Induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. more...

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It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.

The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science.

Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics. (See Problem of induction.) In fact, mathematical induction is a form of deductive reasoning and is fully rigorous.

History

The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite and in Bhaskara's "cyclic method". The earliest implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum formula for integral cubes. His proof was the first to make use of the two basic components of an inductive proof, "namely the truth of the statement for n = 1 (1 = 1³) and the deriving of the truth for n = k from that of n = k − 1." Shortly afterwards, Ibn al-Haytham (Alhazen) used the inductive method to prove the sum of fourth powers, and by extension, the sum of any integral powers, which was an important result in integral calculus. He only stated it for particular integers, but his proof for those integers was by induction and generalizable. Ibn Yahyā al-Maghribī al-Samaw'al came closest to a modern proof by mathematical induction in pre-modern times, which he used to extend the proof of the binomial theorem and Pascal's triangle previously given by al-Karaji. Al-Samaw'al's inductive argument was only a short step from the full inductive proof of the general binomial theorem.

None of these ancient mathematicians, however, explicitly stated the inductive hypothesis. The first rigorous explicit exposition of the principle of induction was given by Francesco Maurolico, in his Arithmeticorum libri duo (1575), who used the technique to prove that the sum of the first n odd integers is n². The inductive hypothesis was also discovered independently by the Swiss Jakob Bernoulli, and the Frenchmen Pascal and Fermat.

Read more at Wikipedia.org