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Rearrangement proof of the Pythagorean theorem. (The area of the white space remains constant throughout the translation rearrangement of the triangles. At all moments in time, the area is always c 2. And likewise, at all moments in time, the area is always a 2 + b 2.)
Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy: Lemmermeyer (2000) has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature ...
This is perhaps the simplest known proof, requiring the least mathematical background. It is an attractive example of a combinatorial proof (a proof that involves counting a collection of objects in two different ways). The proof given here is an adaptation of Golomb's proof. [1] To keep things simple, let us assume that a is a positive integer.
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
This is called the generalized associative law. The number of possible bracketings is just the Catalan number , C n {\displaystyle C_{n}} , for n operations on n+1 values. For instance, a product of 3 operations on 4 elements may be written (ignoring permutations of the arguments), in C 3 = 5 {\displaystyle C_{3}=5} possible ways:
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The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. To gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected. The concept of proof is formalized in the field of mathematical logic. [12]
Proof without words of the Nicomachus theorem (Gulley (2010)) that the sum of the first n cubes is the square of the n th triangular number. In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text.