connected space 0K0GE6BCYW
• Topologist's sine curve 0K3T4A659R
• Confidence interval 0K4NMMJBG2
• Complement of an open set is closed 0K67Q60Z8P
• Isolated zeros theorem 0K83KEJ43P
• Compactness and sequential compactness agree in metric spaces 0K8SGQSV7G
• Continuous image of a compact set is compact 0K9AAJ8G51
• p-norm on R^n 0KNJWEVSMG
• Nowhere dense set 0KSM23SN3H
• Variance under scaling 0KVQ9D86BM
• Modulus of a complex number 0KW2X0TFBX
• Discrete uniform distribution 0KYP951QWH
• Quadratic formula 0KYZXYDGQZ
• Universal property of the product topology 0M6W4XNST6
• Approximation property of the supremum 0M8TGGP9JP
• Schwarz lemma 0MB48XDBY3
• Epsilon characterisation of Cauchy sequences 0MCK8K2D7Z
• One-point compactification 0MM45BT6HC
• Compact Hausdorff spaces are normal 0MQQQ5NA2S
• Contraction mapping 0MT2S0ZX4Y
• Equicontinuity 0MW2WXTHMC
• Hausdorff space 0MXB4CKHFW
• Coupon collector's problem 0MXCP7MXWJ
• Cumulative distribution function 0MXMS3AE13
• Open connected subsets of R^n are path-connected 0MXN7AFBPE
• Characteristic function 0MYDNRT7PP
• Compact subsets of a Hausdorff space are closed 0MYFJVMN7M
• Completeness via Cauchy sequences 0N1ZWFJDN5
• Cauchy distribution 0NA6Q82YR7
• exp(1/z) has an essential singularity at 0 0NB31CC4SN
• Principal part 0NDV5R6Z6X
• Variance as the mean of the square minus the square of the mean 0NMF95YK8W
• Epsilon-net characterisation of total boundedness 0NMGJJ19HD
• Moment-generating function 0NNRNGKMFM
• Student's t-distribution 0NQ5798SB1
• Connected components are closed 0NRT0E2QT2
• Characteristic function of the normal distribution 0NS0TDA4E2
• Meromorphic function 0NSD1TPHR0
• Open cover 0NVPNC12JY
• Sum of independent normal random variables 0NXGPJWNFN
• Closed map 0NYZ2M45G9
• Removable singularity 0P18JC72RE
• Compactness via open covers 0P6PMJB4MA
• Linearity of expectation 0P7PTC8DDE
• Bounded set 0P953MG05E
• Metric spaces are Hausdorff 0P9DCDES73
• Sup norm on bounded continuous functions 0PARCKE8WP
• Moment 0PH0VHTQS0
• Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
• Projections are continuous 0PM88Z9W4M
• Argument principle 0PN2BXRS8G
• Interior is open 0PNWCM559P
• Beta distribution 0PP5ZAJPP7
• Expectation with respect to the binomial distribution 0PVCTCKD3C
• Cumulant generating function 0Q2QC81VTW
• Holomorphic functions with nonzero derivative are conformal 0Q7M6K6VD7
• Joint probability distribution 0QAR0XDZH4
• Probability distribution 0QB913WM5E
• Holomorphic functions are infinitely differentiable 0QEBN8DGVE
• Stochastic process 0QH5ENKZM1
• Sample mean 0QNRBHWV35
• Compact space 0QPHW703PV
• The reals form an ordered field 0QSPNASE50
• Connected set 0R37DFJKFW
• T0 space 0R3B7DX55J
• Type I and type II errors 0R41PVFA39
• Compact subsets of a metric space are bounded 0R7BTEWCJ2
• Characterization of the floor 0R9B0B5458
• Generating function of a sum of independent random variables 0REMT01N36
• Chi-squared distribution 0RFMMBHFTV
• Limit of the cumulative distribution function at infinity 0RJ5TZH6G0
• Cantor set is homeomorphic to the space of binary sequences 0RQ8DHHR8V
• Continuous linear map 0S3SMRH0Z5
• Path component 0SB9FNZECG
• Closure is the smallest closed superset 0SHC6Z304Z
• Hypergeometric distribution 0SPBPW3K61
• Branch of the logarithm 0SQ1F2BEDS
• Open map 0SQ2PC596Y
