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 ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; 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 at a 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 this is not a complete solution to the problem. 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 draw special attention to 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. Let us apply mathematical set theory to 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 this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is 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 the large number 12345, I don’t want to fool my head, let’s consider 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, then this 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: a minus sign, the number four, a designation of degrees). 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 "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
A parallelepiped is a geometric figure, all 6 faces of which are parallelograms.
Depending on the type of these parallelograms, the following types of parallelepiped are distinguished:
- straight;
- inclined;
- rectangular.
A right parallelepiped is a quadrangular prism whose edges make an angle of 90° with the plane of the base.
A rectangular parallelepiped is a quadrangular prism, all of whose faces are rectangles. A cube is a type of quadrangular prism in which all faces and edges are equal to each other.
The features of a figure predetermine its properties. These include the following 4 statements:
It is simple to remember all the above properties, they are easy to understand and are derived logically based on the type and characteristics of the geometric body. However, simple statements can be incredibly useful when solving typical USE tasks and will save the time needed to pass the test.
Parallelepiped formulas
To find answers to the problem, it is not enough to know only the properties of the figure. You may also need some formulas for finding the area and volume of a geometric body.
The area of the bases is found in the same way as the corresponding indicator of a parallelogram or rectangle. You can choose the base of the parallelogram yourself. As a rule, when solving problems it is easier to work with a prism, the base of which is a rectangle.
The formula for finding the lateral surface of a parallelepiped may also be needed in test tasks.
Examples of solving typical Unified State Exam tasks
Exercise 1.
Given: a rectangular parallelepiped with dimensions of 3, 4 and 12 cm.
Necessary find the length of one of the main diagonals of the figure.
Solution: Any solution to a geometric problem must begin with the construction of a correct and clear drawing, on which “given” and the desired value will be indicated. The figure below shows an example of the correct execution of task conditions.
Having examined the drawing made and remembering all the properties of the geometric body, we come to the only correct method of solution. Applying the 4th property of a parallelepiped, we obtain the following expression:
After simple calculations we get the expression b2=169, therefore b=13. The answer to the task has been found; you need to spend no more than 5 minutes searching for it and drawing it.
a, towards the base of the PP;
with its height.
- what do we need to know, what data do we have?
- what properties does a rectangular parallelepiped have?
- does the Pythagorean Theorem apply here? How?
- Is there enough data to apply the Pythagorean theorem, or are some other calculations needed?
Diagonal square, of a square parallelepiped (see properties of a square parallelepiped) is equal to the sum of the squares of its three different sides (width, height, thickness), and, accordingly, the diagonals of a square parallelepiped are equal to the root of this sum.
I remember the school curriculum in geometry, we can say this: the diagonal of a parallelepiped is equal to the square root obtained from the sum of its three sides (they are designated by small letters a, b, c).
The length of the diagonal of a rectangular parallelepiped is equal to the square root of the sum of the squares of its sides.
As far as I know from the school curriculum, grade 9, if I’m not mistaken, and if memory serves, the diagonal of a rectangular parallelepiped is equal to the square root of the sum of the squares of all three sides.
the square of the diagonal is equal to the sum of the squares of the width, height and length, based on this formula we get the answer, the diagonal is equal to the square root of the sum of its three different dimensions, with letters they denote ncz abc
A rectangular parallelepiped (PP) is nothing more than a prism, the base of which is a rectangle. For a PP, all diagonals are equal, which means that any of its diagonals is calculated using the formula:
Another definition can be given by considering the Cartesian rectangular coordinate system:
The PP diagonal is the radius vector of any point in space specified by x, y and z coordinates in the Cartesian coordinate system. This radius vector to the point is drawn from the origin. And the coordinates of the point will be the projections of the radius vector (diagonals of the PP) onto the coordinate axes. The projections coincide with the vertices of this parallelepiped.
A rectangular parallelepiped is a type of polyhedron consisting of 6 faces, at the base of which is a rectangle. A diagonal is a line segment that connects opposite vertices of a parallelogram.
The formula for finding the length of a diagonal is that the square of the diagonal is equal to the sum of the squares of the three dimensions of the parallelogram.
