Real numbers
Description
The real numbers are the numbers that measure continuous quantities, such as lengths. We picture them as the points of a line. We write for the set of real numbers. With addition, multiplication and its order, is an ordered field. This ordered field is complete: each nonempty subset of that is bounded above has a least upper bound. Up to a unique isomorphism, is the only complete ordered field.
Remarks
The set contains the rational numbers and also irrational numbers such as and . Two common constructions of from use Dedekind cuts or Cauchy sequences. The set is uncountable.
Where this leads
920 pages build on this one.
- Closure is the set of limits of sequences
- Variance
- Finite intersection property
- Modulus of the conjugate
- Compact set
- Absolute value
- Chernoff bound
- Residue at a simple pole
- Derivative of the complex exponential
- Chapman–Kolmogorov equation
- Bias of an estimator
- Totally disconnected space
- Urysohn's lemma
- Triangle inequality for a metric
- Cauchy's integral theorem
- Isolated point
- Cantor set is closed
- Arguments add under multiplication
- Identically distributed random variables
- Open set
- Law of the unconscious statistician
- Density of the normal distribution
- Exponential distribution
- Holomorphic functions on a disc have primitives
- Covariance matrix
- Open balls form a basis of the metric topology
- Casorati–Weierstrass theorem
- Chebyshev's inequality
- Isometries are injective
- Characteristic function of a sum of independent random variables
- Stationary distribution
- Closure is closed
- Riemann mapping theorem
- Bernoulli process
- The real part of a holomorphic function is harmonic
- Residue theorem
- Markov chain
- Formula for the modulus
- Statistic
- Arzelà–Ascoli theorem
- Law of total variance
- Variance of the normal distribution
- Triangle inequality
- Equivalence of norms in finite dimension
- Torus
- Cauchy–Schwarz inequality for expectations
- Open mapping theorem
- Lebesgue number lemma
- Entire function
- Continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Uniform limits of holomorphic functions are holomorphic
- Product metric
- Every real number has a decimal expansion
- Maximum modulus principle
- Correlation coefficient
- Variance under translation
- Density of the irrationals
- Almost sure convergence
- Law of total expectation
- Probability density function
- Compact iff complete and totally bounded
- The nth roots of unity
- Expectation of a product of independent random variables
- Closed ball
- Likelihood function
- T1 space
- Homeomorphism
- Composition of continuous maps is continuous
- Markov's inequality
- Standard score
- Poisson point process
- Gamma distribution
- Metric space
- Order of a zero
- Inclusion of a subspace is continuous
- Repeating decimals are rational
- De Moivre–Laplace theorem
- Closed subsets of compact sets are compact
- Existence of square roots of nonnegative reals
- Closure in a metric space
- Diameter of a set
- Heine–Cantor theorem
- Branch cut
- Convergence of a sequence in a metric space
- Existence of Laurent expansions on an annulus
- Decimal expansion
- Kolmogorov's zero–one law
- Independent random variables are uncorrelated
- Convolution of probability distributions
- Lipschitz maps are uniformly continuous
- Continuity at a point of a topological space
- Heine–Borel theorem
- Normal distribution
- Distance from a point to a set
- Irrational number
- Holomorphic functions are analytic
- Absolutely convergent series in a Banach space converge
- Mean of the Poisson distribution
- Isometry
- Statistical hypothesis test
- Complete metric space
- Sequence spaces lp
- Picard's great theorem
- Independence means the joint law is the product of the marginals
- Topological embedding
- Median
- Riesz's lemma
- Domain in the complex plane
- Intermediate value theorem for connected spaces
- Power series of the complex logarithm
- Open sets characterised by neighbourhoods
- Closed sets contain their limits
