The earliest known evidence of using mathematical induction appeared in the book Arithmetic by Francisco Maurorico (1575). Maurolico proved that the sum of the first n odd numbers is n 2.
The simplest and most common proof of mathematical induction is to prove that when n belongs to all natural numbers, an expression becomes. The method comprises the following two steps:
The basis of recursion: prove that the expression holds when n = 1
The basis of recursion: prove that it is true when n = m, and it is also true when n = m+ 1 (The "if" in the recursive basis is defined as inductive hypothesis. Don't call the whole second step inductive hypothesis. )
The principle of this method is to first prove that the initial value in the expression is valid, and then prove that the proof process from one value to the next is effective. If these two steps are proved, then the proof of any value can be included in the repeated process. Perhaps it is easier to understand the domino effect; If you have a long row of upright dominoes, then if you can be sure:
The first domino will fall.
As long as a domino falls, the next domino will fall.
Then you can infer that all the dominoes are going to fall.
The principle of mathematical induction, as an axiom of natural numbers, is usually prescribed (see article 5 of Piano's axiom). But it can be proved by some logical methods; For example, if the following axiom:
Natural number set is orderly.
Be used.
It is worth noting that some other axioms are actually the formulation of alternatives in the principle of mathematical induction. More precisely, the two are equivalent.
Steps to prove by mathematical induction:
(1) (inductive basis) proves that the proposition holds when the first value is taken; It is proved that in the first step, the recursive basis is obtained, but this step alone cannot explain the universality of the conclusion. In the first step, it is enough to examine the smallest positive integer of the conclusion, and it is not necessary to examine several positive integers. Even if the proposition holds true for these positive integers, there is no guarantee that the proposition holds true for other positive integers.
(2) (inductive recursion) The proposition holds when assuming, which proves that the proposition also holds when assuming; It is proved that the second step leads to the foundation of recursion, but without the first step, the foundation of recursion is lost. Only by combining the first step with the second step can we draw a general conclusion.
(3) Conclusion: This proposition holds for all positive integers from the beginning.
note:
When (1) is proved by mathematical induction, the two steps of "inductive basis" and "inductive recursion" are indispensable;
(2) In the second step, before recursion, we are not sure whether the conclusion of time is true, so we use the word hypothesis. The essence of this step is to prove that the correctness of the proposition pair can be passed on to time. With this step, we can know that the proposition pair is also true by connecting the conclusion of the first step (the proposition pair is true), and then we can know that it is also true from the second step. In this way, we can know that it holds for all positive integers not less than. In this step, the time proposition is established and can be used as a condition, while the time situation needs to be proved by inductive hypothesis, known definition, formula and theorem, and cannot be directly substituted into the proposition.