How to extract the root of an unextractable number. How to find the square root of a number manually

Bibliographic description: Pryastanov S. M., Lysogorova L. V. Extraction methods square root// Young scientist. 2017. No. 2.2. P. 76-77..02.2019).



Keywords : square root, square root extraction.

In mathematics lessons, I became acquainted with the concept of a square root, and the operation of extracting a square root. I became interested in whether extracting the square root is possible only using a table of squares, using a calculator, or is there a way to extract it manually. I found several ways: formula Ancient Babylon, through solving equations, the method of discarding full square, Newton's method, geometric method, graphic method(, ), method of selection by guessing, method of deduction of an odd number.

Consider the following methods:

Let's decompose into prime factors, using the divisibility criteria 27225=5*5*3*3*11*11. Thus

1. TO Canadian method. This quick method was discovered by young scientists at one of Canada's leading universities in the 20th century. Its accuracy is no more than two to three decimal places.

where x is the number from which the root must be extracted, c is the number of the nearest square), for example:

=5,92

1. In a column. This method allows you to find the approximate value of the root of any real number with any predetermined accuracy. The disadvantages of this method include the increasing complexity of the calculation as the number of digits found increases. To manually extract the root, a notation similar to long division is used

Square Root Algorithm

1. We divide the fractional part and the integer part separately from the comma on the verge of two digits in each face ( kiss part - from right to left; fractional- from left to right). It is possible that the integer part may contain one digit, and the fractional part may contain zeros.

2. Extraction starts from left to right, and we select a number whose square does not exceed the number in the first face. We square this number and write it under the number on the first side.

3. Find the difference between the number on the first face and the square of the selected first number.

4. We add the next edge to the resulting difference, the resulting number will be divisible. Let's educate divider. We double the first selected digit of the answer (multiply by 2), we get the number of tens of the divisor, and the number of units should be such that its product by the entire divisor does not exceed the dividend. We write down the selected number as an answer.

5. We take the next edge to the resulting difference and perform the actions according to the algorithm. If this face turns out to be a face of a fractional part, then we put a comma in the answer. (Fig. 1.)

Using this method, you can extract numbers with different precisions, for example, up to thousandths. (Fig.2)

Considering various ways extracting the square root, we can conclude: in each specific case, you need to decide on the choice of the most effective one in order to spend less time solving

Literature:

1. Kiselev A. Elements of algebra and analysis. Part one.-M.-1928

Key words: square root, square root.

Annotation: The article describes methods for extracting square roots and provides examples of extracting roots.

In mathematics, the question of how to extract a root is considered relatively simple. If we square numbers from the natural series: 1, 2, 3, 4, 5...n, then we get the following series of squares: 1, 4, 9, 16...n 2. The row of squares is infinite, and if you look closely at it, you will see that there are not very many integers in it. We'll explain why this is a little later.

Root of a number: calculation rules and examples

So, we squared the number 2, that is, multiplied it by itself and got 4. How to extract the root of the number 4? Let's say right away that the roots can be square, cubic and any degree to infinity.

Root degree – always natural number, that is, it is impossible to solve such an equation: a root to the power of 3.6 of n.

Square root

Let's return to the question of how to extract the square root of 4. Since we squared the number 2, we will also extract the square root. In order to correctly extract the root of 4, you just need to choose the right number that, when squared, would give the number 4. And this, of course, is 2. Look at the example:

• 2 2 =4
• Root of 4 = 2

This example is quite simple. Let's try to extract the square root of 64. What number when multiplied by itself gives 64? Obviously it's 8.

• 8 2 =64
• Root of 64=8

Cube root

As was said above, roots are not only square, using an example we will try to explain more clearly how to extract cube root or third root. The principle of extracting a cube root is the same as that of a square root, the only difference is that the required number was initially multiplied by itself not once, but twice. That is, let's say we took the following example:

• 3x3x3=27
• Naturally, the cube root of 27 is three:
• Root 3 of 27 = 3

Let's say you need to find the cube root of 64. To solve this equation, it is enough to find a number that, when raised to the third power, would give 64.