- Completeness of the space of bounded continuous functions
- Method of moments
- Metrizable space
- Finite intersection of open sets is open
- Mean of the normal distribution
- Intersection of closed sets is closed
- Product of compact sets is compact
- Open sets of a metric space
- The reals are Archimedean
- Metric topology
- Baire category theorem
- Finite-dimensional normed spaces are complete
- Metrizable spaces are normal
- Annulus
- Normal space
- Independent and identically distributed random variables
- Ceiling function
- Marginal distribution
- Completeness axiom
- Moment-generating function of the normal distribution
- Open balls are open
- Quantile function
- Continuity via closed sets
- Connected subsets of the real line are intervals
- Banach space
- Weak law of large numbers
- Goursat's theorem
- Polar form
- Riemann's theorem on removable singularities
- Borel sigma-algebra
- Random walk
- Linear maps on finite-dimensional spaces are continuous
- Random variable with a density
- Interior
- Uncorrelated random variables that are not independent
- Reverse triangle inequality
- Kurtosis
- Harmonic conjugate
- Martingale
- Zero of a holomorphic function
- exp(2 pi i/n) is a primitive root of unity
- Dense set
- Complex differentiability
- A set is closed iff it equals its closure
- Rouché's theorem
- Baire space
- Variance of a sum
- Limits are unique in Hausdorff spaces
- Connected space
- Mean of the geometric distribution
- Open ball
- Expected value
- Strong law of large numbers
- Holomorphic function
- Cauchy's integral formula
- Covariance
- Random variable
- Closure
- Irrationality of the square root of 2
- Number of nth roots of unity
- Independence of two random variables
- Existence of the infimum
- Neyman–Pearson lemma
- Convergence in probability
- Liouville's theorem
- P-value
- Epsilon-delta continuity in metric spaces
- Chi-squared test
- Continuous image of a connected set is connected
- Convergence in probability implies convergence in distribution
- Central limit theorem
- Locally compact space
- 68–95–99.7 rule
- Cantor's intersection theorem
- Cauchy's integral formula for derivatives
- Transition matrix
- Conditional variance
- Bounded linear maps are continuous
- Closed subsets of complete spaces are complete
- Universal property of the quotient topology
- The reals form a field
- Epsilon-delta uniform continuity
- Lévy's continuity theorem
- Convergence of random variables
- Meagre set
- Inverse of a complex number
- Maximum likelihood estimation
- The complex numbers form a field
- The complex exponential is entire
- Completion of a metric space
- Multivariate normal distribution
- Closed set
- Separable space
- Absolute value is multiplicative
- Skewness
- Pseudometric space
- Galton–Watson process
- Interior is the largest open subset
- Markov property
- Limit point
- Union of open sets is open
- First-countable space
- Modulus is multiplicative
- Student's t-test
- Mean value property
- Equivalent norms
- Discrete random variable
- Bernoulli distribution
- Convergence in distribution
- The cumulative distribution function determines the distribution
- Regular space
- Residue
- Boundary
- Cauchy's estimates
- Sample variance
- Fundamental theorem of algebra
- Tychonoff's theorem
- Bayesian inference
- Path-connected space
- Topologist's sine curve
- Confidence interval
- Complement of an open set is closed
- Isolated zeros theorem
- Compactness and sequential compactness agree in metric spaces
- Continuous image of a compact set is compact
- Nowhere dense set
- Variance under scaling
- Modulus of a complex number
- Discrete uniform distribution
- Quadratic formula
- Universal property of the product topology
- Approximation property of the supremum
- Schwarz lemma
- Epsilon characterisation of Cauchy sequences
- One-point compactification
- Compact Hausdorff spaces are normal
- Contraction mapping
- Equicontinuity
- Hausdorff space
- Coupon collector's problem
- Cumulative distribution function