• Principal value of the argument 0SQSQP748R
• Hurwitz's theorem 0SSKBQN58M
• Moments are derivatives of the moment-generating function at zero 0STK8KEY9G
• Density of the rationals 0STX0WRDNG
• Almost sure convergence implies convergence in probability 0SVN6X695K
• Continuous map of topological spaces 0SY9ECQ19Q
• Subspace metric 0T2WYDTRP2
• Cauchy–Riemann equations 0T71SSYZ39
• The characteristic function determines the distribution 0T8BYAFEXX
• Lipschitz map 0TMZ21WKCN
• Law of a random variable 0TST4MHA3A
• Convergence in mean 0TTA57SEJG
• Estimator 0TTNRP5PFQ
• Equivalent metrics 0TWQ5PRKWS
• Isolated singularity 0TYB3HM9KE
• Variance of a sum of independent random variables 0V0778DCW9
• Cauchy principal value 0V3RDQV3ZX
• Continuous real function on a compact set attains its bounds 0V4DET59NW
• Mean squared error 0V4VSCBXP3
• Square root 0VB8BFRABH
• Morera's theorem 0VDD7W0EKZ
• Poisson distribution 0VEFSSKX18
• Continuity is equivalent to sequential continuity in metric spaces 0VGPKZ8MRK
• Banach fixed point theorem 0VK6VEH345
• Isometric isomorphism 0VRD8GC5BM
• Binomial distribution 0VRWZDWW6B
• Standard deviation 0VTN5RKNC5
• Compact closed unit ball implies finite dimension 0VZVV8VME5
• Complete subspaces are closed 0W41XG4W60
• Totally bounded set 0WE5BBHGM6
• Probability mass function 0WF4EJP91W
• Gambler's ruin 0WFMKK7DFZ
• Continuous uniform distribution 0WGDPYR30Q
• Circle as a quotient of the real line 0WH1GVBJNH
• There is a natural number n with 1/n below any positive epsilon 0WH916TJNY
• Hoeffding's inequality 0WPDNXGSSN
• Pole 0WPQYW8MTE
• Moment-generating function of a sum of independent random variables 0WRZ170YEH
• Hölder continuity 0WZTM0EFR6
• Complex logarithm 0X2FY491PF
• Countable intersection of dense open sets is dense 0X2T63GFQQ
• Tietze extension theorem 0X52TNMTCJ
• Independence of random variables 0XCE070V8B
• Geometric distribution 0XCVV5Q500
• Monotonicity of the cumulative distribution function 0XD22PBMWA
• Consistent estimator 0XJVF81QF6
• Poisson limit theorem 0XKTZRWTXX
• Negative binomial distribution 0XN7KEVSSS
• The principal logarithm is holomorphic off the branch cut 0XVGJE7Q04
• A root of an integer is an integer or irrational 0XWCKJZRYH
• Tail sum formula for expectation 0Y6QNF1J49
• Conditional expectation 0YAMJRK04E
• Absorbing Markov chain 0YB93GWTKK
• Identity theorem 0YBVKMYX0W
• Continuity via open sets 0YF82GBTND
• Sum of independent Poisson random variables 0YH1M56SBN
• Irrationality of the square root of a prime 0YHFMXJJAZ
• Order of a pole 0YK27TJ4WF
• Essential singularity 0YKDJBV26S
• Principal branch 0YKY4M3KRQ
• Existence of nth roots of complex numbers 0YM6P6C206
• Power series are holomorphic in the disc of convergence 0YMP521ZR5
• Fundamental theorem of contour integration 0YQ458D7PF
• Closed balls are closed 0YS40B38JD
• Conditional probability distribution 0YVDC5PYDW
• Floor function 0YVRNEDSVG
• Proper metric space 0Z4V98VN5E
• The complex conjugate is not holomorphic 0Z8ATBPM5W
• Probability-generating function 0Z8PSSAVBA
• Memorylessness 0ZAZZ34WQS
• A complex number times its conjugate 0ZB7J071WQ
• Path-connected spaces are connected 0ZMT1GP4KM
• Central moment 0ZT24EC90K
• Operator norm 0ZTEH055YF
• Complex power 0ZV9A7EV0F
• Law of large numbers 0ZVEAYVRC3
• Argument of a complex number 0ZZRSX91QR