What are the best ideas in mathematics? Any list is a matter of taste and personal preferences. Due to the vasteness of mathematics, it is quite challenging to build a list, because one tends to focus on things which one uses most. Here is a first shot written in 31. December 2007)

Formulas

• The quadratic equation formula by Brahmagupta. Completion of squares goes back to the Babilonians and Chinese mathematicians
• Eulers formula exp(i x) = cos(x) + i sin(x) is the key to complex numbers and a lot of trigonometry.
• Eulers formula V-E+F=2 for polyhedra is a prototype of an index theorem.
• Taylors formula. A prototype of an approximation result.
• The quadratic reciprocity law.

Theorems

• Pythagoras theorem. Used in so many geometric proofs or computations.
• The fundamental theorem of calculus and its generalizations. Descartes, Fermat, Newton, Barrow and Leibniz, at the core of calculus to Green, Stokes and Gauss in multi variable calculus to Ito's formula in stochastic calculus.
• Zorn's lemma in the form of Tychonov or Banach's theorem.
• The fundamental equations of calculus of variations.
• The Chinese remainder theorem in elementary number theory.
• Banach's fixed point theorem. A constructive method. Prototype of many more fixed point theorems.
• Browers fixed point theorem. A prototype for many other nonconstructive fixed point theorems including fixed point theorems based on index methods.
• Gauss-Bonnet theorem in differential geometry.
• Birkhoff's ergodic theorem with other limit theorems as special cases. Like the law of large numbers.

Algorithms

• Newtons method to find roots. Is a fantastic tool to solve equations and goes as far as KAM theorem. More generally, tools to solve equations including gradient methods or averaging methods.
• Gaussian elimination. It's fundamental to solve linear equations.
• Fourier approximation. So basic in differential equations and probability theory. The beginning of harmonic analysis. Goes on to exponential sums, wavelets.
• Euclid's algorithm. Fundamental to do number theoretical computations.
• LLL algorithm is prevalent in many number theoretical and cryptological applications.
• Continued fraction expansion. Natural and useful for example to solve Diophantine equations or to classify numbers.
• Simplex method in convex optimization.
• Bubble sort as an example of a sorting.

Concepts

• Logarithms both in the reals and indices in number theory.
• Indirect proofs like the irrationality of sqrt(2) and Euclids proof of the infinity of prime numbers.
• Baire category in topology. Countless of elegant existence theorems like nonalgebraic numbers, Liouville numbers.
• Diagonal arguments prototyped by Cantor, computability by Turing or decidability questions by Goedel.
• Topological spaces. The foundation not only put many facts on a sound foundation it also exploded. Prototype of an axiomaticlly clean setup.
• Galois theory as a prototype to bridge different areas of mathematics and to settle so many quests.
• Algebraic topology to use algebra to solve problems in topology.
• Group theory to classify geometry (Erlanger program).
• The methodology of statistics of extremizing in a space of mathematical models.
• Invariants like index theorems. Prototyped by Gauss-Bonnet
• Graph theory to solve combinatorial problems. Relations with linear algebra and spectral theory.
• Generating functions. A fundamental tool in combinatorics, number theory, probability theory.
• Matrix theory. Started with determinants and theory of linear equations. Motivates most of the rest of linear algebra.
• Differential equations, up to partial differential equations and stochastic differential equations.
• Noncommutative geometry. Extends measure theory, topology and geometry to larger setup.