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:
counting zero place notation 100, 1000, ? the four arithmetic operations + ? * / fractions 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 polynomials first order predicate logic infinity irrational numbers Demorgan?s laws statistical independence the notion of a function square root (cube root, ?) inequalities power laws (i.e. a^b a^c = a^( b+c ) Cartesian coordinate plane basic set theory random variable probability distribution histogram the mean, expected value & strong law of large numbers the graph of a function standard deviation Pythagorean theorem vector spaces limits real numbers as limits of fractions, the least upper bound continuity Rn, Euclidean Space, and Hilbert spaces derivative correlation central limit theorem integrals 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 symmetry Euler?s Formula e^(i?)?1=0 Fourier transform fundamental theorem of calculus logarithms matrices 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 Jacobian fundamental theorem of arithmetic duality - (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, ?) intermediate value theorem eigenvalues median entropy KL distance binomial distribution Bayes? theorem 23.32?10 compactness, Heine ? Borel theorem metric space, Triangle Inequality Projections, Best Approximation 1/(1?X)=1+X+X2+? 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, ?