System of inequalities It is customary to call any set of two or more inequalities containing an unknown quantity.

This formulation is clearly illustrated, for example, by the following systems of inequalities:

Solve the system of inequalities - means to find all values ​​of an unknown variable at which each inequality of the system is realized, or to justify that such do not exist .

This means that for each individual system inequalities We calculate the unknown variable. Next, from the resulting values, selects only those that are true for both the first and second inequalities. Therefore, when substituting the selected value, both inequalities of the system become correct.

Let's look at the solution to several inequalities:

Let's place a pair of number lines one below the other; put the value on the top x, for which the first inequality about ( x> 1) become true, and at the bottom - the value X, which are the solution to the second inequality ( X> 4).

By comparing the data on number lines, note that the solution for both inequalities will X> 4. Answer, X> 4.

Example 2.

Calculating the first inequality we get -3 X< -6, или x> 2, second - X> -8, or X < 8. Затем делаем по аналогии с предыдущим примером. На верхнюю числовую прямую наносим все те значения X, at which the first is realized inequality system, and to the lower number line, all those values X, at which the second inequality of the system is realized.

Comparing the data, we find that both inequalities will be implemented for all values X, placed from 2 to 8. Set of values X denote double inequality 2 < X< 8.

Example 3. We'll find

One of the topics that requires maximum attention and perseverance from students is solving inequalities. So similar to equations and at the same time very different from them. Because solving them requires a special approach.

Properties that will be needed to find the answer

All of them are used to replace an existing entry with an equivalent one. Most of them are similar to what was in the equations. But there are also differences.

• A function that is defined in the ODZ, or any number, can be added to both sides of the original inequality.
• Likewise, multiplication is possible, but only by a positive function or number.
• If this action is performed with a negative function or number, then the inequality sign must be replaced with the opposite one.
• Functions that are non-negative can be raised to a positive power.

Sometimes solving inequalities is accompanied by actions that provide extraneous answers. They need to be eliminated by comparing the DL domain and the set of solutions.

Using the Interval Method

Its essence is to reduce the inequality to an equation in which there is a zero on the right side.

1. Determine the area where the permissible values ​​of the variables, that is, the ODZ, lie.
2. Transform the inequality using mathematical operations so that the right side has a zero.
3. Replace the inequality sign with “=” and solve the corresponding equation.
4. On the numerical axis, mark all the answers that were obtained during the solution, as well as the OD intervals. In case of strict inequality, the points must be drawn as punctured. If there is an equal sign, then they should be painted over.
5. Determine the sign of the original function on each interval obtained from the points of the ODZ and the answers dividing it. If the sign of the function does not change when passing through a point, then it is included in the answer. Otherwise, it is excluded.
6. The boundary points for ODZ need to be further checked and only then included or not in the answer.
7. The resulting answer must be written in the form of combined sets.

A little about double inequalities

They use two inequality signs at once. That is, some function is limited by conditions twice at once. Such inequalities are solved as a system of two, when the original is divided into parts. And in the interval method, the answers from solving both equations are indicated.

To solve them, it is also permissible to use the properties indicated above. With their help, it is convenient to reduce inequality to zero.

What about inequalities that have a modulus?

In this case, the solution to the inequalities uses the following properties, and they are valid for a positive value of “a”.

If “x” takes on an algebraic expression, then the following replacements are valid:

• |x|< a на -a < х < a;
• |x| > a to x< -a или х >a.

If the inequalities are not strict, then the formulas are also correct, only in them, in addition to the greater or less sign, “=” appears.

How is a system of inequalities solved?

This knowledge will be required in cases where such a task is given or there is a record of double inequality or a module appears in the record. In such a situation, the solution will be the values ​​of the variables that would satisfy all the inequalities in the record. If there are no such numbers, then the system has no solutions.

The plan according to which the solution of the system of inequalities is carried out:

• solve each of them separately;
• depict all intervals on the number axis and determine their intersections;
• write down the system’s response, which will be a combination of what happened in the second paragraph.

What to do with fractional inequalities?

Since solving them may require changing the sign of inequality, you need to very carefully and carefully follow all the points of the plan. Otherwise, you may get the opposite answer.

