Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (positive integers). 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 in this extended sense is closely related to recursion.

Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics (see Problem of induction for more information). In fact, mathematical induction is a form of rigorous deductive reasoning.[1]

Contents [hide]
1 History
2 Description
3 Axiom of induction
4 Example
5 Variants
5.1 Starting at some other number
5.2 Building on n = 2
5.2.1 Example: product rule for the derivative
5.2.2 Example: Pólya's proof that there is no "horse of a different color"
5.3 Induction on more than one counter
5.4 Infinite descent
6 Complete induction
6.1 Transfinite induction
7 Proof of mathematical induction
8 See also
9 Notes
10 References
[edit]History

In 370 BC, Plato's Parmenides may have contained an early example of an implicit inductive proof.[2] The earliest implicit traces of mathematical induction can be found in Euclid's [3] proof that the number of primes is infinite and in Bhaskara's "cyclic method".[4] An opposite iterated technique, counting down rather than up, is found in the Sorites paradox, where one argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap.

An 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 and properties of Pascal's triangle.

None of these ancient mathematicians, however, explicitly stated the inductive hypothesis. Another similar case (contrary to what Vacca has written, as Freudenthal carefully showed) was that of 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 first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665). Another Frenchman, Fermat, made ample use of a related principle, indirect proof by infinite descent. The inductive hypothesis was also employed by the Swiss Jakob Bernoulli, and from then on it became more or less well known. The modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole,[5] Augustus de Morgan, Charles Sanders Peirce,[6] Giuseppe Peano, and Richard Dedekind.[4]

[edit]Description

The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:

The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1.
The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.
The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.

The choice between n = 0 and n = 1 in the base case is specific to the context of the proof: If 0 is considered a natural number, as is common in the fields of combinatorics and mathematical logic, then n = 0. If, on the other hand, 1 is taken as the first natural number, then the base case is given by n = 1.

This method works by first proving the statement is true for a starting value, and then proving that the process used to go from one value to the next is valid. If these are both proven, then any value can be obtained by performing the process repeatedly. It may be helpful to think of the domino effect; if one is presented with a long row of dominoes standing on end, one can be sure that:

The first domino will fall
Whenever a domino falls, its next neighbor will also fall, so it is concluded that all of the dominoes will fall, and that this fact is inevitable.

[edit]Axiom of induction

The basic assumption or axiom of induction is, in logical symbols,

where P is any proposition and k and n are both natural numbers.

In other words, the basis P(0) being true along with the inductive case ("P(k) is true implies P(k + 1) is true" for all natural k) being true together imply that P(n) is true for any natural number n. A proof by induction is then a proof that these two conditions hold, thus implying the required conclusion.

This works because k is used to represent an arbitrary natural number. Then, using the inductive hypothesis, i.e. that P(k) is true, show P(k + 1) is also true. This allows us to "carry" the fact that P(0) is true to the fact that P(1) is also true, and carry P(1) to P(2), etc., thus proving P(n) holds for every natural number n.

Note that the first quantifier in the axiom ranges over predicates rather than over individual numbers. This is called a second-order quantifier, which means that the axiom is stated in second-order logic. Axiomatizing arithmetic induction in first-order logic requires an axiom schema containing a separate axiom for each possible predicate. The article Peano axioms contains further discussion of this issue.