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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Graphics
Visual component of modern single and online games are textures of high quality, animation, three-dimensional character models. 3D-modeling technology is developed taking into account mathematical concepts such as: vectors, matrices.

The knowledge is used by programmers to create realistic lighting, believable three-dimensional models of people, animals. Similar technologies are used in the development of video slots – entertainment at online casinos https://onlinecasinozonder.com/. Gambling games acquire features similar to the rating titles with multimillion-dollar budgets.

Artificial intelligence
The games use mathematical principles, as far as non-player characters – NPCs. This manifests itself in the gameplay when the player interacts with the NPC – controlled by a program. The laws of logic, as well as certain regularities, are the basis of the technology.

Features of non-player characters and computing algorithms:

-React to player action. Characters interact with the user: respond, perform an action. This is due to the laws of algorithms and logic.
-Scripts. NPCs’ actions are explained by the work of a script, which was originally programmed by the developer. This means that the character can not make an independent decision.
-Task. Characters serve a specific purpose in the game, as well as being primary or secondary.

There are non-player characters that are not controlled by the script, but by a representative of the development team. This approach is used in online projects.

Calculation and analysis of data
Mathematics in computer games remains an effective tool in the collection of analytical information. Data is needed to fix software bugs, bugs, improve the performance of the game on weak devices.

Mathematical models are added to automate the process of collecting information. This allows to speed up the work, as well as to avoid the drawback of the human factor. Modern technology is also used to analyze information when complaints and reports of different types are automatically categorized.

Developers thus work with the prepared material, which concerns the stability of the game on different systems. Calculations are based on basic mathematical laws. Without the use of formulas, the system would not work correctly.

Physics
Physical laws are used in most modern and classic game projects. This allows for a realistic gameplay experience. Most games with high scores on aggregators are made with an emphasis on realism – the laws of physics, nature.

Realism manifests itself in features such as:

-First-person shooters. Projects reliably recreated the principle of firearms and throwing weapons. This is due to the physical laws of motion, energy.
-Simulators. The most complex game projects, if we consider the issue in terms of physical laws. The authors are working to reliably recreate the movement of objects. Players are convinced of realism, which concerns the geometric shapes of objects: trees, mountains.
-The genre of sci-fi games. With the appropriate setting developers have to use the laws of physics, which are associated with the movement of objects in space. Accounted for the behavior of the body in weightlessness, the influence of gravity.

Physics in the games and refers to the destruction. There are several projects, which thoroughly implemented the destruction of objects: boxes, furniture, buildings, glass and other materials. The impact of a bullet of a firearm on an object is studied.

In racing physical laws are associated with damage to the car, with the impact of high speed on the object. In modern car simulators, vehicles “react” to the road surface, to the level of wheel inflation, to the weather conditions, and to the shape of the exterior elements of the car body.

Conclusion
Mathematics in games is the basis for the correct operation of scripts, item physics, and behavioral algorithms. Without the benefits of exact science, project development would take dozens of times longer, as well as require a lot of human resources. The use of mathematical formulas has made it possible to automate the process of improving games, eliminating bugs and errors.

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One of the first to realize this was Alan Turing, who gave a precise definition and the most convincing analysis of the concept of a computable function in 1936. Other scientists who, along with Turing, came up with the same ideas around the same time were Alonzo Church and Emile Post. These and other researchers in the 1930s created the beginnings of algorithm theory, which became the basis for understanding the operation and construction of computing devices in the 40s and 50s. In particular, the idea of the universal Turing machine was later technically realized in the “von Neumann” computer architecture, according to which the program is stored in the memory of the device and can be modified during its operation. All operating systems are based on this idea.

The problem that the pioneers of algorithm theory sought to solve was the question posed by Hilbert, which he called in German Entscheidungsproblem, “the solution problem. The question was to find an algorithm that, for a given statement written in the language of predicate logic, would answer whether or not there was a formal proof of that statement. If such an algorithm existed, then all mathematical problems would, in a sense, have a purely mechanical solution: as mentioned above, almost any mathematical statement can be formulated in the language of predicate logic, such as the famous hypothesis about the infinity of the number of pairs of prime twin numbers. Then the question of whether this hypothesis is deducible from the axioms of set theory would be reduced to checking the provability of some statement in predicate calculus. Not surprisingly, the researchers of the Entscheidungsproblem sought to show that the required algorithm could not exist in principle.

If it is possible to prove the existence of a particular algorithm by presenting it explicitly, in order to argue that such an algorithm does not exist, it is necessary to have an exact mathematical description of the class of problems that admit an algorithmic solution. The answer to this question required the development of formal languages to describe algorithms even before computers appeared. Moreover, since the goal of such development was more theoretical than practical, researchers sought to formulate the simplest for description and at the same time universal computational models. The first such models were Gödel-Erbrand recursive functions, Church lambda calculus and Turing machines.

