Pythagoras’ Theorem
Rectangular triangles with sides equal to 3,4,5 were known in ancient Egypt; it was often used by people engaged in stringing. Pythagoras used algebraic methods to find the Pythagorean triples, an ordered set of three natural values. This information was generally accepted, and already Plato had given his hypothesis of finding where the Pythagorean triples combine algebra and geometry. Henceforth, the resulting theory received the name Pythagoras’ theorem, in honor of its creator.

According to the theorem, in a right-angled triangle, the length of the hypotenuse (the side lying against the angle) is equal to the sum of the lengths of the square cathetuses (the sides that form the angle). The formula is as follows:

a2+b2=c2

More than 400 proofs of the theorem were later presented, denoting its fundamental meaning. The most common rhyme used in school folklore is “Pythagoras’ pants are equal in all directions,” a name given by the comic opera Ivanov Paul.

Pythagoras’ theorem is often used to solve problems in geometry, algebra, and physics. In life, it is often used to calculate building and architectural structures.

Thales theorem
Thales’ theorem is that a pair of secant lines always forms equal segments to a pair of straight lines. This conclusion was reached by the Greek mathematician Thales of Miletus, who, according to legend, calculated the height of the pyramid of Cheops by measuring the shadow on the ground and its length. The formulation of Thales’ theorem is as follows:

(A1A2)/(B1B2)=(A2A3)/(B2B3), etc.

The Argentinean musical group “Les Luthiers” even dedicated a song to this theorem. Today, when designing various objects or models, design engineers often turn to this theorem for help.

The Sinus Theorem
The first mention of the sine theorem was in a chapter of the Almagest, but not a direct statement. Of the first ancient proofs that have come down to us on the plane, Nasir ad-Din At-Tust’s book “Treatise on the Total Quadrilateral”, written in the 13th century, is believed to be the first.

The sides of triangles are directly proportional to the sines of the opposing angles, in practice it looks like this:

(a/sin a)=(b/sin B)=(c/sin Y)

The trigonometric theory of sines is still used to this day, and is used by auto mechanics, factory workers, and even girls who draw eyebrows with a pencil.

Menelaus’ Theorem
Menelaus’s Theorem or Quadrilateral Theorem was proved in the third book of Spherika by the ancient Greek mathematician Menelaus of Alexandria. The original proof was presented for the flat case, and it was not until some time later that Menelaus transferred it to the sphere. Most theorems in project geometry are based on Menelaus’ theorem, which is formulated like this: if the points A1, B1 and C1 lie on the sides BC, CA and AB of triangle ABC then they are collinear. There are a huge number of variations of the theory, where it takes on a form depending on the direction of use:

-trigonometric equivalent;
-spherical geometry;
-Lobachevsky geometry.

The use of Menelaus’ theorem will simplify the solution of many problems and calculate the areas of figures for estimators.

The Viette Theorem
Thanks to the Vieth theorem, the coefficients of a polynomial and its roots are connected. The formulas are great for checking the correctness of finding the roots of a polynomial, as well as for composing a polynomial according to the given roots. This theorem was discovered by the French scholar François Viet while in the royal service as an advisor. The formulation of his theory is as follows:

If C1, C2, C3 are roots of a polynomial, then xn=a1xn-1+a2xn-2+an

In mathematics, Viet’s theorem is often used to solve quadratic or cubic equations using the system method. In life, when calculating apartment buildings, only specialists use it not on their own, but through special programs that perform the necessary calculations.

Many people, studying geometry at school and at universities, believe that these are rules nobody needs, although in fact they are found in various fields. Without this fundamental knowledge it would be difficult for many professionals to work, so you should not neglect the knowledge and carefully study geometry and its most common theories.

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Whether you embrace it or dread it, math is a necessity in our lives. True, the subject may not be thrilling for everyone, but its utility makes up for what it lacks in excitement. You can apply math skills to calculate your miles when running at the park, budget your summer job paychecks, and eventually manage your own checkbook – all of which culminates into an invaluable ability that pays off over time.

For many students, math can be especially difficult. If you experience any difficulties while doing homework, you can get help from the online assignment writing service. A math expert will help you with tasks of any difficulty. Also, such assistance is designed to allow students to understand the material better.

Tips for doing math homework

For those who find their minds frequently drifting away, staying on top of math is no easy feat. Fortunately, these simple tips can help you remain ahead in this critical subject matter – even if it’s during summer school or as classes resume come September! With a bit of extra effort and focus, anyone can excel in mathematics.

