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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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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Do you feel a chill of anxiety run through your body when it comes to tackling equations and formulas? Perhaps you think that being competent in mathematics is impossible for someone like yourself, no matter how hard they try. If this sounds familiar, don’t worry! You are not alone; many people have been successful in understanding the basics of geometry or algebra.

I understand how intimidating math can be, and the complexity only increases as you go deeper into your studies. Don’t let a fear of numbers stop you from excelling in math! With the correct approach and tactics, anyone can become proficient in mathematics. You don’t have to love calculations – just take the right steps toward understanding the subject, and your grades will improve drastically!

In addition, the knowledge of mathematics will stay with you far beyond when you finish your educational studies. From budgeting funds to filing taxes and even baking desserts, having faith in numerical abilities can be beneficial each day.

Are you looking for the key to long-term success in mathematics? Look no further – with a few simple steps, you can increase your math skills and ability substantially. Conquer your anxiousness and cultivate confidence with these tips, so you can face any challenge that comes your way- whether it be in the classroom or in life!

Connect math problems to everyday scenarios.

When you gaze upon a complex math equation, what do you envision? Is it simply an unorganized mess of characters and values? No wonder you’re feeling overwhelmed by math! It’s understandable – after all, it can be a challenging topic to tackle.

If you want to become better at math, the key is understanding how it relates and applies to your everyday life. When a teacher puts up an equation on the chalkboard that looks intimidating, ask yourself what practical use it has in reality. Memorization of mathematics becomes easier when we understand its purpose; making connections will help bring context and meaning to each problem-solving situation.

Perfect the fundamentals of mathematics.

Studying math can feel as intimidating and daunting as learning a foreign language. If that’s the case, why not treat it like one? To get started in Spanish, you don’t attempt to have an entire conversation right away. Instead, you learn some essential words first, such as hola or gracias. So when learning math, it all begins with building a basic knowledge of core concepts before getting into more complex topics.

Just like stringing a few words together to form sentences, complex math equations rely on basic building blocks in order to be successful. Ensure that you have an unwavering grasp of fractions and decimals before moving onto more advanced levels — even if it takes extra effort through additional practice problems. With consistent effort and dedication, you’ll soon discover your own potential as a mathematician!

Dissect complex problems.

As you continue your academic pursuits, it’s ultimately inevitable that you will come across some difficult math problems. It can be disheartening when presented with challenging questions, but remember they are just an extension of concepts and skills that you already know!

Are you stumped by this math problem? Take a deep breath and ask yourself: How can I divide up this challenging equation into more manageable pieces? Carefully read the question several times, as it is important to grasp its entirety before starting your solution.

Are you familiar with any element of the larger issue? If you decide to break down a complicated problem into smaller pieces, it will help reduce your sense of being overwhelmed.

If this tactic doesn’t work, use simpler numbers in place of the original ones. For example, replace 10,000 or 37/52 with 10 and 2/3, respectively. Remember to switch back after solving the equation!

When confronted by challenging problems either inside or outside math class, try simplifying them as much as possible – you might be astonished at how easily solvable they become!

Maximize your potential by taking practice tests.

Although good grades are not the only measure of success in education, exams remain an integral component throughout your school years. Completing practice tests is a powerful way to ensure you perform well on upcoming assessments. However, there exists one problem – obtaining reliable and relevant materials for these mock examinations can be difficult.

Students often work hard to develop a strategy for solving difficult questions when they practice months before the big test – only then to forget how to approach similar problems on the actual exam.

To optimize your math test performance, I highly recommend the following approach. Place a sticky note on any challenging questions you encounter and get help to understand it. Then, review all such practice problems at regular intervals in the days ahead of taking the exam. This strategy, added to your existing examination preparation techniques, will undoubtedly result in grade improvements!

