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Online encyclopedia
PlanetMath is a free, collaborative, mathematics online encyclopedia. Intended to be comprehensive, the project is currently hosted by the University of
PlanetMath
Hyperbolic analogues of trigonometric functions
Encyclopedia of Mathematics, EMS Press, 2001 [1994] Hyperbolic functions on PlanetMath GonioLab: Visualization of the unit circle, trigonometric and hyperbolic
Hyperbolic_functions
Seven mathematical problems with a US$1 million prize for each solution
ISBN 978-0-8218-3679-8. This article incorporates material from Millennium Problems on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Millennium_Prize_Problems
Mathematical model for deduction or proof systems
Philosophy. Metaphysics Research Lab, Stanford University. "Formal System". PlanetMath. Rapaport, William J. (25 March 2010). "Syntax & Semantics of Formal Systems"
Formal_system
Copyleft license primarily for free software documentation
Last.fm – artist descriptions are under GFDL Marxists Internet Archive PlanetMath (now uses CC BY-SA license) Rosetta Code SourceWatch The specification
GNU Free Documentation License
GNU_Free_Documentation_License
Pair of logical equivalences
Weisstein, Eric W. "de Morgan's Laws". MathWorld. de Morgan's laws at PlanetMath. Duality in Logic and Language, Internet Encyclopedia of Philosophy.
De_Morgan's_laws
Coordinate system using perpendicular axes
system", Encyclopedia of Mathematics, EMS Press "Cartesian coordinates". planetmath.org. Retrieved 25 August 2024. Hughes-Hallett, McCallum & Gleason 2013
Cartesian_coordinate_system
Branch of mathematics
[1994] Weisstein, Eric W. "Calculus". MathWorld. Topics on Calculus at PlanetMath. Calculus Made Easy (1914) by Silvanus P. Thompson Full text in PDF Calculus
Calculus
Horizontal and vertical axes/coordinate numbers of a 2D coordinate system or graph
"Ordinate". MathWorld. Retrieved 14 July 2013. "Cartesian coordinates". PlanetMath. Archived from the original on 2025-02-21. Retrieved 2025-04-02. 'applicate'
Abscissa_and_ordinate
Coordinate system
(solid geometry) Orthant Ray (geometry) Wikimedia Commons has media related to Quadrants. Weisstein, Eric W. "Quadrant". MathWorld. Quadrant at PlanetMath.
Quadrant_(plane_geometry)
Statement that is taken to be true
dictionary. Wikisource has the text of the 1911 Encyclopædia Britannica article "Axiom". Axiom at PhilPapers Axiom at PlanetMath. Metamath axioms page
Axiom
Number in {..., –2, –1, 0, 1, 2, ...}
"Integer". MathWorld. This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Integer
Large number defined as ten to the 100th power
the free dictionary. Weisstein, Eric W. "Googol". MathWorld. Googol at PlanetMath. Padilla, Tony; Symonds, Ria. "Googol and Googolplex". Numberphile. Brady
Googol
Topological space that is homeomorphic to a metric space
no. 2, 293–300. This article incorporates material from Metrizable on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Metrizable_space
90° angle (π/2 radians)
Heath p. 181 Wentworth p. 9 Heath pp. 200–201 for the paragraph "octant". PlanetMath.org. 2013-03-22. Retrieved 2024-10-21. Wentworth, G.A. (1895). A Text-Book
Right_angle
Part of a line that is bounded by two distinct end points; line with two endpoints
dictionary. Weisstein, Eric W. "Line segment". MathWorld. Line Segment at PlanetMath Copying a line segment with compass and straightedge Dividing a line segment
Line_segment
Spiral with constant distance from itself
Weisstein, Eric W. "Archimedes' Spiral". MathWorld. archimedean spiral at PlanetMath. Page with Java application to interactively explore the Archimedean spiral
Archimedean_spiral
Type of probability distribution
London: Springer. 2005. ISBN 978-1-85233-896-1. OCLC 262680588. "Joint continuous density function". PlanetMath. Mathworld: Joint Distribution Function
Joint probability distribution
Joint_probability_distribution
Unproved conjecture in mathematics
"Swinnerton-Dyer Conjecture". MathWorld. "Birch and Swinnerton-Dyer Conjecture". PlanetMath. The Birch and Swinnerton-Dyer Conjecture: An Interview with Professor
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Online mathematics reference work
This case made a wave of headlines in online publishing circles. The PlanetMath project was a result of MathWorld's being unavailable. List of online
MathWorld
Algebraic structure with an associative operation and an identity element
S2CID 56410568. Zbl 0865.06010. "Monoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Monoid". MathWorld. Monoid at PlanetMath.
