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In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy.There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or ...
2 + 2 = 5 or two plus two equals five is a mathematical falsehood which is used as an ... when those who dared to say that 2 + 2 = 4 rather than 2 + 2 = 5 were ...
Bertrand's postulate and a proof; Estimation of covariance matrices; Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational
A direct proof is the simplest form of proof there is. The word ‘proof’ comes from the Latin word probare, [3] which means “to test”. The earliest use of proofs was prominent in legal proceedings.
Dirichlet's proof for n = 5 is divided into the two cases (cases I and II) defined by Sophie Germain. In case I, the exponent 5 does not divide the product xyz. In case II, 5 does divide xyz. Case I for n = 5 can be proven immediately by Sophie Germain's theorem(1823) if the auxiliary prime θ = 11.
The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle. Visual proof for the (3,4,5) triangle as in the Zhoubi Suanjing 500–200 BCE.
Animation demonstrating the smallest Pythagorean triple, 3 2 + 4 2 = 5 2. A Pythagorean triple consists of three positive integers a, b, and c, such that a 2 + b 2 = c 2. Such a triple is commonly written (a, b, c), a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.
Reflexivity: for every a, one has a = a.; Symmetry: for every a and b, if a = b, then b = a.; Transitivity: for every a, b, and c, if a = b and b = c, then a = c. [7] [8]Substitution: Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning.