In which expression the first action is subtraction? The order of actions. Fill in the missing number - examples with brackets. Training apparatus
- 1+2*3/4-5=?
- 1*3/(2+4)?
- 1+2*(3-1*5)=?
- If there are no parentheses 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.
- 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.
- If the example contains parentheses, the actions in the parentheses are performed first.
If we compare the functions 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 order from left to right.
It should be remembered that the actions in parentheses have special priority in the example. Thus, even if there is multiplication outside the brackets and addition inside the brackets, you should add first and then multiply.
To understand this topic, you can consider all cases one by one.
Let us immediately take into account that our expressions do not have parentheses.
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 the examples, in addition to multiplication and division, there is addition and subtraction, then multiplication and division are done first, and then addition and subtraction.
In the case of addition and subtraction, it also makes no difference which of these actions is done first. The order is observed 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 parentheses, and then only do the multiplication and division.
And so you need to remember that in any formula, operations such as multiplication and division are performed first, and then only subtraction and addition.
Also, with 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 operations will be: multiplication and division.
Only then are addition and subtraction performed.
However, if there is a parenthesis, then the actions that are in them will be executed first. Even if it's addition and subtraction.
For example:
In this example, we will first multiply, 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 the example contains all 4 operations, then first there is multiplication and division, 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 action 3+8 (11).
All 4 operations can be divided into two main groups, in one - addition and subtraction, in the other - multiplication and division. The first will be the action that is the first in the example, that is, the leftmost one.
Example: 60-7+9=62, first you need 60-7, then what happens is (53) +9;
Example: 5*8:2=20, first you need 5*8, then what happens is (40) :2.
When THERE ARE BRACKETS in an example, the actions in the bracket are performed first (according to the above rules), and then the rest are performed as usual.
Example: 2+(9-8)*10:2=7.
1 action 9-8 (1);
2nd action 1*10 (10);
Act 3 10:2 (5);
4 action 2+5 (7).
It depends on how the expression is written, let’s look at the simplest numerical 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 = 36
If this is an expression with parentheses, then perform the operations in parentheses, then multiplication or division and finally addition/subtraction, for example:
(18-6) : 3 + 10 x 2 = 12:3 + 20 = 4+20=24
Everything is correct: first perform multiplication and division, then addition and subtraction.
If there are no parentheses in the example, then multiplication and division in order are performed first, and then addition and subtraction are performed, 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, the operations 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 come one after another, then they are performed in order from left to right.
If you use parentheses when writing a mathematical expression, then first of all you should perform the actions indicated in the parentheses. Parentheses help change the order when it is necessary to perform addition or subtraction first, and then multiplication and division.
Any parentheses can be expanded and then the order of execution will again be correct:
6*(45+15) = 6*45 +6*15
Better immediately in examples:
In this case, we perform multiplication first, since it is to the left of division. Then division. Then addition, because of the more left-hand location, and at the end subtraction.
First we do the calculation in parentheses, then multiplication and division.
First we do the operations in brackets: multiplication, then subtraction. This is followed by multiplication outside the brackets and addition at the end.
Multiplication and division come first. If there are parentheses in the example, then the action in the parentheses is considered at the beginning. Whatever the sign may be!
Here you need to remember a few basic rules:
For example, 5+8-5=8 (we do everything in order - add 8 to 5, and then subtract 5)
For example, 5+8*3=29 (first multiply 8 by 3 and then add 5)
For example, 3*(5+8)=39 (first 5+8, and then multiply by 3)
multiply in any order.
Methodologically, this rule aims to prepare the child to become familiar with the methods of multiplying numbers ending in zeros, so they are introduced to it only in the fourth grade. In reality, this property of multiplication allows you to rationalize mental calculations in both 2nd and 3rd grade.
For example:
Calculate: (7 2) 5 = ...
In this case, it is much easier to calculate the option
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 comes down to multiplying and dividing 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 method of selecting the quotient.
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 in the form of digit units was used, which reduces the case under consideration to a tabular one (8:2). In the second case, the quotient figure is found by selection and checked by multiplication. In the second case, the child may not immediately select the correct number of the quotient, which means that the check will be performed more than once.
2. Method of multiplying a two-digit number by a single-digit number: 23 4; 4-23
When multiplying a two-digit number by a one-digit number, the following knowledge and skills are updated:
In the case of multiplication of the form 4 23, the permutation of factors is first applied, and then the same multiplication scheme as above is applied.
