Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
- When you submit an application on the site, we may collect various information, including your name, telephone number, address email etc.
How we use your personal information:
- The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and communications.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose the information received from you to third parties.
Exceptions:
- If necessary - in accordance with the law, judicial procedure, legal proceedings, and/or based on public requests or requests from government bodies on the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
- In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
Respecting your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes...were involved in the study of the issue mathematical analysis, set theory, new physical and philosophical approaches; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs with constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it's not complete solution problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Applicable mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” from shamans that mathematicians use. But that's not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, it means it has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: minus sign, number four, degree designation). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is the "pooping man" or the number "twenty-six" in hexadecimal system Reckoning. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
While studying the algebra course for grades 7-9, we have so far dealt with algebraic functions, i.e. functions defined analytically by expressions in which algebraic operations on numbers and variables were used (addition, subtraction, multiplication, division, exponentiation, extraction square root). But mathematical models of real situations are often associated with functions of a different type, not algebraic. We will get acquainted with the first representatives of the class of non-algebraic functions - trigonometric functions - in this chapter. You will study trigonometric functions and other types of non-algebraic functions (exponential and logarithmic) in more detail in high school.
For introduction trigonometric functions we will need a new one mathematical model- a number circle that you have not yet encountered, but you are very familiar with the number line. Recall that the number line is a straight line on which the starting point O, the scale (unit segment) and the positive direction are given. We can compare any real number with a point on a line and vice versa.
How to find the corresponding point M on a line using the number x? The number 0 corresponds to the starting point O. If x > 0, then, moving along a straight line from point 0 in the positive direction, you need to go n^th of length x; the end of this path will be the desired point M(x). If x< 0, то, двигаясь по прямой из точки О в отрицательном направлении, нужно пройти путь 1*1; конец этого пути и будет искомой точкой М(х). Число х - координата точки М.
And how did we solve the inverse problem, i.e. How did you find the x coordinate of a given point M on the number line? We found the length of the segment OM and took it with the sign “+” or * - “depending on which side of the point O the point M is located on the straight line.
But in real life You have to move not only in a straight line. Quite often, movement along circle. Here concrete example. Let us consider the stadium running track to be a circle (in fact, it is, of course, not a circle, but remember, as sports commentators usually say: “the runner has run a circle”, “there is half a circle left to run before the finish”, etc.), its length is 400 m. The start is marked - point A (Fig. 97). A runner from point A moves around a circle counterclockwise. Where will he be in 200 m? in 400 m? in 800 m? in 1500 m? Where should he draw the finish line if he is running a marathon distance of 42 km 195 m?
After 200 m, he will be at point C, diametrically opposite to point A (200 m is the length of half the treadmill, i.e. the length of half a circle). After running 400 m (i.e., “one lap,” as the athletes say), he will return to point A. After running 800 m (i.e., “two laps”), he will again be at point A. What is 1500 m ? This is “three circles” (1200 m) plus another 300 m, i.e. 3
Treadmill - the finish of this distance will be at point 2) (Fig. 97).
We just have to deal with the marathon. After running 105 laps, the athlete will cover a distance of 105-400 = 42,000 m, i.e. 42 km. There are 195 m left to the finish line, which is 5 m less than half the circumference. This means that the finish of the marathon distance will be at point M, located near point C (Fig. 97).
Comment. You, of course, understand the convention of the last example. No one runs a marathon distance around the stadium; the maximum is 10,000 m, i.e. 25 laps.
You can run or walk any length along the stadium treadmill. This means that any positive number corresponds to some point - the “finish of the distance”. Moreover, anyone can negative number match a point on the circle: you just need to make the athlete run in the opposite direction, i.e. start from point A not in a counter-clockwise direction, but in a clockwise direction. Then the stadium running track can be considered as a number circle.
In principle, any circle can be considered as a numerical circle, but in mathematics it was agreed to use the unit circle for this purpose - a circle with radius 1. This will be our “ treadmill" The length b of a circle with radius K is calculated by the formula The length of a half circle is n, and the length of a quarter circle is AB, BC, SB, DA in Fig. 98 - equal Let's agree to call the arc AB the first quarter unit circle, arc BC - the second quarter, arc CB - the third quarter, arc DA - the fourth quarter (Fig. 98). In this case, we are usually talking about an Open arc, i.e. about an arc without its ends (something like an interval on a number line).