1. Location is everything. Identify any possible distractions that may be preventing your focus, whether it’s requesting a seat in the front near the board or away from windows and friends alike.
2. Don’t be afraid to use your voice! If you don’t understand something in class, ask the teacher. It’s likely others are having difficulty understanding too. Furthermore, is there a particular time of day when you feel more focused? Request from your guidance counselor whether it would be possible for you to arrange math classes at that hour so that what is being taught can really sink into your brain. Asking questions and speaking up will help ensure success in mathematics!
3. Strengthen your math skills by actively working to solve problems. Don’t just try to figure it out; you must actually work the equations and formulas in order to truly understand them. Make sure that you are engaging with these topics regularly, no matter what time of year or if homework is assigned – consistency will be key! With consistent effort, even complex mathematics becomes easier and more manageable for everyone involved.
4. Combat the boredom that comes with math tasks. Instead of sitting in front of your math homework for hours, join forces with peers and make studying fun! Forming a study group in the summer can help you stay ahead when September lessons come around – other people will be able to keep you engaged by making mathematics interactive. Don’t let the doldrums set in; liven up your learning process today!

Generally, the more complex the material is, the longer you should devote to it. Consider having your group meet at least one or two times a week so that by the end of the course, there will be no difficulty in catching up with lessons. Additionally, find creative ways to use math concepts learned from class outside of lectures – for instance, try adding numbers related to budgeting instead of random figures! This way, the math may become even more enjoyable.

How to prepare for tests

Although English and History classes require students to compose papers, math grades are more dependent on exams. While it may be too soon for a test this semester, you should still take proactive steps toward conquering any test-taking anxiety now! It’s not uncommon for some students to feel apprehensive when it comes to taking tests in mathematics — here are some methods that can help reduce the stress:

1. To achieve success on your upcoming exams, be sure to study steadily rather than waiting until the last minute. Doing so will help you move facts from short-term to long-term memory — a resource that will prove invaluable when taking tests in spite of any butterflies in your stomach. Be proactive and get support from teachers who may be able to fortify those grades hovering between Cs and Bs if they see how hard you’ve worked!
2. Advocate for yourself. Discover the accommodations accessible at school and request them. Perhaps you’d like to be able to use a calculator during tests? If finishing on time causes anxiety, ask for extra time—if distractions are your problem, then inquire about being in a quiet room when taking exams apart from the rest of the class.
3. To excel on a test, make sure you’re getting adequate rest. A study conducted by Trent University in Peterborough, Ontario, indicated that having an established sleep schedule is crucial to retaining information and performing optimally. Skimping out on weekends can have a 30% adverse effect when it comes to comprehending complex concepts! Therefore, if you want the most out of your studying sessions and results from exams, proper rest is critical to success!

Conclusion

Math is a subject that requires dedication and effort, but with the right strategies in place, you can make it less overwhelming. Make sure you take the time to ask questions when needed and form study groups to tackle concepts collaboratively. Don’t be afraid to advocate for yourself; your teacher may be able to assist in helping you reach your full potential! Lastly, remember that rest is essential – having an established sleep schedule will help ensure that information stays with you and improve test performance. By putting all of these pieces together, math becomes far more manageable! 

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The need to accept axioms without proof follows from an inductive consideration: any proof has to rely on some statements, and if one requires its own proof for each of them, the chain “statement – proof” would become infinite. To avoid infinity, it is necessary to break this chain somewhere, that is, to accept some statements without proofs as initial. Exactly such statements, accepted as initial, are called axioms.

Typical examples of axioms:

Some expression of the symbolic language of the calculus, if by further reasoning is understood to use its conclusions within that calculus. In this case, the reason for accepting axioms is the very definition of the calculus in question. In this case, doubts about the acceptance of the axioms are meaningless.
Some empirical hypothesis, if by further reasoning we mean, for example, a section of physics systematically developed on its basis. In this case the reason for accepting an axiom is the belief in the regularity of nature expressed by this hypothesis. In this case, doubts about the acceptance of the axiom are not only meaningful, but also desirable.
The agreement to understand the terms involved in the formulation of some judgement as one wishes, but still in such a way that with this understanding the formulation in question expresses a true judgement. This is the case in which further reasoning is understood as the derivation of knowingly true consequences from an ambiguously understood initial judgment. In this case, doubts about accepting the axiom are meaningless. When this kind of axiom is used within a scientific theory, it is often called a postulate of meaning.

A statement evaluated as necessarily true (apodictic) if further reasoning is understood as some systematically developed doctrine that claims to be epistemologically perfect. In this case, the reason for accepting an axiom is evidence of a special cognitive capacity (intuition) to directly discern certain (often called self-evident) truths. Within this claim, doubts about the acceptance of an axiom are meaningless, but the question of the validity of the claim itself is one of the most significant problems in philosophy (see Philosophy).
Axioms emerge in the long and complex development of scientific cognition. From antiquity until the nineteenth century, axioms were seen not simply as a starting point of evidence, but as intuitively obvious or a priori true propositions. The importance of axioms was substantiated by Aristotle who believed that axioms do not require proof because they are clear and simple because they “possess the highest degree of generality and represent the beginning of everything”.

