"Trade" revolution
Komkov Sergey 12/26/2012
Against the backdrop of Russia's just accession to the WTO, the destruction of the RGTEU - the leading Russian university in the system of trade (and, first of all, foreign trade) relations, as well as the dismissal of its rector, the well-known politician Sergei Baburin, look not just like stupidity. All this is very similar to a pre-planned provocation.
It seems that the World Trade Organization and, mainly, the United States, which plays a key role in it, turned out to be seriously concerned about the possible consequences of Russia's entry into this organization.
But then they remembered in time that in Russia for a long time and successfully the organization grown and nurtured by them has been operating - Graduate School Economics. It was it that was created in 1992 with the money of the World Bank in order to destroy the entire intellectual potential of the nation in our country. It is under her leadership that the main collective “agent of influence” in this area, the Ministry of Education and Science of Russia, operates today.
You can talk a lot and endlessly about the stupidity and incompetence of the newly-minted minister - Mr. Livanov, who hardly distinguishes between the types and directions of education. But in itself, Mr. Livanov is an absolute zero without a wand. From the mouth of which, every time they are opened, some next nonsense will certainly jump out. More colorful figures loom behind him. For example, the main "ideologist" of all economic transformations in our country, US citizen Yevgeny Yasin, and his henchman, the rector of HSE Yaroslav Kuzminov.
It was they who, at the suggestion of American advisers from the World Bank, who are actively working on the basis of the HSE, concocted the criteria for the so-called “monitoring” of Russian universities.
And it's no secret to anyone that, in accordance with these "criteria", the most significant Russian higher educational institutions fell into the category of "inefficient". Universities with rich history and traditions with great creative potential. For example, Moscow Architectural Institute, Russian State University for the Humanities, Literary Institute.
The Russian State Trade and Economic University - RGTEU fell into this category. Although, according to many of its indicators, this university can give a hundred points of odds to the very “Pleshka”, to which it was so suddenly decided to join. And, first of all, in matters of training specialists for the system foreign trade.
RGTEU not only has huge international connections. It thoroughly studies the features of the trade development of foreign countries. Within the walls of this university, leading economic and political figures of the world, ambassadors foreign countries. Honorary doctors of this university are the world's leading leaders. For example, Fidel Castro and Hugo Chavez.
And these, as you know, are the "sworn friends" of America. So the tools to destroy such a dangerous educational institution. So that Russia, God forbid, does not turn off the “true path” and betray the interests of American customers.
And the personality of the rector himself - a well-known politician and scientist in Russia and far beyond its borders - stood in our American uncles like a bone in the throat.
Sergey Baburin was not just one of the leaders of the parliamentary opposition, occupying the place of vice-speaker in the previous composition of the State Duma of Russia. He was an active supporter of Russia's new policy throughout the post-Soviet space. It was he who in 2006 actively helped the people of Abkhazia to get out of the deepest political crisis. In which, by the way, he was driven again by all the same stupid and obedient to the will of American advisers officials of the government and the presidential administration of Russia.
Thanks to the efforts of Sergei Baburin, the progressive forces headed by Sergei Bagapsh then took over in Abkhazia. And since 2008, Abkhazia has become Russia's main strategic partner in the North Caucasus.
Such a position is an expression of sound, balanced patriotism. Therefore, for a number of years, Baburin has been the head of the Russian All-People's Union and is the organizer of the annual traditional Russian Marches. Not the ones with swastikas and fascist slogans “Russia is only for Russians!” And speeches that are quite understandable for the entire population of the country with demands to observe Russian national interests in matters foreign policy and fulfill the social promises made to their own people.
But this is exactly what the American henchmen entrenched in the offices of the Russian government do not like. Because for them the requirement to respect our national interests is like a knife to the heart.
So it came to someone's mind to kill two birds with one stone at once: both the university that trains specialists for the successful foreign trade of Russia, and its patriotic rector.
