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  2. Mathematical induction - Wikipedia

    en.wikipedia.org/wiki/Mathematical_induction

    Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes. [9] In India, early implicit proofs by mathematical induction appear in Bhaskara's "cyclic method". [10] None of these ancient mathematicians, however, explicitly stated the induction hypothesis.

  3. List of mathematical proofs - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_proofs

    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; Proof that the sum of the reciprocals of the primes diverges

  4. Proofs involving the addition of natural numbers - Wikipedia

    en.wikipedia.org/wiki/Proofs_involving_the...

    We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0) Each equation follows by definition [A1]; the first with a + b, the second with b. Now, for the induction. We assume the induction hypothesis, namely we assume that for some ...

  5. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    For example, direct proof can be used to prove that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for some integers a and b. Then the sum is x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence ...

  6. Transfinite induction - Wikipedia

    en.wikipedia.org/wiki/Transfinite_induction

    Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor). Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.

  7. Structural induction - Wikipedia

    en.wikipedia.org/wiki/Structural_induction

    Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural numbers and can be further generalized to arbitrary Noetherian induction .

  8. De Moivre's formula - Wikipedia

    en.wikipedia.org/wiki/De_Moivre's_formula

    See angle sum and difference identities. We deduce that S(k) implies S(k + 1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos(0x) + i sin(0x) = 1 + 0i = 1. Finally, for the negative integer cases, we consider an exponent of −n for natural n.

  9. Squared triangular number - Wikipedia

    en.wikipedia.org/wiki/Squared_triangular_number

    The sum within each gmonon is a cube, so the sum of the whole table is a sum of cubes. Visual demonstration that the square of a triangular number equals a sum of cubes. In the more recent mathematical literature, Edmonds (1957) provides a proof using summation by parts.