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How to decide on actions. Procedure for performing actions - Knowledge Hypermarket

20.04.2023

When we work with various expressions, including numbers, letters and variables, we have to perform a large number of arithmetic operations. When we do a transformation or calculate a value, it is very important to follow the correct order of these actions. In other words, arithmetic operations have their own special execution order.

In this article, we will tell you what actions should be done first and which after. First, let's look at a few simple expressions that contain only variables or numeric values, as well as division, multiplication, subtraction, and addition signs. Then we will take examples with brackets and consider in what order they should be evaluated. In the third part, we will give the correct order of transformations and calculations in those examples that include the signs of roots, powers, and other functions.

Definition 1

In the case of expressions without brackets, the order of actions is determined unambiguously:

All actions are performed from left to right.
First of all, we perform division and multiplication, and secondly, subtraction and addition.

The meaning of these rules is easy to understand. The traditional writing order from left to right determines the basic sequence of calculations, and the need to first multiply or divide is explained by the very essence of these operations.

Let's take a few tasks for clarity. We have used only the simplest numerical expressions so that all calculations can be done mentally. So you can quickly remember the desired order and quickly check the results.

Example 1

Condition: calculate how much 7 − 3 + 6 .

Solution

There are no brackets in our expression, multiplication and division are also absent, so we perform all the actions in the specified order. First, subtract three from seven, then add six to the remainder, and as a result we get ten. Here is a record of the entire solution:

7 − 3 + 6 = 4 + 6 = 10

Answer: 7 − 3 + 6 = 10 .

Example 2

Condition: in what order should the calculations be performed in the expression 6:2 8:3?

Solution

To answer this question, we reread the rule for expressions without parentheses, which we formulated earlier. We only have multiplication and division here, which means we keep the written order of calculations and count sequentially from left to right.

Answer: first, we divide six by two, multiply the result by eight, and divide the resulting number by three.

Example 3

Condition: calculate how much will be 17 − 5 6: 3 − 2 + 4: 2.

Solution

First, let's determine the correct order of operations, since we have here all the basic types of arithmetic operations - addition, subtraction, multiplication, division. The first thing we need to do is divide and multiply. These actions do not have priority over each other, so we perform them in the written order from right to left. That is, 5 must be multiplied by 6 and get 30, then 30 divided by 3 and get 10. After that we divide 4 by 2 , that's 2 . Substitute the found values into the original expression:

17 - 5 6: 3 - 2 + 4: 2 = 17 - 10 - 2 + 2

There is no division or multiplication here, so we do the remaining calculations in order and get the answer:

17 − 10 − 2 + 2 = 7 − 2 + 2 = 5 + 2 = 7

Answer:17 - 5 6: 3 - 2 + 4: 2 = 7.

Until the order of performing actions is firmly learned, you can put numbers over the signs of arithmetic operations, indicating the order of calculation. For example, for the problem above, we could write it like this:

If we have literal expressions, then we do the same with them: first we multiply and divide, then we add and subtract.

What are steps one and two

Sometimes in reference books all arithmetic operations are divided into operations of the first and second stages. Let us formulate the required definition.

The operations of the first stage include subtraction and addition, the second - multiplication and division.

Knowing these names, we can write the rule given earlier regarding the order of actions as follows:

Definition 2

In an expression that does not contain parentheses, first perform the actions of the second step in the direction from left to right, then the actions of the first step (in the same direction).

Order of evaluation in expressions with brackets

Parentheses themselves are a sign that tells us the desired order in which to perform actions. In this case, the desired rule can be written as follows:

Definition 3

If there are brackets in the expression, then the action in them is performed first, after which we multiply and divide, and then add and subtract in the direction from left to right.

As for the parenthesized expression itself, it can be considered as a component of the main expression. When calculating the value of the expression in brackets, we keep the same procedure known to us. Let's illustrate our idea with an example.

Example 4

Condition: calculate how much 5 + (7 − 2 3) (6 − 4) : 2.