Monoid
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. "Leyland number". planetmath.org. Retrieved 2025-05-20. Sloane, N. J. A. (ed.). "Sequence A006958 (Number
10,000,000
Ancient Egyptian texts
from University College, London A Kahun Mathematical Fragment, a paper by John A.R. Legon Kahun Papyrus and Arithmetic Progressions at PlanetMath. v t e
Kahun_Papyri
Middle quantile of a data set or probability distribution
Mechanics This article incorporates material from Median of a distribution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Median
Geometric figure which has infinite surface area but finite volume
60–65. doi:10.5951/MT.82.1.0060. JSTOR 27966098. Torricelli's trumpet at PlanetMath Weisstein, Eric W. "Gabriel's Horn". MathWorld. "Gabriel's Horn" by John
Gabriel's_horn
Branch of mathematics
Algebraic Geometry by Ravi Vakil, 808 pp. Algebraic geometry entry on PlanetMath English translation of the van der Waerden textbook Dieudonné, Jean (March
Algebraic_geometry
Algebraic structure of set algebra
York. https://doi.org/10.1007/978-1-4757-2545-2 "Algebra of sets", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Sigma Algebra from PlanetMath.
Σ-algebra
Method of solution to differential equations
for differential operator at PlanetMath. Green's function at PlanetMath. Green functions and conformal mapping at PlanetMath. Introduction to the Keldysh
Green's_function
Generalization of metric spaces in mathematics
Pseudometric space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. "Example of pseudometric space". PlanetMath.
Pseudometric_space
Distance from zero to a number
"Absolute value". Encyclopedia of Mathematics. EMS Press. 2001 [1994]. absolute value at PlanetMath. Weisstein, Eric W. "Absolute Value". MathWorld.
Absolute_value
National encyclopedia of Canada
Encyclopedia Encyclopedia of Canadian Biography Encyclopaedia Metallum PlanetMath List of online encyclopedias National historic significance Events of
The_Canadian_Encyclopedia
Subclass of matrices
Cambridge University Press. ISBN 0-521-38632-2. PlanetMath: Diagonal dominance definition PlanetMath: Properties of diagonally dominant matrices Mathworld
Diagonally_dominant_matrix
Simple curve of Euclidean geometry
"Circle". Encyclopedia of Mathematics. EMS Press. 2001 [1994]. Circle at PlanetMath. Weisstein, Eric W. "Circle". MathWorld. "Interactive Java applets". for
Circle
Bound lattice in which every element has a complement
"Complemented lattice". PlanetMath. "Relative complement". PlanetMath. "Uniquely complemented lattice". PlanetMath. "Orthocomplemented lattice". PlanetMath.
Complemented_lattice
Polynomial of the elements of a matrix
Mathematical Monthly, 72 (6): 577–591, doi:10.2307/2313846, JSTOR 2313846 Permanent at PlanetMath. Van der Waerden's permanent conjecture at PlanetMath.
Permanent_(mathematics)
Non-orientable mathematical surface
October 2022. This article incorporates material from Klein bottle on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Klein_bottle
Probability distribution
populations Seeing Around Corners (Artificial societies turn up Zipf's law) PlanetMath article on Zipf's law Distributions de type "fractal parabolique" dans
Zipf's_law
Result in combinatorics and graph theory
article incorporates material from proof of Hall's marriage theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Hall's_marriage_theorem
Non-zero element of a matrix selected by an algorithm
row echelon form. This article incorporates material from Pivoting on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Pivot_element
article incorporates material from Isomorphism-closed subcategory on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Isomorphism-closed subcategory
Isomorphism-closed_subcategory
Formula for systems of linear equations
Wikibooks has a book on the topic of: Linear Algebra/Cramer's Rule Proof of Cramer's Rule at PlanetMath. Weisstein, Eric W. "Cramer's rule". MathWorld.
