Date: Nov 27, 2012 1:24 PM
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 + ? * /
basic propositional logic (Modus ponens, If-then, and, or, ?)
equality & substitution
basic algebra ? idea of variables, equations, ?
the idea of probability
commutative and associative properties
powers (squared, cubed,?), - compound interest (miracle of)
scientific notation 1.3e6 = 1,300,000
first order predicate logic
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
the mean, expected value & strong law of large numbers
the graph of a function
real numbers as limits of fractions, the least upper bound
Rn, Euclidean Space, and Hilbert spaces
central limit theorem
sine cosine tangent
circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,?
the number e and the exponential function
the notion of linearity
injective (one-to-one) / surjective (onto) functions
Euler?s Formula e^(i?)?1=0
fundamental theorem of calculus
binomial theorem - Pascal?s triangle
ordinary differential equation (ODE)
mode (maximum likelihood estimator)
fundamental theorem of arithmetic
duality - (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, ?)
intermediate value theorem
compactness, Heine ? Borel theorem
metric space, Triangle Inequality
Projections, Best Approximation
partial differential equations
Reisz representation theorem
the ides of groups, semigroups, monoids, rings, ?
Singular Value Decomposition
numeric integration - trapezoidal rule, ?
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, ?