The order of performing mathematical operations without parentheses. Procedure for performing actions - Knowledge Hypermarket

In the fifth century BC ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia “Achilles and the Tortoise.” 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 ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

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

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

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

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

Another interesting aporia of Zeno tells about a flying arrow:

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

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, July 4, 2018

The differences between set and multiset are described very well on Wikipedia. Let's see.

As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level 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 interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.

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

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

Sunday, March 18, 2018

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

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

Let's figure out what and how we do in order to find the sum of 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. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols into numbers. This is not a mathematical operation.

4. Add the resulting numbers. Now this is mathematics.

The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.

From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.

As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.

Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units 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 such a work of design art flashes before your eyes several times a day,

Then it’s not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is 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.

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:

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:

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 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.

Wednesday, July 4, 2018

The differences between set and multiset are described very well on Wikipedia. Let's see.

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: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...

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.

Sunday, March 18, 2018

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.

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

2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols into numbers. This is not a mathematical operation.

4. Add the resulting numbers. Now this is mathematics.

The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.

From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems 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 the large number 12345, I don’t want to fool my head, let’s consider 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.

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:

Female... The halo on top and the arrow down are male.

If such a work of design art flashes before your eyes several times a day,

Then it’s not surprising that you suddenly find a strange icon in your car:

October 24th, 2017 admin

Lopatko Irina Georgievna

Target: formation of knowledge about the order of performing arithmetic operations in numerical expressions without brackets and with brackets, consisting of 2-3 actions.

Tasks:

Educational: to develop in students the ability to use the rules of the order of actions when calculating specific expressions, the ability to apply an algorithm of actions.

Developmental: develop skills of working in pairs, mental activity of students, the ability to reason, compare and contrast, calculation skills and mathematical speech.

Educational: cultivate interest in the subject, tolerant attitude towards each other, mutual cooperation.

Type: learning new material

Equipment: presentation, visuals, handouts, cards, textbook.

Methods: verbal, visual and figurative.

DURING THE CLASSES

Organizing time

Greetings.

We came here to study

Don't be lazy, but work.

We work diligently

Let's listen carefully.

Markushevich said great words: “Whoever studies mathematics from childhood develops attention, trains his brain, his will, cultivates perseverance and perseverance in achieving goals.” Welcome to math lesson!

Updating knowledge

The subject of mathematics is so serious that no opportunity should be missed to make it more entertaining.(B. Pascal)

I suggest you complete logical tasks. You are ready?

Which two numbers, when multiplied, give the same result as when added? (2 and 2)

From under the fence you can see 6 pairs of horse legs. How many of these animals are there in the yard? (3)

A rooster standing on one leg weighs 5 kg. How much will he weigh standing on two legs? (5kg)

There are 10 fingers on the hands. How many fingers are there on 6 hands? (thirty)

The parents have 6 sons. Everyone has a sister. How many children are there in the family? (7)

How many tails do seven cats have?

How many noses do two dogs have?

How many ears do 5 babies have?

Guys, this is exactly the kind of work I expected from you: you were active, attentive, and smart.

Assessment: verbal.

Verbal counting

BOX OF KNOWLEDGE

Product of numbers 2 * 3, 4 * 2;

Partial numbers 15: 3, 10:2;

Sum of numbers 100 + 20, 130 + 6, 650 + 4;

The difference between numbers is 180 – 10, 90 – 5, 340 – 30.

Components of multiplication, division, addition, subtraction.

Assessment: students independently evaluate each other

Communicating the topic and purpose of the lesson

“To digest knowledge, you need to absorb it with appetite.”(A. Franz)

Are you ready to absorb knowledge with appetite?

Guys, Masha and Misha were offered such a chain

24 + 40: 8 – 4=

Masha decided it like this:

24 + 40: 8 – 4= 25 correct? Children's answers.

And Misha decided like this:

24 + 40: 8 – 4= 4 correct? Children's answers.

What surprised you? It seems that both Masha and Misha decided correctly. Then why do they have different answers?

They counted in different orders; they did not agree in what order they would count.

What does the calculation result depend on? From order.

What do you see in these expressions? Numbers, signs.

What are signs called in mathematics? Actions.

What order did the guys not agree on? About the procedure.

What will we study in class? What is the topic of the lesson?

We will study the order of arithmetic operations in expressions.

Why do we need to know the procedure? Perform calculations correctly in long expressions

"Basket of Knowledge". (The basket hangs on the board)

Students name associations related to the topic.

Learning new material

Guys, please listen to what the French mathematician D. Poya said: “The best way to study something is to discover it for yourself.” Are you ready for discoveries?

180 – (9 + 2) =

Read the expressions. Compare them.

How are they similar? 2 actions, same numbers

What is the difference? Parentheses, different actions

Rule 1.

Read the rule on the slide. Children read the rule aloud.

In expressions without parentheses containing only addition and subtraction or multiplication and division, operations are performed in the order they are written: from left to right.

What actions are we talking about here? +, — or : , ·

From these expressions, find only those that correspond to rule 1. Write them down in your notebook.

Calculate the values of expressions.

Examination.

180 – 9 + 2 = 173

Rule 2.

Read the rule on the slide.

Children read the rule aloud.

In expressions without parentheses, multiplication or division are performed first, in order from left to right, and then addition or subtraction.

:, · and +, — (together)

Are there parentheses? No.

What actions will we perform first? ·, : from left to right

What actions will we take next? +, — left, right

Find their meanings.

Examination.

180 – 9 * 2 = 162

Rule 3

In expressions with parentheses, first evaluate the value of the expressions in parentheses, thenmultiplication or division are performed in order from left to right, and then addition or subtraction.

What arithmetic operations are indicated here?

:, · and +, — (together)

Are there parentheses? Yes.

