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.

http://artent.net/blog/2012/11/27/100-most-useful-theorems-and-ideas-in-mathematics/

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, ?