Usually fools are most suitable for this kind of action. For, as you know, they do not know what they are actually doing. But in this particular case, a very serious blunder could result, fraught with grave social consequences for the entire country.
Our officials, snickering at state-owned grubs and considering themselves completely right in any unrighteous deed, have forgotten the simplest truth: they have no power over youthful souls and youthful impulses.
It was impulses of this kind that swept away the government of General de Gaulle in France at the end of the 1960s. There, too, it all began with seemingly harmless things. And ended with general chaos, riots, burning cars and offices.
Young people (especially organized student youth) are not a bunch of bankrupt opposition politicians who have been in power and, therefore, are very offended by it. Student youth has always and at all times been one of the main driving forces revolution. And today's youth is no exception to the rule. Rather the opposite. It is today's youth, who are especially sensitive to the social injustice and inequality that have arisen in society, that are capable of taking the steepest and most radical steps. And if the government tries to use force, it will be deadly for it. Because the youth will never forgive her for this.
When Mr. Livanov and Co. announced their intention to start solving the problem by force higher education By closing and merging universities, they actually signed their own verdict. They did not even bother to think about what deep forces they raise. And this will end tragically not only for those who today find themselves in leading positions in the Ministry of Education and Science, but also for the entire Russian leadership as a whole. For, even a locally suppressed youth rebellion does not go into oblivion. He matures with renewed vigor. But where and when it will strike, no one can predict.
So the events at RGTEU only at first glance look like some kind of “trade revolution”. In fact, they are the harbingers of another - tougher and bloodier social war, in which there will be no winners.
The loser is already known. This is our Motherland. A country that we still sometimes call Russia with some pride.
Therefore, today's actions of the leadership of the Ministry of Education and Science in relation to a single educational institution and in relation to a single rector can be regarded as inciting a social war in the name and for the benefit of another state.
And this is called: National Treason.
The problem of finding a root in mathematics is the inverse problem of raising a number to a power. Roots are different: roots of the second degree, roots of the third degree, roots of the fourth degree, and so on. It depends on what power the number was originally raised to. The root is denoted by the symbol: √ is the square root, that is, the root of the second degree, if the root has a degree greater than the second, then the corresponding degree is assigned above the sign of the root. The number that is under the radical sign is a radical expression. When finding the root, there are several rules that will help you not to make a mistake in finding the root:
- An even root (if the exponent is 2, 4, 6, 8, and so on) of a negative number does NOT exist. If the radical expression is negative, but the root of an odd degree (3, 5, 7, and so on) is sought, then the result will be negative.
- The root of any degree of unity is always one: √1 = 1.
- The root of zero is zero: √0 = 0.
How to find the root of 100
If the task does not say what degree root is to be found, then it is usually understood that it is necessary to find the root of the second degree (square).
Let's find √100 = ? We need to find such a number, when raised to the second power, the number 100 will be obtained. Obviously, the number 10 is such a number, since: 10 2 \u003d 100. Therefore, √100 \u003d 10: the square root of 100 is 10.
Quite often, when solving problems, we are faced with large numbers from which we need to extract Square root. Many students decide that this is a mistake and start resolving the whole example. Under no circumstances should this be done! There are two reasons for this:
- The roots of large numbers do occur in problems. Especially in text;
- There is an algorithm by which these roots are considered almost verbally.
We will consider this algorithm today. Perhaps some things will seem incomprehensible to you. But if you pay attention to this lesson, you will get the most powerful weapon against square roots .
So the algorithm:
- Limit the desired root above and below to multiples of 10. Thus, we will reduce the search range to 10 numbers;
- From these 10 numbers, weed out those that definitely cannot be roots. As a result, 1-2 numbers will remain;
- Square these 1-2 numbers. That of them, the square of which is equal to the original number, will be the root.
Before applying this algorithm works in practice, let's look at each individual step.
Roots constraint
First of all, we need to find out between which numbers our root is located. It is highly desirable that the numbers be a multiple of ten:
10 2 = 100;
20 2 = 400;
30 2 = 900;
40 2 = 1600;
...
