Zero is a very interesting figure in itself. By itself, it means emptiness, lack of meaning, and next to another number increases its significance by 10 times. Any numbers in the zero degree always give 1. This sign was used in the Mayan civilization, and it also denoted the concept of "beginning, cause". Even the calendar started from day zero. And this figure is also associated with a strict ban.

Ever since the elementary school years, we all have clearly learned the rule “you cannot divide by zero”. But if in childhood you take a lot on faith and the words of an adult rarely raise doubts, then over time, sometimes you still want to understand the reasons, to understand why certain rules were established.

Why can't you divide by zero? I would like to get a clear logical explanation for this question. In the first grade, the teachers could not do this, because in mathematics the rules are explained using equations, and at that age we had no idea what it was. And now it's time to figure it out and get a clear logical explanation of why you can't divide by zero.

The fact is that in mathematics, only two of the four basic operations (+, -, x, /) with numbers are recognized as independent: multiplication and addition. The rest of the operations are considered to be derivatives. Let's look at a simple example.

Tell me, how much will it turn out if you subtract 18 from 20? Naturally, the answer immediately arises in our head: it will be 2. And how did we come to such a result? To some, this question will seem strange - after all, everything is already clear that it will turn out 2, someone will explain that he took 18 from 20 kopecks and he got two kopecks. Logically, all these answers are not in doubt, but from the point of view of mathematics, this problem should be solved in a different way. Let us remind once again that the main operations in mathematics are multiplication and addition, and therefore in our case the answer lies in the solution of the following equation: x + 18 = 20. From which it follows that x = 20 - 18, x = 2. It would seem, why describe everything in such detail? After all, everything is elementary simple. However, without this, it is difficult to explain why one cannot divide by zero.

Now let's see what happens if we wish to divide 18 by zero. Let's make the equation again: 18: 0 = x. Since the division operation is a derivative of the multiplication procedure, transforming our equation we get x * 0 = 18. This is where the dead end begins. Any number in the place of x when multiplied by zero will give 0 and we will not be able to get 18 in any way. Now it becomes very clear why one cannot divide by zero. Zero itself can be divided by any number, but on the contrary - alas, it cannot be.

What happens if zero is divided by itself? It can be written like this: 0: 0 = x, or x * 0 = 0. This equation has countless solutions. So the end result is infinity. Therefore, the operation does not make sense in this case either.

Division by 0 is at the root of many supposed mathematical jokes that can be used to puzzle any ignorant person if desired. For example, consider the equation: 4 * x - 20 = 7 * x - 35. Let's take out 4 in the left part, and in the right part 7. We get: 4 * (x - 5) = 7 * (x - 5). Now we multiply the left and right sides of the equation by the fraction 1 / (x - 5). The equation will take this form: 4 * (x - 5) / (x - 5) = 7 * (x - 5) / (x - 5). Reduce the fractions by (x - 5) and we get that 4 = 7. From this we can conclude that 2 * 2 = 7! Of course, the catch here is that it is equal to 5 and it was impossible to cancel fractions, since this led to division by zero. Therefore, when reducing fractions, you must always check so that zero does not accidentally end up in the denominator, otherwise the result will turn out to be completely unpredictable.

If we can rely on other laws of arithmetic, then this separate fact can be proved.

Suppose there is a number x for which x * 0 = x ", and x" is not zero (for simplicity, we will assume that x "> 0)

Then, on the one hand, x * 0 = x ", on the other hand, x * 0 = x * (1 - 1) = x - x

It turns out that x - x = x ", whence x = x + x", that is, x> x, which cannot be true.

This means that our assumption leads to a contradiction and there is no such number x for which x * 0 would not be equal to zero.

the assumption cannot be true because it is just an assumption! no one can explain in simple language or is at a loss! if 0 * x = 0 then 0 * x = (0 + 0) * x = 0 * x + 0 * x and as a result we have reduced the right to left 0 = 0 * x this is like a mathematical proof! but such nonsense with this zero is terribly contradictory and in my opinion 0 should not be a number, but only just an abstract concept! So that mere mortals do not burn in the brain by the fact that the physical presence of objects, when miraculously multiplied by nothing, gave rise to nothing!