- Open connected subsets of R^n are path-connected
- Characteristic function
- Compact subsets of a Hausdorff space are closed
- Completeness via Cauchy sequences
- Cauchy distribution
- exp(1/z) has an essential singularity at 0
- Principal part
- Variance as the mean of the square minus the square of the mean
- Epsilon-net characterisation of total boundedness
- Moment-generating function
- Student's t-distribution
- Connected components are closed
- Characteristic function of the normal distribution
- Meromorphic function
- Open cover
- Sum of independent normal random variables
- Closed map
- Removable singularity
- Compactness via open covers
- Linearity of expectation
- Bounded set
- Metric spaces are Hausdorff
- Sup norm on bounded continuous functions
- Moment
- Heine–Borel theorem in finite-dimensional normed spaces
- Projections are continuous
- Argument principle
- Interior is open
- Beta distribution
- Expectation with respect to the binomial distribution
- Cumulant generating function
- Holomorphic functions with nonzero derivative are conformal
- Joint probability distribution
- Probability distribution
- Holomorphic functions are infinitely differentiable
- Stochastic process
- Sample mean
- Compact space
- The reals form an ordered field
- Connected set
- T0 space
- Type I and type II errors
- Compact subsets of a metric space are bounded
- Characterization of the floor
- Generating function of a sum of independent random variables
- Chi-squared distribution
- Limit of the cumulative distribution function at infinity
- Cantor set is homeomorphic to the space of binary sequences
- Continuous linear map
- Path component
- Closure is the smallest closed superset
- Hypergeometric distribution
- Branch of the logarithm
- Open map
- Principal value of the argument
- Hurwitz's theorem
- Moments are derivatives of the moment-generating function at zero
- Density of the rationals
- Almost sure convergence implies convergence in probability
- Continuous map of topological spaces
- Subspace metric
- Cauchy–Riemann equations
- The characteristic function determines the distribution
- Lipschitz map
- Law of a random variable
- Convergence in mean
- Estimator
- Equivalent metrics
- Isolated singularity
- Variance of a sum of independent random variables
- Cauchy principal value
- Continuous real function on a compact set attains its bounds
- Mean squared error
- Square root
- Morera's theorem
- Poisson distribution
- Continuity is equivalent to sequential continuity in metric spaces
- Banach fixed point theorem
- Isometric isomorphism
- Binomial distribution
- Standard deviation
- Compact closed unit ball implies finite dimension
- Complete subspaces are closed
- Totally bounded set
- Probability mass function
- Gambler's ruin
- Continuous uniform distribution
- Circle as a quotient of the real line
- There is a natural number n with 1/n below any positive epsilon
- Hoeffding's inequality
- Pole
- Moment-generating function of a sum of independent random variables
- Hölder continuity
- Complex logarithm
- Countable intersection of dense open sets is dense
- Tietze extension theorem
- Independence of random variables
- Geometric distribution
- Monotonicity of the cumulative distribution function
- Consistent estimator
- Poisson limit theorem
- Negative binomial distribution
- The principal logarithm is holomorphic off the branch cut
- A root of an integer is an integer or irrational
- Tail sum formula for expectation
- Conditional expectation
- Absorbing Markov chain
- Identity theorem
- Continuity via open sets
- Sum of independent Poisson random variables
- Irrationality of the square root of a prime
- Order of a pole
- Essential singularity
- Principal branch
- Existence of nth roots of complex numbers
- Power series are holomorphic in the disc of convergence
- Fundamental theorem of contour integration
- Closed balls are closed
- Conditional probability distribution
- Floor function
- Proper metric space
- The complex conjugate is not holomorphic
- Probability-generating function