3. Method of dividing a two-digit number by a single-digit number: 48:3; 48:2
When dividing a two-digit number by a single-digit number, the following knowledge and skills are updated:
4. Method of dividing a two-digit number by a two-digit number: 68: 17
When dividing a two-digit number by a two-digit number, the following knowledge and skills are required:
The difficulty 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 quite complex calculations. Many children spend a lot of time performing calculations of this type, because they begin not so much to select the appropriate quotient number, but rather to sort through all the factors in a row, starting with two.
To facilitate calculations, two techniques can be used:
1) orientation to the last digit of the dividend;
2) rounding method.
First appointment assumes that when selecting a possible digit of a quotient, 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 the number 68 is 8, which means there is no point in multiplying 17 by 3, the last digit of the divisor still does not match. Let's try the number 4 in the quotient - multiply 7 4 = 28. The last digit matches, so it makes sense to find the product 17 4.
Second appointment involves rounding the divisor and selecting the quotient digit based on the rounded divisor.
For example, 68:17, the divisor of 17 is rounded to 20. The approximate quotient of 3 gives, when checked, 20 3 = 60< 68, значит имеет смысл сразу проверять в качестве цифры частного 4:17 4 = 68.
These techniques allow you to reduce the cost of effort and time when performing calculations of this type, but require a 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, the numbers 12, 13, 14 should be rounded to 10. The numbers 62, 63, 64 should be rounded 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 to 20. The numbers 45,47, 49 are rounded to 50.
Order of operations in expressions containing multiplication and division
Rules for the order of actions specify the main characteristics of expressions that should be used when calculating their values.
The first rules defining the order of operations in arithmetic expressions specified the order of actions in expressions containing addition and subtraction operations:
1. In expressions without parentheses containing only addition and subtraction operations, the actions are performed in the order they are written: from left to right.
2. Actions in brackets are performed first.
3. If an expression contains only addition actions, then two adjacent terms can always be replaced by their sum (combinative property of addition).
In grade 3, new rules for the order of performing actions in expressions containing multiplication and division are studied:
4. In expressions without parentheses containing only multiplication and division, the actions are performed in the order they are written: from left to right.
5. In expressions without parentheses, multiplication and division are performed before addition and subtraction.
In this case, the setting to perform the action in brackets first is preserved. Possible cases of violation of this setting were discussed earlier.
Rules for the order of actions are general rules for calculating the values of mathematical expressions (examples), which are maintained throughout the entire period of studying mathematics at school. In this regard, developing in a child a clear understanding of the algorithm for performing actions is an important successive task of teaching mathematics in primary school. The problem is that the rules for the order of actions are quite variable and not always clearly defined.
For example, in the expression 48-3 + 7 + 8, as a general rule, rule 1 should be applied for an expression without parentheses containing addition and subtraction operations. At the same time, as an option for rational calculations, you can use the technique 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 in the elementary grades, since there are fears that with an inadequate understanding of this approach, the child will use 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 significant flexibility of thinking, a good understanding of the meaning of mathematical actions, the sequence of mental actions, mathematical “feeling” and intuition (mathematicians call this “number sense”). In reality, it is much easier to teach a child to strictly adhere to a clearly established procedure for analyzing a numerical expression from the point of view of the features that each rule is focused on.
When determining the course of action, think like this:
1) If there are parentheses, I perform the action written in parentheses first.
2) I perform multiplication and division in order.
3) I perform addition and subtraction in order.
This algorithm sets the order of actions quite unambiguously, although with minor variations.
In these expressions, the order of action is uniquely determined by the algorithm and is the only possible one. Let's give other examples
After performing multiplication and division in this example, you could immediately add 6 to 54, and subtract 9 from 18, and then add the results. Technically, it would be much easier than the path determined by the algorithm; an initially different order of actions in the example is possible:
Thus, the question of developing the ability to determine the order of actions in expressions in elementary school in a certain way contradicts the need to teach the child methods of rational calculations.
For example, in this case, the order of actions is absolutely unambiguously determined by the algorithm, and requires a series of complex mental calculations with transitions through the digits: 42 - 7 and 35 + 8.
If, after performing the division 21:3, you perform the addition 42 + 8 = 50, and then subtract 50 - 7 = 43, which is much easier technically, the answer will be the same. This calculation path contradicts the setting given in the textbook
And when calculating the values of expressions, actions are performed in a certain order, in other words, you must observe order of actions.
In this article, we will figure out which actions should be performed first and which ones after them. Let's start with the simplest cases, when the expression contains only numbers or variables connected by plus, minus, multiply and divide signs. Next, we will explain what order of actions should be followed in expressions with brackets. Finally, let's look at the order in which actions are performed in expressions containing powers, roots, and other functions.