90 2 = 8100;
100 2 = 10 000.
We get a series of numbers:
100; 400; 900; 1600; 2500; 3600; 4900; 6400; 8100; 10 000.
What do these numbers give us? It's simple: we get boundaries. Take, for example, the number 1296. It lies between 900 and 1600. Therefore, its root cannot be less than 30 and greater than 40:
[Figure caption]
The same is with any other number from which you can find the square root. For example, 3364:
[Figure caption]Thus, instead of an incomprehensible number, we get a very specific range in which the original root lies. To further narrow the scope of the search, go to the second step.
Elimination of obviously superfluous numbers
So, we have 10 numbers - candidates for the root. We received them very quickly, without complex thinking and multiplication in a column. It's time to move on.
Believe it or not, now we will reduce the number of candidate numbers to two - and again without any complex calculations! It is enough to know the special rule. Here it is:
The last digit of the square depends only on the last digit original number.
In other words, it is enough to look at the last digit of the square - and we will immediately understand where the original number ends.
There are only 10 digits that can be in last place. Let's try to find out what they turn into when they are squared. Take a look at the table:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 0 |
| 1 | 4 | 9 | 6 | 5 | 6 | 9 | 4 | 1 | 0 |
This table is another step towards calculating the root. As you can see, the numbers in the second line turned out to be symmetrical with respect to the five. For example:
2 2 = 4;
8 2 = 64 → 4.
As you can see, the last digit is the same in both cases. And this means that, for example, the root of 3364 necessarily ends in 2 or 8. On the other hand, we remember the restriction from the previous paragraph. We get:
[Figure caption] The red squares show that we don't know this figure yet. But after all, the root lies between 50 and 60, on which there are only two numbers ending in 2 and 8:
[Figure caption]That's all! Of all the possible roots, we left only two options! And this is in the most difficult case, because the last digit can be 5 or 0. And then the only candidate for the roots will remain!
Final Calculations
So, we have 2 candidate numbers left. How do you know which one is the root? The answer is obvious: square both numbers. The one that squared will give the original number, and will be the root.
For example, for the number 3364, we found two candidate numbers: 52 and 58. Let's square them:
52 2 \u003d (50 +2) 2 \u003d 2500 + 2 50 2 + 4 \u003d 2704;
58 2 \u003d (60 - 2) 2 \u003d 3600 - 2 60 2 + 4 \u003d 3364.
That's all! It turned out that the root is 58! At the same time, in order to simplify the calculations, I used the formula of the squares of the sum and difference. Thanks to this, you didn’t even have to multiply the numbers in a column! This is another level of optimization of calculations, but, of course, it is completely optional :)
Root Calculation Examples
Theory is good, of course. But let's test it in practice.
[Figure caption]
First, let's find out between which numbers the number 576 lies:
400 < 576 < 900
20 2 < 576 < 30 2
Now let's look at the last number. It is equal to 6. When does this happen? Only if the root ends in 4 or 6. We get two numbers:
It remains to square each number and compare with the original:
24 2 = (20 + 4) 2 = 576
Great! The first square turned out to be equal to the original number. So this is the root.
Task. Calculate the square root:
[Figure caption]
900 < 1369 < 1600;
30 2 < 1369 < 40 2;
Let's look at the last number:
1369 → 9;
33; 37.
Let's square it:
33 2 \u003d (30 + 3) 2 \u003d 900 + 2 30 3 + 9 \u003d 1089 ≠ 1369;
37 2 \u003d (40 - 3) 2 \u003d 1600 - 2 40 3 + 9 \u003d 1369.
Here is the answer: 37.
Task. Calculate the square root:
[Figure caption]
We limit the number:
2500 < 2704 < 3600;
50 2 < 2704 < 60 2;
Let's look at the last number:
2704 → 4;
52; 58.
Let's square it:
52 2 = (50 + 2) 2 = 2500 + 2 50 2 + 4 = 2704;
We got the answer: 52. The second number will no longer need to be squared.
