Update August 11, 2018: in the summer of 2018, I wrote down a list of theorems. Here is the top ten list:
The document (PDF) Here on Arxiv for a time stamp on July 23, 2018 is likely to be extended a bit more in the future as time permits.

What are the best ideas/theorems/formulas in mathematics? Any list is a matter of taste and personal preferences. Due to the vastness 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

• Completion of squares and quadratic formula by Brahmagupta. Babylonians and Chinese mathematicians
• Euler's formula exp(i x) = cos(x) + i sin(x) as a key to geometry and trigonometry.
• Euler's formula V-E+F=2 for polyhedra is a prototype of an index theorem.
• Taylors formula. A prototype of an approximation result.

Theorems

• Pythagoras theorem. Used in so many geometric proofs or computations.
• The fundamental theorem of calculus and generalizations to Stokes and 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.
• Brower's fixed point theorem as archetype for other nonconstructive fixed point and index arguments.
• 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 solve equations up to KAM theorem. More general Gradient methods or averaging methods.
• Gaussian elimination. It's fundamental to solve linear equations.
• Fourier approximation. Basic differential equations and probability theory. Starts harmonic analysis.
• Euclid's algorithm. Fundamental to do number theoretical computations.
• LLL algorithm is prevalent in many number theoretical and cryptological applications.
• Continued fraction expansion. For example to solve Diophantine equations or to classify numbers.
• Simplex method in convex optimization.
• Bubble sort as an example of a sorting.

Concepts

• Exponential maps in differential and Lie geometry. 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 generalizing metric spaces. Prototype of an axiomatically 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: Klein's Erlanger program.
• Scaling and renormalization arguments as an other symmetry.
• The methodology of statistics to extremize in a space of mathematical models.
• Invariants like index theorems. Archetype: Gauss-Bonnet. Leading to invariants.
• 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.
• Non-commutative geometry. Extends measure theory, topology and geometry to larger setup.

Oliver Knill, Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA. SciCenter 432 Tel: (617) 495 5549, Email: knill@math.harvard.edu Quantum calculus blog, Twitter, Youtube, Vimeo, Linkedin, Scholar Harvard, Academia, Google plus, Google Scholar, ResearchGate, Slashdot, Ello, Webcam, Fall 2018 office hours: TBA Mon-Fri 11:30-12:30 AM and by appointment.