P / s is not entirely clear to me, not a mathematician, but a mere mortal where did you get units in the equation-reasoning (like 0 is the same as 1-1)

I bastard with reasoning, like there is some kind of X and let it be any number

is in the equation 0 and when multiplying by it we reset all numerical values

hence X is a numerical value, and 0 is the number of actions performed on the number X (and actions, in turn, are also displayed in numerical format)

EXAMPLE on apples)):

Kolya had 5 apples, he took these apples and went to the market in order to increase the capital, but the day was rainy, the cloudy trade did not work out and Kalek returned home with nothing. In mathematical terms, the story about Kolya and apples looks like this

5 apples * 0 sales = got 0 profit 5 * 0 = 0

Before going to the bazaar, Kolya went and plucked 5 apples from the tree, and tomorrow he went to pick but didn't get there for some reason ...

Apples 5, tree 1, 5 * 1 = 5 (Kolya collected 5 apples on the 1st day)

Apples 0, tree 1, 0 * 1 = 0 (actually the result of Kolya's work on the second day)

The scourge of mathematics is the word "Suppose"

To answer

And if in another way, 5 apples to 0 apples = how many apples, according to mathematics, there should be zero, and so

In fact, any numbers make sense only when they are associated with material objects, such as 1 cow, 2 cows, or whatever, and an account has appeared in order to count objects and not just like that, and there is a paradox if I do not have a cow , and the neighbor has a cow, and we multiply my absence by the neighbor's cow, then his cow should disappear, multiplication is generally invented to facilitate the addition of large quantities of identical objects when it is difficult to count them by the addition method, for example, money was added in columns of 10 coins, and then the number of columns was multiplied by the number of coins in a column, much easier than adding. but if the number of columns is multiplied by zero of coins, then naturally it will turn out to be zero, but if there are columns and coins, then how do not multiply them by zero, the coins will not go anywhere because they are there, and even if it is one coin, then the column is composed of one coin, so you can't go anywhere, so zero when multiplied by zero is obtained only under certain conditions, that is, in the absence of a material component, and if I have 2 socks, don't multiply them by zero, they won't go anywhere ...

The number 0 can be thought of as a kind of border that separates the world of real numbers from imaginary or negative ones. Due to the ambiguous position, many operations with this numerical value do not obey mathematical logic. The impossibility of dividing by zero is a prime example of this. And permitted arithmetic operations with zero can be performed using generally accepted definitions.

The story of zero

Zero is the point of reference in all standard numeric systems. Europeans began to use this number relatively recently, but the sages of ancient India used zero for a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Mayan number system. This American people used the duodecimal system of number, and they began with a zero on the first day of each month. Interestingly, the Maya sign for "zero" exactly coincided with the sign for "infinity." Thus, the ancient Maya concluded that these values ​​were identical and unknowable.

Math operations with zero

Standard math operations with zero can be boiled down to a few rules.

Addition: if you add zero to an arbitrary number, then it will not change its value (0 + x = x).

Subtraction: When subtracting zero from any number, the value of the subtracted remains unchanged (x-0 = x).

Multiplication: Any number multiplied by 0 gives 0 in the product (a * 0 = 0).

Division: zero can be divided by any number other than zero. In this case, the value of such a fraction will be 0. And division by zero is prohibited.

Exponentiation. This action can be performed with any number. An arbitrary number raised to the zero power will give 1 (x 0 = 1).

Zero to any power is equal to 0 (0 a = 0).

In this case, a contradiction immediately arises: the expression 0 0 has no meaning.

Paradoxes of mathematics

Many people know that division by zero is impossible from school. But for some reason it is not possible to explain the reason for such a ban. Indeed, why does the formula for division by zero not exist, but other actions with this number are quite reasonable and possible? The answer to this question is given by mathematicians.

The thing is that the usual arithmetic operations that schoolchildren learn in primary grades, in fact, are not nearly as equal as we think. All simple operations with numbers can be reduced to two: addition and multiplication. These actions are the essence of the very concept of number, and the rest of the operations are based on the use of these two.

