Not only every student, but every self-respecting educated person should know what theorem and proof of theorems. Maybe such notions are not meet in real life, but to structure a knowledge as well as to make inferences they will really help. That is why we consider in this article the methods of proofs of theorems, as well as get acquainted with such a famous theorem of Pythagoras.
If we consider the school course of mathematics, very often there are such scientific terms as theorem, axiom, definition and proof. In order to navigate the program, you need to read each of these definitions. Now we consider what theorem and proof of theorems.
So, theorem – this is a statement that requires proof. To consider the concept of need in parallel with the axiom, since the latter proof is not required. Its definition is already true, so taken for granted.
It is a mistake to think that the theorems only apply in mathematics. Actually it is not so. For example, there are an incredible number of theorems in physics, allowing in detail and from all sides to consider some phenomena and concepts. This can be attributed to the theorem of ampere, Steiner and many others. The proof of these theorems enable a good deal in the moments of inertia, statics, dynamics, and many other concepts of physics.
It is Hard to imagine such a science as mathematics without theorems and proofs. For example, proofs of theorems of the triangle, allow to study in detail all properties of the figure. It is very important to understand the signs of similarity, properties of an isosceles triangle, and many other things.
The proof of the theorem of the square allows us to understand what is the easiest way to calculate the area of shapes based on some data. Because, as you know, there are a large number of formulas describing how to find the area of a triangle. But before to use them, it is very important to prove that it is possible and rational in the specific case.
Each student must know what a theorem, and theorem proving. In fact, to prove any claim not so easy. For this you need to operate on many data and be able to make logical conclusions. Of course, if you know the information about a particular scientific discipline, then to prove the theorem, you will not be difficult. The main thing - to perform the proof procedure in a logical sequence.
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In order to learn how to prove theorems in such scientific disciplines as geometry and algebra, you need to have good knowledge and know the algorithm proof. If you master this procedure, you solve math problems later you will not be difficult.
What is the theorem and proof of theorems? This is a question that worries many people in modern society. It is very important to learn how to prove mathematical theorems, it will help you in the future to build a logical chain and to come to a conclusion.
So, in order to prove the theorem correct, it is very important to make the right picture. It will display all the data that was specified in the condition. It is also very important to record all information that was provided in the task. This will help you to correctly perform the task and understand exactly what value it is given. And only after such procedures, you can proceed to the proof. To do this you need to build a logical chain of thoughts using other theorems, axioms or definitions. A summary of the evidence must be the result, the truth of which is unquestionable.
In the school course of mathematics there are two ways how to prove the theorem. Often in problems using the direct method and the method of proof by contradiction. In the first case, just analyze the available data and, based on them, make respective conclusions. Also a very often used method by contradiction. In this case, we assume the opposite and prove that it is not true. Based on this, we get the opposite result and say that our judgment was wrong, which means that the specified in the condition information is correct.
In fact, many mathematical problems can have multiple solutions. For example, Fermat's last theorem has several proofs. Of course, some are treated only one way, but, for example, the Pythagorean theorem, you can consider several of them.
Of Course, every schoolboy knows that the Pythagorean theorem concerns the right triangle. It sounds like this: “the square of the hypotenuse is equal to the sum of the squares of the legs». Despite the name of this theorem is opened, it was not by Pythagoras, and even before him. Thereseveral ways to prove this assertion, we will look at some of them.
According to statistics, in the beginning it was considered a rectangular equilateral triangle. Then build squares on all sides. The square constructed on the hypotenuse, will consist of four equal triangles. While the figures constructed on the sides will consist of two such triangles. This proof of the Pythagorean theorem is the easiest.
Consider another proof of this theorem. It is necessary to use the knowledge not only of geometry but also algebra. In order to prove this theorem in this way, we need to build four similar right triangle, and sign them as a, b and C.
To Build these triangles need so that in the end we got two squares. External of them will have sides (a+b), but the interior – p. to find the area of the inner square, we need to find a product with*s. But in order to find the area of the big square, you need to fold the square in small squares and add square received rectangular triangles. Now, after making some algebraic operations, we can obtain the following formula:
In fact, there are plenty of methods of proof of theorems. Perpendicular, triangle, square or any other shapes and their properties can be considered by applying different theorems and proofs. The Pythagorean theorem is the only proof of that.
It is Very important be able to formulate theorems as well as prove it. Of course, this procedure is rather complicated, since its implementation must not only be able to process a large amount of information, but also to build a logical chain. Mathematics – this is a very interesting science that has no end or edge.
Start to study it, and you will not only increase the level of your intellect, but also get a huge amount of interesting information. Loans to their education today. Once you understand the basics of math proofs, you will be able to spend time with great benefit.
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