Task. Calculate the square root:
[Figure caption]
We limit the number:
3600 < 4225 < 4900;
60 2 < 4225 < 70 2;
Let's look at the last number:
4225 → 5;
65.
As you can see, after the second step, only one option remains: 65. This is the desired root. But let's still square it and check:
65 2 = (60 + 5) 2 = 3600 + 2 60 5 + 25 = 4225;
Everything is correct. We write down the answer.
Conclusion
Alas, no better. Let's take a look at the reasons. There are two of them:
- It is forbidden to use calculators at any normal math exam, be it the GIA or the Unified State Examination. And for carrying a calculator into the classroom, they can easily be kicked out of the exam.
- Don't be like stupid Americans. Which are not like roots - they are two prime numbers cannot fold. And at the sight of fractions, they generally get hysterical.
I looked again at the plate ... And, let's go!
Let's start with a simple one:
Wait a minute. this, which means we can write it like this:
Got it? Here's the next one for you:
The roots of the resulting numbers are not exactly extracted? Don't worry, here are some examples:
But what if there are not two multipliers, but more? The same! The root multiplication formula works with any number of factors:
Now completely independent:
Answers: Well done! Agree, everything is very easy, the main thing is to know the multiplication table!
Root division
We figured out the multiplication of the roots, now let's proceed to the property of division.
Recall that the formula general view looks like that:
And that means that the root of the quotient is equal to the quotient of the roots.
Well, let's look at examples:
That's all science. And here's an example:
Everything is not as smooth as in the first example, but as you can see, there is nothing complicated.
What if the expression looks like this:
You just need to apply the formula in reverse:
And here's an example:
You can also see this expression:
Everything is the same, only here you need to remember how to translate fractions (if you don’t remember, look at the topic and come back!). Remembered? Now we decide!
I am sure that you coped with everything, everything, now let's try to build roots in a degree.
Exponentiation
What happens if the square root is squared? It's simple, remember the meaning square root of number is the number whose square root is .
So, if we square a number whose square root is equal, then what do we get?
Well, of course, !
Let's look at examples:
Everything is simple, right? And if the root is in a different degree? It's OK!
Stick to the same logic and remember the properties and possible actions with degrees.
Read the theory on the topic "" and everything will become extremely clear to you.
For example, here's an expression:
In this example, the degree is even, but what if it is odd? Again, apply the power properties and factor everything:
With this, everything seems to be clear, but how to extract the root from a number in a degree? Here, for example, is this:
Pretty simple, right? What if the degree is greater than two? We follow the same logic using the properties of degrees:
Well, is everything clear? Then solve your own examples:
And here are the answers:
Introduction under the sign of the root
What we just have not learned to do with the roots! It remains only to practice entering the number under the root sign!
It's quite easy!
Let's say we have a number
What can we do with it? Well, of course, hide the triple under the root, while remembering that the triple is the square root of!
Why do we need it? Yes, just to expand our capabilities when solving examples:
How do you like this property of roots? Makes life much easier? For me, that's right! Only we must remember that we can only enter positive numbers under the square root sign.
Try this example for yourself:
Did you manage? Let's see what you should get:
Well done! You managed to enter a number under the root sign! Let's move on to something equally important - consider how to compare numbers containing a square root!
Root Comparison
Why should we learn to compare numbers containing a square root?
Very simple. Often, in large and long expressions encountered in the exam, we get an irrational answer (do you remember what it is? We already talked about this today!)
We need to place the received answers on the coordinate line, for example, to determine which interval is suitable for solving the equation. And this is where the snag arises: there is no calculator on the exam, and without it, how to imagine which number is larger and which is smaller? That's it!
For example, determine which is greater: or?
You won't say right off the bat. Well, let's use the parsed property of adding a number under the root sign?
Then forward:
Well, obviously, the larger the number under the sign of the root, the larger the root itself!
Those. if means .
From this we firmly conclude that And no one will convince us otherwise!
