When you were small and played with cubes, you may have added the figures shown in Figure 154. These figures give an idea of cuboid. The shape of a rectangular parallelepiped is, for example, a box of chocolates, a brick, Matchbox, packing box, juice package.
Figure 155 shows a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 .
cuboid limited to six faces. Each face is a rectangle, i.e. the surface of a cuboid consists of six rectangles.
The sides of the faces are called edges of a rectangular parallelepiped, face vertices − vertices of a rectangular parallelepiped. For example, the segments AB, BC, A 1 B 1 are edges, and the points B, A 1 , C 1 are the vertices of the parallelepiped ABCDA 1 B 1 C 1 D 1 (Fig. 155).
A cuboid has 8 vertices and 12 edges.
Faces AA 1 B 1 B and DD 1 C 1 C do not have common vertices. Such edges are called opposite. The parallelepiped ABCDA 1 B 1 C 1 D 1 has 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 cuboid are equal.
In figure 155, face ABCD is called basis cuboid 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 cuboid, it suffices to consider any three edges that have a common vertex. The lengths of these edges are called measurements rectangular parallelepiped. To distinguish between them, use the names: length, width, height(Fig. 156).
A rectangular parallelepiped in which all dimensions are equal is called cube(Fig. 157). The surface of a cube consists of six equal squares.
If the box, which has the shape of a rectangular parallelepiped, is opened ( fig. 158) and cut along four vertical edges ( fig. 159), and then deployed, then 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 sweep, 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.
A cuboid 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 have met one of the seven wonders of the world - the Egyptian pyramids.
Figure 163 shows the pyramids MABC, MABCD, MABCDE. The surface of the pyramid is side faces− triangles having a common vertex, and grounds(Fig. 164). The common vertex of the side faces is called the edges of the base of the pyramid, and the sides of the side faces that do not belong to the base − lateral ribs of the pyramid.
Pyramids can be classified according to the number of sides of the base: triangular, quadrangular, pentagonal (see fig. 163), etc.
The surface of a 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, in which all faces are equilateral triangles.
Polyhedra are examples geometric bodies.
Figure 167 shows familiar geometric bodies that are not polyhedra. You will learn more about these bodies in the 6th grade.
A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will parallelograms.
Each parallelepiped can be considered as a prism with three different ways, since every two opposite faces can be taken as bases (in Fig. 5, faces ABCD and A "B" C "D", or ABA "B" and CDC "D", or BC "C" and ADA "D") .
The body under consideration has twelve edges, four equal and parallel to each other.
Theorem 3
. The diagonals of the parallelepiped intersect at one point, coinciding with the midpoint of each of them.
The parallelepiped ABCDA"B"C"D" (Fig. 5) has four diagonals AC", BD", CA", DB". We must prove that the midpoints of any two of them, for example, AC and BD, coincide. This follows from the fact that the figure ABC "D", which has equal and parallel sides AB and C "D", is a parallelogram.
Definition 7
. A right parallelepiped is a parallelepiped that is also a straight prism, that is, a parallelepiped whose side edges are perpendicular to the base plane.
Definition 8
. A rectangular parallelepiped is a right parallelepiped whose base is a rectangle. In this case, all its faces will be rectangles.
A rectangular parallelepiped is a right prism, no matter which of its faces we take as the base, since each of its edges is perpendicular to the edges coming out of the same vertex with it, and will, therefore, be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, box can be viewed as a right prism in only one way.
Definition 9
. The lengths of three edges of a cuboid, of which no two are parallel to each other (for example, three edges coming out of the same vertex), are called its dimensions. Two |rectangular parallelepipeds having correspondingly equal dimensions are obviously equal to each other.
Definition 10
A cube is a rectangular parallelepiped, all three dimensions of which are equal to each other, so that all its faces are squares. Two cubes whose edges are equal are equal. 
Definition 11
. An inclined parallelepiped in which all edges are equal and the angles of all faces are equal or complementary is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (The shape of a rhombohedron is found in some crystals of great importance, such as crystals of Iceland spar.) In a rhombohedron one can find such a vertex (and even two opposite vertices) that all angles adjacent to it are equal to each other.
Theorem 4
. The diagonals of a rectangular parallelepiped are equal to each other. The square of the diagonal is equal to the sum of the squares of three dimensions.
In a rectangular parallelepiped ABCDA "B" C "D" (Fig. 6), the diagonals AC "and BD" are equal, since the quadrilateral ABC "D" is a rectangle (line AB is perpendicular to the plane BC "C", in which lies BC ") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the hypotenuse square theorem. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC "2 \u003d AB 2 + AA" 2 + A "D" 2 \u003d AB 2 + AA "2 + AD 2.
Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition involves applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply the constant units of time to the reciprocal. FROM physical point To the eye, it looks like time slowing down until it stops completely at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs with constant speed. Each subsequent segment of its 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 infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces 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 insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of 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 the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What do I want to point to 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 exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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 desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: different coins available different amount dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interesting question: where is the boundary beyond which 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 here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us about either 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can 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 elementarily.
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's say we 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. We write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up 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 used by mathematicians. But that's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, 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. FROM a large number 12345 I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have 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 like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. Because we can't 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 of a mathematical action does not depend on the value of the number, the unit of measurement used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
It will be useful for high school students to learn how to solve USE tasks to find the volume and other unknown parameters of a rectangular parallelepiped. The experience of previous years confirms the fact that such tasks are quite difficult for many graduates.
At the same time, high school students with any level of training should understand how to find the volume or area of a rectangular parallelepiped. Only in this case they will be able to count on getting competitive scores based on the results of passing the unified state exam in mathematics.
Key points to remember
- The parallelograms that make up the parallelepiped are its faces, their sides are edges. The vertices of these figures are considered to be the vertices of the polyhedron itself.
- All diagonals of a cuboid are equal. Since this is a straight polyhedron, the side faces are rectangles.
- Since a parallelepiped is a prism with a parallelogram at its base, this figure has all the properties of a prism.
- Side ribs rectangular parallelepiped are perpendicular to the base. Therefore, they are its heights.
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You will find the necessary basic information in the "Theoretical reference" section. You can also immediately start solving problems on the topic "Rectangular parallelepiped" online. In the section "Catalogue" is presented big selection exercises of varying difficulty. The base of tasks is regularly updated.
Check if you can easily find the volume of a cuboid right now. Disassemble any task. If the exercise is easy for you, move on to more complex tasks. And if there are certain difficulties, we recommend that you plan your day in such a way that your schedule includes classes with the Shkolkovo remote portal.