Cramer's_rule
Scalar-valued bilinear function
Press, 2001 [1994] "Bilinear form". PlanetMath. This article incorporates material from Unimodular on PlanetMath, which is licensed under the Creative
Bilinear_form
Mathematics web site
resource in history of mathematics". Mathematics Genealogy Project MathWorld PlanetMath Hughes, Barnabas (July 2007), "St Andrews History of Mathematics Archive"
MacTutor History of Mathematics Archive
MacTutor_History_of_Mathematics_Archive
Theorem in topology
Theorem for Triangles at cut-the-knot Brouwer fixed-point theorem at PlanetMath. Reconstructing Brouwer at MathPages Brouwer Fixed Point Theorem at Math
Brouwer_fixed-point_theorem
Plane curve
at Wikiquote Media related to Ellipses at Wikimedia Commons ellipse at PlanetMath. Weisstein, Eric W. "Ellipse". MathWorld. Weisstein, Eric W. "Ellipse
Ellipse
Arithmetic operation
Archived 2019-08-17 at the Wayback Machine Retrieved October 23, 2018 Planetmath division Division on a Japanese abacus selected from Abacus: Mystery of
Division_(mathematics)
Function space of all functions whose derivatives are rapidly decreasing
article incorporates material from Space of rapidly decreasing functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Schwartz_space
Approximation for factorials
n! Weisstein, Eric W., "Stirling's Approximation", MathWorld{{cite web}}: CS1 maint: overridden setting (link) Stirling's approximation at PlanetMath.
Stirling's_approximation
Two matrices placed in the diagonal of a larger matrix
McGraw-Hill Education. ISBN 9781260011449. Direct sum of matrices at PlanetMath. Abstract nonsense: Direct Sum of Linear Transformations and Direct Sum
Direct_sum_of_matrices
American mathematician and encyclopedist (born 1969)
MathWorld on November 6, 2001. This ultimately led to the creation of PlanetMath. Wolfram Research, Stephen Wolfram, and Eric Weisstein settled with the
Eric_W._Weisstein
British mathematician
the European Mathematical Society (33): 8–9. "William Timothy Gowers". PlanetMath. Listen to Timothy Gowers on The Forum from BBC World Service Radio Timothy
Timothy_Gowers
American number theorist (b.1965)
19: 35–39. This article incorporates material from Judy Holdener on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Judy_A._Holdener
Type of prime number
questions", American Mathematical Monthly, 109 (6): 554–559, doi:10.2307/2695445, JSTOR 2695445. https://planetmath.org/pillaiprime, PlanetMath v t e
Pillai_prime
Additional mathematical object
1170–1173. doi:10.1126/science.260.5111.1170. PMID 17806355. "Structure". PlanetMath. (provides a model theoretic definition.) Mathematical structures in computer
Mathematical_structure
Set of philosophical problems
Philosophy. This article incorporates material from Zeno's paradox on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Zeno's_paradoxes
Subset of incomparable elements
connected", SIAM Journal on Computing, 12 (4): 777–788, doi:10.1137/0212053, MR 0721012 Weisstein, Eric W. "Antichain". MathWorld. "Antichain". PlanetMath.
Antichain
Concept in number theory
Mathematics book series. Vol. 1467. Springer. doi:10.1007/BFb0098246. ISBN 978-3-540-47374-9. S2CID 118143570. Dirichlet's Approximation Theorem at PlanetMath.
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Mathematical concept in algebra
Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 Nilpotent matrix and nilpotent transformation on PlanetMath.
