method of mathematical induction can be equated to progress.So, starting from the lowest level, researchers with the help of logical thinking to go higher.Any self-respecting man constantly strives for progress and the ability to think logically.That is why nature created by inductive reasoning.
term "induction" in translation into Russian means guidance, so the inductive assumed certain findings of experiments and observations, which are obtained by forming from the particular to the general.
example might be contemplating the sunrise.Observe this phenomenon for several days in a row, we can say that the sun will rise in the east tomorrow, and the day after, etc.
inductive inference is widely used in experimental and applied sciences.So, with the help of these provisions can be formulated on the basis of which are already using deductive methods can be made further deductions.With some confidence we can assert that the "three pillars" of theoretical mechanics - Newton's laws of motion - are themselves the result of experiments with private summing grand total.But Kepler's law of planetary motion was put to them on the basis of long-term observations of T. Brahe, Danish astronomer.It is in these cases induction has played a positive role to clarify and summarize the assumptions made.
Despite the expansion of its field of application method of induction, unfortunately, takes a little time in the school curriculum.However, in today's world it is a childhood need to teach the younger generation to think inductively, not just to solve problems in a certain pattern or a predetermined formula.
method of mathematical induction can be widely used in algebra, arithmetic and geometry.These sections should be carried out proof of the truth of a set of numbers, which depends on natural variables.
principle of induction is based on the proof of the truth of proposals A (n) for all values of the variable and consists of two phases:
1. true sentence A (n) is proved for n = 1.
2. Where the proposal A(n) keeps truth for n = k (k - a positive integer), it will be true for the following values of n = k + 1.
This principle and formulate a method mat.induction.Often, it is accepted as an axiom that defines the series of numbers, and is without evidence.
There are times when the method of mathematical induction in some cases, subject to proof.So, when you want to prove the truth offers a variety of A (n) for all positive integers n, you must:
- to check on the truth of the statement A (1);
- prove the truth of statements A (k + 1), while taking into account the truth of A (k).
In case of a successful proof of the validity of this offer for any positive integer k recognized as a true sentence A (n) for all values of n, in accordance with this principle.
Powered induction method is widely used in the proofs of identities, theorems, inequalities.It can also be used in solving the problems of geometric nature and divisibility.
However, we should not think that this ends the use of a method of induction in mathematics.For example, do not necessarily experimentally verify all theorems are logically derived from the axioms.But while these axioms have the opportunity to formulate a large number of claims.And that is the choice of the allegations and suggested by using induction.With this method, you can share all of the theorem on the necessary science and practice, and not very much.