Euclid considered axioms accepted by him within his geometrical system as self-evident truths, sufficient to deduce all other truths of geometry. On the basis of the accumulated by that time facts and knowledge he singled out and formulated several axiomatic statements (postulates), accepted without proofs, from which their logical consequences were deduced in the form of theorems. At the same time axioms were often treated as eternal and immutable truths, known before any experience and not depending on it, the attempt to justify which could only undermine their obviousness. Kant’s doctrine of the a priori nature of axioms, that is, that they precede all experience and do not depend on it, was the culmination of such views on axioms.

The rethinking of axioms is connected with the discovery in the 19th century of non-Euclidean geometry (C. F. Gauss, N. I. Lobachevsky, J. Boiai); the appearance in abstract algebra of new number systems, and their whole families at once; the appearance of variable structures like groups; finally, the wide discussion of questions like “which geometry is true?” All this contributed to the realization of two new statuses of axioms: axioms as descriptions (classes of possible universes of reasoning) and axioms as assumptions rather than self-evident assertions.

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Algebra is a branch of mathematics in which numbers are replaced by letters, and an algebraic equation is a scale in which what is done on one side of the scale is also done on the other side of the scale, and the numbers act as constants. Algebra can include real numbers , complex numbers, matrices, vectors, and many other forms of mathematical representation.

The field of algebra can be further broken down into basic concepts known as elementary algebra, or the more abstract study of numbers and equations known as abstract algebra, where the former is used in much of mathematics, science, economics, medicine, and engineering, while the latter is mostly used only in higher mathematics.

Practical applications of elementary algebra
Elementary algebra is taught in all schools in the United States, beginning in the seventh and ninth grades, and continuing through high school and even college. The subject is widely used in many fields, including medicine and accounting, but it can also be used to solve everyday problems when it comes to the unknown variables in mathematical equations.

One such practical application of algebra would be if you were trying to determine how many balloons you started the day with if you sold 37 but had 13 left. The algebraic equation for this problem would be x – 37 = 13, where the number of balloons you started with is represented by x, the unknown we are trying to solve.

The goal of algebra is to find the unknown, and to do that in this example, you have to manipulate the scale of the equation to isolate x on one side of the scale by adding 37 to both parts, which leads to the equation x = 50 meaning that you started the day with 50 balloons if you had 13 after selling 37 of them.

Why Algebra Matters
Even if you don’t think you’ll need algebra outside the hallowed halls of high school, managing a budget, paying bills, and even determining health care costs and planning for future investments will require a basic understanding of algebra.

Along with the development of critical thinking, especially logic, patterns, problem solving , deductive and inductive reasoning, an understanding of basic algebra concepts can help people better handle complex tasks involving numbers, especially when they enter the workplace where real life scenarios involve unknown variables. to costs and profits require employees to use algebraic equations to determine missing factors.

Ultimately, the more one knows about mathematics, the better chance one has of succeeding in engineering, actuarial, physics, programming, or any other field related to technology, and algebra and other higher mathematics courses are usually required for university admission. most colleges and universities.

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Many objects are rectangular, others are round, others are triangular. There are also more complex shapes.

If you look more closely, you can see that the same rectangle consists of four segments, which form its sides. That is, we can say that most shapes are made up of simpler shapes. All figures consist of points. Therefore, a point can be considered the simplest element.

When describing figures, it is important not only to specify the geometrical primitives of which it consists, but also the “relations” between them. For example, a rectangle does not simply consist of four segments, but they must be joined together; the angles formed by the joined segments must be straight; besides, the segments must be equal in pairs, and the segments of equal length must be on opposite sides.

At the same time, rectangles can be different. One is more elongated on one side and more like a bar, the other has a width and length do not differ much, and such a rectangle is similar to a square. And understandably, rectangles can vary in size. All this says that by the term “rectangle” we mean a set of figures which satisfy certain requirements.
Geometry is an ancient science. It originated about four or five thousand years ago. People since ancient times have needed to measure land, distances, various objects, to make measurements when constructing buildings. The word “geometry” in translation from Greek means “surveying”.

At first, history accumulated the rules of various geometric constructions. Then, in ancient Greece, there were scientists who brought a lot of new things to geometry. In particular, they began to pay more attention to reasoning, on the basis of which new facts and regularities could be discovered. We can say that geometry as a science was formed by the beginning of our era.

The practical importance of geometry is great. In addition, it teaches one to reason, to see the world of forms in their interconnection and interaction.

The science of geometry is divided into two large sections, planimetry and stereometry. Planimetry studies shapes in the plane. These are rectangles, triangles, circles, trapezoids, and other quadrilaterals. Stereometry studies shapes in three-dimensional space. These include a ball, cube, cylinder, pyramid, and many others.

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