Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, it can be thought of as a rectangle, with one side representing lettuce and the other side representing water. The sum of these two sides will indicate borscht. The diagonal and area of ​​such a “borscht” rectangle are purely mathematical concepts and are never used in borscht recipes.


How do lettuce and water turn into borscht from a mathematical point of view? How can the sum of two line segments become trigonometry? To understand this, we need linear angular functions.


You won't find anything about linear angular functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.

Linear angular functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.

Is it possible to do without linear angular functions? It’s possible, because mathematicians still manage without them. The trick of mathematicians is that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We don’t know other problems and we don’t know how to solve them. What should we do if we only know the result of the addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angular functions. Next, we ourselves choose what one term can be, and linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. IN Everyday life We can do just fine without decomposing the sum; subtraction is enough for us. But when scientific research laws of nature, decomposing a sum into its components can be very useful.

Another law of addition that mathematicians don't like to talk about (another of their tricks) requires that the terms have the same units of measurement. For salad, water, and borscht, these could be units of weight, volume, value, or unit of measure.

The figure shows two levels of difference for mathematical . The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the field of units of measurement, which are shown in square brackets and indicated by the letter U. This is what physicists do. We can understand the third level - differences in the area of ​​​​the objects being described. Different objects can have the same number of identical units of measurement. How important this is, we can see in the example of borscht trigonometry. If we add subscripts to the same unit designation for different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or due to our actions. Letter W I will denote water by the letter S I'll designate the salad with a letter B- borsch. This is what linear angular functions for borscht will look like.

If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What were we taught to do then? We were taught to separate units of measurement from numbers and add numbers. Yes, any one number can be added to any other number. This is a direct path to the autism of modern mathematics - we do it incomprehensibly what, incomprehensibly why, and very poorly understand how this relates to reality, because of the three levels of difference, mathematicians operate with only one. It would be more correct to learn how to move from one unit of measurement to another.

Bunnies, ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar problem for adults. What do you get when you add bunnies and money? There are two possible solutions here.

First option. We determine the market value of the bunnies and add it to the available amount of money. We got total cost our wealth in monetary terms.

Second option. You can add the number of bunnies to the number of banknotes we have. We will receive the amount of movable property in pieces.

As you can see, the same law of addition allows us to obtain different results. It all depends on what exactly we want to know.

But let's get back to our borscht. Now we can see what will happen when different meanings angle of linear angular functions.

The angle is zero. We have salad, but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. There can be zero borscht with zero salad (right angle).


For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This happens because addition itself is impossible if there is only one term and the second term is missing. You can feel about this however you like, but remember - everything mathematical operations mathematicians themselves came up with zero, so throw away your logic and stupidly cram the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero”, “beyond the puncture of the point is zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never again have a question whether zero is a natural number or not, because such a question loses all meaning: how can something that is not a number be considered a number? It's like asking what color an invisible color should be classified as. Adding a zero to a number is the same as painting with paint that is not there. We waved a dry brush and told everyone that “we painted.” But I digress a little.

The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but not enough water. As a result, we will get thick borscht.

The angle is forty-five degrees. We have equal quantities of water and salad. This is the perfect borscht (forgive me, chefs, it's just math).

The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You will get liquid borscht.

Right angle. We have water. All that remains of the salad are memories, as we continue to measure the angle from the line that once marked the salad. We can't cook borscht. The amount of borscht is zero. In this case, hold on and drink water while you have it)))

Here. Something like this. I can tell other stories here that would be more than appropriate here.

Two friends had their shares in general business. After killing one of them, everything went to the other.

The emergence of mathematics on our planet.

All these stories are told in the language of mathematics using linear angular functions. Some other time I'll show you real place these functions in the structure of mathematics. In the meantime, let's return to borscht trigonometry and consider projections.

Saturday, October 26, 2019

Wednesday, August 7, 2019

Concluding the conversation about, we need to consider an infinite set. The point is that the concept of “infinity” affects mathematicians like a boa constrictor affects a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:

The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take the infinite set as an example natural numbers, then the considered examples can be presented as follows:

To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantasy story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite number of such corridors. Moreover, the “infinite hotel” has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.