Definition. A unit circle is given, and the starting point A is marked on it - the right end of the horizontal diameter (Fig. 98). Let's match each one real number I point of the circle according to the following rule:
1) if x > 0, then, moving from point A in a counterclockwise direction (the positive direction of going around the circle), we will describe a path along the circle with length and the end point M of this path will be the desired point: M = M(x);
2) if x< 0, то, двигаясь из точки А в направлении по часовой стрелке (отрицательное направление обхода окружности), опишем по окружности путь длиной и |; конечная точка М этого пути и будет искомой точкой: М = М(1);
Let us associate point A with 0: A = A(0).
A unit circle with an established correspondence (between real numbers and points on the circle) will be called a number circle.
Example 1. Find on the number circle 
Since the first six of the given seven numbers are positive, then to find the corresponding points on the circle, you need to walk a path of a given length along the circle, moving from point A in the positive direction. Let us take into account that
The number 2 corresponds to point A, since, having passed along the circle a path of length 2, i.e. exactly one circle, we will again get to the starting point A So, A = A(2).
What's happened 
This means that moving from point A in a positive direction, you need to go through a whole circle.
Comment. When we are in 7th and 8th grades worked with the number line, then we agreed, for the sake of brevity, not to say “the point on the line corresponding to the number x,” but to say “point x.” We will adhere to exactly the same agreement when working with the number circle: “point f” - this means that we are talking about a point on the circle that corresponds to the number
Example 2.
Dividing the first quarter AB into three equal parts by points K and P, we get: 
Example 3. Find points on the number circle that correspond to numbers 
We will make constructions using Fig. 99. Depositing arc AM (its length is -) from point A five times in the negative direction, we obtain point!, - the middle of arc BC. So,
Comment. Please note some of the liberties we take in using mathematical language. It is clear that the arc AK and the length of the arc AK are different things (the first concept is geometric figure, and the second concept is number). But both are designated the same way: AK. Moreover, if points A and K are connected by a segment, then both the resulting segment and its length are denoted in the same way: AK. It is usually clear from the context what meaning is intended in the designation (arc, arc length, segment or segment length).
Therefore, two number circle layouts will be very useful to us.
FIRST LAYOUT
Each of the four quarters of the number circle is divided into two equal parts, and near each of the available eight points their “names” are written (Fig. 100).
SECOND LAYOUT Each of the four quarters of the number circle is divided into three equal parts, and near each of the available twelve points their “names” are written (Fig. 101).
Please note that on both layouts we could assign other “names” to the given points.
Have you noticed that in all the examples of arc lengths
expressed by some fractions of the number n? This is not surprising: after all, the length of a unit circle is 2n, and if we divide a circle or its quarter into equal parts, we get arcs whose lengths are expressed in fractions of the number and. Do you think it is possible to find a point E on the unit circle such that the length of the arc AE is equal to 1? Let's figure it out:
Reasoning in a similar way, we conclude that on the unit circle one can find point Eg, for which AE = 1, and point E2, for which AEr = 2, and point E3, for which AE3 = 3, and point E4, for which AE4 = 4, and point Eb, for which AEb = 5, and point E6, for which AE6 = 6. In Fig. 102 the corresponding points are marked (approximately) (for orientation, each of the quarters of the unit circle is divided by dashes into three equal parts).
Example 4. Find the point on the number circle corresponding to the number -7.
We need, starting from point A(0) and moving in a negative direction (clockwise direction), to go along a circle with a path of length 7. If we go through one circle, we get (approximately) 6.28, which means we still need to go through ( in the same direction) a path of length 0.72. What kind of arc is this? Slightly less than half a quarter circle, i.e. its length less number -. 
So, on a number circle, like on a number line, each real number corresponds to one point (only, of course, it is easier to find it on a line than on a circle). But for a straight line the opposite is also true: each point corresponds singular. For a number circle, such a statement is not true; we have repeatedly seen this above. The following statement is true for the number circle.
If point M of the number circle corresponds to the number I, then it also corresponds to a number of the form I + 2k, where k is any integer (k e 2).
In fact, 2n is the length of the numerical (unit) circle, and the integer |th| can be considered as the number of complete rounds of the circle in one direction or another. If, for example, k = 3, then this means that we make three rounds of the circle in the positive direction; if k = -7, then this means that we make seven (| k | = | -71 = 7) rounds of the circle in the negative direction. But if we are at point M(1), then, having also completed | to | complete rounds of the circle, we will again find ourselves at point M.
A.G. Mordkovich Algebra 10th grade
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year methodological recommendations discussion programs Integrated LessonsCoordinates x points lying on the circle are equal to cos(θ), and the coordinates y correspond to sin(θ), where θ is the magnitude of the angle.