- Memorylessness
- A complex number times its conjugate
- Path-connected spaces are connected
- Central moment
- Operator norm
- Complex power
- Law of large numbers
- Argument of a complex number
- Function with no Laplace transform
- Doob's upcrossing inequality
- Relative CW complex
- Weierstrass factorization theorem
- Riemann domain over C^n
- Univalent function
- Riesz projector
- Translation surface
- Bernstein–von Mises theorem
- Busemann G-space
- Cramér–Rao bound
- Stone–Weierstrass theorem
- Pettis measurability theorem
- Mahler measure
- Efficiency of an estimator
- Log-rank test
- Arakelyan's theorem
- Functional data analysis
- Conformal invariance of harmonic functions
- Poisson regression
- Sequential analysis
- Shell theorem
- Locally compact group
- Ergodic decomposition
- Nonexistence of infinite-dimensional Lebesgue measure
- Köthe–Toeplitz dual
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Martingale representation theorem
- Phragmén–Lindelöf principle for subharmonic functions
- Matrix Wiener–Hopf factorization
- Rectifiable set
- Repair radii and villainy
- Classification of surfaces
- Absorbing set
- Namioka--Phelps
- Mathematical statistics
- F-test
- Nevanlinna characteristic
- Stationary process
- Cauchy–Pompeiu formula
- Weierstrass elliptic function
- Weierstrass zeta and sigma functions
- L2 convergence of Fourier series
- Joukowsky transform
- Cheng–Yau gradient estimate
- Schwarzian derivative
- Removable singularities for harmonic functions
- Cramér–Wold theorem
- Minimum principle for harmonic functions
- Brouwer's characterization of the Cantor space
- Lyusternik's theorem on tangent spaces
- Boolean model
- Convergence of the Neumann series for Volterra kernels
- Transfer principle
- Exponential of a bounded operator
- Tarski–Seidenberg theorem
- Durand–Kerner method
- Hardy's uncertainty principle
- Kolmogorov–Arnold–Moser theorem
- Schwarz–Christoffel mapping
- Stochastic partial differential equation
- Series in Banach spaces
- Jensen's formula
- Teichmüller space
- Minkowski question-mark function
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Doob's martingale inequality
- Seminorm
- Sequential probability ratio test
- Nevanlinna–Pick interpolation
- Céa's lemma
- Stieltjes function
- Modular lambda function
- Brownian bridge
- Consistency of the maximum likelihood estimator
- Monodromy theorem
- Wiener process
- Empirical Bayes method
- Artin–Schreier theorem
- Perron's irreducibility criterion
- Elliptic function
- Karhunen–Loève theorem
- Pettis integral
- Strongly measurable function
- Uniformization theorem
- Brent's method
- Mergelyan's theorem
- Itô isometry
- Matrix Riemann–Hilbert problem
- CW complex
- High indices theorem
- Stein interpolation theorem
- Kobayashi metric
- Hille–Yosida theorem
- Bayes factor
- Pickands–Balkema–De Haan theorem
- Normed field
- Global attractor
- Wirtinger derivatives
- Runge's theorem
- Carlson's theorem
- Isoperimetric problem of the calculus of variations
- Basu's theorem
- Harmonic majorant
- Bump function
- Vanishing mean oscillation
- Spectral decomposition theorem
- Banach bundle
- Variogram
- Robbins–Monro algorithm
- Gamma function as Mellin transform of the exponential
- Contiguity of measures
- Analyticity of the Laplace transform in its half-plane
- Order statistic
- Besicovitch covering theorem
- Balanced set
- Absolute neighborhood retract
- Lindley's paradox
- Splitting circle method
- Tychonoff space
- M-estimator
- Fixed-point iteration
- Ornstein–Uhlenbeck process
- The exponential map from the additive reals to the positive reals
- Identity theorem in several variables
- Kernel regression
- Hilbert's twenty-first problem
- Log-modulus of a holomorphic function is subharmonic
- Bisection method
- Interior algebra
- Adapted process
- Score test
- Kingman's subadditive ergodic theorem
- Multivariate analysis of variance
- Lax–Milgram theorem
- Holomorphic map between Riemann surfaces
- McDiarmid's inequality
- Cauchy-type integral