Option one. “Let us be given” one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:

I recorded the actions in algebraic system notation and in the notation system adopted in set theory, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same unit is added.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. Let's take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. This is what we get:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If you add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.

You can accept or not accept my reasoning - it is your own business. But if you ever encounter mathematical problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).

pozg.ru

Sunday, August 4, 2019

I was finishing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... rich theoretical basis The mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it difficult for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics is not holistic in nature and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I won’t go far to confirm my words - it has a language and conventions that are different from the language and symbols many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious mistakes of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.

May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we use the usual school math. Look what happened.

After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that essentially everything was done correctly; it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I will tell you about this.

As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.

As you can see, units of measurement and ordinary mathematics make set theory a relic of the past. A sign that all is not well with set theory is that for set theory mathematicians invented own language and own notations. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.

In conclusion, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

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, to reach a common opinion about the essence of paradoxes scientific community so far it has not been possible... 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 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 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 not be sought endlessly 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.
I'll show you the process with an example. We select the “red solid in a pimple” - this is our “whole”. At the same time, we see that these things are with a bow, and there are without a bow. After that, we select part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.

Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting sets “with a bow” and “red” the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so it will be.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. This is what it looks like.

The letter "a" with different indices means different units measurements. The units of measurement by which the “whole” is distinguished at the preliminary stage are highlighted in brackets. The unit of measurement by which the set is formed is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing that it is “obvious,” because units of measurement are not part of their “scientific” arsenal.

Using units of measurement, it is very easy to split one set or combine several sets into one superset. Let's take a closer look at the algebra of this process.

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parallelepiped, parallelepiped photo
Parallelepiped(ancient Greek παραλληλ-επίπεδον from ancient Greek παρ-άλληλος - “parallel” and other Greek ἐπί-πεδον - “plane”) - a prism, the base of which is a parallelogram, or (equivalently) polyhedron, which has six faces and each of them - parallelogram.

Types of parallelepiped

Rectangular parallelepiped

There are several types of parallelepipeds:

  • A cuboid is a parallelepiped whose all faces are rectangles.
  • An inclined parallelepiped is a parallelepiped whose side faces are not perpendicular to the bases.

Essential elements

Two faces of a parallelepiped that do not have a common edge are called opposite, and those that have a common edge are called adjacent. Two vertices of a parallelepiped that do not belong to the same face are called opposite. The segment connecting opposite vertices is called the diagonal of a parallelepiped. The lengths of three edges of a rectangular parallelepiped that have a common vertex are called its dimensions.

Properties

  • The parallelepiped is symmetrical about the middle of its diagonal.
  • Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it.
  • The opposite faces of a parallelepiped are parallel and equal.
  • Square of the diagonal length of a cuboid equal to the sum squares of its three dimensions.

Basic formulas

Right parallelepiped

Lateral surface area Sb=Po*h, where Po is the perimeter of the base, h is the height

Total surface area Sp=Sb+2So, where So is the base area

Volume V=Sо*h

Rectangular parallelepiped

Main article: Rectangular parallelepiped

Side surface area Sb=2c(a+b), where a, b are the sides of the base, c is the side edge of the rectangular parallelepiped

Total surface area Sp=2(ab+bc+ac)

Volume V=abc, where a, b, c are the dimensions of a rectangular parallelepiped.

Cube

Surface area:
Volume: , where is the edge of the cube.

Any parallelepiped

The volume and ratios in an inclined parallelepiped are often determined using vector algebra. The volume of the parallelepiped is absolute value mixed product three vectors defined by the three sides of the parallelepiped emanating from one vertex. The relationship between the lengths of the sides of the parallelepiped and the angles between them gives the statement that the Gram determinant of the indicated three vectors is equal to the square of their mixed product: 215.

In mathematical analysis

In mathematical analysis, an n-dimensional rectangular parallelepiped is understood as a set of points of the form

Notes

  1. Dvoretsky's Ancient Greek-Russian Dictionary “παραλληλ-επίπεδον”
  2. Gusyatnikov P.B., Reznichenko S.V. Vector algebra in examples and problems. - M.: graduate School, 1985. - 232 p.