Extracting roots from large numbers
Before that, we introduced a factor under the sign of the root, but how to take it out? You just need to factor it out and extract what is extracted!
It was possible to go the other way and decompose into other factors:
Not bad, right? Any of these approaches is correct, decide how you feel comfortable.
Factoring is very useful when solving such non-standard tasks as this one:
We don't get scared, we act! We decompose each factor under the root into separate factors:
And now try it yourself (without a calculator! It will not be on the exam):
Is this the end? We don't stop halfway!
That's all, it's not all that scary, right?
Happened? Well done, you're right!
Now try this example:
And an example is a tough nut to crack, so you can’t immediately figure out how to approach it. But we, of course, are in the teeth.
Well, let's start factoring, shall we? Immediately, we note that you can divide a number by (recall the signs of divisibility):
And now, try it yourself (again, without a calculator!):
Well, did it work? Well done, you're right!
Summing up
- The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal.
. - If we just take the square root of something, we always get one non-negative result.
- Properties arithmetic root:
- When comparing square roots, it must be remembered that the larger the number under the sign of the root, the larger the root itself.
How do you like the square root? All clear?
We tried to explain to you without water everything you need to know in the exam about the square root.
It's your turn. Write to us whether this topic is difficult for you or not.
Did you learn something new or everything was already so clear.
Write in the comments and good luck on the exams!
When solving various problems from the course of mathematics and physics, pupils and students are often faced with the need to extract roots of the second, third or nth degree. Of course, in the century information technologies It will not be difficult to solve such a problem using a calculator. However, there are situations when it is impossible to use an electronic assistant.
For example, it is forbidden to bring electronics to many exams. In addition, the calculator may not be at hand. In such cases, it is useful to know at least some methods for manually calculating radicals.
Extracting the square root using the table of squares
One of the simplest ways to calculate roots is to using a special table. What is it and how to use it correctly?
Using the table, you can find the square of any number from 10 to 99. At the same time, the rows of the table contain tens values, and the columns contain unit values. The cell at the intersection of a row and a column contains the square of a two-digit number. In order to calculate the square of 63, you need to find a row with a value of 6 and a column with a value of 3. At the intersection, we find a cell with the number 3969.
Since extracting the root is the inverse operation of squaring, to perform this action, you must do the opposite: first find the cell with the number whose radical you want to calculate, then determine the answer from the column and row values. As an example, consider the calculation of the square root of 169.
We find a cell with this number in the table, horizontally we determine the tens - 1, vertically we find the units - 3. Answer: √169 = 13.
Similarly, you can calculate the roots of the cubic and n-th degree, using the appropriate tables.
The advantage of the method is its simplicity and the absence of additional calculations. The disadvantages are obvious: the method can only be used for a limited range of numbers (the number for which the root is found must be between 100 and 9801). In addition, it will not work if the given number is not in the table.
Prime factorization
If the table of squares is not at hand or with its help it was impossible to find the root, you can try decompose the number under the root into prime factors . Prime factors are those that can be completely (without remainder) divided only by itself or by one. Examples would be 2, 3, 5, 7, 11, 13, etc.
Consider the calculation of the root using the example √576. Let's decompose it into simple factors. We get the following result: √576 = √(2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3) = √(2 ∙ 2 ∙ 2)² ∙ √3². Using the main property of the roots √a² = a, we get rid of the roots and squares, after which we calculate the answer: 2 ∙ 2 ∙ 2 ∙ 3 = 24.
What to do if any of the factors does not have its own pair? For example, consider the calculation of √54. After factoring, we get the result in the following form: The non-removable part can be left under the root. For most problems in geometry and algebra, such an answer will be counted as the final one. But if there is a need to calculate approximate values, you can use the methods that will be discussed later.
Heron's method
What to do when you need to know at least approximately what the extracted root is (if it is impossible to get an integer value)? A quick and fairly accurate result is obtained by applying the Heron method.. Its essence lies in the use of an approximate formula:
√R = √a + (R - a) / 2√a,
where R is the number whose root is to be calculated, a is the nearest number whose root value is known.
