Not only every student, but also every educated self-respecting person should know what a theorem and proof of theorems are. Maybe such concepts will not be found in real life, but they will definitely help structure many knowledge, as well as make conclusions. That is why we will consider in this article the methods of proving theorems, and also get acquainted with such a famous Pythagorean theorem.
What is a theorem?
If we consider the school course of mathematics, then very often it contains such scientific terms as a theorem, axiom, definition and proof. In order to navigate the program, you need to familiarize yourself with each of these definitions. Now we consider what a theorem and proof of theorems are.
So, a theorem is a certain statement that requires proof. It is necessary to consider this concept in parallel with the axiom, since the latter does not require proof. Its definition is already true, therefore it is taken for granted.
Scope of Theorems
It is a mistake to think that theorems apply only in mathematics. In fact, this is far from the case. For example, there is simply an incredible number of theorems in physics that allow us to examine in detail and from all sides some phenomena and concepts. These include the theorems of Ampere, Steiner, and many others. The proofs of such theorems allow a good understanding of the moments of inertia, statics, dynamics, and many other concepts of physics.
Using Theorems in Math
It is hard to imagine a science like mathematics without theorems and proofs. For example, the proofs of triangle theorems allow us to study in detail all the properties of a figure. After all, it is very important to understand the signs of similarity, the properties of an isosceles triangle and many other things.
The proof of the area theorem allows us to understand how it is easiest to calculate the area of a figure, based on some data. After all, as you know, there are a large number of formulas that describe how you can find the area of a triangle. But before using them, it is very important to prove that this is possible and rational in a particular case.
How to prove the theorems
Every student should know what a theorem is and a proof of theorems. In fact, proving a statement is not so simple. To do this, you need to operate with many data and be able to draw logical conclusions. Of course, if you have good knowledge of information on a certain scientific discipline, then proving a theorem will not be difficult for you. The main thing is to carry out the proof procedure in a certain logical sequence.
In order to learn how to prove theorems in such scientific disciplines as geometry and algebra, you need to have a good knowledge base, as well as know the proof algorithm itself. If you master this procedure, then solving math problems later on will not be difficult for you.
What you need to know about proving theorems
What is a 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, this will help you build logical chains in the future and come to a certain conclusion.
So, in order to prove the theorem correctly, it is very important to make a correct drawing. On it, display all the data that was specified in the condition. It is also very important to record all the information that was provided in the task. This will help you analyze the task correctly and understand exactly what quantities are given in it. And only after carrying out such procedures can one proceed to the proof itself. To do this, you need to logically build a chain of thoughts using other theorems, axioms or definitions. The result of the proof should be a result whose truth is not in doubt.
Basic methods for proving theorems
In a school course in mathematics, there are two ways to prove a theorem. Most often, a direct method is used in problems, as well as a method of proof from the contrary. In the first case, they simply analyze the available data and, drawing on them, draw the appropriate conclusions. Also very often used and the method of the contrary. In this case, we assume the opposite statement and prove that it is false. Based on this, we get the opposite result and say that our judgment was incorrect, which means that the information specified in the condition is correct.
In fact, many mathematical problems can have several solutions. For example, Fermat's theorem has several proofs. Of course, some are considered in only one way, but, for example, several of them can be considered at once in the Pythagorean theorem.
What is the Pythagorean theorem
Of course, every schoolchild knows that the Pythagorean theorem concerns precisely a right-angled triangle. And 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, it was not discovered by Pythagoras himself, but long before him. There are several ways to prove this statement, and we will consider some of them.
According to scientific evidence, an equilateral right triangle was considered at the very beginning. Then squares were built on all its sides. A square built on hypotenuse will consist of four triangles equal to each other. While the figures built on the legs, will consist of only two of the same triangles. Such a proof of the Pythagorean theorem is the simplest.
We consider one more proof of this theorem. It needs to use knowledge not only from geometry, but also from algebra. In order to prove this theorem in this way, we need to construct four similar right-angled triangles, and sign their sides as a, b, and c.
We need to construct these triangles in such a way that as a result we get two squares. The outer of them will have sides (a + c), but the inner one will have sides. In order to find the area of the inner square, we need to find the product s * s. But in order to find the area of a large square, you need to add up the area of small squares and add the area of the resulting right-angled triangles. Now, having performed some algebraic operations, we can obtain the following formula:
and 2 + to 2 = s 2
In fact, there are a huge number of methods for proving theorems. A perpendicular, triangle, square or any other figures and their properties can be considered using various theorems and proofs. The Pythagorean theorem only confirms this.
Instead of a conclusion
It is very important to be able to formulate theorems, as well as correctly prove them. Of course, such a procedure is quite complicated, since for its implementation it is necessary not only to be able to operate with a large amount of information, but also to build logical chains. Mathematics is a very interesting science, which has no end, no edge.
Start to study it, and you will not only increase your level of intelligence, but also get a huge amount of interesting information. Take care of your education today. Having understood the basic principles of the proof of theorems, you can spend your time with great benefit.