I found a good diagram-table on the Internet with a complete listing of everything that is in the parallelepiped. There is a formula to find the diagonal, which is denoted by d.
There is an image of the edge, vertex and other important things for the parallelepiped.
If the length, height and width (a,b,c) of a rectangular parallelepiped are known, then the formula for calculating the diagonal will look like this:
Typically, teachers do not offer their students a bare formula, but make efforts so that they can derive it on their own by asking leading questions:
Usually, after answering the questions posed, students can easily derive this formula on their own.
The diagonals of a rectangular parallelepiped are equal. As well as the diagonals of its opposite faces. The length of the diagonal can be calculated by knowing the length of the edges of the parallelogram emanating from one vertex. This length is equal to the square root of the sum of the squares of the lengths of its edges.
A cuboid is one of the so-called polyhedra, which consists of 6 faces, each of which is a rectangle. A diagonal is a segment that connects opposite vertices of a parallelogram. If the length, width and height of a rectangular parallelepiped are taken to be a, b, c, respectively, then the formula for its diagonal (D) will look like this: D^2=a^2+b^2+c^2.
Diagonal of a rectangular parallelepiped is a segment connecting its opposite vertices. So we have cuboid with diagonal d and sides a, b, c. One of the properties of a parallelepiped is that the square diagonal length d is equal to the sum of the squares of its three dimensions a, b, c. Hence the conclusion is that diagonal length can be easily calculated using the following formula:
Also:
How to find the height of a parallelepiped?
A cuboid is a type of polyhedron consisting of 6 faces, each of which is a rectangle. In turn, a diagonal is a segment that connects the opposite vertices of a parallelogram. Its length can be detected in two ways.
You will need
- Knowing the lengths of all sides of a parallelogram.
Instructions
1. Method 1. Given a rectangular parallelepiped with sides a, b, c and diagonal d. According to one of the properties of a parallelogram, the square of the diagonal is equal to the sum of the squares of its 3 sides. It follows that the length of the diagonal itself can be calculated by extracting the square from a given sum (Fig. 1).
2. Method 2. It is possible that the rectangular parallelepiped is a cube. A cube is a rectangular parallelepiped in which every face is represented by a square. Consequently, all its sides are equal. Then the formula for calculating the length of its diagonal will be expressed as follows: d = a*?3
A parallelepiped is a special case of a prism, in which all six faces are parallelograms or rectangles. A parallelepiped with rectangular faces is also called rectangular. A parallelepiped has four intersecting diagonals. If three edges a, b, c are given, you can find all the diagonals of a rectangular parallelepiped by performing additional constructions.
Instructions
1. Draw a rectangular parallelepiped. Write down the known data: three edges a, b, c. First construct one diagonal m. To determine it, we use the quality of a rectangular parallelepiped, according to which all its angles are right.
2. Construct the diagonal n of one of the faces of the parallelepiped. Carry out the construction so that the famous edge, the desired diagonal of the parallelepiped and the diagonal of the face together form a right triangle a, n, m.
3. Find the constructed diagonal of the face. It is the hypotenuse of another right triangle b, c, n. According to the Pythagorean theorem, n² = c² + b². Calculate this expression and take the square root of the resulting value - this will be the diagonal of face n.
4. Find the diagonal of the parallelepiped m. To do this, in the right triangle a, n, m, find an unfamiliar hypotenuse: m² = n² + a². Substitute the known values, then calculate the square root. The resulting result will be the first diagonal of the parallelepiped m.
5. Similarly, draw all the other three diagonals of the parallelepiped in steps. Also, for all of them, perform additional construction of diagonals of adjacent faces. By looking at the right triangles formed and applying the Pythagorean theorem, discover the values of the remaining diagonals of the cuboid.
Video on the topic
Many real objects have a parallelepiped shape. Examples are the room and the pool. Parts with this shape are not uncommon in industry. For this reason, the task of finding the volume of a given figure often arises.