Let's see how the method works in practice and evaluate how accurate it is. Let's calculate what √111 is equal to. The nearest number to 111, the root of which is known, is 121. Thus, R = 111, a = 121. Substitute the values in the formula:
√111 = √121 + (111 - 121) / 2 ∙ √121 = 11 - 10 / 22 ≈ 10,55.
Now let's check the accuracy of the method:
10.55² = 111.3025.
The error of the method was approximately 0.3. If the accuracy of the method needs to be improved, you can repeat the steps described earlier:
√111 = √111,3025 + (111 - 111,3025) / 2 ∙ √111,3025 = 10,55 - 0,3025 / 21,1 ≈ 10,536.
Let's check the accuracy of the calculation:
10.536² = 111.0073.
After repeated application of the formula, the error became quite insignificant.
Calculation of the root by division into a column
This method of finding the square root value is a little more complicated than the previous ones. However, it is the most accurate among other calculation methods without a calculator..
Let's say that you need to find the square root with an accuracy of 4 decimal places. Let's analyze the calculation algorithm using the example of an arbitrary number 1308.1912.
- Divide the sheet of paper into 2 parts with a vertical line, and then draw another line from it to the right, slightly below the top edge. We write the number on the left side, dividing it into groups of 2 digits, moving to the right and left of the decimal point. The very first digit on the left can be without a pair. If the sign is missing on the right side of the number, then 0 should be added. In our case, we get 13 08.19 12.
- Let's pick the most big number, whose square will be less than or equal to the first group of digits. In our case, this is 3. Let's write it on the top right; 3 is the first digit of the result. At the bottom right, we indicate 3 × 3 = 9; this will be needed for subsequent calculations. Subtract 9 from 13 in a column, we get the remainder 4.
- Let's add the next pair of numbers to the remainder 4; we get 408.
- Multiply the number on the top right by 2 and write it on the bottom right, adding _ x _ = to it. We get 6_ x _ =.
- Instead of dashes, you need to substitute the same number, less than or equal to 408. We get 66 × 6 \u003d 396. Let's write 6 on the top right, since this is the second digit of the result. Subtract 396 from 408, we get 12.
- Let's repeat steps 3-6. Since the numbers carried down are in the fractional part of the number, it is necessary to put a decimal point on the top right after 6. Let's write the doubled result with dashes: 72_ x _ =. A suitable number would be 1: 721 × 1 = 721. Let's write it down as an answer. Let's subtract 1219 - 721 = 498.
- We perform the sequence of actions given in the previous paragraph three more times to get required amount decimal places. If there are not enough signs for further calculations, two zeros must be added to the current number on the left.
As a result, we get the answer: √1308.1912 ≈ 36.1689. If you check the action with a calculator, you can make sure that all the characters were determined correctly.
Bitwise calculation of the square root value
The method is highly accurate. In addition, it is quite understandable and it does not require memorizing formulas or a complex algorithm of actions, since the essence of the method is to select the correct result.
Let's extract the root from the number 781. Let's consider in detail the sequence of actions.
- Find out which digit of the square root value will be the highest. To do this, let's square 0, 10, 100, 1000, etc. and find out between which of them the root number is located. We get that 10²< 781 < 100², т. е. старшим разрядом будут десятки.
- Let's take the value of tens. To do this, we will take turns raising to the power of 10, 20, ..., 90, until we get a number greater than 781. In our case, we get 10² = 100, 20² = 400, 30² = 900. The value of the result n will be within 20< n <30.
- Similarly to the previous step, the value of the units digit is selected. We alternately square 21.22, ..., 29: 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784. We get that 27< n < 28.
- Each subsequent digit (tenths, hundredths, etc.) is calculated in the same way as shown above. Calculations are carried out until the required accuracy is achieved.
Video
From the video you will learn how to extract square roots without using a calculator.