Conclusion

Improving math skills requires dedication and consistency. Don’t get discouraged when you encounter difficult problems; break them down into smaller pieces, and take practice tests to ensure your success! Lastly, remember that understanding the purpose behind all of these equations will add meaning to each problem-solving situation. With hard work and determination, anyone can become a master mathematician!

Good luck in your mathematical endeavors!

The post Strategies to Improve Your Math Skills appeared first on 2IMO18.

Before the mathematical concept of randomness was developed, people believed that everything that occurred was a result of the will of gods, supernatural entities that watched over human affairs and made decisions to “tilt the balance” in one direction or another to affect events. 

As a result, rituals and sacrifices were carried out to ascertain the “intent of the gods” or to try to have an impact on human affairs. This notion is still widely held, and many people still use lucky charms, follow superstitious customs, consult horoscopes, and hold some sort of conviction that there are such means of influencing their life. 

The Universe and the Earth

Other additional rituals attempted to combat nature’s and man’s condition’s randomness. Geomancy, the nine-square grid (also known as the magic square), and temple designs, the forerunners of board games, are a few of these that developed a particular mathematical interest.

Geomancy

Geomancy, which translates to “earth divination” or “divination of or from the earth,” is a scheme of 16 mathematically connected configurations of stones, beans, or other readily available tiny objects intended to make choices, provide answers to problems, or predict the future. The stones are thrown on the ground, and the resulting pattern is deciphered. Many binary “opposites” are represented by symbols, including male or female, good and evil, and sadness and happiness. Even and odd numbers can be represented by combining these opposites.

Each of these figures, like with all methods of astrology, has a variety of meanings based on how it relates to the other figures displayed and several other factors, such as the daytime, the climate, and the type of person posing the question.

The Nine-Square Grid

The 9-Square Grid is thought to have originated from an old method of land division, presumably from medieval India. The nine-square pattern, which in China consisted of eight farmer’s fields circling a central well, was thought to be the best layout. In Tibetan sacred diagrams, the grid of nine squares or a circle divided into 9 sections by parallel lines frequently occurs as the main form. 

In this way, the nine-square grid evolved from utilitarian uses in various cultures to the mystic significance and a portrayal of divine order and control by the gods.

Squares of Magic

They are directly connected to the Divine Grid, with the alleged numerical riddle that underpins their physical appearance being the Sacred Grid. The square of nine, which is attributed to Saturn, is the simplest magic square. Its rows and columns add up to 15, making it the square of nine; its diagonals add up to 30, and its rows and columns add up to 45. Jupiter is represented by the 4×4 square with columns and rows number 34, and Mars by the 5×5 square with columns or rows number 65.

Game boards

They are unmistakably connected to astrology, divination, and sacred geometries, and the patterns of the decks can reveal their esoteric or spiritual beginnings. The outcome of the dice-controlled game “snakes and ladders” now refers to good and bad “luck” as the game progresses; the stairs and snakes originally signifying good and evil fortune. Some of the designs on the boards resemble the layouts of cathedrals and holy places with a central “sacred area.”

Magic and Mathematics

Few people in the past could comprehend even the most basic geometry and arithmetic, and there is a long history of conflating mathematics with magic.

The positions of the planets and the precise times that astronomical phenomena appeared in particular parts of the sky might be predicted by those who were familiar with the regular motions of the cosmos. These highly competent workers, known as “priests,” worked in both the scientific and religious fields in earlier civilizations. After Christianity arrived in Europe, the religious component of these customs was derided as superstition.

These methods involved numbers, thus anyone who did so was viewed with a great deal of mistrust. Since the terms “astrologer,” “mathematician,” and “conjurer” were practically interchangeable, legitimate mathematicians were viewed with mistrust by the uninformed.

The Evolution of Probability

It’s thought that since cubes were used for gambling, religious rituals, and divination, persons who used them had a strong intuitive sense of the frequency of certain number combinations. The Latin poetry De Vetula, which depicts all the permutations for the collapse of three dice and is said to have been composed in the early thirteenth century, is the first printed example of the alternatives with three dice. Although it is mentioned in the poem, the use of binomial coefficients to determine the possibilities is not pursued until much later.

