100 Most useful Theorems and Ideas in Mathematics

So I have been thinking about which ideas in mathematics I use most often and I have listed them below.  Please feel free to comment because I would like to use this list as a starting point for a future list of “The most useful ideas in Mathematics for Scientists and Engineers”.
$\ $
I showed this list to Carl and he said his list would be completely different.  (He choked a bit when he saw primes at #71.)  Hopefully we can post his list as well as others.
$\ $
  1. counting
  2. zero
  3. integer decimal positional notation 100, 1000, …
  4. the four arithmetic operations + – * /
  5. fractions
  6. decimal notation  0.1, 0.01, …
  7. basic propositional logic (Modus ponens, contrapositive, If-then, and, or, nand, …)
  8. negative numbers
  9. equivalence classes
  10. equality & substitution
  11. basic algebra – idea of variables, equations, …
  12. the idea of probability
  13. commutative and associative properties
  14. distributive property
  15. powers (squared, cubed,…),  – compound interest (miracle of)
  16. scientific notation 1.3e6 = 1,300,000
  17. polynomials
  18. first order predicate logic
  19. infinity
  20. irrational numbers
  21. De Morgan’s laws
  22. statistical independence
  23. the notion of a function
  24. square root  (cube root, …)
  25. inequalities (list of inequalities)
  26. power laws (i.e. $a^b a^c = a^{b+c}$ )
  27. Cartesian coordinate plane
  28. basic set theory
  29. random variable 
  30. probability distribution
  31. histogram
  32. the meanexpected value & strong law of large numbers
  33. the graph of a function
  34. standard deviation
  35. Pythagorean theorem
  36. vectors and vector spaces
  37. limits
  38. real numbers as limits of fractions, the least upper bound
  39. continuity
  40. $R^n$, Euclidean Space,  and Hilbert spaces (inner or dot product)
  41. derivative
  42. correlation
  43. central limit theorem, Gaussian Distribution, Properties of Guassains.
  44. integrals
  45. chain rule
  46. modular arithmetic
  47. sine cosine tangent
  48. $\pi$circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,…
  49. linear regression
  50. Taylor’s theorem
  51. the number e and the exponential function
  52. Rolle’s theoremKarush–Kuhn–Tucker conditions, derivative is zero at the maximum
  53. the notion of linearity
  54. Big O notation
  55. injective (one-to-one) / surjective (onto) functions
  56. imaginary numbers
  57. symmetry
  58. Euler’s Formula $e^{i \pi} + 1 = 0$
  59. Fourier transform, convolution in time domain is the product in the frequency domain (& vice versa), the FFT
  60. fundamental theorem of calculus
  61. logarithms
  62. matrices
  63. conic sections
  64. Boolean algebra
  65. Cauchy–Schwarz inequality
  66. binomial theorem – Pascal’s triangle
  67. the determinant
  68. ordinary differential equation (ODE)
  69. mode (maximum likelihood estimator)
  70. cosine law
  71. prime numbers
  72. linear independence
  73. Jacobian
  74. fundamental theorem of arithmetic
  75. duality – (polyhedron faces & pointsgeometry lines and pointsDual Linear Programdual space, …)
  76. intermediate value theorem
  77. eigenvalues
  78. median
  79. entropy
  80. KL distance
  81. binomial distribution
  82. Bayes’ theorem
  83. $2^{10} \approx 1000$
  84. compactnessHeine – Borel theorem
  85. metric space, Triangle Inequality
  86. ProjectionsBest Approximation
  87. $1/(1-X) = 1 + X + X^2 + \ldots$
  88. partial differential equations
  89. quadratic formula
  90. Reisz representation theorem
  91. Fubini’s theorem
  92. the ideas of groups, semigroups, monoids, rings, …
  93. Singular Value Decomposition
  94. numeric integration – trapezoidal rule, Simpson’s rule, …
  95. mutual information
  96. Plancherel’s theorem
  97. matrix condition number
  98. integration by parts
  99. Euler’s method for numerical integration of ODEs (and improved EulerRunge–Kutta)
  100. pigeon hole principle
There is a long list of mathematical ideas that I use less often.  Here’s a sampling: Baire category theorem, Banach SpacesBrouwer Fixed Point Theorem, Carathéodory’s Theorem, Category TheoryCauchy integral formula, calculus of variations, closed graph theoremChinese remainder theorem, Clifford algebra (quaternions), Context Free Grammarscountable vs uncountable infinityCramer’s RulecohomologyEuclidean algorithm, fundamental group, Gauss’ LawGrassmannian algebra , Graph TheoryHahn-Banach Theorem, homology, Hairy Ball Theorem, Hölder’s inequality, inclusion-exclusion, Jordan Decomposition, Kalman FiltersMarkov Chains (Hidden Markov Models), modules, non-associative algebras, Picard’s Great TheoremPlatonic/Euclidean solids, Principle of InductionProbabilistic Graphical Models (Bayesian Networks, Markov Random Fields), Pontryagain duality, QuaternionsSpectral TheoremSylow p subgroup, repeating decimals equal a fraction, ring ideals, sine law, tensorstessellation, transcendental numbers, Uniform Boundedness TheoremWeierstrass approximation theorem, …


  1. antianticamper’s avatar

    Interesting list, especially since geometry plays a distinctly secondary role. I’d add:

    partially ordered set
    dynamical system

  2. hundalhh’s avatar

    Vance wrote; “I ran through the list very quickly. I am sure there are items missing. For example:


    Also, there is a sort of taxonomy that is implicit. Many of these items are at a different level from others. I would suggest looking for that taxonomy. Math Reviews? So for example, Calculus, Geometry, Algebra are high level nodes.”

  3. hundalhh’s avatar

    antianticamper ,
    I really need to add “the idea of mathematical proof” and ‘the definition of algorithm”. I don’t think I use partially ordered sets consciously. Nor do I use Topological sorting much. I may have to add ‘dynamical systems’ because I do a little control theory. (forgot that)

  4. tropicalbats’s avatar

    Might be buried in there, as there is a lot in there, but a basic math bit is factors. Pi should probably be clearly stated. And maybe I missed it, but the ordinal rules for doing a calculation (as in, things in parentheses get done first, etc.) seem pretty important.

  5. hundalhh’s avatar

    I missed Pi !! And basic geometry needs to be in there somehow. Hmm

Comments are now closed.