Solving fractional inequalities also uses the interval method. And the action plan will be like this:

• Using the described properties, give the fraction such a form that only zero remains to the right of the sign.
• Replace the inequality with “=” and determine the points at which the function will be equal to zero.
• Mark them on the coordinate axis. In this case, the numbers obtained as a result of calculations in the denominator will always be punched out. All others are based on the condition of inequality.
• Determine the intervals of constancy of sign.
• In response, write down the union of those intervals whose sign corresponds to that in the original inequality.

Situations when irrationality appears in inequality

In other words, there is a mathematical root in the notation. Since in the school algebra course most of the tasks are for the square root, this is what will be considered.

The solution to irrational inequalities comes down to obtaining a system of two or three that will be equivalent to the original one.

Original inequality	condition	equivalent system
√ n(x)< m(х)	m(x) less than or equal to 0	no solutions
	m(x) greater than 0	n(x) is greater than or equal to 0 n(x)< (m(х)) 2
√ n(x) > m(x)		m(x) is greater than or equal to 0 n(x) > (m(x)) 2
		n(x) is greater than or equal to 0 m(x) less than 0
√n(x) ≤ m(x)	m(x) less than 0	no solutions
	m(x) is greater than or equal to 0	n(x) is greater than or equal to 0 n(x) ≤ (m(x)) 2
√n(x) ≥ m(x)		m(x) is greater than or equal to 0 n(x) ≥ (m(x)) 2
		n(x) is greater than or equal to 0 m(x) less than 0
√ n(x)< √ m(х)		n(x) is greater than or equal to 0 n(x) less than m(x)
√n(x) * m(x)< 0		n(x) greater than 0 m(x) less than 0
√n(x) * m(x) > 0		n(x) greater than 0 m(x) greater than 0
√n(x) * m(x) ≤ 0		n(x) greater than 0
		n(x) equals 0 m(x) - any
√n(x) * m(x) ≥ 0		n(x) greater than 0
		n(x) equals 0 m(x) - any

Examples of solving different types of inequalities

In order to add clarity to the theory about solving inequalities, examples are given below.

First example. 2x - 4 > 1 + x

Solution: To determine the ADI, all you have to do is look closely at inequality. It is formed from linear functions, therefore defined for all values ​​of the variable.

Now you need to subtract (1 + x) from both sides of the inequality. It turns out: 2x - 4 - (1 + x) > 0. After the brackets are opened and similar terms are given, the inequality will take the following form: x - 5 > 0.

Equating it to zero, it is easy to find its solution: x = 5.

Now this point with the number 5 must be marked on the coordinate ray. Then check the signs of the original function. On the first interval from minus infinity to 5, you can take the number 0 and substitute it into the inequality obtained after the transformations. After calculations it turns out -7 >0. under the arc of the interval you need to sign a minus sign.

On the next interval from 5 to infinity, you can choose the number 6. Then it turns out that 1 > 0. There is a “+” sign under the arc. This second interval will be the answer to the inequality.

Answer: x lies in the interval (5; ∞).

Second example. It is required to solve a system of two equations: 3x + 3 ≤ 2x + 1 and 3x - 2 ≤ 4x + 2.

Solution. The VA of these inequalities also lies in the region of any numbers, since linear functions are given.

The second inequality will take the form of the following equation: 3x - 2 - 4x - 2 = 0. After transformation: -x - 4 =0. This produces a value for the variable equal to -4.

These two numbers need to be marked on the axis, depicting intervals. Since the inequality is not strict, all points need to be shaded. The first interval is from minus infinity to -4. Let the number -5 be chosen. The first inequality will give the value -3, and the second 1. This means that this interval is not included in the answer.

The second interval is from -4 to -2. You can choose the number -3 and substitute it into both inequalities. In the first and second, the value is -1. This means that under the arc “-”.

On the last interval from -2 to infinity, the most best number is zero. You need to substitute it and find the values ​​of the inequalities. The first of them produces a positive number, and the second a zero. This gap must also be excluded from the answer.

Of the three intervals, only one is a solution to the inequality.

Answer: x belongs to [-4; -2].