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One of the most beneficial aspects of creating bespoke mathematical essays is that they help individuals fully grasp math ideas. It’s similar to how expressing arithmetic in prose helps a pupil actually comprehend.

Let’s go into the details of the Top 7 Most Effective Mathematics Essay Writing Tips by easy essay service.

1. Select a topic

The preferred technique for choosing a topic is to choose a math idea that interests you. It’s a smart option to choose an issue from school. Students would be asked to investigate the topic further throughout the essay. To have a simple experience studying the issue, one must first enjoy the issue.

2. While creating a mathematics essay, keep your intended audience in mind. 

The audience has mostly been made up of learners who possess a rudimentary comprehension of your chosen topic. One must use rhetorical appeals to persuade your audience to accept the presented analytical process.

3. Arrange your essay in the same way that you would prepare any other work. 

While writing about that now, include an explanation of the notion theory. Create the strategy in the next paragraphs by providing evidence that supports your statements. Lastly, the essay should contain a finish. Writing an excellent essay about mathematics demonstrates that you comprehend the arithmetic idea you are focused on in your essay.

4. Outline for a thorough mathematics essay

An effective essay requires a decent plan. When creating an outline, you must select how many points you will cover in the essay. The set of questions should indeed be kept to a minimum. Too many or only a few questions in the essay may not yield good results. So, arrange a moderate level of questions to explain the subject.

Once you’ve actually written a reasonable number of problems, organize them according to the way you want to answer each in the paper. Most importantly, the essay outline must include a clear introduction, body, and conclusion. Never neglect to clarify each question you’ve written in the essay’s body section.

5. It is required to communicate less about the broad implications of laws and mathematical principles. 

Students must explain the situation to the audience and reveal key information about how the given concepts are implemented. Readers must understand how the legislation or rules function and the alternatives for implementing them.

6. For something like a better comprehension of the content, charts, and illustrations must be used. 

Visual analytics is a crucial component of every mathematical assignment. The reader can get through the writing if they have a look at the visuals. They can gain a better understanding of the subject through the images they perceive. There are numerous sorts of charts for different situations, and every one of them may be generated using Microsoft Office capabilities.

7. In the conclusion, summarize the original thesis definitively and persuasively so that the significant bit it was beneficial to study the article all the way through. 

You can guide the reader in summarizing his views through the conclusion. A math essay is necessary because it allows pupils to show how well they understand mathematical ideas.

A math essay is necessary because it allows pupils to show how well they understand mathematical ideas. The essay provides more details on what essential mathematics class teachings may be lacking. The end of the essay is just as important as the rest of the work. The conclusion must be consistent with the information presented throughout the essay.

It is of utmost importance that the student must not try to introduce a new idea or any tip in the conclusion.

It’s wrap

When writing an essay on mathematics, the aforementioned guidelines can help you comprehend it better. Before you begin writing, use these ideas to strengthen your position and clarity. The aforementioned seven suggestions are carefully described and investigated to your core pleasure and need. These suggestions can help your essay rank higher on the list of excellent essays.

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A credible event is an event that, as a result of a test (the implementation of certain actions, a certain set of conditions) is bound to happen. For example, in terms of gravity, a coin tossed will certainly fall down.

An impossible event is an event that knowingly will not happen as a result of the test. An example of an impossible event: Under the conditions of terrestrial gravitation, a flipped coin will fly upwards.

Finally, an event is called random if, as a result of the test, it may or may not happen, and there must be a fundamental criterion of randomness: a random event is a consequence of random factors, the impact of which is impossible or extremely difficult to predict. Example: as a result of a coin toss an “eagle” will fall out. In the case considered, random factors are the shape and physical characteristics of the coin, the force/direction of the throw, air resistance, etc.