Instructions
1. A parallelepiped is a prism whose base is a parallelogram. A parallelepiped has faces - all the planes that form this figure. Each of them has six faces, and all of them are parallelograms. Its opposite sides are equal and parallel to each other. In addition, it has diagonals that intersect at one point and bisect at it.
2. There are 2 types of parallelepiped. For the first, all faces are parallelograms, and for the second, they are rectangles. The final one is called a rectangular parallelepiped. All its faces are rectangular, and the side faces are perpendicular to the base. If a rectangular parallelepiped has faces whose bases are squares, then it is called a cube. In this case, its faces and edges are equal. An edge is a side of any polyhedron, which includes a parallelepiped.
3. In order to find the volume of a parallelepiped, you need to know the area of its base and height. The volume is found based on which particular parallelepiped appears in the conditions of the problem. An ordinary parallelepiped has a parallelogram at its base, while a rectangular one has a rectangle or a square, which invariably has right angles. If there is a parallelogram at the base of a parallelepiped, then its volume is found as follows: V = S * H, where S is the area of the base, H is the height of the parallelepiped. The height of a parallelepiped is usually its lateral edge. At the base of a parallelepiped there can also be a parallelogram that is not a rectangle. From the course of planimetry it is known that the area of a parallelogram is equal to: S=a*h, where h is the height of the parallelogram, a is the length of the base, i.e. :V=a*hp*H
4. If the 2nd case occurs, when the base of the parallelepiped is a rectangle, then the volume is calculated using the same formula, but the area of the base is found in a slightly different way: V=S*H,S=a*b, where a and b are the sides, respectively rectangle and parallelepiped edge.V=a*b*H
5. To find the volume of a cube, one should be guided by primitive logical methods. Since all the faces and edges of the cube are equal, and at the base of the cube there is a square, guided by the formulas indicated above, we can derive the following formula: V = a^3
A closed geometric figure formed by two pairs of parallel segments of identical length lying opposite each other is called a parallelogram. A parallelogram, all angles of which are equal to 90°, is also called a rectangle. In this figure, you can draw two segments of identical length connecting opposite vertices - diagonals. The length of these diagonals is calculated by several methods.
Instructions
1. If the lengths of 2 adjacent sides are known rectangle(A and B), then the length of the diagonal (C) is very simple to determine. Proceed from the fact that diagonal lies opposite the right angle in the triangle formed by it and these two sides. This allows us to apply the Pythagorean theorem in calculations and calculate the length of the diagonal by finding the square root of the sum of the squared lengths of the leading sides: C = v (A? + B?).
2. If the length of only one side is known rectangle(A), as well as the size of the angle (?), the one that forms with it diagonal, then to calculate the length of this diagonal (C) you will have to use one of the direct trigonometric functions - cosine. Divide the length of the leading side by the cosine of the famous angle - this will be the desired length of the diagonal: C=A/cos(?).
3. If a rectangle is given by the coordinates of its vertices, then the task of calculating the length of its diagonal will be reduced to finding the distance between two points in this coordinate system. Apply the Pythagorean theorem to the triangle that forms the projection of the diagonal on each of the coordinate axes. It is possible that a rectangle in two-dimensional coordinates is formed by the vertices A(X?;Y?), B(X?;Y?), C(X?;Y?) and D(X?;Y?). Then you need to calculate the distance between points A and C. The length of the projection of this segment onto the X axis will be equal to the modulus of the coordinate difference |X?-X?|, and the projection onto the Y axis – |Y?-Y?|. The angle between the axes is 90°, from which it follows that these two projections are legs, and the length of the diagonal (hypotenuse) is equal to the square root of the sum of the squares of their lengths: AC=v((X?-X?)?+(Y?- Y?)?).
4. To find the diagonal rectangle in a three-dimensional coordinate system, proceed in the same way as in the previous step, only adding to the formula the length of the projection onto the third coordinate axis: AC=v((X?-X?)?+(Y?-Y?)?+(Z?- Z?)?).
Video on the topic
A mathematical joke remains in the memory of many: Pythagorean pants are equal in all directions. Use it to calculate diagonal rectangle .
You will need
- A sheet of paper, a ruler, a pencil, a calculator with a function for calculating roots.