Nilpotent_matrix
Open problem on 3x+1 and x/2 functions
Weisstein, Eric W. "Collatz Problem". MathWorld. Collatz Problem at PlanetMath. Nochella, Jesse. "Collatz Paths". Wolfram Demonstrations Project. Eisenbud
Collatz_conjecture
General theory of mathematical structures
channel on YouTube, a channel about category theory. Category theory at PlanetMath.. Video archive of recorded talks relevant to categories, logic and the
Category_theory
Unique positive real number which when multiplied by itself gives 3
this case, Φv2 = −120° and Φv3 = −240° or 120°.) "square root of 3". planetmath.org. Retrieved 2025-07-23. Komsta, Łukasz (December 2013). "Computations
Square_root_of_3
Average uncertainty in variable's states
ISBN 978-1-4757-4321-0. This article incorporates material from Shannon's entropy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Entropy_(information_theory)
Branch of mathematics that studies dynamical systems
EMS Press This article incorporates material from ergodic theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Ergodic_theory
Zbl 0061.39807. This article incorporates material from Development on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Development_(topology)
Number whose square ends in the same digits
institution membership required.) examples of 1-automorphic numbers at PlanetMath. Weisstein, Eric W. "Automorphic number". MathWorld. Weisstein, Eric W
Automorphic_number
Result in ring theory
Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 Goldie's theorem at PlanetMath. Goldie Ring at PlanetMath. v t e
Goldie's_theorem
About maxima and minima of functions
1007/978-3-319-12772-9. ISBN 978-3-319-12771-2. "Fermat's Theorem (stationary points)". PlanetMath. "Proof of Fermat's Theorem (stationary points)". PlanetMath.
Interior_extremum_theorem
including formulas, related sequences, programs, etc. Free/CC BY-NC 3.0 PlanetMath English Mathematics related topics and articles Free/GNU Stacks Project
List_of_online_encyclopedias
Geometric relation between a triangle's side lengths and cevian length
Cochran "Proposition II" Russell 1905, p. 3 Proof of Stewart's Theorem at PlanetMath. Coxeter, H.S.M.; Greitzer, S.L. (1967), Geometry Revisited, New Mathematical
Stewart's_theorem
Mathematical binary relation
ISBN 0-521-58519-8. This article incorporates material from subsequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Subsequence
Concept in mathematical group theory
by Leonard L. Scott. New York-Heidelberg: Springer-Verlag. doi:10.1007/978-1-4684-9458-7. ISBN 978-0-387-90190-9. MR 0450380. Character at PlanetMath.
Character_theory
Infinite products of functions indexed by primes
1000-digit values This article incorporates material from Euler product on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Euler_product
Mnemonic device for calculating 3 by 3 matrix determinants
Elements of Linear Algebra. CRC Press. ISBN 978-0-412-55280-9. Sarrus' rule at Planetmath Linear Algebra: Rule of Sarrus of Determinants at khanacademy.org
Rule_of_Sarrus
Geometric concept of a 2D space with "points at infinity" adjoined
of Small Order, (2003) Ch. Weibel: Survey of Nondesarguesian planes Weisstein, Eric W., "Projective plane", MathWorld "Projective plane" at PlanetMath.
Projective_plane
Theorem in complex analysis
New York. This article incorporates material from Hardy's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Hardy's_theorem
Right inverse of a fiber bundle map
ISBN 0-201-10096-7. Husemöller, Dale (1994), Fibre Bundles, Springer Verlag, ISBN 0-387-94087-1 Fiber Bundle, PlanetMath Weisstein, Eric W. "Fiber Bundle". MathWorld.
Section_(fiber_bundle)
Type of incidence structure
1992. Eric Moorhouse: Incidence Geometry. Lecture notes (archived) partial linear space at the University of Kiel Partial linear space at PlanetMath.
