Famous Theorems of Mathematics

Not all of mathematics deals with proofs, as mathematics involves a rich range of human experience, including ideas, problems, patterns, mistakes and corrections. However, proofs are a very big part of modern mathematics, and today, it is generally considered that whatever statement, remark, result etc. one uses in mathematics, it is considered meaningless until is accompanied by a rigorous mathematical proof. This book is intended to contain the proofs (or sketches of proofs) of many famous theorems in mathematics in no particular order. It should be used both as a learning resource, a good practice for acquiring the skill for writing your own proofs is to study the existing ones, and for general references.

It is not however intended as a companion to any other wikibook or wikipedia articles but can complement them by providing them with links to the proofs of the theorems they contain.

One note here. There are usually many ways to solve a problem. Many times the proof used comes down to the primary definitions of terms involved. We will follow the definition given by the first major contributor.

High school

 * /Pythagoras theorem/
 * /Euclid's proof of the infinitude of primes/
 * /√2 is irrational/
 * sin2Θ+cos2Θ=1

Undergraduate

 * /e is irrational/
 * /π is irrational/
 * /Fermat's little theorem/
 * /Fermat's theorem on sums of two squares/
 * /Sum of the reciprocals of the primes diverges/
 * /Bertrand's postulate/
 * /Law of large numbers/
 * /Spectral Theorem/
 * /L'Hôpital's rule/
 * /Four color theorem/
 * eπi+1=0
 * /Gaussian integral/
 * /Gamma function/
 * /Zeta function/

Postgraduate

 * /Fermat's last theorem/
 * /Brouwer fixed-point theorem/
 * /Prime number theorem/
 * /Riemann hypothesis/

Old table of contents
Proofs and definitions will be arranged according to the fields of mathematics:


 * /Algebra/
 * /Analysis/
 * /Applied Mathematics/
 * /Geometry/
 * /Logic/
 * /Number Theory/
 * /Set Theory/
 * /Boy's surface/

Manual of style

 * Proof style - Style guide for proofs.

External links etc

 * Proofs on the wikipedia project
 * List of mathematical proofs
 * List of theorems
 * Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review
 * The Hundred Greatest Theorems
 * ProofWiki.org - The online compendium of mathematical proofs

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