Gödel, though he was the first to actually formulate a universal programming language, did not believe that the concept he (and the French logician Erbrand) found was universal in the sense of being able to program any algorithm. Alonzo Church was the first to make a thesis about the universality of his computational model. He also presented a proof of the impossibility of solving the Entscheidungsproblem within this model. Church’s calculus was very simple in form, but it looked more like a formal logical calculus than a real computing machine. Turing machines were in this sense closer to future reality, and it was much easier to believe Turing’s thesis that any problem with an algorithmic solution could be solved by a Turing machine than a similar thesis by Church (Turing’s work is known to have convinced Gödel of the validity of this thesis). Turing also showed that his machines were equivalent to lambda-calculus in terms of computational capability, which was indirect evidence for the validity of the Church-Turing thesis, as it is now commonly called.

Subsequently, many researchers proposed their own computational models in the hope of extending the class of computable functions first described by Church and Turing. All such attempts failed to expand this class, which turned out to be very stable. Today, the Church-Turing thesis–understood in the sense of any of the equivalent computational models–is one of the cornerstones on which algorithm theory is based.

As for lambda-calculus, it has long been on the margins of mathematical logic, being displaced from the theory of algorithms by more convenient and intuitive models. However, in the second half of XX century lambda-calculus and systems based on it have found serious practical applications. Lambda-calculus has become a prototype of so called functional programming languages (such as the modern Haskell language), which have a number of advantages over traditional imperative languages and are now very actively developing.

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Among these kinds of problems, the most famous is Hilbert’s 10th problem about recognizing the solvability of dioffant equations. A Diophant equation is an equation of the form , where is a polynomial with integer coefficients on the variables , …, . It is required to find out from a given polynomial , whether there are integers , …, , satisfying this equation.

Hilbert’s question of constructing a general algorithm that works for all Diophantine equations looked hopeless from the very beginning. With the advent of algorithm theory, researchers began to make efforts to prove the unsolvability of this problem. The American logicians J. Robinson, M. Davis, and H. Putnam obtained intermediate results in this direction, and the final solution of the problem was obtained only in 1970 by the Leningrad mathematician Yu.

Nowadays algorithmic problems occupy an important place in mathematics. Mathematical logic has taught us that not every such question is solvable. Moreover, even if an algorithm for solving this or that problem exists in principle, it is not always possible to talk about its applicability in practice. For example, the execution of the algorithm may require too much time or too much computer memory. This kind of issues is dealt with in a separate area of algorithm theory – the theory of computational complexity, which is discussed in detail in another article of this collection.

Gödel’s Theorems and Unprovable Assertions. Another discovery made by the brilliant Austrian logician Kurt Gödel in 1931 was the phenomenon of incompleteness of axiomatic systems. Gödel’s famous incompleteness theorems not only had a great influence on the development of mathematical logic and gave rise to the theory of algorithms, but also became a cultural phenomenon, affecting even the work of writers and artists. Gödel was named one of the 100 most influential personalities of the 20th century by Time magazine. However, the prominence of Gödel’s theorems also leads to the fact that they are often interpreted in an overly expansive, metaphorical sense.

Gödel’s theorems belong to the class of axiomatic systems that satisfy two natural and broad requirements. First, the concept of a natural number and the operations of addition and multiplication must be at least expressible in the formal language in question. At first sight, this requirement seems very special, but natural numbers are one of the basic mathematical objects, and languages that pretend to formalize a large part of mathematics must allow one to talk about them.

Second, there must be an algorithm that recognizes whether a given text is an axiom of the theory in question or not. (If the axioms of the theory are unrecognizable, it is not clear how one can build a proof in such a system.)

Gödel showed that if these requirements are met, any system of axioms is either contradictory or incomplete. Moreover, for any non-contradictory system one can explicitly specify a proposition concerning the arithmetic of natural numbers that cannot be either proved or disproved in a given system (such statements are usually called independent of a given system of axioms). In particular, this means that the system of axioms of formal arithmetic cannot be “extended” in any consistent way: there will always be arithmetic statements independent of it. This is the content of the so-called first Gödel theorem.

Gödel’s second theorem says that a statement expressing the consistency of a given axiomatic system is not provable in the system itself, if that system is indeed consistent. If one assumes that standard mathematical methods fit within an axiomatic set theory, then it follows, for example, from this theorem that standard mathematical methods cannot establish the consistency of set theory (and thus their own consistency).

Gödel’s theorems made it possible to construct the first examples of independent assertions for strong systems of axioms, such as arithmetic or even set theory. Since Gödel’s work, such examples have been found among open problems in various fields of mathematics. One of the most famous discovered problems in mathematics was the Cantor continuum hypothesis, according to which every infinite subset of a set of real numbers is either countable (equal to the set of natural numbers) or continuum-like (equal to the set of real numbers). In 1938 Gödel managed to prove that this hypothesis cannot be disproved in set theory, and in 1961 the American mathematician Paul Cohen established its unprovability.