- Lévy's characterization of Brownian motion
- Asymptotic normality of the maximum likelihood estimator
- Mann–Whitney U test
- Neumann series
- Catenary problem
- Kolmogorov automorphism
- Standard map
- Score
- Non-Archimedean absolute value
- Maximum modulus principle in several variables
- F-distribution
- Schur's inequality
- Δ-complex
- Weierstrass product
- Lebesgue covering dimension
- Edgeworth expansion
- Weyl's lemma
- Zeros and poles
- Continuous functional calculus
- Monotone likelihood ratio
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Constructive analysis
- Bromwich integral
- Nash–Moser theorem
- Poisson–Jensen formula
- Metropolis–Hastings algorithm
- Conditionally complete lattice
- Minimum total potential energy principle
- Birkhoff's axioms
- Continuous-time Markov chain
- Modulus of continuity
- Banach limit
- Kriging
- Baire function
- Branched covering
- Minimal sufficient statistic
- Whittle likelihood
- Logistic regression
- Carathéodory's theorem on boundary extension
- Smoothness and analyticity of harmonic functions
- Green's function and the Riemann map
- Convergence to equilibrium of ergodic Markov chains
- Band-limited function
- Cramér–von Mises criterion
- Azuma's inequality
- Fréchet manifold
- Gaussian free field
- Kernel density estimation
- Hardy–Littlewood maximal inequality
- Arnold's cat map
- Landau's constants
- Implicit function theorem in Banach spaces
- Sub-mean value property
- Erdős–Kac theorem
- Hilbert's sixteenth problem
- Blaschke product
- Borel–Ritt theorem
- Detailed balance
- Coupling from the past
- Robust statistics
- Euler's elastica
- Kakutani's solution of the Dirichlet problem
- Lipschitz domain
- Hölder's theorem on Archimedean groups
- Hidden Markov model
- Finite-dimensional distributions
- Continuous mapping theorem
- Brennan conjecture
- Bias–variance decomposition
- Legendre duplication formula
- Lévy process
- Converse of the mean value property
- Wald's equation
- Random ordinary differential equation
- Local existence of harmonic conjugates
- Topological ring
- Pólya's theorem on integer-valued entire functions
- Suslin's problem
- Post's inversion formula
- Hadamard gap theorem
- Monte Carlo integration
- Time series
- Optional stopping theorem
- Earnshaw's theorem
- Real closed field
- Conformal radius
- Measurable space
- Singular simplicial set
- Breakdown point
- Delta method
- Nehari's theorem on univalence
- Euler's sine product
- Regular surface
- Sokhotski–Plemelj formulae
- Minimum description length
- Completeness of the real numbers
- Noncentral chi-squared distribution
- Hájek–Le Cam convolution theorem
- Standard part function
- Nontangential limit
- Repair radii and villainy
- Hartogs's theorem on separate holomorphicity
- Harmonic measure
- Behrens–Fisher problem
- FK-space
- Anosov flow
- Bochner integral
- Submartingale
- Superharmonic potential
- Hodges–Lehmann estimator
- Kruskal–Wallis test
- Stochastic difference equation
- Borel–Carathéodory theorem
- Dido's problem
- Sampling distribution
- Vitali covering lemma
- Nested radical
- Information geometry
- Cauchy integral formula on a polydisc
- Banach–Tarski paradox
- Recurrence and transience of states
- Approximation of a kernel by degenerate kernels
- Complex p-adic numbers
- Positivstellensatz
- Continuous function with divergent Fourier series
- Malliavin calculus
- Wishart distribution
- Birth–death process
- Hahn embedding theorem
- de Rham curve
- Statistical inference
- Phragmén–Lindelöf principle
- Fredholm map
- Nonstandard analysis
- Minimum-variance unbiased estimator
- Strip of convergence of the bilateral transform
- Sign test
- Bounded sequences ℓ∞
- Sequence space
- Krasner's lemma
- Action-angle coordinates
- Slutsky's theorem
- Tanaka's formula
- Kramers–Kronig relations
- Reduction of a first-kind Volterra equation to the second kind
- Space of convergent sequences
- Second main theorem of Nevanlinna theory
- Sinai billiard
- Liouville–Neumann series
- Geometric Brownian motion
