When you were little and played with cubes, you may have made the shapes shown in Figure 154. These figures give an idea of rectangular parallelepiped. The shape of a rectangular parallelepiped is, for example, a box of chocolates, a brick, Matchbox, packing box, juice pack.
Figure 155 shows a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1.
Rectangular parallelepiped limited to six edges. Each face is a rectangle, i.e. The surface of a rectangular parallelepiped consists of six rectangles.
The sides of the faces are called edges of a rectangular parallelepiped, vertices of faces − vertices of a rectangular parallelepiped. For example, segments AB, BC, A 1 B 1 are edges, and points B, A 1, C 1 are vertices of the parallelepiped ABCDA 1 B 1 C 1 D 1 (Fig. 155).
A rectangular parallelepiped has 8 vertices and 12 edges.
The faces AA 1 B 1 B and DD 1 C 1 C do not have common vertices. Such edges are called opposite. In the parallelepiped ABCDA 1 B 1 C 1 D 1 there are two more pairs of opposite faces: rectangles ABCD and A 1 B 1 C 1 D 1, as well as rectangles AA 1 D 1 D and BB 1 C 1 C.
Opposite faces of a rectangular parallelepiped are equal.
In Figure 155, the face ABCD is called basis rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 .
The surface area of a parallelepiped is the sum of the areas of all its faces.
To have an idea of the dimensions of a rectangular parallelepiped, it is enough to consider any three edges that have a common vertex. The lengths of these edges are called measurements rectangular parallelepiped. To distinguish them, they use names: length, width, height(Fig. 156).
A rectangular parallelepiped in which all dimensions are equal is called cube(Fig. 157). The surface of the cube consists of six equal squares.
If a box in the shape of a rectangular parallelepiped is opened (Fig. 158) and cut along four vertical edges (Fig. 159), and then unfolded, we get a figure consisting of six rectangles (Fig. 160). This figure is called development of a rectangular parallelepiped.
Figure 161 shows a figure consisting of six equal squares. It is the development of a cube.
Using a development, you can make a model of a rectangular parallelepiped.
This can be done, for example, like this. Draw its outline on paper. Cut it out, bend it along the segments corresponding to the edges of the rectangular parallelepiped (see Fig. 159), and glue it together.
A rectangular parallelepiped is a type of polyhedron - a figure whose surface consists of polygons. Figure 162 shows polyhedra.
One type of polyhedron is pyramid.
This figure is not new to you. Studying the course Ancient world, you got acquainted with one of the seven wonders of the world - the Egyptian pyramids.
Figure 163 shows the pyramids MABC, MABCD, MABCDE. The surface of the pyramid consists of side faces− triangles having a common vertex, and grounds(Fig. 164). The common vertex of the lateral faces is called edges of the base of the pyramid, and the sides of the side faces that do not belong to the base are lateral edges of the pyramid.
Pyramids can be classified according to the number of sides of the base: triangular, quadrangular, pentagonal (see Fig. 163), etc.
Surface triangular pyramid consists of four triangles. Any of these triangles can serve as the base of a pyramid. This base is a type of pyramid, any face of which can serve as its base.
Figure 165 shows a figure that can serve sweep quadrangular pyramid . It consists of a square and four equal isosceles triangles.
Figure 166 shows a figure consisting of four equal equilateral triangles. Using this figure, you can make a model of a triangular pyramid, all of whose faces are equilateral triangles.
Polyhedra are examples geometric bodies.
Figure 167 shows familiar ones geometric bodies, which are not polyhedra. You will learn more about these bodies in 6th grade.
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... we 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, mathematical apparatus The use of 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. WITH physical point From a perspective, it 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? Stay in constant units measurements of time and do not go to reciprocal quantities. 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. For the next time interval, equal to 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 is 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 in different moments time, but distance cannot be determined from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed 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 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 “screw 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. Apply 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 stack and hand it to the mathematician" mathematical set salaries." We explain to the mathematics 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: on different coins there is different quantities mud, crystal structure and the arrangement of atoms in each coin is unique...
And now I have the most interest Ask: 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 the help of 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 numbers 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 systems In calculus, 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 measurements. 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 mathematical operation does not depend on the size of the number, the unit of measurement used and who performs the 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 stupid, no knowledgeable in physics. She just has an arch stereotype of perception 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.
In geometry key concepts are plane, point, straight line and angle. Using these terms, you can describe any geometric figure. Polyhedra are usually described in terms of more simple figures, which lie in the same plane, such as a circle, triangle, square, rectangle, etc. In this article we will look at what a parallelepiped is, describe the types of parallelepipeds, its properties, what elements it consists of, and also give the basic formulas for calculating the area and volume for each type of parallelepiped.
Definition
Parallelepiped in three-dimensional space is a prism, all sides of which are parallelograms. Accordingly, she can only have three pairs parallel parallelograms or six sides.
To visualize a parallelepiped, imagine an ordinary standard brick. Brick - good example a rectangular parallelepiped that even a child can imagine. Other examples include multi-storey panel houses, cabinets, storage containers food products appropriate form, etc.
Varieties of figure
There are only two types of parallelepipeds:
- Rectangular, all side faces of which are at an angle of 90° to the base and are rectangles.
- Sloping, the side edges of which are located at a certain angle to the base.
What elements can this figure be divided into?
- Just like any other geometric figure, in a parallelepiped, any 2 faces with a common edge are called adjacent, and those that do not have it are parallel (based on the property of a parallelogram, which has pairs of parallel opposite sides).
- The vertices of a parallelepiped that do not lie on the same face are called opposite.
- The segment connecting such vertices is a diagonal.
- The lengths of the three edges of a cuboid that meet at one vertex are its dimensions (namely, its length, width and height).
Shape Properties
- It is always built symmetrically with respect to the middle of the diagonal.
- The intersection point of all diagonals divides each diagonal into two equal segments.
- Opposite faces are equal in length and lie on parallel lines.
- If you add the squares of all dimensions of a parallelepiped, the resulting value will be equal to the square of the length of the diagonal.
Calculation formulas
The formulas for each particular case of a parallelepiped will be different.
For an arbitrary parallelepiped it is true that its volume is equal to absolute value triple dot product vectors of three sides emanating from one vertex. However, there is no formula for calculating the volume of an arbitrary parallelepiped.
For a rectangular parallelepiped the following formulas apply:
- V=a*b*c;
- Sb=2*c*(a+b);
- Sp=2*(a*b+b*c+a*c).
- V - volume of the figure;
- Sb - lateral surface area;
- Sp - area full surface;
- a - length;
- b - width;
- c - height.
Another special case of a parallelepiped in which all sides are squares is a cube. If any of the sides of the square is designated by the letter a, then the following formulas can be used for the surface area and volume of this figure:
- S=6*a*2;
- V=3*a.
- S- area of the figure,
- V is the volume of the figure,
- a is the length of the figure's face.
The last type of parallelepiped we are considering is a straight parallelepiped. What is the difference between a right parallelepiped and a cuboid, you ask. The fact is that the base of a rectangular parallelepiped can be any parallelogram, but the base of a straight parallelepiped can only be a rectangle. If we mark the perimeter of the base, equal to the sum lengths of all sides as Po, and the height is designated by the letter h, we have the right to use the following formulas to calculate the volume and areas of the full and lateral surfaces.
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