Solution

This expression has parentheses, so let's start with them. First of all, let's calculate how much 7 − 2 · 3 will be. Here we need to multiply 2 by 3 and subtract the result from 7:

7 − 2 3 = 7 − 6 = 1

We consider the result in the second brackets. There we have only one action: 6 − 4 = 2 .

Now we need to substitute the resulting values into the original expression:

5 + (7 − 2 3) (6 − 4) : 2 = 5 + 1 2: 2

Let's start with multiplication and division, then subtract and get:

5 + 1 2:2 = 5 + 2:2 = 5 + 1 = 6

This completes the calculations.

Answer: 5 + (7 − 2 3) (6 − 4) : 2 = 6.

Do not be alarmed if the condition contains an expression in which some brackets enclose others. We only need to apply the rule above consistently to all parenthesized expressions. Let's take this task.

Example 5

Condition: calculate how much 4 + (3 + 1 + 4 (2 + 3)).

Solution

We have brackets within brackets. We start with 3 + 1 + 4 (2 + 3) , namely 2 + 3 . It will be 5 . The value will need to be substituted into the expression and calculate that 3 + 1 + 4 5 . We remember that we must first multiply, and then add: 3 + 1 + 4 5 = 3 + 1 + 20 = 24. Substituting the found values into the original expression, we calculate the answer: 4 + 24 = 28 .

Answer: 4 + (3 + 1 + 4 (2 + 3)) = 28.

In other words, when evaluating the value of an expression involving parentheses within parentheses, we start with the inner parentheses and work our way to the outer ones.

Let's say we need to find how much will be (4 + (4 + (4 - 6: 2)) - 1) - 1. We start with the expression in the inner brackets. Since 4 − 6: 2 = 4 − 3 = 1 , the original expression can be written as (4 + (4 + 1) − 1) − 1 . Again we turn to the inner brackets: 4 + 1 = 5 . We have come to the expression (4 + 5 − 1) − 1 . We believe 4 + 5 − 1 = 8 and as a result we get the difference 8 - 1, the result of which will be 7.

The order of calculation in expressions with powers, roots, logarithms and other functions

If we have an expression in the condition with a degree, root, logarithm or trigonometric function (sine, cosine, tangent and cotangent) or other functions, then first of all we calculate the value of the function. After that, we act according to the rules specified in the previous paragraphs. In other words, functions are equal in importance to the expression enclosed in brackets.

Let's look at an example of such a calculation.

Example 6

Condition: find how much will be (3 + 1) 2 + 6 2: 3 - 7 .

Solution

We have an expression with a degree, the value of which must be found first. We consider: 6 2 \u003d 36. Now we substitute the result into the expression, after which it will take the form (3 + 1) 2 + 36: 3 − 7 .

(3 + 1) 2 + 36: 3 - 7 = 4 2 + 36: 3 - 7 = 8 + 12 - 7 = 13

Answer: (3 + 1) 2 + 6 2: 3 − 7 = 13.

In a separate article devoted to calculating the values of expressions, we provide other, more complex examples of calculations in the case of expressions with roots, degrees, etc. We recommend that you familiarize yourself with it.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Which action is performed first: multiplication and division or addition and ...?

If we compare the functions of addition and subtraction with multiplication and division, then multiplication and division are always calculated first.

In the example, two functions such as addition and subtraction, as well as multiplication and division, are equivalent to each other. The order of execution is determined in turn order from left to right.

It should be remembered that the actions taken in parentheses have special precedence in the example. Thus, even if there is multiplication outside the brackets, and addition in brackets, you should first add, and only then multiply.

To understand this topic, you can consider all cases in turn.

Immediately take into account that our expressions do not have brackets.

So, if in the example the first action is multiplication, and the second is division, then we perform the multiplication first.

If in the example the first action is division, and the second is multiplication, then we do division first.

In such examples, actions are performed in order from left to right, regardless of which numbers are used.

If, in addition to multiplication and division, there are addition and subtraction in the examples, then multiplication and division are done first, and then addition and subtraction.

In the case of addition and subtraction, it also does not matter which of these operations is done first. The order is from left to right.

Let's consider different options:

In this example, the first action that needs to be performed is multiplication, and then addition.