Partial_linear_space
Theorem on modular exponentiation
Elementary Number Theory, New York: Chelsea Weisstein, Eric W. "Euler's Totient Theorem". MathWorld. Euler-Fermat Theorem at PlanetMath Portal: Mathematics
Euler's_theorem
Natural number
Explorer". Numbermatics. Apr 30, 2024. Retrieved Apr 30, 2024. "Blum Number". PlanetMath. Apr 30, 2024. Retrieved Apr 30, 2024. Sloane, N. J. A. (May 1, 2024)
309_(number)
Theory of a class of elliptic curves
14015. Complex multiplication from PlanetMath.org Examples of elliptic curves with complex multiplication from PlanetMath.org Ribet, Kenneth A. (October 1995)
Complex_multiplication
Probability Measures on Polish Spaces. STOCHASTICS MONOGRAPHS. London: CRC Press. p. 13. ISBN 0415273870. "Measurable projection theorem", PlanetMath
Projection_(measure_theory)
on scholarpedia This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike
Expansive_homeomorphism
Number of subsets of a given size
Retrieved 2013-09-03. This article incorporates material from the following PlanetMath articles, which are licensed under the Creative Commons Attribution/Share-Alike
Binomial_coefficient
Extension of the Brouwer fixed-point theorem
Encyclopedia of Mathematics, EMS Press, 2001 [1994] "Schauder fixed point theorem". PlanetMath. "proof of Schauder Fixed Point Theorem". PlanetMath..
Schauder_fixed-point_theorem
Disjoint, equal-size subsets of a group's underlying set
"Coset in a group", Encyclopedia of Mathematics, EMS Press Coset at PlanetMath. Illustrated examples "Coset". groupprops. The Group Properties Wiki.
Coset
Counting technique in combinatorics
article incorporates material from principle of inclusion–exclusion on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Inclusion–exclusion_principle
Natural number, cyclic number
14 (2): 105–109. doi:10.2307/3026586. JSTOR 3026586. Cyclic Number at PlanetMath. Hogan, Kathryn (August 2005). "Go figure (cyclic numbers)". Australian
142857
Mathematical proof technique using contradiction
theorem in the case n = 4", Mathematical Gazette 91, July 2007, 260–262. Infinite descent at PlanetMath. Example of Fermat's last theorem at PlanetMath.
Proof_by_infinite_descent
Chaotic model of atmospheric convection
attractor by Rob Morris, Wolfram Demonstrations Project. Lorenz Equation at PlanetMath. Synchronized Chaos and Private Communications, with Kevin Cuomo. The
Lorenz_system
Mathematical problem
"Erdös-Ginzburg-Ziv theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] PlanetMath Erdős, Ginzburg, Ziv Theorem Sun, Zhi-Wei, "Covering Systems, Restricted
Zero-sum_problem
Matrix normal form
form. MP274: Linear Algebra, Lecture Notes, University of Queensland, 1991. Smith normal form at PlanetMath. Example of Smith normal form at PlanetMath.
Smith_normal_form
Point which a function/system returns to after some time or iterations
mappings This article incorporates material from hyperbolic fixed point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Periodic_point
Theorem for proving more complex theorems
a Thousand Theorems This article incorporates material from Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike
Lemma_(mathematics)
Orthonormalization of a set of vectors
of n vectors of order m routine Proof: Raymond Puzio, Keenan Kidwell. "proof of Gram-Schmidt orthogonalization algorithm" (version 8). PlanetMath.org.
Gram–Schmidt_process
Theorem in mathematics
Encyclopedia of Mathematics, EMS Press, 2001 [1994] Mean value theorem at PlanetMath. Weisstein, Eric W. "Mean-Value Theorem". MathWorld. Weisstein, Eric W
Mean_value_theorem
Sum of an (infinite) geometric progression
Weisstein, Eric W. "Geometric Series". MathWorld. Geometric Series at PlanetMath. Peppard, Kim. "College Algebra Tutorial on Geometric Sequences and Series"
Geometric_series
Mathematical function
functions". MathWorld. "Mangoldt summatory function". PlanetMath. "Chebyshev functions". PlanetMath. Riemann's Explicit Formula, with images and movies
Chebyshev_function
to provide fixes and information on all PC games 45,508 CC NC-SA 3.0 PlanetMath Misc – Mathematics Free wiki-style mathematical encyclopedia Proteopedia
List_of_wikis
Function in algebra
"Valuation", Encyclopedia of Mathematics, EMS Press Discrete valuation at PlanetMath. Valuation at PlanetMath. Weisstein, Eric W. "Valuation". MathWorld.
Valuation_(algebra)
Theorem in calculus
originally based on the GFDL article from PlanetMath at https://web.archive.org/web/20021029094728/http://planetmath.org/encyclopedia/Divergence.html
Divergence_theorem
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