# 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”. This is only a draft, so I expect to make a number of revisions over the next few days.
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I showed this list to Carl and he said his list would be completely different.  (He choked a bit when he saw primes at #70.)  Hopefully we can post his list as well as others.
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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. Demorgan’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
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

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

proof
fractal
graph
optimization
model
partially ordered set
algorithm
dynamical system

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

http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

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