- If you find it difficult to remember this rule, just remember that in the pair (cos; sin) “the sine comes last.”
- This rule can be derived by considering right triangles and determination of these trigonometric functions (the sine of an angle is equal to the ratio of the length of the opposite, and the cosine - of the adjacent leg to the hypotenuse).
Write down the coordinates of four points on the circle. A “unit circle” is a circle whose radius is equal to one. Use this to determine the coordinates x And y at four points of intersection of the coordinate axes with the circle. Above, for clarity, we designated these points as “east”, “north”, “west” and “south”, although they do not have established names.
- "East" corresponds to the point with coordinates (1; 0) .
- "North" corresponds to the point with coordinates (0; 1) .
- "West" corresponds to the point with coordinates (-1; 0) .
- "South" corresponds to the point with coordinates (0; -1) .
- This is similar to a regular graph, so there is no need to memorize these values, just remember the basic principle.
Remember the coordinates of the points in the first quadrant. The first quadrant is located in the upper right part of the circle, where the coordinates x And y take positive values. These are the only coordinates you need to remember:
- the point π / 6 has coordinates () ;
- the point π/4 has coordinates () ;
- the point π / 3 has coordinates () ;
- Note that the numerator only takes three values. If you move in a positive direction (from left to right along the axis x and from bottom to top along the axis y), the numerator takes the values 1 → √2 → √3.
Draw straight lines and determine the coordinates of the points of their intersection with the circle. If you draw straight horizontal and vertical lines from the points of one quadrant, the second points of intersection of these lines with the circle will have the coordinates x And y with the same absolute values, but different signs. In other words, you can draw horizontal and vertical lines from the points of the first quadrant and label the points of intersection with the circle with the same coordinates, but at the same time leave space on the left for correct sign("+" or "-").
- For example, you can draw a horizontal line between the points π/3 and 2π/3. Since the first point has coordinates ( 1 2 , 3 2 (\displaystyle (\frac (1)(2)),(\frac (\sqrt (3))(2)))), the coordinates of the second point will be (? 1 2 , ? 3 2 (\displaystyle (\frac (1)(2)),?(\frac (\sqrt (3))(2)))), where instead of the "+" or "-" sign there is a question mark.
- Use the simplest method: pay attention to the denominators of the coordinates of the point in radians. All points with a denominator of 3 have the same absolute coordinate values. The same applies to points with denominators 4 and 6.
To determine the sign of the coordinates, use the rules of symmetry. There are several ways to determine where to place the "-" sign:
- Remember the basic rules for regular charts. Axis x negative on the left and positive on the right. Axis y negative from below and positive from above;
- start with the first quadrant and draw lines to other points. If the line crosses the axis y, coordinate x will change its sign. If the line crosses the axis x, the sign of the coordinate will change y;
- remember that in the first quadrant all functions are positive, in the second quadrant only the sine is positive, in the third quadrant only the tangent is positive, and in the fourth quadrant only the cosine is positive;
- Whichever method you use, you should get (+,+) in the first quadrant, (-,+) in the second, (-,-) in the third, and (+,-) in the fourth.
Check if you made a mistake. Below is full list coordinates of “special” points (except for four points on the coordinate axes), if you move along the unit circle counterclockwise. Remember that to determine all these values, it is enough to remember the coordinates of the points only in the first quadrant:
- first quadrant: ( 3 2 , 1 2 (\displaystyle (\frac (\sqrt (3))(2)),(\frac (1)(2)))); (2 2 , 2 2 (\displaystyle (\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (1 2 , 3 2 (\displaystyle (\frac (1)(2)),(\frac (\sqrt (3))(2))));
- second quadrant: ( − 1 2 , 3 2 (\displaystyle -(\frac (1)(2)),(\frac (\sqrt (3))(2)))); (− 2 2 , 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (− 3 2 , 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),(\frac (1)(2))));
- third quadrant: ( − 3 2 , − 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),-(\frac (1)(2)))); (− 2 2 , − 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (− 1 2 , − 3 2 (\displaystyle -(\frac (1)(2)),-(\frac (\sqrt (3))(2))));
- fourth quadrant: ( 1 2 , − 3 2 (\displaystyle (\frac (1)(2)),-(\frac (\sqrt (3))(2)))); (2 2 , − 2 2 (\displaystyle (\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (3 2 , − 1 2 (\displaystyle (\frac (\sqrt (3))(2)),-(\frac (1)(2)))).