In this case, you first multiply the values, then divide, and only then add.

In this case, you must first do all the operations in the brackets, and then only do the multiplication and division.

And so it must be remembered that in any formula, operations are first performed as multiplication and division, and then only subtraction and addition.

Also, with the numbers that are in brackets, you need to count them in brackets, and only then do various manipulations, remembering the sequence described above.

The first will be the following actions: multiplication and division.

Only then are addition and subtraction performed.

However, if there is a bracket, then the actions that are in them will be performed first. Even if it's addition and subtraction.

For example:

In this example, first we perform the multiplication, then 4 by 5, then add 4 to 20. We get 24.

But if it is like this: (4 + 5) * 4, then first we perform the addition, we get 9. Then we multiply 9 by 4. We get 36.

If all 4 actions are present in the example, then multiplication and division come first, and then addition and subtraction.

Or in the example of 3 different actions, then the first will be either multiplication (or division), and then either addition (or subtraction).

When there are NO BRACKETS.

Example: 4-2*5:10+8=11,

1 action 2*5 (10);

act 2 10:10 (1);

3 action 4-1 (3);

4 act 3+8 (11).

All 4 actions can be divided into two main groups, in one - addition and subtraction, in the other - multiplication and division. The first action will be the one that is the first in a row in the example, that is, the leftmost one.

Example: 60-7+9=62, first you need 60-7, then what happens (53) +9;

Example: 5*8:2=20, first you need 5*8, then what you get (40) :2.

When there are BRACKETS in the example, then the actions that are in the bracket are performed first (according to the above rules), and then the rest as usual.

Example: 2+(9-8)*10:2=7.

1 act 9-8 (1);

2 action 1*10 (10);

Act 3 10:2(5);

4 act 2+5 (7).

It depends on how the expression is written, consider the simplest numeric expression:

18 - 6:3 + 10x2 =

First, we perform operations with division and multiplication, then in turn, from left to right, with subtraction and addition: 18-2 + 20 \u003d 36

If it's a parenthesized expression, then do the parentheses, then multiply or divide, and finally add/subtract, like so:

(18-6): 3 + 10 x 2 = 12:3 + 20 = 4+20=24

Sun is correct: first perform multiplication and division, then addition and subtraction.

If there are no brackets in the example, then multiplication and division in order are performed first, and then addition and subtraction, the same in order.

If the example contains only multiplication and division, then the actions will be performed in order.

If the example contains only addition and subtraction, then the actions will also be performed in order.

First of all, actions in brackets are performed according to the same rules, that is, first multiplication and division, and only then addition and subtraction.

22-(11+3x2)+14=19

The order of performing arithmetic operations is strictly prescribed so that there are no discrepancies when performing the same type of calculations by different people. First of all, multiplication and division are performed, then addition and subtraction, if actions of the same order go one after the other, then they are performed in turn order from left to right.

If brackets are used when writing a mathematical expression, then first of all, you should perform the actions indicated in brackets. Parentheses help to change the order, if necessary, first perform addition or subtraction, and only after multiplication and division.

Any brackets can be opened and then the execution order will again be correct:

6*(45+15) = 6*45 +6*15

Better with examples:

1+2*3/4-5=?

In this case, we perform the multiplication first, since it is to the left of the division. Then division. Then addition, because of the more left-hand location, and finally subtraction.

1*3/(2+4)?

first we do the calculation in brackets, then the multiplication and division.

1+2*(3-1*5)=?

First, we do the actions in brackets: multiplication, then subtraction. After that comes the multiplication outside the brackets and the addition at the end.

Multiplication and division come first. If there are brackets in the example, then the action in brackets is considered at the beginning. Whatever the sign is!

Here you need to remember a few basic rules:

If there are no brackets in the example and there are operations - only addition and subtraction, or only multiplication and division - in this case, all actions are carried out in order from left to right.

For example, 5 + 8-5 = 8 (we do everything in order - add 8 to 5, and then subtract 5)

If the example contains mixed operations - addition, subtraction, multiplication, and division, then first of all we perform the operations of multiplication and division, and then only addition or subtraction.