Links

Wiktionary has an article "parallelepiped"
  • Rectangular parallelepiped
  • Parallelepiped, educational film

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Parallelepiped Information About

Translated from Greek language parallelogram means plane. A parallelepiped is a prism with a parallelogram at its base. There are five types of parallelogram: oblique, straight and cuboid. The cube and rhombohedron also belong to the parallelepiped and are its variety.

Before moving on to the basic concepts, let's give some definitions:

  • The diagonal of a parallelepiped is a segment that unites the vertices of the parallelepiped that are opposite each other.
  • If two faces have a common edge, then we can call them adjacent edges. If there is no common edge, then the faces are called opposite.
  • Two vertices that do not lie on the same face are called opposite.

What properties does a parallelepiped have?

  1. The faces of a parallelepiped lying on opposite sides are parallel to each other and equal to each other.
  2. If you draw diagonals from one vertex to another, then the point of intersection of these diagonals will divide them in half.
  3. The sides of the parallelepiped lying at the same angle to the base will be equal. In other words, the angles of the co-directed sides will be equal to each other.

What types of parallelepiped are there?

Now let's figure out what kind of parallelepipeds there are. As mentioned above, there are several types of this figure: straight, rectangular, inclined parallelepiped, as well as cube and rhombohedron. How do they differ from each other? It's all about the planes that form them and the angles they form.

Let's look in more detail at each of the listed types of parallelepiped.

  • As is already clear from the name, an inclined parallelepiped has inclined faces, namely those faces that are not at an angle of 90 degrees in relation to the base.
  • But for a right parallelepiped, the angle between the base and the edge is exactly ninety degrees. It is for this reason that this type of parallelepiped has such a name.
  • If all the faces of the parallelepiped are identical squares, then this figure can be considered a cube.
  • A rectangular parallelepiped received this name because of the planes that form it. If they are all rectangles (including the base), then this is a cuboid. This type of parallelepiped is not found very often. Translated from Greek, rhombohedron means face or base. This is the name given to a three-dimensional figure whose faces are rhombuses.



Basic formulas for a parallelepiped

The volume of a parallelepiped is equal to the product of the area of ​​the base and its height perpendicular to the base.

The area of ​​the lateral surface will be equal to the product of the perimeter of the base and the height.
Knowing the basic definitions and formulas, you can calculate the base area and volume. The base can be chosen at your discretion. However, as a rule, a rectangle is used as the base.

Theorem. In any parallelepiped, opposite faces are equal and parallel.

Thus, the faces (Fig.) BB 1 C 1 C and AA 1 D 1 D are parallel, because two intersecting lines BB 1 and B 1 C 1 of one face are parallel to two intersecting lines AA 1 and A 1 D 1 of the other. These faces are equal, since B 1 C 1 =A 1 D 1, B 1 B=A 1 A (as opposite sides parallelograms) and ∠BB 1 C 1 = ∠AA 1 D 1 .

Theorem. In any parallelepiped, all four diagonals intersect at one point and are bisected at it.

Let's take (Fig.) some two diagonals in the parallelepiped, for example, AC 1 and DB 1, and draw straight lines AB 1 and DC 1.


Since the edges AD and B 1 C 1 are respectively equal and parallel to the edge BC, then they are equal and parallel to each other.

As a result, the figure ADC 1 B 1 is a parallelogram in which C 1 A and DB 1 are diagonals, and in a parallelogram the diagonals intersect in half.

This proof can be repeated for every two diagonals.

Therefore, diagonal AC 1 intersects BD 1 in half, diagonal BD 1 intersects A 1 C in half.

Thus, all diagonals intersect in half and, therefore, at one point.

Theorem. In a rectangular parallelepiped, the square of any diagonal is equal to the sum of the squares of its three dimensions.

Let (Fig.) AC 1 be some diagonal of a rectangular parallelepiped.


Drawing AC, we get two triangles: AC 1 C and ACB. Both of them are rectangular:


the first because the parallelepiped is straight, and therefore edge CC 1 is perpendicular to the base,

the second because the parallelepiped is rectangular, which means that there is a rectangle at its base.

From these triangles we find:

AC 2 1 = AC 2 + CC 2 1 and AC 2 = AB 2 + BC 2


Therefore, AC 2 1 = AB 2 + BC 2 + CC 2 1 = AB 2 + AD 2 + AA 2 1

Consequence. In a rectangular parallelepiped all diagonals are equal.