Addition and multiplication

Let's take a standard example of subtraction: 10-2 = 8. At school, it is considered simply: if two are taken away from ten subjects, eight will remain. But mathematicians look at this operation in a completely different way. After all, such an operation as subtraction does not exist for them. This example can be written in another way: x + 2 = 10. For mathematicians, the unknown difference is simply a number that needs to be added to two to make eight. And no subtraction is required here, you just need to find a suitable numeric value.

Multiplication and division are treated the same way. In example 12: 4 = 3, you can understand that we are talking about dividing eight objects into two equal piles. But in reality this is just an inverted formula for writing 3x4 = 12. There are endless examples of division.

Division by 0 examples

This is where it becomes a little clear why it is impossible to divide by zero. Multiplication and division by zero obeys their own rules. All examples of the division of this quantity can be formulated as 6: 0 = x. But this is the inverted notation of the expression 6 * x = 0. But, as you know, any number multiplied by 0 gives in the product only 0. This property is inherent in the very concept of a zero value.

It turns out that such a number that, when multiplied by 0, gives some tangible value, does not exist, that is, this problem has no solution. You should not be afraid of such an answer, it is a natural answer for problems of this type. It's just that the 6-0 record doesn't make any sense, and it can't explain anything. In short, this expression can be explained by the immortal "division by zero is impossible."

Is there a 0: 0 operation? Indeed, if the operation of multiplying by 0 is legal, can zero be divided by zero? After all, the equation of the form 0x 5 = 0 is completely legal. Instead of the number 5, you can put 0, the product will not change from this.

Indeed, 0x0 = 0. But you still can't divide by 0. As said, division is simply the inverse of multiplication. Thus, if in the example 0x5 = 0, you need to determine the second factor, we get 0x0 = 5. Or 10. Or infinity. Dividing infinity by zero - how do you like it?

But if any number fits into the expression, then it does not make sense, we cannot choose one from the infinite set of numbers. And if so, it means that the expression 0: 0 does not make sense. It turns out that even zero itself cannot be divided by zero.

Higher mathematics

Division by zero is headache for school mathematics. Studied in technical universities mathematical analysis slightly expands the concept of problems that have no solution. For example, to the already known expression 0: 0, new ones are added that have no solution in school courses mathematics:

• infinity divided by infinity: ∞: ∞;
• infinity minus infinity: ∞ − ∞;
• one raised to an infinite power: 1 ∞;
• infinity times 0: ∞ * 0;
• some others.

It is impossible to solve such expressions by elementary methods. But higher mathematics thanks to the additional possibilities for a number of similar examples, it provides final solutions. This is especially evident in the consideration of problems from the theory of limits.

Disclosure of uncertainty

In the theory of limits, the value 0 is replaced by the conditional infinitesimal variable... And expressions in which, when the desired value is substituted, division by zero is obtained, are converted. Below is a standard example of limit expansion using ordinary algebraic transformations:

As you can see in the example, a simple reduction of the fraction leads its value to a completely rational answer.

When considering the limits trigonometric functions their expressions tend to be reduced to the first remarkable limit. When considering the limits in which the denominator goes to 0 when the limit is substituted, a second remarkable limit is used.

Lopital's method

In some cases, the limits of expressions can be replaced by the limit of their derivatives. Guillaume L'Hôpital - French mathematician, founder of the French school of mathematical analysis. He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. In mathematical notation, his rule is as follows.

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, this 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 represent borscht. The diagonal and area of ​​such a "borsch" rectangle are pure mathematical concepts and are never used in borscht recipes.

How do lettuce and water turn into borscht mathematically? How can the sum of two line segments turn into trigonometry? To understand this, we need linear angle functions.

You won't find anything about linear angle functions in mathematics 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 angle functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.

Can linear angle functions be dispensed with? You can, because mathematicians still do 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. Everything. We do not know other tasks and are not able to solve them. What to do if we only know the result of addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angle functions. Then we ourselves choose what one term can be, and the 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 perfectly manage without the decomposition of the sum, subtraction is enough for us. But in scientific research of the laws of nature, the decomposition of the sum into terms can be very useful.