• 4 3 =64
• Root 3 of 64 = 4

Extract the root of a number on a calculator

Of course, it is best to learn to extract square, cube and other roots through practice, by solving many examples and memorizing tables of squares and cubes of small numbers. In the future, this will greatly facilitate and reduce the time required to solve equations. Although, it should be noted that sometimes it is necessary to extract the root of such a large number that it is impossible to find correct number, squared, will cost a lot great work, if at all possible. A regular calculator will come to the rescue in extracting the square root. How to extract the root on a calculator? Very simply enter the number from which you want to find the result. Now take a close look at the calculator buttons. Even the simplest of them has a key with a root icon. By clicking on it, you will immediately get the finished result.

Not every number can be extracted whole root, consider the following example:

Root of 1859 = 43.116122…

You can simultaneously try to solve this example on a calculator. As you can see, the resulting number is not an integer; moreover, the set of digits after the decimal point is not finite. A more accurate result can be given by special engineering calculators, but the full result simply does not fit on a regular display. And if you continue the series of squares that you started earlier, you will not find the number 1859 in it precisely because the number that was squared to obtain it is not an integer.

If you need to extract the third root on a simple calculator, then you need to double-click on the button with the root sign. For example, take the number 1859 used above and take the cube root from it:

Root 3 of 1859 = 6.5662867…

That is, if the number 6.5662867... is raised to the third power, then we get approximately 1859. Thus, extracting roots from numbers is not difficult, you just need to remember the above algorithms.

When deciding various tasks In mathematics and physics courses, 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 technology It won’t be difficult to solve this problem using a calculator. However, situations arise when it is impossible to use the electronic assistant.

For example, many exams do not allow you to bring electronics. In addition, you may not have a calculator at hand. In such cases, it is useful to know at least some methods for calculating radicals manually.

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. The rows of the table contain the values ​​of tens, and the columns contain the values ​​of units. The cell at the intersection of a row and a column contains a square double 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 will 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 use the values ​​of the column and row to determine the answer. As an example, consider calculating the square root of 169.

We find a cell with this number in the table, horizontally we determine tens - 1, vertically we find units - 3. Answer: √169 = 13.

Similarly, you can calculate cube and nth roots 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 in the range from 100 to 9801). Moreover, it will not work if given number not in the table.

Prime factorization

If the table of squares is not at hand or it turned out to be impossible to find the root with its help, you can try factor the number under the root into prime factors. Prime factors are those that can be completely (without remainder) divisible only by themselves or by one. Examples could be 2, 3, 5, 7, 11, 13, etc.

Let's look at calculating the root using √576 as an example. Let's break it down into prime factors. We get the following result: √576 = √(2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3) = √(2 ∙ 2 ∙ 2)² ∙ √3². Using the basic property of roots √a² = a, we will get rid of roots and squares, and then calculate the answer: 2 ∙ 2 ∙ 2 ∙ 3 ​​= 24.

What to do if any of the multipliers does not have its own pair? For example, consider the calculation of √54. After factorization, we obtain the result in the following form: √54 = √(2 ∙ 3 ​​∙ 3 ∙ 3) = √3² ∙ √(2 ∙ 3) = 3√6. The non-removable part can be left under the root. For most geometry and algebra problems, this will count as the final answer. But if there is a need to calculate approximate values, you can use methods that will be discussed below.

Heron's method

What to do when you need to at least approximately know what the extracted root is equal to (if it is impossible to obtain an integer value)? A quick and fairly accurate result is obtained by using the Heron method. Its essence is to use an approximate formula:

√R = √a + (R - a) / 2√a,

where R is the number whose root needs to be calculated, a is the nearest number whose root value is known.

Let's look at how the method works in practice and evaluate how accurate it is. Let's calculate what √111 is equal to. The number closest to 111, the root of which is known, is 121. Thus, R = 111, a = 121. Substitute the values ​​into 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 previously described steps:

√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 re-applying the formula, the error became completely insignificant.