Third example. |1 - x| > 2 |x - 1|.

Solution. The first step is to determine the points at which the functions vanish. For the left one this number will be 2, for the right one - 1. They need to be marked on the beam and the intervals of constancy of sign determined.

On the first interval, from minus infinity to 1, the function from the left side of the inequality takes positive values, and from the right - negative. Under the arc you need to write two signs “+” and “-” side by side.

The next interval is from 1 to 2. On it, both functions take positive values. This means there are two pluses under the arc.

The third interval from 2 to infinity will give the following result: the left function is negative, the right function is positive.

Taking into account the resulting signs, you need to calculate the inequality values ​​for all intervals.

At the first, we get the following inequality: 2 - x > - 2 (x - 1). The minus before the two in the second inequality is due to the fact that this function is negative.

After transformation, the inequality looks like this: x > 0. It immediately gives the values ​​of the variable. That is, from this interval only the interval from 0 to 1 will be answered.

On the second: 2 - x > 2 (x - 1). The transformations will give the following inequality: -3x + 4 is greater than zero. Its zero will be x = 4/3. Taking into account the inequality sign, it turns out that x must be less than this number. This means that this interval is reduced to an interval from 1 to 4/3.

The latter gives the following inequality: - (2 - x) > 2 (x - 1). Its transformation leads to the following: -x > 0. That is, the equation is true when x is less than zero. This means that on the required interval the inequality does not provide solutions.

In the first two intervals, the limit number turned out to be 1. It needs to be checked separately. That is, substitute it into the original inequality. It turns out: |2 - 1| > 2 |1 - 1|. Calculation shows that 1 is greater than 0. This is true statement, so one is included in the answer.

Answer: x lies in the interval (0; 4/3).

Inequalities and systems of inequalities are one of the topics covered in algebra in high school. In terms of difficulty level, it is not the most difficult, since it has simple rules (more on them a little later). As a rule, schoolchildren learn to solve systems of inequalities quite easily. This is also due to the fact that teachers simply “train” their students on this topic. And they cannot help but do this, because it is studied in the future using other mathematical quantities, and is also tested on the Unified State Exam and the Unified State Exam. In school textbooks, the topic of inequalities and systems of inequalities is covered in great detail, so if you are going to study it, it is best to resort to them. This article only summarizes larger material and there may be some omissions.

The concept of a system of inequalities

If we turn to scientific language, we can define the concept of “system of inequalities”. This is a mathematical model that represents several inequalities. This model, of course, requires a solution, and this will be the general answer for all the inequalities of the system proposed in the task (usually it is written like this, for example: “Solve the system of inequalities 4 x + 1 > 2 and 30 - x > 6... "). However, before moving on to the types and methods of solutions, you need to understand something else.

Systems of inequalities and systems of equations

In the process of studying new topic very often misunderstandings arise. On the one hand, everything is clear and you want to start solving tasks as soon as possible, but on the other hand, some moments remain in the “shadow” and are not fully understood. Also, some elements of already acquired knowledge may be intertwined with new ones. As a result of this “overlapping”, errors often occur.

Therefore, before we begin to analyze our topic, we should remember the differences between equations and inequalities and their systems. To do this, we need to once again explain what these mathematical concepts represent. An equation is always an equality, and it is always equal to something (in mathematics this word is denoted by the sign "="). Inequality is a model in which one value is either greater or less than another, or contains a statement that they are not the same. Thus, in the first case, it is appropriate to talk about equality, and in the second, no matter how obvious it may sound from the name itself, about the inequality of the initial data. Systems of equations and inequalities practically do not differ from each other and the methods for solving them are the same. The only difference is that in the first case equalities are used, and in the second inequalities are used.

Types of inequalities

There are two types of inequalities: numerical and with an unknown variable. The first type represents provided quantities (numbers) that are unequal to each other, for example, 8 > 10. The second are inequalities that contain an unknown variable (denoted by a letter of the Latin alphabet, most often X). This variable needs to be found. Depending on how many there are, the mathematical model distinguishes between inequalities with one (they make up a system of inequalities with one variable) or several variables (they make up a system of inequalities with several variables).