For example, 5+8*3=29 (first multiply 8 by 3 and then add 5)

If the example contains parentheses, then the actions in the parentheses are performed first.

For example, 3*(5+8)=39 (first 5+8 and then multiply by 3)

The original source is located. Alpha denotes a 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 an infinite set of natural numbers as an example, then the considered examples can be represented as follows:

To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the 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? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an "infinite hotel"? An infinity inn is an inn that always has any number of vacancies, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, 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, on the other hand, are not able to move away from banal everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, 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 exist. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on a 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. What if you really want to? No problem. We can take a unit 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 have written down the operations in algebraic notation and in set theory notation, listing the elements of the set in detail. 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 is subtracted from it and the same one is added.

Option two. We have many different infinite sets of natural numbers on the 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:

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

The set of natural numbers is used for counting in the same way as a ruler for measurements. Now imagine that you have added 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 - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, they deprive us of free thinking).

pozg.ru

Sunday, August 4, 2019

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

We read: "... the rich theoretical basis of Babylonian mathematics 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 weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:

The rich theoretical basis of modern mathematics does not have a holistic character and is reduced 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 conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole cycle of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.

May we have many A consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter A, the subscript with a number will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, 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, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question, how correctly applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.

As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.

As you can see, units of measurement and common math make set theory a thing of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.

Finally, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about 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 universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

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

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

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

Another interesting aporia of Zeno tells of a flying arrow:

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

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

Wednesday, July 4, 2018

I already told you that, with the help of which shamans try to sort "" realities. How do they do it? How does the formation of the set actually take place?

Let's take a closer look at the definition of a set: "a collection of different elements, conceived as a single whole." Now feel the difference between the two phrases: "thinkable as a whole" and "thinkable as a whole." The first phrase is the end result, the multitude. The second phrase is a preliminary preparation for the formation of the set. At this stage, reality is divided into separate elements ("whole") from which a multitude ("single whole") will then be formed. At the same time, the factor that allows you to combine the "whole" into a "single whole" is carefully monitored, otherwise the shamans will not succeed. After all, shamans know in advance exactly what set they want to demonstrate to us.

I will show 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 without a bow. 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 trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received sets "with a bow" and "red" the same set or 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 formed a set of "red solid pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.

The letter "a" with different indices denotes different units of measurement. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to which the set is formed, is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it with “obviousness”, because units of measurement are not included in their “scientific” arsenal.

With the help of units of measurement, it is very easy to break down one
It is today that everything that we do not take belongs to some set (as mathematicians assure us). By the way, did you see in the mirror on your forehead a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of sets to which this thing belongs. Sets are all inventions of shamans. How do they do it? Let's look a little deeper into history and see how the elements of the set looked before the mathematicians-shamans pulled them apart into their sets.

A long time ago, when no one had heard of mathematics yet, and only trees and Saturn had rings, huge herds of wild elements of sets roamed the physical fields (after all, shamans had not yet invented mathematical fields). They looked like this.

Yes, do not be surprised, from the point of view of mathematics, all elements of the sets are most similar to sea urchins - from one point, like needles, units of measurement stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any quantity can be represented as a bundle of segments sticking out in different directions from one point. This point is the zero point. I will not draw this work of geometric art (no inspiration), but you can easily imagine it.

What units of measurement form an element of the set? Any that describe this element from different points of view. These are the ancient units of measurement used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are units of measurement unknown to us, which our descendants will come up with and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. And what about physics? Units of measurement - this is the direct connection between mathematics and physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. I personally can’t imagine a real science of mathematics without units of measurement. That is why, at the very beginning of the story about set theory, I spoke of it as the Stone Age.

But let's move on to the most interesting - to the algebra of elements of sets. Algebraically, any element of the set is a product (the result of multiplication) of different quantities. It looks like this.

I deliberately did not use the conventions adopted in set theory, since we are considering an element of a set in a natural habitat before the advent of set theory. Each pair of letters in brackets denotes a separate value, consisting of the number indicated by the letter " n" and units of measurement, indicated by the letter " a". Indexes near the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of values \u200b\u200b(as long as we and our descendants have enough imagination). Each bracket is geometrically represented by a separate segment. In the example with the sea urchin one bracket is one needle.