Page navigation.
First multiplication and division, then addition and subtraction
The school gives the following a rule that determines the order in which actions are performed in expressions without parentheses:
- actions are performed in order from left to right,
- Moreover, multiplication and division are performed first, and then addition and subtraction.
The stated rule is perceived quite naturally. Performing actions in order from left to right is explained by the fact that it is customary for us to keep records from left to right. And the fact that multiplication and division are performed before addition and subtraction is explained by the meaning that these actions carry.
Let's look at a few examples of how this rule applies. For examples, we will take the simplest numerical expressions so as not to be distracted by calculations, but to focus specifically on the order of actions.
Example.
Follow steps 7−3+6.
Solution.
The original expression does not contain parentheses, and it does not contain multiplication or division. Therefore, we should perform all actions in order from left to right, that is, first we subtract 3 from 7, we get 4, after which we add 6 to the resulting difference of 4, we get 10.
Briefly, the solution can be written as follows: 7−3+6=4+6=10.
Answer:
7−3+6=10 .
Example.
Indicate the order of actions in the expression 6:2·8:3.
Solution.
To answer the question of the problem, let's turn to the rule indicating the order of execution of actions in expressions without parentheses. The original expression contains only the operations of multiplication and division, and according to the rule, they must be performed in order from left to right.
Answer:
At first We divide 6 by 2, multiply this quotient by 8, and finally divide the result by 3.
Example.
Calculate the value of the expression 17−5·6:3−2+4:2.
Solution.
First, let's determine in what order the actions in the original expression should be performed. It contains both multiplication and division and addition and subtraction. First, from left to right, you need to perform multiplication and division. So we multiply 5 by 6, we get 30, we divide this number by 3, we get 10. Now we divide 4 by 2, we get 2. We substitute the found value 10 into the original expression instead of 5·6:3, and instead of 4:2 - the value 2, we have 17−5·6:3−2+4:2=17−10−2+2.
The resulting expression no longer contains multiplication and division, so it remains to perform the remaining actions in order from left to right: 17−10−2+2=7−2+2=5+2=7 .
Answer:
17−5·6:3−2+4:2=7.
At first, in order not to confuse the order in which actions are performed when calculating the value of an expression, it is convenient to place numbers above the action signs that correspond to the order in which they are performed. For the previous example it would look like this: .
The same order of operations - first multiplication and division, then addition and subtraction - should be followed when working with letter expressions.
Actions of the first and second stages
In some mathematics textbooks there is a division of arithmetic operations into operations of the first and second stages. Let's figure this out.
Definition.
Actions of the first stage addition and subtraction are called, and multiplication and division are called second stage actions.
In these terms, the rule from the previous paragraph, which determines the order of execution of actions, will be written as follows: if the expression does not contain parentheses, then in order from left to right, first the actions of the second stage (multiplication and division) are performed, then the actions of the first stage (addition and subtraction).
Order of arithmetic operations in expressions with parentheses
Expressions often contain parentheses to indicate the order in which actions should be performed. In this case a rule that specifies the order of execution of actions in expressions with parentheses, is formulated as follows: first, the actions in brackets are performed, while multiplication and division are also performed in order from left to right, then addition and subtraction.
So, the expressions in brackets are considered as components of the original expression, and they retain the order of actions already known to us. Let's look at the solutions to the examples for greater clarity.
Example.
Follow these steps 5+(7−2·3)·(6−4):2.
Solution.
The expression contains parentheses, so let's first perform the actions in the expressions enclosed in these parentheses. Let's start with the expression 7−2·3. In it you must first perform multiplication, and only then subtraction, we have 7−2·3=7−6=1. Let's move on to the second expression in brackets 6−4. There is only one action here - subtraction, we perform it 6−4 = 2.
We substitute the obtained values into the original expression: 5+(7−2·3)·(6−4):2=5+1·2:2. In the resulting expression, we first perform multiplication and division from left to right, then subtraction, we get 5+1·2:2=5+2:2=5+1=6. At this point, all actions are completed, we adhered to the following order of their implementation: 5+(7−2·3)·(6−4):2.
Let's write down a short solution: 5+(7−2·3)·(6−4):2=5+1·2:2=5+1=6.
Answer:
5+(7−2·3)·(6−4):2=6.
It happens that an expression contains parentheses within parentheses. There is no need to be afraid of this; you just need to consistently apply the stated rule for performing actions in expressions with brackets. Let's show the solution of the example.
Example.
Perform the operations in the expression 4+(3+1+4·(2+3)) .