The last two types, according to the degree of their construction and the level of complexity of the solution, are divided into simple and complex. Simple ones are also called linear inequalities. They, in turn, are divided into strict and non-strict. Strict ones specifically “say” that one quantity must necessarily be either less or more, so this is in pure form inequality. Several examples can be given: 8 x + 9 > 2, 100 - 3 x > 5, etc. Non-strict ones also include equality. That is, one value can be greater than or equal to another value (the “≥” sign) or less than or equal to another value (the “≤” sign). Even in linear inequalities, the variable is not at the root, square, or divisible by anything, which is why they are called “simple.” Complex ones involve unknown variables that require more math to find. They are often located in a square, cube or under a root, they can be modular, logarithmic, fractional, etc. But since our task is the need to understand the solution of systems of inequalities, we will talk about a system of linear inequalities. However, before that, a few words should be said about their properties.

Properties of inequalities

The properties of inequalities include the following:

1. The inequality sign is reversed if an operation is used to change the order of the sides (for example, if t 1 ≤ t 2, then t 2 ≥ t 1).
2. Both sides of the inequality allow you to add the same number to itself (for example, if t 1 ≤ t 2, then t 1 + number ≤ t 2 + number).
3. Two or more inequalities with a sign in the same direction allow their left and right sides to be added (for example, if t 1 ≥ t 2, t 3 ≥ t 4, then t 1 + t 3 ≥ t 2 + t 4).
4. Both sides of the inequality can be multiplied or divided by the same positive number (for example, if t 1 ≤ t 2 and a number ≤ 0, then the number · t 1 ≥ number · t 2).
5. Two or more inequalities that have positive terms and a sign in the same direction allow themselves to be multiplied by each other (for example, if t 1 ≤ t 2, t 3 ≤ t 4, t 1, t 2, t 3, t 4 ≥ 0 then t 1 · t 3 ≤ t 2 · t 4).
6. Both sides of the inequality can be multiplied or divided by the same thing negative number, but the sign of the inequality changes (for example, if t 1 ≤ t 2 and the number ≤ 0, then the number · t 1 ≥ number · t 2).
7. All inequalities have the property of transitivity (for example, if t 1 ≤ t 2 and t 2 ≤ t 3, then t 1 ≤ t 3).

Now, after studying the basic principles of the theory related to inequalities, we can proceed directly to the consideration of the rules for solving their systems.

Solving systems of inequalities. General information. Solutions

As mentioned above, the solution is the values ​​of the variable that are suitable for all the inequalities of the given system. Solving systems of inequalities is the implementation of mathematical operations that ultimately lead to a solution to the entire system or prove that it has no solutions. In this case, the variable is said to belong to an empty numerical set (written as follows: letter denoting a variable∈ (sign “belongs”) ø (sign “empty set”), for example, x ∈ ø (read: “The variable “x” belongs to the empty set”). There are several ways to solve systems of inequalities: graphical, algebraic, substitution method. It is worth noting that they are among those mathematical models, which have several unknown variables. In the case where there is only one, the interval method is suitable.

Graphic method

Allows you to solve a system of inequalities with several unknown quantities (from two and above). Thanks to this method, a system of linear inequalities can be solved quite easily and quickly, so it is the most common method. This is explained by the fact that plotting a graph reduces the amount of writing mathematical operations. It becomes especially pleasant to take a little break from the pen, pick up a pencil with a ruler and begin further actions with their help when a lot of work has been done and you want a little variety. However this method some people don’t like it because they have to break away from the task and switch their mental activity to drawing. However, this is a very effective method.

To solve a system of inequalities using a graphical method, it is necessary to transfer all terms of each inequality to their left side. The signs will be reversed, zero should be written on the right, then each inequality needs to be written separately. As a result, functions will be obtained from inequalities. After this, you can take out a pencil and a ruler: now you need to draw a graph of each function obtained. The entire set of numbers that will be in the interval of their intersection will be a solution to the system of inequalities.

Algebraic way

Allows you to solve a system of inequalities with two unknown variables. Also, inequalities must have the same inequality sign (that is, they must contain either only the “greater than” sign, or only the “less than” sign, etc.) Despite its limitations, this method is also more complex. It is applied in two stages.