How do shamans form sets from different elements? In fact, by units of measurement or by numbers. Understanding nothing in mathematics, they take different sea urchins and carefully examine them in search of that single needle by which they form a set. If there is such a needle, then this element belongs to the set; if there is no such needle, this element is not from this set. Shamans tell us fables about mental processes and a single whole.

As you may have guessed, the same element can belong to a variety of sets. Next, I will show you how sets, subsets and other shamanic nonsense are formed.

multiply in any order.

Methodically, this rule aims to prepare the child to get acquainted with the methods of multiplying into a column of numbers ending in zeros, therefore they get acquainted with it only in the fourth grade. In reality, this property of multiplication makes it possible to rationalize oral calculations both in grades 2 and 3.

For example:

Calculate: (7 2) 5 = ...

In this case, it is much easier to calculate the variant

7 (2 5) = 7 10 - 70.

Calculate: 12 (5 7) = ...

8 in this case it is much easier to calculate the option (12-5)-7 = 60-7 = 420.

Calculation techniques

1. Multiplication and division of numbers ending in zero: 20 3; 3 20; 60:3; 80:20

The computational technique in this case is reduced to the multiplication and division of single-digit numbers expressing the number of tens in given numbers. For example:

20 3 =... 3 20 =... 60:3 = ...

2 dec. 3 = 20 3 = 60 b dec: 3 = 2 dec.

20 - 3 = 60 3 20 = 60 60: 3 = 20

For the 80:20 case, two calculation methods can be used: the one used in previous cases, and the quotient selection method.

For example: 80:20=... 80:20=...

8 dec.: 2 dec. = 4 or 20 4 = 80

80: 20 = 4 80: 20 = 4

In the first case, the technique of representing two-digit tens as bit units was used, which reduces the case under consideration to a tabular one (8:2). In the second case, the quotient is found by selection and verified by multiplication. In the second case, the child may not immediately select the correct quotient digit, which means that the check will be performed more than once.

2. Reception of multiplying a two-digit number by a single one: 23 4; 4-23

When multiplying a two-digit number by a single number, the following knowledge and skills are updated:

In the case of multiplication of the form 4 23, a permutation of the factors is first applied, and then the same multiplication scheme as above.

3. Reception of dividing a two-digit number by a single one: 48:3; 48:2

When dividing a two-digit number by a single number, the following knowledge and skills are updated:

4. Reception of dividing a two-digit number by two-digit: 68: 17

When dividing a two-digit number by a two-digit number, the following knowledge and skills are required:

The complexity of the last technique is that the child cannot immediately select the desired digit of the quotient and performs several checks of the selected digits, which requires rather complex calculations. Many children spend a lot of time doing this kind of calculation, as they begin not so much to find the right quotient digit as they go through all the factors in a row, starting with two.

In order to facilitate calculations, two methods can be used:

1) orientation to the last digit of the dividend;

2) rounding off.

First reception suggests that when selecting a possible quotient digit, the child is guided by knowledge of the multiplication table, immediately multiplying the selected digit (number) and the last digit of the divisor.

For example, 3-7 = 21. The last digit of 68 is 8, so it makes no sense to multiply 17 by 3, the last digit of the divisor still does not match. We try in a private number 4 - we multiply 7 4 \u003d 28. The last digit matches, so it makes sense to find the product 17 4.

Second reception involves rounding the divisor and selecting the quotient with a reference to the rounded divisor.

For example, 68:17 divisor 17 is rounded up to 20. An approximate quotient of 3 gives 20 when tested 3 = 60< 68, значит имеет смысл сразу проверять в качестве цифры частного 4:17 4 = 68.

These techniques allow you to reduce the effort and time spent when performing calculations of this type, but require good knowledge of the multiplication table and the ability to round numbers.

Integers ending in 0,1,2,3,4 are rounded to the nearest whole ten, discarding those digits.