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

The figure shows two levels of difference for math. 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 area 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 described objects. Different objects can have the same number of identical units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same designation of units of measurement of different objects, we can say exactly which mathematical value describes a particular object and how it changes over time or in connection with our actions. By letter W I will designate water, with the letter S I will designate the salad and the letter B- Borsch. This is what the linear angular functions for borsch would look like.

If we take some of the water and some of the salad, together they will turn into one portion of borscht. Here I suggest you take a 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 then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we do not understand what, it is not clear why, and we very poorly understand how this relates to reality, because of the three levels of difference, mathematics operates only one. It would be more correct to learn how to switch from one measurement unit to another.

And bunnies, and ducks, and animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a childish version of the problem. Let's take a look at a similar problem for adults. What happens if 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 the total value of 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 number of movable property in pieces.

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

But 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 cannot cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. Zero borscht can be at 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 is because the addition itself is impossible if there is only one term and there is no second term. You can relate to this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "for the knock-out point zero" and other delirium. It is enough to remember once that zero is not a number, and you will never have a question whether zero is natural number or not, because such a question generally loses any meaning: how can that which is not a number be considered a number? It's like asking what color an invisible color should be. Adding zero to a number is like painting with paint that doesn't exist. We waved with a dry brush and told everyone that "we have painted". But I digress a little.

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

The angle is forty-five degrees. We have equal amounts of water and salad. This is the perfect borscht (yes, the cooks will forgive me, 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 get liquid borscht.

Right angle. We have water. From the salad, only memories remain, as we continue to measure the angle from the line that once stood for the salad. We cannot 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 will be more than appropriate here.

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

The emergence of mathematics on our planet.

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

Saturday, 26 October 2019

I watched an interesting video about Grandi row One minus one plus one minus one - Numberphile... Mathematicians lie. They did not perform the equality test in the course of their reasoning.

This echoes my reasoning about.

Let's take a closer look at the signs of deceiving us by mathematicians. At the very beginning of reasoning, mathematicians say that the sum of a sequence DEPENDS on whether the number of elements in it is even or not. This is an OBJECTIVELY DETERMINED FACT. What happens next?

Then mathematicians subtract a sequence from one. What does this lead to? This leads to a change in the number of elements in the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we have added one element equal to one to the sequence. Despite all the external similarities, the sequence before conversion is not equal to the sequence after conversion. Even if we are talking about an infinite sequence, it must be remembered that an infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.

Putting an equal sign between two sequences that differ in the number of elements, mathematicians argue that the sum of the sequence DOES NOT DEPEND on the number of elements in the sequence, which contradicts an OBJECTIVELY DETERMINED FACT. Further reasoning about the sum of an infinite sequence is false, since it is based on false equality.

If you see that mathematicians in the course of proofs place parentheses, rearrange the elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card magicians, mathematicians distract your attention with various expression manipulations in order to end up slipping you a false result. If you cannot repeat the card trick without knowing the secret of deception, then in mathematics everything is much simpler: you do not even suspect anything about deception, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result, just like when something convinced you.

Question from the audience: And what about infinity (as the number of elements in sequence S), is it even or odd? How can you change the parity of something that has no parity?

Infinity for mathematicians, like the Kingdom of Heaven for priests - no one has ever been there, but everyone knows exactly how everything works there))) I agree, after death you will be absolutely indifferent whether you have lived an even or an odd number of days, but ... just one day at the beginning of your life, we will get a completely different person: his surname, name and patronymic are exactly the same, only the date of birth is completely different - he was born one day before you.

And now, in essence))) Suppose a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must also lose parity. We do not see this. The fact that we cannot say for sure whether the number of elements in an infinite sequence is even or odd does not mean at all that parity has disappeared. Parity, if it exists, cannot disappear without a trace into infinity, as in the sleeve of a sharpie. There is a very good analogy for this case.