- Empirical likelihood
- Dvoretzky–Kiefer–Wolfowitz inequality
- Bayesian information criterion
- Dvoretzky–Rogers theorem
- Berry–Esseen theorem
- Hilbert metric
- Constructive analysis
- Hardy–Littlewood–Sobolev inequality
- Gabrielov's theorem of the complement
- Nevanlinna theory
- Hunt process
- Cochran's theorem
- Stochastic approximation
- Kronecker's approximation theorem
- Iterated Laplace transform equals Stieltjes transform
- Jackknife
- Quantile regression
- Markov chain mixing time
- Hilbert's seventeenth problem
- Birkhoff's ergodic theorem
- Akaike information criterion
- Schauder fixed-point theorem for integral equations
- Tarski's axioms
- Milnor's isospectral flat tori
- Poincaré inequality
- Hewitt–Savage zero–one law
- Order of a zero or pole
- Berkovich space dynamics
- Bramble–Hilbert lemma
- Reflection principle
- Wiman–Valiron theory
- Credible interval
- Conformal modulus
- Semialgebraic set
- Contraction mapping principle for integral equations
- Density of trigonometric polynomials in continuous functions
- Completion of a valued field
- Orbifold
- Ill-posedness of first-kind equations
- Koebe's theorem on the mean value property
- Independence of sample mean and sample variance for normal samples
- Self-similar process
- Gibbs sampling
- Estimating equations
- Bloch's theorem
- Spectral radius formula
- Hadamard factorization theorem
- Supermartingale
- Anderson–Darling test
- Spearman's rank correlation coefficient
- Inverse function theorem in Banach spaces
- Family-wise error rate
- Backward Euler method
- Torus knot
- Markov chain Monte Carlo
- Suslin line
- Nested intervals theorem
- Compactness of integral operators
- Hankel operator
- Dickey–Fuller test
- Galerkin method
- Wiener–Hopf method
- Generalized method of moments
- Doob's martingale convergence theorems
- Random integral equation
- Uniformly most powerful test
- Compactness of families of harmonic functions
- Bernstein inequalities
- Kendall rank correlation coefficient
- Semimartingale
- Resurgent function
- Empirical distribution function
- Bootstrap
- Gradient estimates for harmonic functions
- Osgood's lemma
- Influence function
- Orlicz–Pettis theorem
- Oseledets theorem
- Itô calculus
- Ancillary statistic
- Commutative Gelfand–Naimark theorem
- Stopped process
- Carathéodory metric
- Fourier basis of L2 of the circle
- Brody hyperbolicity
- Hyperbolic set
- Schwarz reflection principle
- Moments of the log-normal distribution
- Hilbert manifold
- Jentzsch's theorem on positive kernels
- Laguerre–Pólya class
- Complete statistic
- Horvitz–Thompson estimator
- Nörlund–Rice integral
- Littlewood subordination theorem
- Quasisymmetric map
- Liouville–Arnold theorem
- U-statistic
- Beurling–Ahlfors transform
- Fractional Brownian motion
- L-estimator
- Power of a test
- Fatou's theorem
- Lagrange multipliers on Banach spaces
- Banach–Stone theorem
- Banach–Mazur theorem
- Frobenius theorem on real division algebras
- Ahlfors theory
- Poles of the gamma function
- First main theorem of Nevanlinna theory
- Randomization
- Liouville's theorem for harmonic functions
- Moore–Osgood theorem
- Sufficient statistic
- Cauchy–Riemann equations in several variables
- Gaussian process
- Normal family
- Missing data mechanisms
- Picard's little theorem
- Liouville's theorems on elliptic functions
- Cauchy–Riemann operator
- Likelihood-ratio test
- Cardinal characteristics of the continuum
- Karlin–Rubin theorem
- Sokhotski–Plemelj theorem
- Harmonic functions on Riemann surfaces
- Noncommutative geometry
- Riemann–Hilbert problem
- Schauder basis
- Axiom A
- Tsirelson space
- Conjugate prior
- Kolmogorov–Smirnov test
- Lebesgue differentiation theorem
- Denjoy–Carleman–Ahlfors theorem
- F-space
- Ladyzhenskaya–Babuška–Brezzi condition
- Singular homology
- Euler–Gauss limit formula
- Delta derivative on a time scale
- Expectation–maximization algorithm
- Carleman's singular integral equation
- Hotelling's T-squared distribution
- Burkholder–Davis–Gundy inequality