The underlined criterion of randomness is very important – so, for example, a card cheat can very cleverly imitate randomness and let the victim win, but we are not talking about any random factors affecting the final result.

Any result of a trial is called an outcome, which, in fact, is the occurrence of a certain event. In particular, when a coin is tossed, 2 outcomes (random events) are possible: heads will fall and tails will fall. Naturally, it is assumed that this test is conducted under such conditions that the coin cannot get on the edge or, say, hang in weightlessness.

Events (any) are denoted by capital Latin letters or by the same letters with subscripts, e.g.: . The exception is the letter , which is reserved for other needs.

Let us write down the following random events:

• A coin toss will result in an “eagle”;
• A roll of the die will result in 5 points;
• a card of the suit of clubs will be drawn from the deck (by default, the deck is full).

Yes, events are written in this way in practical problems, and it is convenient to use “talking” subscripts in appropriate cases (though one can do without them).

It should be emphasized for the third time that random events necessarily satisfy the above criterion of randomness. In this sense the 3rd example is again illustrative: if all cards of the clubs suit are initially removed from the deck, the event becomes impossible. On the contrary, if the tester knows that, for example, the queen of clubs lies at the bottom, he can make the event credible if he wants to =) Thus, this example assumes that the cards are well mixed and their shirts are indistinguishable, i.e., the deck is not mottled. And, here, “speck” doesn’t even mean “skillful hands” that eliminate the randomness of your winnings, but visible flaws in the cards. For example, the shirt of the queen of clubs can be dirty, torn, taped up with scotch tape… man, it’s some kind of manual for beginner chicatillos =)

Thus, when drawing an important lot it always makes sense to casually look to see if the faces of the coins are the same

Another important characteristic of events is that they are equally possible. Two or more events are called equiprobable if none of them is more possible than the others. For example:

A roll of heads or tails on a coin toss;
1, 2, 3, 4, 5, or 6 on a die roll;
drawing a card of clubs, spades, diamonds or hearts from the deck.

This assumes that the coin and die are homogeneous and geometrically correct, and that the deck is well mixed and “perfect” in terms of the indistinguishability of the card shirts.

Can the same events not be equally possible? They can! For example, if a coin or die has its center of gravity displaced, then quite certain faces are much more likely to fall out. As they say, another loophole for cheaters. Events-extracting clubs, spades, hearts, or diamonds are also equiprobable. However, equiprobability is easily broken by a trickster who, while shuffling a deck (even a “perfect” one), will cleverly peek and hide in his sleeve, such as the ace of clubs. Here it becomes less possible that the opponent will be dealt a club, and more importantly, less possible that he will be dealt an ace.

Nevertheless, in the three cases considered, the loss of equal possibility still preserves the randomness of events.

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Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas go back to earlier mathematicians. The first results of analysis were implicitly present at the dawn of ancient Greek mathematics. For example, an infinite geometric sum is implicit in Zeno’s dichotomy paradox. Later Greek mathematicians such as Eudoxus and Archimedes made a more explicit but informal use of the concepts of limits and convergence when they used the . exhaustion method to calculate the area and volume of areas and solids. Explicit use of infinitesimals appears in Archimedes’ Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chunzhi developed a method that would later be called the Cavalieri principle to determine the volume of a sphere in the 5th century. The Indian mathematician Bhaskara II gave examples of the derivative and used what is now known as Rolle’s theorem in the 12th century.

In the 14th century, Madhava of Sangamagrama developed an infinite series of expansions, such as the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent. In addition to developing the Taylor series of trigonometric functions, he also estimated the magnitude of the errors arising from truncation of these series and gave a rational approximation of the infinite series. His followers in the Kerala school of astronomy and mathematics extended his work into the 16th century.