Have you ever asked a cuckoo sitting in a clock in which direction the clock hand rotates? For her, the arrow rotates in reverse direction what we call "clockwise". As paradoxical as it sounds, the direction of rotation depends solely on which side we observe the rotation from. And so, we have one wheel that turns. We cannot say in which direction the rotation occurs, since we can observe it both from one side of the plane of rotation, and from the other. We can only attest to the fact that there is rotation. Complete analogy with the parity of an infinite sequence S.

Now let's add a second spinning wheel, whose plane of rotation is parallel to the plane of rotation of the first spinning wheel. We still cannot say for sure in which direction these wheels rotate, but we can absolutely say for sure whether both wheels rotate in the same direction or in opposite directions. Comparing two endless sequences S and 1-S, I showed with the help of mathematics that these sequences have different parity and it is an error to put an equal sign between them. Personally, I believe in mathematics, I do not trust mathematicians))) By the way, for a complete understanding of the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity"... This will need to be drawn.

Wednesday, 7 August 2019

Concluding the conversation about, there is an infinite number to consider. The result is that the concept of "infinity" acts on mathematicians like a boa constrictor on a rabbit. The tremulous fear of infinity robs mathematicians 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 as an example an infinite set of natural numbers, then the considered examples can be presented in the following form:

For a visual proof of their correctness, mathematicians have come up with many different methods. Personally, I look at all these methods as dancing shamans with tambourines. Essentially, they all boil down to the fact that either some of the rooms are not occupied 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 fantastic 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 the century. Of course, the time factor can be stupidly ignored, but this will already be from the category "the law is not written for fools." It all depends on what we are doing: adjusting reality to match mathematical theories or vice versa.

What is an "endless hotel"? An endless hotel is a hotel that always has any number of vacant places, no matter how many rooms are occupied. If all the rooms in the endless visitor corridor are occupied, there is another endless corridor with the 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, however, are not able to distance themselves from commonplace everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that you can "shove the stuff in."

I will demonstrate the logic of my reasoning to you on 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, in Nature there are no numbers. Yes, Nature is excellent at counting, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers there are. Consider both options, as befits a real scientist.

Option one. "Let us be given" a 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 there is nowhere to take them. We cannot add one to this set, since we already have it. And 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 a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write 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 subtracts from it and adds the same unit.

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. We 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. Here's what we get:

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

Lots of natural numbers are used for counting in the same way as a ruler for measurements. Now imagine adding one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - it's your own business. But if you ever run into mathematical problems, think about whether you are not following the path of false reasoning trodden by generations of mathematicians. After all, doing mathematics, first of all, form a stable stereotype of thinking in us, and only then add mental abilities to us (or, on the contrary, deprive us of free thinking).

pozg.ru

Sunday, 4 August 2019

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

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

Wow! How smart we are and how well we can see the shortcomings of others. Is it hard 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 and comes down to a set of disparate sections devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and legend many other areas of mathematics. The same names in different areas of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, 3 August 2019

How to subdivide a set? To do this, it is necessary to enter a new unit of measurement that is present for some of the elements of the selected set. Let's look at an example.

Let us have many 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, a subscript with a digit will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sex" and denote it by the letter b... Since sexual characteristics are inherent in all people, we multiply each element of the set A by gender b... Note that now our multitude of "people" has become a multitude of "people with sex characteristics." After that, we can divide the sex characteristics into masculine bm and women bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sex characteristics, it does not matter which one is 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 apply the usual school mathematics. See what happened.

After multiplication, reduction and rearrangement, we got two subsets: a subset of men Bm and a subset of women Bw... Mathematicians think about the same when they apply set theory in practice. But they do not devote us to the details, but give a 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 is the mathematics applied in the above transformations? I dare to assure you, in fact, everything was done correctly, it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? I'll tell you about it some other time.

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

As you can see, units and common mathematics make set theory a thing of the past. An indication that set theory is not all right is that mathematicians have come up with their own language and notation for set theory. Mathematicians did what shamans once did. Only shamans know how to "correctly" apply their "knowledge". They teach us this "knowledge".

Finally, I want to show you how mathematicians manipulate with
Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.

