Mathematical induction
Mathematical induction is a method for proving that a statement is true for every natural number, that is, that the infinitely many cases all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:
A proof by induction consists of two cases. The first, the base case, proves the statement for without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case, then it must also hold for the next case. These two steps establish that the statement holds for every natural number. The base case does not necessarily begin with, but often with, and possibly with any fixed natural number, establishing the truth of the statement for all natural numbers.
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 is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs.
Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy, in which the examination of many cases results in a probable conclusion. The mathematical method examines infinitely many cases to prove a general statement, but it does so by a finite chain of deductive reasoning involving the variable, which can take infinitely many values. The result is a rigorous proof of the statement, not an assertion of its probability.
History
In 370 BC, Plato's Parmenides may have contained traces of an early example of an implicit inductive proof, however, the earliest implicit proof by mathematical induction was written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle. Whilst the original work was lost, it was later referenced by Al-Samawal al-Maghribi in his treatise al-Bahir fi'l-jabr in around 1150 AD.Katz says in his history of mathematics
In India, early implicit proofs by mathematical induction appear in Bhaskara's "cyclic method".
None of these ancient mathematicians, however, explicitly stated the induction hypothesis. Another similar case was that of Francesco Maurolico in his Arithmeticorum libri duo, who used the technique to prove that the sum of the first odd integers is.
The earliest rigorous use of induction was by Gersonides. The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. Another Frenchman, Fermat, made ample use of a related principle: indirect proof by infinite descent.
The induction hypothesis was also employed by the Swiss Jakob Bernoulli, and from then on it became well known. The modern formal treatment of the principle came only in the 19th century, with George Boole, Augustus De Morgan, Charles Sanders Peirce, Giuseppe Peano, and Richard Dedekind.
Description
The simplest and most common form of mathematical induction infers that a statement involving a natural number holds for all values of. The proof consists of two steps:- The ' : prove that the statement holds for 0, or 1.
- The ' : prove that for every, if the statement holds for, then it holds for. In other words, assume that the statement holds for some arbitrary natural number, and prove that the statement holds for.
Authors who prefer to define natural numbers to begin at 0 use that value in the base case; those who define natural numbers to begin at 1 use that value.
Examples
Sum of consecutive natural numbers
Mathematical induction can be used to prove the following statement for all natural numbers :This states a general formula for the sum of the natural numbers less than or equal to a given number; in fact an infinite sequence of statements:,,, etc.
Proposition. For every, we have that
Proof. Let be the statement We give a proof by induction on.
Base case: Show that the statement holds for the smallest natural number.
is clearly true:
Induction step: Show that for every, if holds, then also holds.
Assume the induction hypothesis that for a particular, the single case holds, meaning is true:
It follows that:
Algebraically, the right hand side simplifies as:
Equating the extreme left hand and right hand sides, we deduce that: That is, the statement also holds true, establishing the induction step.
Conclusion: Since both the base case and the induction step have been proved as true, by mathematical induction the statement holds for every natural number. Q.E.D.
A trigonometric inequality
Induction is often used to prove inequalities. As an example, we prove that for any real number and natural number.At first glance, it may appear that a more general version, for any real numbers, could be proven without induction; but the case shows it may be false for non-integer values of. This suggests we examine the statement specifically for natural values of, and induction is the readiest tool.
Proposition. For any and,.
Proof. Fix an arbitrary real number, and let be the statement. We induce on.
Base case: The calculation verifies.
Induction step: We show the implication for any natural number. Assume the induction hypothesis: for a given value, the single case is true. Using the angle addition formula and the triangle inequality, we deduce:
The inequality between the extreme left-hand and right-hand quantities shows that is true, which completes the induction step.
Conclusion: The proposition holds for all natural numbers Q.E.D.
Variants
In practice, proofs by induction are often structured differently, depending on the exact nature of the property to be proven.All variants of induction are special cases of transfinite induction; see [|below].
Base case other than 0 or 1
If one wishes to prove a statement, not for all natural numbers, but only for all numbers greater than or equal to a certain number, then the proof by induction consists of the following:- Showing that the statement holds when.
- Showing that if the statement holds for an arbitrary number, then the same statement also holds for.
In this way, one can prove that some statement holds for all, or even for all. This form of mathematical induction is actually a special case of the previous form, because if the statement to be proved is then proving it with these two rules is equivalent with proving for all natural numbers with an induction base case.
Example: forming dollar amounts by coins
Assume an infinite supply of 4- and 5-dollar coins. Induction can be used to prove that any whole amount of dollars greater than or equal to can be formed by a combination of such coins. Let denote the statement " dollars can be formed by a combination of 4- and 5-dollar coins". The proof that is true for all can then be achieved by induction on as follows:Base case: Showing that holds for is simple: take three 4-dollar coins.
Induction step: Given that holds for some value of , prove that holds, too. Assume is true for some arbitrary. If there is a solution for dollars that includes at least one 4-dollar coin, replace it by a 5-dollar coin to make dollars. Otherwise, if only 5-dollar coins are used, must be a multiple of 5 and so at least 15; but then we can replace three 5-dollar coins by four 4-dollar coins to make dollars. In each case, is true.
Therefore, by the principle of induction, holds for all, and the proof is complete.
In this example, although also holds for, the above proof cannot be modified to replace the minimum amount of dollar to any lower value. For, the base case is actually false; for, the second case in the induction step will not work; let alone for even lower.
Induction on more than one counter
It is sometimes desirable to prove a statement involving two natural numbers, and, by iterating the induction process. That is, one proves a base case and an induction step for, and in each of those proves a base case and an induction step for. See, for example, the proof of commutativity accompanying addition of natural numbers. More complicated arguments involving three or more counters are also possible.Infinite descent
The method of infinite descent is a variation of mathematical induction which was used by Pierre de Fermat. It is used to show that some statement is false for all natural numbers. Its traditional form consists of showing that if is true for some natural number, it also holds for some strictly smaller natural number. Because there are no infinite decreasing sequences of natural numbers, this situation would be impossible, thereby showing that cannot be true for any.The validity of this method can be verified from the usual principle of mathematical induction. Using mathematical induction on the statement defined as " is false for all natural numbers less than or equal to ", it follows that holds for all, which means that is false for every natural number.
Limited mathematical induction
If one wishes to prove that a property holds for all natural numbers less than or equal to a fixed, proving that satisfies the following conditions suffices:- holds for 0,
- For any natural number less than, if holds for, then holds for