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.

Here is the list without the TeX and HTML Formatting:

place notation 100, 1000, ?
the four arithmetic operations + ? * /
decimal notation
basic propositional logic (Modus ponens, If-then, and, or, ?)
negative numbers
equivalence classes
equality & substitution
basic algebra ? idea of variables, equations, ?
the idea of probability
commutative and associative properties
distributive property
powers (squared, cubed,?), - compound interest (miracle of)
scientific notation 1.3e6 = 1,300,000
first order predicate logic
irrational numbers
Demorgan?s laws
statistical independence
the notion of a function
square root (cube root, ?)
power laws (i.e. a^b a^c = a^( b+c )
Cartesian coordinate plane
basic set theory
random variable
probability distribution
the mean, expected value & strong law of large numbers
the graph of a function
standard deviation
Pythagorean theorem
vector spaces
real numbers as limits of fractions, the least upper bound
Rn, Euclidean Space, and Hilbert spaces
central limit theorem
chain rule
modular arithmetic
sine cosine tangent
circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,?
linear regression
Taylor?s theorem
the number e and the exponential function
Rolle?s theorem
the notion of linearity
injective (one-to-one) / surjective (onto) functions
imaginary numbers
Euler?s Formula e^(i?)?1=0
Fourier transform
fundamental theorem of calculus
conic sections
Boolean algebra
Cauchy?Schwarz inequality
binomial theorem - Pascal?s triangle
the determinant
ordinary differential equation (ODE)
mode (maximum likelihood estimator)
cosine law
prime numbers
linear independence
fundamental theorem of arithmetic
duality - (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, ?)
intermediate value theorem
KL distance
binomial distribution
Bayes? theorem
compactness, Heine ? Borel theorem
metric space, Triangle Inequality
Projections, Best Approximation
partial differential equations
quadratic formula
Reisz representation theorem
Fubini?s theorem
the ides of groups, semigroups, monoids, rings, ?
Singular Value Decomposition
numeric integration - trapezoidal rule, ?
mutual information
Plancherel?s theorem
matrix condition number
integration by parts
Euler?s method for numerical integration of ODEs (and improved Euler & Runge?Kutta)
countable vs uncountable infinity
pigeon hole principle

There 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, ?