The first involves actions to get rid of one of the unknown variables. First you need to select it, then check for the presence of numbers in front of this variable. If they are not there (then the variable will look like a single letter), then we do not change anything, if there are (the type of the variable will be, for example, 5y or 12y), then it is necessary to make sure that in each inequality the number in front of the selected variable is the same. To do this, you need to multiply each term of the inequalities by a common factor, for example, if 3y is written in the first inequality, and 5y in the second, then you need to multiply all the terms of the first inequality by 5, and the second by 3. You get 15y and 15y, respectively.

Second stage of solution. It is necessary to transfer the left side of each inequality to their right sides, changing the sign of each term to the opposite, and write zero on the right. Then comes the fun part: getting rid of the selected variable (otherwise known as “reduction”) while adding the inequalities. This results in an inequality with one variable that needs to be solved. After this, you should do the same thing, only with another unknown variable. The results obtained will be the solution of the system.

Substitution method

Allows you to solve a system of inequalities if it is possible to introduce a new variable. Typically, this method is used when the unknown variable in one term of the inequality is raised to the fourth power, and in the other term it is squared. Thus, this method is aimed at reducing the degree of inequalities in the system. The sample inequality x 4 - x 2 - 1 ≤ 0 is solved in this way. A new variable is introduced, for example t. They write: “Let t = x 2,” then the model is rewritten in a new form. In our case, we get t 2 - t - 1 ≤0. This inequality needs to be solved using the interval method (more on that a little later), then back to the variable X, then do the same with the other inequality. The answers received will be the solution of the system.

Interval method

This is the simplest way to solve systems of inequalities, and at the same time it is universal and widespread. It is used in secondary schools and even in higher schools. Its essence lies in the fact that the student looks for intervals of inequality on a number line, which is drawn in a notebook (this is not a graph, but just an ordinary line with numbers). Where the intervals of inequalities intersect, the solution to the system is found. To use the interval method, you need to follow these steps:

1. All terms of each inequality are transferred to the left side with the sign changing to the opposite (zero is written on the right).
2. The inequalities are written out separately, and the solution to each of them is determined.
3. The intersections of inequalities on the number line are found. All numbers located at these intersections will be a solution.

Which method should I use?

Obviously the one that seems easiest and most convenient, but there are cases when tasks require a certain method. Most often they say that you need to solve either using a graph or the interval method. The algebraic method and substitution are used extremely rarely or not at all, since they are quite complex and confusing, and besides, they are more used for solving systems of equations rather than inequalities, so you should resort to drawing graphs and intervals. They bring clarity, which cannot but contribute to the efficient and fast execution of mathematical operations.

If something doesn't work out

While studying a particular topic in algebra, naturally, problems may arise with its understanding. And this is normal, because our brain is designed in such a way that it is not able to understand complex material at one time. Often you need to reread a paragraph, take help from a teacher, or practice solving standard tasks. In our case, they look, for example, like this: “Solve the system of inequalities 3 x + 1 ≥ 0 and 2 x - 1 > 3.” Thus, personal desire, help from outsiders and practice help in understanding any complex topic.

Solver?

A solution book is also very suitable, not for copying homework, but for self-help. In them you can find systems of inequalities with a solution, look at them (as templates), try to understand exactly how the author of the solution coped with the task, and then try to do the same on your own.

Conclusions

Algebra is one of the most difficult subjects in school. Well, what can you do? Mathematics has always been like this: for some it is easy, but for others it is difficult. But in any case, it should be remembered that the general education program is structured in such a way that any student can cope with it. In addition, one must keep in mind the huge number of assistants. Some of them have been mentioned above.

In the fifth century BC 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; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes...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, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells about a flying arrow:

A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to 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 of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.

Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: on different coins there is different quantities dirt, crystal structure and atomic arrangement of each coin is unique...

And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.

Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.

2. 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's 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 of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, it means it has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.

Sign on the door He opens the door and says:

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 something like this flashes before your eyes several times a day design art,

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: minus sign, number four, degree designation). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.

1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.