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On the other hand, there were paradoxes in Cantor’s set theory itself that indicated that there was something wrong with this theory at the most basic level. The simplest paradox of this kind, known in folklore as the barber’s paradox, was invented by Bertrand Russell: Consider the set of all those sets which do not contain themselves as an element. Then if and only if , a contradiction.

This state of affairs led many prominent mathematicians and philosophers of the era (Peano, Frege, Russell, Hilbert, Poincaré, Brauer, Weil, etc.) to think about the foundations of mathematics. They were concerned with such fundamental questions as:

When we talk about the truthfulness and provability of a mathematical statement, do we mean the same thing?
In parallel, new standards of rigor began to take root in mathematics. The major areas of mathematics-analysis, algebra, geometry-were put on an axiomatic basis. The great mathematician David Hilbert (1862-1943) was a brilliant proponent and promoter of the axiomatic method. Under his influence the now generally accepted system of axioms of set theory, free of obvious paradoxes, was constructed. This axiomatics was proposed in 1908 by E. Zermelo and later supplemented by J. von Neumann and A. Frenkel. But where is the real guarantee that the resulting system does not contain contradictions? How can this be established?

These questions turned out to be much more complicated than Hilbert had imagined at the time. They required an in-depth study of axiomatic systems and their formalization; they led to an exact analysis of the structure of mathematical statements, the first formulations of rigorous mathematical models of phenomena such as provability, expressibility, truth, and made their study by mathematical methods possible. Thus emerged mathematical logic, a special field of research within mathematics. Within the framework of this discipline, an exact language and mathematical apparatus were created for investigating a whole stratum of phenomena that previously belonged to purely humanitarian knowledge. (In this role, mathematical logic can be compared to such an area of modern mathematics as probability theory, which was not strictly a mathematical discipline back in the early 20th century.)

Formal languages. From a modern point of view, the field of interest of mathematical logic is much broader than the science of correct reasoning; it can be roughly described, with caveats and refinements, as the construction and study of formal languages and systems by mathematical methods. Note that if we drop the word “formal” in this definition, instead of logic, we get essentially mathematical linguistics, which indicates a certain affinity between the two disciplines. The key difference between mathematical logic and logic in the broad sense is precisely the use of mathematical methods applied to exact formal models.

Formal and natural languages have things in common: both have syntax (the way we speak or write), semantics (the meaning of what is written), and pragmatics (how what is written is used). The main difference is that – at least ideally – the syntax and semantics of formal languages can be defined at the level of mathematical rigor and are therefore in principle amenable to analysis by purely mathematical methods.

Nowadays, formal languages can be found in every electronic device available to us, like cell phones, and some of them – programming languages – are even taught at school. So you don’t have to go far to find examples. However, in the middle of 19th century, when the process of mathematical logic began, formal languages did not yet exist, they were just to be created.

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We will talk about how and why mathematical logic arose, what it studies, what its achievements and modern applications are.

The foundations of the doctrine of correct reasoning were laid by Aristotle. He observed that correct inferences follow certain elementary schemes called syllogisms and listed a number of such schemes. (A classic example of a syllogism: “All men are mortal. Socrates is a human being. Consequently, Socrates is mortal.”) The doctrine of syllogisms in turn relied on an in-depth analysis of concepts and their conjunction into statements.

Aristotle’s syllogism was not without its drawbacks, but on the whole it was an outstanding theory and became the basis for the study of logic throughout antiquity and the Middle Ages. In the works of ancient Stoics and medieval scholastics it was modified and supplemented. In this form Aristotelian logic had existed up to the middle of XIX century, where it had met with revolution connected to penetration into logic of mathematical methods.

The emergence of mathematical logic has completely changed the views of scientists, both about the methods of research of logic, and about what constitutes its very subject of study. Nowadays, the statement that logic is the science of correct reasoning seems as true as the statement “mathematics is the science of correct computation”.

The analogy between reasoning and computation is somewhat deeper than it first appears. The emergence of logic as a mathematical science was associated with the work of British scholars George Bull and August de Morgan, who discovered that logical statements could be operated on as algebraic expressions. For example, if addition is read as the logical conjunction “or”, multiplication as “and”, and equality as “equals”, then for any utterance , the laws as well as many other laws of arithmetic that we are familiar with.

This view of the logic of statements and syllogism proved to be both unexpected and fruitful. Nowadays this view has been developed by the field called algebraic logic, and one of its central concepts is the concept of Boolean algebra, named after its discoverer. This field of research, through the concept of relational algebra, which generalizes the Boolean algebra, led in the 1960s to the theory of relational databases, now the basis of the most common query languages such as SQL.

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