Instructions
1. A rectangle is a quadrilateral whose angles are all right. Diagonal rectangle- a straight line segment connecting its two opposite vertices.
2. On a piece of paper supported by a ruler and pencil, draw an arbitrary rectangle ABCD. It’s cooler to do this on a squared notebook sheet - it will be easier to draw right angles. Connect the vertices with a segment rectangle A and C. The resulting segment AC is diagonal Yu rectangle ABCD.
3. Note, diagonal AC divides rectangle ABCD into triangles ABC and ACD. The resulting triangles ABC and ACD are right triangles, because angles ABC and ADC are equal to 90 degrees (by definition rectangle). Remember the Pythagorean theorem - the square of the hypotenuse is equal to the sum of the squares of the legs.
4. The hypotenuse is the side of the triangle opposite the right angle. Legs are the sides of a triangle adjacent to a right angle. In relation to triangles ABC and ACD: AB and BC, AD and DC are legs, AC is the universal hypotenuse for both triangles (desired diagonal). Consequently, AC squared = square AB + square BC or AC squared = square AD + square DC. Substitute the side lengths rectangle into the above formula and calculate the length of the hypotenuse (diagonal rectangle).
5. Let's say the sides rectangle ABCD are equal to the following values: AB = 5 cm and BC = 7 cm. The square of the diagonal AC of a given rectangle calculated using the Pythagorean theorem: AC squared = square AB + square BC = 52+72 = 25 + 49 = 74 sq.cm. Using a calculator, calculate the square root of 74. You should get 8.6 cm (rounded value). Please note that according to one of the properties rectangle, its diagonals are equal. So the length of the 2nd diagonal BD rectangle ABCD is equal to the length of diagonal AC. For the example above, this value is 8.6 cm.
Video on the topic
Tip 6: How to find the diagonal of a parallelogram given the sides
A parallelogram is a quadrilateral whose opposite sides are parallel. The straight lines connecting its opposite angles are called diagonals. Their length depends not only on the lengths of the sides of the figure, but also on the values of the angles at the vertices of this polygon; therefore, without knowing the truth of one of the angles, calculating the lengths of the diagonals is allowed only in exceptional cases. These are special cases of parallelograms - square and rectangle.
Instructions
1. If the lengths of all sides of a parallelogram are identical (a), then this figure can also be called a square. The values of all its angles are equal to 90°, and the lengths of the diagonals (L) are identical and can be calculated using the Pythagorean theorem for a right triangle. Multiply the length of the side of the square by the root of two - the result will be the length of each of its diagonals: L=a*?2.
2. If it is known about a parallelogram that it is a rectangle with the length (a) and width (b) indicated in the conditions, then in this case the lengths of the diagonals (L) will be equal. And here, too, use the Pythagorean theorem for a triangle in which the hypotenuse is the diagonal, and the legs are two adjacent sides of the quadrilateral. Calculate the desired value by taking the root of the sum of the squared width and height of the rectangle: L=?(a?+b?).
3. For all other cases, the skill of the side lengths alone is sufficient only to determine a value that includes the lengths of both diagonals at once - the sum of their squares, by definition, is equal to twice the sum of the squares of the side lengths. If, in addition to the lengths of the two adjacent sides of the parallelogram (a and b), the angle between them (?) is also known, then this will allow us to calculate the lengths of any segment connecting the opposite corners of the figure. Find the length of the diagonal (L?), lying opposite the given angle, using the cosine theorem - add the squares of the lengths of adjacent sides, subtract from the total the product of the same lengths by the cosine of the angle between them, and from the resulting value take the square root: L? = ?(a?+b?-2*a*b*cos(?)). To find the length of another diagonal (L?), you can use the property of a parallelogram given at the beginning of this step - double the sum of the squares of the lengths of 2 sides, subtract the square of the calculated diagonal from the total, and take the root from the resulting value. In general, this formula can be written as follows: L? = ?(a?+b?- L??) = ?(a?+b?-(a?+b?-2*a*b*cos(?))) = ?(a?+b?- a?-b?+2*a*b*cos(?)) = ?(2*a*b*cos(?)).