This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... the discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them has become a generally accepted solution to the question ..."[Wikipedia, Zeno's Aporia"]. 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 magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn over the logic we are used to, 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 infinitely quickly catch up with the turtle."

How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:

During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". 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 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 rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single 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 are needed, taken from the same point at different points in time, but the distance cannot be determined from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but it is impossible to determine the fact of movement from them (of course, additional data are still needed 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.
Let me show you the process with an example. We select "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 no bows. After that we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

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

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

The letter "a" with different indices denotes different units of measurement. In brackets are the units of measurement for which the "whole" is allocated at the preliminary stage. The unit of measurement, by which the set is formed, is taken out of the 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 dancing shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it “by evidence,” because units of measurement are not included in their “scientific” arsenal.

It is very easy to use units to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Back in school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!"- but all the same, a lot of controversy constantly arises around him. Someone just memorized the rule and does not bother with the question “why?”. "You can't and that's it, because they said so at school, a rule is a rule!" Someone can write half a notebook with formulas, proving this rule or, conversely, its illogicality.

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Who is right in the end

During these disputes, both people who have opposite points of view look at each other like a ram and prove with all their might their innocence. Although, if you look at them from the side, you can see not one, but two rams resting their horns against each other. The only difference between them is that one is slightly less educated than the other.

More often than not, those who believe this rule to be incorrect try to invoke logic in this way:

I have two apples on my table, if I put zero apples to them, that is, I don’t put a single one, then my two apples will not disappear from this! The rule is illogical!

Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 = 2. So we discard such a conclusion right away - it is illogical, although it has the opposite purpose - to call to logic.

What is multiplication

The original rule of multiplication was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies the naturalness of the number. Thus, any number with multiplication can be reduced to this equation:

1. 25 × 3 = 75
2. 25 + 25 + 25 = 75
3. 25 × 3 = 25 + 25 + 25

The conclusion follows from this equation, that multiplication is simplified addition.

What is zero

Any person from childhood knows: zero is emptiness, Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient oriental scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw deep meaning in this number. After all, zero, which has the meaning of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy over multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty digits in decimal fractions, this is done both before and after the comma.

Is it possible to multiply by emptiness

You can multiply by zero, but it's useless, because, whatever one may say, but even with multiplication negative numbers it will still be zero. It is enough just to remember this simple rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as the ancient scientists believed. The most logical explanation will be given below that this multiplication is useless, because when a number is multiplied by it, the same thing will still be obtained - zero.

Going back to the very beginning, to the argument about two apples, 2 times 0 looks like this:

• If you eat two apples five times, then you eat 2 × 5 = 2 + 2 + 2 + 2 + 2 = 10 apples
• If you eat them two three times, then 2 × 3 = 2 + 2 + 2 = 6 apples are eaten
• If you eat two apples zero times, then nothing will be eaten - 2 × 0 = 0 × 2 = 0 + 0 = 0

After all, to eat an apple 0 times means not to eat a single one. Even the smallest child will understand this. Whatever one may say - 0 will come out, a two or three can be replaced with absolutely any number and absolutely the same will come out. To put it simply, then zero is nothing and when you have there is nothing, then no matter how much you multiply, it doesn't matter will be zero... There is no magic, and nothing will come out of an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to dissipate, and everything falls into place.

Division

Another important rule follows from all of the above:

You cannot divide by zero!

This rule has also been stubbornly hammered into our heads since childhood. We just know that it's impossible and that's all, without stuffing our heads with unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the simplest question from school curriculum because there are not so many controversies and contradictions around this rule.

Everyone just memorized the rule and did not divide by zero, not suspecting that the answer lies on the surface. Addition, multiplication, division and subtraction are unequal, only multiplication and addition are complete from the above, and all other manipulations with numbers are built from them. That is, writing 10: 2 is an abbreviation of the equation 2 * x = 10. So, writing 10: 0 is the same abbreviation from 0 * x = 10. It turns out that dividing by zero is a task to find a number, multiplying it by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be incorrect a priori.

Let me tell you

To not divide by 0!

Cut 1 as you want, lengthwise,

Just don't divide by 0!