The modern foundations of mathematical analysis were laid in 17th-century Europe. Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed the calculus of infinitesimals, which, through the stimulus of applied work that continued into the 18th century, evolved into such analytic topics as the calculus of variations, ordinary and partial derivative equations, Fourier analysis and derivative functions. During this period, calculus methods were used to approximate discrete problems by continuous ones.

In the 18th century, Euler introduced the concept of the mathematical function. Real analysis began to become a subject in its own right when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano’s work did not become widely known until the 1870s. In 1821, Cauchy began to put the calculus on a firm logical footing, rejecting the principle of generality of algebra that had been widely used in earlier works, notably by Euler. Instead, Cauchy formulated the calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required that an infinitesimal change in x corresponded to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence and began the formal theory of complex analysis. Poisson, Liouville, Fourier, and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of the limit approach, thus beginning the modern field of mathematical analysis.

In the middle of the 19th century, Riemann introduced his theory of integration. The last third of the century was marked by the arithmetization of analysis by Weierstrass, who believed that geometric reasoning was initially misleading, and introduced the definition of “epsilon-delta” from limit. Mathematicians then began to worry that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed real numbers using Dedekind’s abbreviations, in which irrational numbers are formally defined, which serve to fill the “gaps” between rational numbers, thus creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, attempts to refine theorems from Riemann integration led to a study of the “size” of the set of discontinuities of real functions.

Also, “monsters” (nowhere continuous functions, continuous but nowhere differentiable functions, curves that fill space ) began to be investigated. In this context, Jordan developed his measure theory, Cantor developed what is now called naive set theory, and Baer proved Baer’s category theorem. In the early 20th century, calculus was formalized with axiomatic set theory. Lebeg solved the measure problem, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of a normalized vector space was in the air, and in the 1920s Banach created functional analysis.

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WHAT IS PROBABILITY THEORY?
The result of research regarding the effects of randomness and uncertainty on social, behavioral, and physical phenomena is a branch of mathematics devoted to probability theory. Quantitatively, probability is defined by a number from 0 to 1, where 0 means the ultimate impossibility of an event and 1 is one hundred percent certainty that the event will occur. The more this number approaches 1, the greater the probability that certain events will occur. Probability is also measured on a scale of 0 to 100%.

A simple example of probability is a toss-up: a toss-up of heads or tails is equal in probability because there are no other outcomes for such a coin flip. In practice, probability theory is used to model situations where, under the same conditions, we have different results due to the same actions.

The outcome of a coin flip is random. Random events cannot be fully predicted, but they all have long-term patterns that we can describe and quantify with probability.

Let’s consider three basic theories.

Equally probable outcomes.
There is no reason to claim that the probability of one outcome of an event takes precedence over other outcomes. Imagine a vessel with identical balls that have been thoroughly mixed. The player is asked to take out one of the marbles, with the probability of choosing each of the marbles being the same. If a given situation has a number of results equal to n, then the probability of each result is 100%.

Frequency theory
According to this theory, probability is the limit on the relative frequency with which an event occurs under repeated conditions. The statement “the probability that A will happen is p%” in this case means the following: if you repeat the experiment over and over again, independently and under approximately the same conditions, the percentage of time that A will happen approaches p. The relative frequency is calculated solely after the experiments are conducted based on the data actually obtained.

If a number of experiments are performed under unchanged conditions, the relative frequency becomes stable, i.e., it varies within marginal differences. For example, a professional archer has fired 100 shots and hit the target 90 times out of them. His probability of hitting the target under certain conditions is 0.9. If he fired 10511 shots in his career, of which he hit the target 9846 times, the relative frequency is 9846/10511=0.9367. This figure will be taken into account to predict the archer’s result in future competitions.

Subjective theory
This type of probability is used in the decision-making process in order to further predict human behavior. It has no statistical characteristic. In this case, the probability is the level of verification of a certain statement. For example, the appropriateness of investing in various risky projects, participation in the lottery, planning drug stocks in medical institutions, etc. Subjective probability is determined by means of appropriate local expertise.

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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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