For example, numbers 12, 13, 14 should be rounded up to 10. Numbers 62, 63, 64 should be rounded up to 60.

Integers ending in 5, 6, 7,8,9 are rounded up to the nearest whole ten.

For example, the numbers 15,16,17,18,19 are rounded up to 20. The numbers 45,47, 49 are rounded up to 50.

Order of operations in expressions containing multiplication and division

The rules for the order of execution of actions set the main features of expressions that should be guided by when calculating their values.

The first rules that determined the order of operations in arithmetic expressions set the order of operations in expressions containing addition and subtraction operations:

1. In expressions without brackets, containing only addition and subtraction operations, the operations are performed in the order they are written: from left to right.

2. Actions in brackets are performed first.

3. If the expression contains only addition operations, then two neighboring terms can always be replaced by their sum (the associative property of addition).

In grade 3, new rules for the order in which operations are performed in expressions containing multiplication and division are studied:

4. In expressions without brackets, containing only multiplication and division, actions are performed in the order they are written: from left to right.

5. In expressions without brackets, multiplication and division are performed before addition and subtraction.

In this case, the setting to perform the action in brackets is the first to be saved. Possible cases of violation of this installation were discussed earlier.

The rules for the order of actions are the general rules for calculating the values of mathematical expressions (examples), which are stored throughout the entire period of studying mathematics at school. In this regard, the formation in the child of a clear understanding of the algorithm for the execution of actions is an important successive task of teaching mathematics in elementary school. The problem is that the rules for the order of actions are quite variable and not always uniquely specified.

For example, in the expression 48-3 + 7 + 8, the general rule should be to apply rule 1 for an expression without brackets that contains addition and subtraction operations. At the same time, as a variant of rational calculations, you can use the method of replacing the sum of the part 7 + 8, since after subtracting the number 3 from 48, you get 45, to which it is convenient to add 15.

However, such an analysis of such an expression is not provided for in primary grades, since there are fears that with an inadequate understanding of this approach, the child will apply it in cases of the form 72 - 9 - 3 + 6. In this case, replacing the expression 3 + 6 with a sum is impossible, it will lead to wrong answer.

Great variability in the application of the entire group of rules and variants of rules in determining the order of actions requires considerable flexibility of thinking, a good understanding of the meaning of mathematical actions, the sequence of mental actions, mathematical "flair" and intuition (mathematicians call this "number sense"). In fact, it is much easier to teach a child to strictly follow a clearly established procedure for analyzing a numerical expression in terms of those features that each rule is focused on.

When determining the course of action, reason like this:

1) If there are brackets, I perform the first action written in brackets.

2) I perform multiplication and division in order.

3) Perform addition and subtraction in order.

This algorithm sets the order of actions quite unambiguously, although with slight variations.

In these expressions, the order of action is uniquely determined by the algorithm and is the only possible one. Here are some other examples

After performing the multiplication and division in this example, you could immediately add 6 to 54, and subtract 9 from 18, after which the results were added. Technically, it would be much easier than the path dictated by the algorithm, but an initially different order of actions in the example is possible:

Thus, the question of the formation of the ability to determine the order of actions in expressions in elementary school in a certain way contradicts the need to teach the child the methods of rational calculations.

For example, in the case, the order of actions is absolutely unambiguously determined by the algorithm, while it requires the child of the most complex calculations in the mind with transitions through the category: 42 - 7 and 35 + 8.

If, after performing the division 21:3, add 42 + 8 = 50, and then subtract 50 - 7 = 43, which is much easier technically, the answer will be the same. This way of calculation contradicts the setting given in the textbook

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Sunday, August 4, 2019

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

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

The rich theoretical basis of modern mathematics does not have a holistic character and is reduced to a set of disparate sections, devoid of a common system and evidence base.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.

As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.

Finally, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

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

Another interesting aporia of Zeno tells of a flying arrow:

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

Wednesday, July 4, 2018

I already told you that, with the help of which shamans try to sort "" realities. How do they do it? How does the formation of the set actually take place?

As you may have guessed, the same element can belong to a variety of sets. Next, I will show you how sets, subsets and other shamanic nonsense are formed.

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