Solution.
This is an expression with brackets, which means that the execution of actions must begin with the expression in brackets, that is, with 3+1+4·(2+3) . This expression also contains parentheses, so you must perform the actions in them first. Let's do this: 2+3=5. Substituting the found value, we get 3+1+4·5. In this expression, we first perform multiplication, then addition, we have 3+1+4·5=3+1+20=24. The initial value, after substituting this value, takes the form 4+24, and all that remains is to complete the actions: 4+24=28.
Answer:
4+(3+1+4·(2+3))=28.
In general, when an expression contains parentheses within parentheses, it is often convenient to perform actions starting with the inner parentheses and moving to the outer ones.
For example, let's say we need to perform the actions in the expression (4+(4+(4−6:2))−1)−1. First, we perform the actions in the inner brackets, since 4−6:2=4−3=1, then after this the original expression will take the form (4+(4+1)−1)−1. We again perform the action in the inner brackets, since 4+1=5, we arrive at the following expression (4+5−1)−1. Again we perform the actions in brackets: 4+5−1=8, and we arrive at the difference 8−1, which is equal to 7.
Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take the infinite set of natural numbers as an example, then the considered examples can be represented in this form:
To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantasy story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.
What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite number of such corridors. Moreover, the “infinite hotel” has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”
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 are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.
Option one. “Let us be given” one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and 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 one from the set we have already taken and return it to the shelf. After that, we can take one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:
I wrote down the actions in algebraic notation and in set theory notation, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same unit is added.
Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. Let's 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. This is 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 you add another infinite set 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 is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.
You can accept or not accept my reasoning - it is your own business. But if you ever encounter mathematical problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).
Sunday, August 4, 2019
I was finishing a postscript to an article about and saw this wonderful text on Wikipedia:
We read: "... the rich theoretical basis of the mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."
Wow! How smart we are and how well we can see the shortcomings of others. Is it difficult for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:
The rich theoretical basis of modern mathematics is not holistic in nature and is reduced to a set of disparate sections, devoid of a common system and evidence base.
I won’t 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 series of publications to the most obvious mistakes of modern mathematics. See you soon.
Saturday, August 3, 2019
How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.
May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we use regular school mathematics. Look what happened.
After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that essentially everything was done correctly; it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I will tell you about this.
As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.
As you can see, units of measurement and ordinary mathematics make set theory a relic 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. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.
In conclusion, 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 “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 ... 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 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 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
I have already told you that with the help of which shamans try to sort ““ reality. How do they do this? How does the formation of a set actually occur?
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 two phrases: “conceivable as a whole” and “conceivable as a whole.” The first phrase is the end result, the set. The second phrase is a preliminary preparation for the formation of a multitude. At this stage, reality is divided into individual elements (the “whole”), from which a multitude will then be formed (the “single whole”). At the same time, the factor that makes it possible 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 show us.
I'll show you the process with an example. We select the “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 part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.
Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting 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 it will be.
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 with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. This is what it looks like.
The letter "a" with different indices indicates different units of measurement. The units of measurement by which the “whole” is distinguished at the preliminary stage are highlighted in brackets. The unit of measurement by 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 of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing that it is “obvious,” because units of measurement are not part of their “scientific” arsenal.
Using units of measurement, it is very easy to split one set or combine several sets into one superset. Let's take a closer look at the algebra of this process.
Saturday, June 30, 2018
If mathematicians cannot reduce a concept to other concepts, then they do not understand anything about mathematics. I answer: how do the elements of one set differ from the elements of another set? The answer is very simple: numbers and units of measurement.
Today, 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 haven't seen such a list. I will say more - not a single thing in reality has a tag with a list of the 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 what the elements of the set looked like before the mathematician shamans took them into their sets.
A long time ago, when no one had ever heard of mathematics, 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 something like this.
Yes, don’t be surprised, from the point of view of mathematics, all elements of 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 bunch of segments sticking out in different directions from one point. This point is point zero. I won’t draw this piece of geometric art (no inspiration), but you can easily imagine it.
What units of measurement form an element of a set? All sorts of things that describe a given element from different points of view. These are ancient units of measurement that our ancestors used and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are also units of measurement unknown to us, which our descendants will come up with and which they will use to describe reality.
We've sorted out the geometry - the proposed model of the elements of the set has a clear geometric representation. What about physics? Units of measurement are 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 the 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 being in the Stone Age.
But let's move on to the most interesting thing - the algebra of elements of sets. Algebraically, any element of a set is a product (the result of multiplication) of different quantities. It looks like this.
