```Date: Nov 27, 2012 1:24 PM
Author: Irchans
Subject: Top 100 most used Theorems and Ideas from Mathematics

I put together a list of the ideas and theorems that I used most and tried to put them in order by frequency of use.  It's hard because I often use a mathematical idea like symmetry without even realizing that I've use it, but this is my best guess.  Feel free comments because I want to revise the list.  I am sure I forgot a lot of things.  It would be great if other people posted their most used theorems.http://artent.net/blog/2012/11/27/100-most-useful-theorems-and-ideas-in-mathematics/Here is the list without the TeX and HTML Formatting: countingzeroplace notation 100, 1000, ?the four arithmetic operations + ? * /fractionsdecimal notationbasic propositional logic (Modus ponens, If-then, and, or, ?)negative numbersequivalence classesequality & substitutionbasic algebra ? idea of variables, equations, ?the idea of probabilitycommutative and associative propertiesdistributive propertypowers (squared, cubed,?),  - compound interest (miracle of)scientific notation 1.3e6 = 1,300,000polynomialsfirst order predicate logicinfinityirrational numbersDemorgan?s lawsstatistical independencethe notion of a functionsquare root  (cube root, ?)inequalitiespower laws (i.e. a^b a^c = a^( b+c )Cartesian coordinate planebasic set theoryrandom variable probability distributionhistogramthe mean, expected value & strong law of large numbersthe graph of a functionstandard deviationPythagorean theoremvector spaceslimitsreal numbers as limits of fractions, the least upper boundcontinuityRn, Euclidean Space,  and Hilbert spaces derivativecorrelationcentral limit theoremintegralschain rulemodular arithmeticsine cosine tangentcircumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,?linear regressionTaylor?s theoremthe number e and the exponential functionRolle?s theoremthe notion of linearityinjective (one-to-one) / surjective (onto) functionsimaginary numberssymmetryEuler?s Formula e^(i?)?1=0Fourier transformfundamental theorem of calculuslogarithmsmatricesconic sectionsBoolean algebraCauchy?Schwarz inequalitybinomial theorem - Pascal?s trianglethe determinantordinary differential equation (ODE)mode (maximum likelihood estimator)cosine lawprime numberslinear independenceJacobianfundamental theorem of arithmeticduality - (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, ?)intermediate value theoremeigenvaluesmedianentropyKL distancebinomial distributionBayes? theorem23.32?10compactness, Heine ? Borel theoremmetric space, Triangle InequalityProjections, Best Approximation1/(1?X)=1+X+X2+?partial differential equationsquadratic formulaReisz representation theoremFubini?s theoremthe ides of groups, semigroups, monoids, rings, ?Singular Value Decompositionnumeric integration - trapezoidal rule, ?mutual informationPlancherel?s theoremmatrix condition numberintegration by partsEuler?s method for numerical integration of ODEs (and improved Euler & Runge?Kutta)countable vs uncountable infinitypigeon hole principleThere is a long list of mathematical ideas that I use less often.  Here?s a sampling: Baire category theorem, Cauchy integral formula, calculus of variations, closed graph theorem, Chinese remainder theorem, Clifford algebra (quaternions), cohomology, Euclidean algorithm, fundamental group, Gauss? Law, Grassmannian algebra, homology, modules, non-associative algebra, Platonic/Euclidean solids, Pontryagain duality, Sylow p subgroup, repeating decimals equal a fraction, ring ideals, sine law, tensors, tessellation, transcendental numbers, Weierstrass approximation theorem, ?
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