What actions will we perform first? In brackets

What actions will we take next? ·, : from left to right

And then? +, — left, right

Write down expressions that relate to the second rule.

Find their meanings.

Examination.

180: (9 * 2) = 10

180 – (9 + 2) = 169

Once again, we all say the rule together.

PHYSMINUTE

Consolidation

“Much of mathematics does not remain in the memory, but when you understand it, then it is easy to remember what you have forgotten on occasion.”, said M.V. Ostrogradsky. Now we will remember what we just learned and apply new knowledge in practice .

Page 52 No. 2

(52 – 48) * 4 =

Page 52 No. 6 (1)

The students collected 700 kg of vegetables in the greenhouse: 340 kg of cucumbers, 150 kg of tomatoes, and the rest - peppers. How many kilograms of peppers did the students collect?

What are they talking about? What is known? What do you need to find?

Let's try to solve this problem with an expression!

700 – (340 + 150) = 210 (kg)

Answer: The students collected 210 kg of pepper.

Work in pairs.

Cards with the task are given.

5 + 5 + 5 5 = 35

(5+5) : 5 5 = 10

Grading:

speed – 1 b
correctness - 2 b
logic - 2 b

Homework

Page 52 No. 6 (2) solve the problem, write the solution in the form of an expression.

Result, reflection

Bloom's Cube

Name it topic of our lesson?

Explain the order of execution of actions in expressions with brackets.

Why Is it important to study this topic?

Continue first rule.

Come up with it algorithm for performing actions in expressions with brackets.

“If you want to participate in great life, then fill your head with mathematics while you have the opportunity. She will then be of great help to you in all your work.”(M.I. Kalinin)

Thanks for your work in class!!!

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The video lesson “Order of Actions” explains in detail an important topic in mathematics - the sequence of performing arithmetic operations when solving an expression. During the video lesson, it is discussed what priority various mathematical operations have, how they are used in calculating expressions, examples are given for mastering the material, and the knowledge gained is generalized in solving tasks where all the considered operations are present. With the help of a video lesson, the teacher has the opportunity to quickly achieve the goals of the lesson and increase its effectiveness. The video can be used as visual material to accompany the teacher’s explanation, as well as as an independent part of the lesson.

Visual material uses techniques that help to better understand the topic, as well as remember important rules. With the help of color and different writing, the features and properties of operations are highlighted, and the peculiarities of solving examples are noted. Animation effects help deliver consistency educational material and also draw students' attention to important points. The video is voiced, so it is supplemented with comments from the teacher, helping the student understand and remember the topic.

The video lesson begins by introducing the topic. Then it is noted that multiplication and subtraction are operations of the first stage, operations of multiplication and division are called operations of the second stage. This definition will need to be operated further, displayed on the screen and highlighted in large color font. Then the rules that make up the order of operations are presented. The first order rule is derived, which indicates that if there are no parentheses in the expression, and there are actions of the same level, these actions must be performed in order. The second order rule states that if there are actions of both stages and there are no parentheses, the operations of the second stage are performed first, then the operations of the first stage are performed. The third rule sets the order of operations for expressions that include parentheses. It is noted that in this case the operations in brackets are performed first. The wording of the rules is highlighted in colored font and is recommended for memorization.

Next, it is proposed to understand the order of operations by considering examples. The solution to an expression containing only addition and subtraction operations is described. The main features that affect the order of calculations are noted - there are no parentheses, there are first-stage operations. Below is a description of how calculations are performed, first subtraction, then addition twice, and then subtraction.

In the second example 780:39·212:156·13 you need to evaluate the expression, performing actions according to the order. It is noted that this expression contains exclusively second-stage operations, without parentheses. IN in this example all actions are performed strictly from left to right. Below we describe the actions one by one, gradually approaching the answer. The result of the calculation is the number 520.

The third example considers a solution to an example in which there are operations of both stages. It is noted that in this expression there are no parentheses, but there are actions of both stages. According to the order of operations, the second stage operations are performed, followed by the first stage operations. Below is a step-by-step description of the solution, in which three operations are performed first - multiplication, division, and another division. Then, first-stage operations are performed with the found values of the product and quotients. During the solution, the actions of each step are combined in curly braces for clarity.

The following example contains parentheses. Therefore, it is demonstrated that the first calculations are performed on the expressions in parentheses. After them, the second stage operations are performed, followed by the first.

The following is a note about in what cases you can not write parentheses when solving expressions. It is noted that this is only possible in the case where eliminating the parentheses does not change the order of operations. An example is the expression with brackets (53-12)+14, which contains only first-stage operations. Having rewritten 53-12+14 with the elimination of parentheses, you can note that the order of searching for the value will not change - first the subtraction 53-12=41 is performed, and then the addition 41+14=55. It is noted below that you can change the order of operations when finding a solution to an expression using the properties of the operations.

At the end of the video lesson, the material studied is summarized in the conclusion that each expression requiring a solution specifies a specific program for calculation, consisting of commands. An example of such a program is presented in the description of the solution complex example, which is the quotient of (814+36·27) and (101-2052:38). The given program contains the following points: 1) find the product of 36 with 27, 2) add the found sum to 814, 3) divide the number 2052 by 38, 4) subtract the result of dividing 3 points from the number 101, 5) divide the result of step 2 by the result of point 4.

At the end of the video lesson there is a list of questions that students are asked to answer. These include the ability to distinguish between actions of the first and second stages, questions about the order of actions in expressions with actions of the same stage and different stages, about the order of actions in the presence of parentheses in the expression.

The video lesson “Order of Actions” is recommended to be used in a traditional school lesson to increase the effectiveness of the lesson. Also, visual material will be useful for distance learning. If a student needs an additional lesson to master a topic or is studying it independently, the video can be recommended for independent study.