I deliberately did not use the conventions of set theory, since we are considering an element of a set in its natural environment before the emergence of set theory. Each pair of letters in brackets denotes a separate quantity, consisting of a number indicated by the letter " n" and the unit of measurement indicated by the letter " a". The indices next to the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of quantities (how much we and our descendants have enough imagination). Each bracket is geometrically depicted as 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. Not understanding anything about mathematics, they take different sea urchins and carefully examine them in search of that single needle, along which they form a set. If there is such a needle, then this element belongs to the set; if there is no such needle, then this element is not from this set. Shamans tell us fables about thought processes and the whole.
As you may have guessed, the same element can belong to very different sets. Next I will show you how sets, subsets and other shamanic nonsense are formed. 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: 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.
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."
When we work with various expressions that include numbers, letters and variables, we have to perform a large number of arithmetic operations. When we do a conversion 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 order of execution.
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In this article we will tell you which actions should be done first and which ones 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 let's take examples with parentheses and consider in what order they should be calculated. In the third part we will give the necessary order of transformations and calculations in those examples that include signs of roots, powers and other functions.
Definition 1In the case of expressions without parentheses, the order of actions is determined unambiguously:
- All actions are performed from left to right.
- We perform division and multiplication first, and subtraction and addition second.
The meaning of these rules is easy to understand. The traditional left-to-right writing order defines the basic sequence of calculations, and the need to multiply or divide first is explained by the very essence of these operations.
Let's take a few tasks for clarity. We used only the simplest numerical expressions so that all calculations could be done mentally. This way you can quickly remember the desired order and quickly check the results.
Example 1
Condition: calculate how much it will be 7 − 3 + 6 .
Solution
There are no parentheses in our expression, there is also no multiplication and division, so we perform all the actions in the specified order. First we subtract three from seven, then add six to the remainder and end up with ten. Here is a transcript 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, let’s reread the rule for expressions without parentheses that 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 it will be 17 − 5 · 6: 3 − 2 + 4: 2.
Solution
First, let's determine the correct order of operations, since we have all the basic types of arithmetic operations here - 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 to get 30, then 30 divided by 3 to get 10. After that, divide 4 by 2, this is 2. Let's substitute the found values into the original expression:
17 − 5 6: 3 − 2 + 4: 2 = 17 − 10 − 2 + 2
There is no longer 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 memorized, you can put numbers above 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 letter expressions, then we do the same with them: first we multiply and divide, then we add and subtract.
What are the first and second stage actions?
Sometimes in reference books all arithmetic operations are divided into actions of the first and second stages. Let us formulate the necessary definition.
The operations of the first stage include subtraction and addition, the second - multiplication and division.
Knowing these names, we can write the previously given rule regarding the order of actions as follows:
Definition 2
In an expression that does not contain parentheses, you must first perform the actions of the second stage in the direction from left to right, then the actions of the first stage (in the same direction).
Order of calculations in expressions with parentheses
The parentheses themselves are a sign that tells us the desired order of actions. In this case, the required rule can be written as follows:
Definition 3
If there are parentheses in the expression, then the first step is to perform the operation in them, after which we multiply and divide, and then add and subtract from left to right.
As for the parenthetical expression itself, it can be considered as an integral part of the main expression. When calculating the value of the expression in brackets, we maintain the same procedure known to us. Let's illustrate our idea with an example.
Example 4
Condition: calculate how much it will be 5 + (7 − 2 3) (6 − 4) : 2.
Solution
There are parentheses in this expression, 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 calculate 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 perform subtraction and get:
5 + 1 2: 2 = 5 + 2: 2 = 5 + 1 = 6
This concludes the calculations.
Answer: 5 + (7 − 2 3) (6 − 4) : 2 = 6.
Don't be alarmed if our condition contains an expression in which some parentheses enclose others. We only need to apply the rule above consistently to all expressions in parentheses. Let's take this problem.
Example 5
Condition: calculate how much it will be 4 + (3 + 1 + 4 (2 + 3)).
Solution
We have parentheses within parentheses. 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 calculated that 3 + 1 + 4 · 5. We remember that we first need to 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 calculating the value of an expression that includes 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 (4 + (4 + (4 − 6: 2)) − 1) − 1 will be. 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. Looking again at the inner parentheses: 4 + 1 = 5. We have come to the expression (4 + 5 − 1) − 1 . We count 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 our condition contains an expression with a power, 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 this, 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 is (3 + 1) · 2 + 6 2: 3 − 7.
Solution
We have an expression with a degree, the value of which must be found first. We count: 6 2 = 36. Now let's 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.
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