- Doob–Meyer decomposition theorem
- Resolvent as Laplace transform of a semigroup
- Cross-validation
- Liouville's theorem for planar harmonic functions
- Maximum principle for harmonic functions
- Inner function
- Wald test
- Outer function
- Extremal length
- Pivotal quantity
- Banach algebra
- Generalized linear model
- Paley–Wiener theorem
- Hyperreal number
- Nevanlinna class
- Iterated function system
- Hadamard three-circle theorem for subharmonic functions
- Tame Fréchet space
- Lindelöf's maximum principle
- Basel problem via Fourier series
- Hadamard three-circle theorem
- Box–Jenkins method
- Kolmogorov continuity theorem
- Schwarz–Pick theorem
- Banach–Lie group
- Krull topology
- Banach manifold
- BK-space
- Differentiation in the transform variable
- Jury stability criterion
- Graph embedding
- Pesin theory
- Thom–Mather stratified space
- Grunsky matrix
- Cantor algebra
- Empirical process
- Exponential type
- Convenient vector space
- Binomial proportion confidence interval
- Pseudoanalytic function
- Lévy's upward theorem
- False discovery rate and the Benjamini–Hochberg procedure
- Cantor space
- Neyman construction
- Gamma function has no zeros
- Space of null sequences
- Harmonic measure as exit distribution
- Natural boundary
- Reflection formula for the gamma function
- Riemann surface
- Unique solvability of Volterra equations of the second kind
- Hadamard three-lines theorem
- Corona theorem
- Wilks' theorem
- Parabolic and hyperbolic Riemann surfaces
- Pseudo-Anosov map
- Anosov diffeomorphism
- Analytic capacity
- Mellin's inverse formula
- Mittag-Leffler's theorem
- Algorithmic randomness
- Lindelöf's theorem
- Branch point
- Sturmian word
- Hadamard's two-constants theorem
- Stability conjecture
- Skorokhod's embedding theorem
- Prime end
- Torus as a group
- Disk algebra
- Bôcher's theorem
- Gauss multiplication formula
- de Branges space
- Permutation test
- Fisher's exact test
- Reciprocal gamma function
- Bonferroni correction
- Automorphism group of a bounded domain is a Lie group
- Doob decomposition theorem
- Circle average
- Rellich–Kondrachov theorem
- Multiplication theorem
- Local asymptotic normality
- Frequentist inference
- Likelihood principle
- Order of an entire function
- Surreal number
- Bunimovich stadium
- Maximal torus
- Fisher–Neyman factorization theorem
- Girsanov theorem
- Euler integral for the gamma function
- Quadratic variation
- Functional equation of the gamma function
- Hartogs lemma for subharmonic functions
- Faber polynomials
- Observed information
- Feller process
- Point estimation
- Montel's theorem
- Local martingale
- Residue calculus for Laplace inversion
Further reading
- Web https://en.wikipedia.org/wiki/Real_number, lead section
- Web https://en.wikipedia.org/wiki/Real_number, Characterizing properties
About this page
| Tag | 08J2M1JYAG |
|---|---|
| URI | m2://08J2M1JYAG |
| Aliases | real-number, reals, set-of-real-numbers, real-line |
| Kind | object |
| Writing | W1 stated |
| MSC | 26A03 |
| Same as |
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What this page needs for W2
- section Invariants: a table of the main invariants (0 of 1)
- edge instance_of out (0 of 1)
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What links here
Used by
- Metric space
- Decimal expansion
- Irrational number
- Completeness axiom
- Borel sigma-algebra
- The reals form a field
- Cauchy principal value
- Transfer principle
- Stieltjes function
- Normed field
- The exponential map from the additive reals to the positive reals
- Non-Archimedean absolute value
- Schur's inequality
- Constructive analysis
- Conditionally complete lattice
- Birkhoff's axioms
- Hilbert's sixteenth problem
- Hölder's theorem on Archimedean groups
- Real closed field
- Standard part function
- Positivstellensatz
- Hahn embedding theorem
- Nonstandard analysis
- Constructive analysis
- Hilbert's seventeenth problem
- Semialgebraic set
- Frobenius theorem on real division algebras
- Hyperreal number