Calculating the root by long division

This method of finding the square root value is a little more complex 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 accurate to 4 decimal places. Let's look at the calculation algorithm using an example any number 1308,1912.

1. 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. Let's write the number on the left side, dividing it into groups of 2 digits, moving to the right and left side from comma. The very first digit on the left may be without a pair. If the sign is missing on the right side of the number, then you should add 0. In our case, the result will be 13 08.19 12.
2. Let's choose the best large number, the square of which will be less than or equal to the first group of digits. In our case it is 3. Let's write it on the top right; 3 is the first digit of the result. On the bottom right we indicate 3×3 = 9; this will be needed for subsequent calculations. From 13 in the column we subtract 9, we get a remainder of 4.
3. Let's assign the next pair of numbers to remainder 4; we get 408.
4. Multiply the number at the top right by 2 and write it down at the bottom right, adding _ x _ = to it. We get 6_ x _ =.
5. Instead of dashes, you need to substitute the same number, less than or equal to 408. We get 66 × 6 = 396. We write 6 from the top right, since this is the second digit of the result. Subtract 396 from 408, we get 12.
6. Let's repeat steps 3-6. Since the digits moved down are in the fractional part of the number, it is necessary to put decimal point on the top right after 6. Let's write down the double result with dashes: 72_ x _ =. A suitable number would be 1: 721×1 = 721. Let's write it down as the answer. Let's subtract 1219 - 721 = 498.
7. Let's perform the sequence of actions given in the previous paragraph three more times to get required quantity decimal places. If there are not enough characters for further calculations, you need to add two zeros to the current number on the left.

As a result, we get the answer: √1308.1912 ≈ 36.1689. If you check the action using a calculator, you can make sure that all signs were identified correctly.

Bitwise square root calculation

The method is highly accurate. In addition, it is quite understandable and does not require memorizing formulas or complex algorithm actions, since the essence of the method is to select the right result.

Let's extract the root of the number 781. Let's look at the sequence of actions in detail.

1. Let's find out which digit of the square root value will be the most significant. To do this, let’s square 0, 10, 100, 1000, etc. and find out which of them is between radical number. We get that 10²< 781 < 100², т. е. старшим разрядом будут десятки.
2. Let's choose 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. For our case, we get 10² = 100, 20² = 400, 30² = 900. The value of the result n will be within 20< n <30.
3. Similar to the previous step, the value of the units digit is selected. Let's square 21.22, ..., 29 one by one: 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784. We get that 27< n < 28.
4. 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.

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 begin to re-solve the entire example. Under no circumstances should you do this! There are two reasons for this:

1. Roots of large numbers do appear in problems. Especially in text ones;
2. There is an algorithm by which these roots are calculated almost orally.

We will consider this algorithm today. Perhaps some things will seem incomprehensible to you. But if you pay attention to this lesson, you will receive a powerful weapon against square roots.

So, the algorithm:

1. Limit the required root above and below to numbers that are multiples of 10. Thus, we will reduce the search range to 10 numbers;
2. From these 10 numbers, weed out those that definitely cannot be roots. As a result, 1-2 numbers will remain;
3. Square these 1-2 numbers. The one whose square is equal to the original number will be the root.

Before putting this algorithm into practice, let's look at each individual step.

Root limitation

First of all, we need to find out between which numbers our root is located. It is highly desirable that the numbers be multiples 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 tell 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:

[Caption for the picture]

The same thing applies to any other number from which you can find the square root. For example, 3364:

Thus, instead of an incomprehensible number, we get a very specific range in which the original root lies. To further narrow the search area, move on to the second step.

Eliminating obviously unnecessary numbers

So, we have 10 numbers - candidates for the root. We got them very quickly, without complex thinking and multiplication in a column. It's time to move on.

Believe it or not, we will now reduce the number of candidate numbers to two - again without any complicated 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, just 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 come in last place. Let's try to find out what they turn into when squared. Take a look at the table:

This table is another step towards calculating the root. As you can see, the numbers in the second line turned out to be symmetrical relative 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. This means that, for example, the root of 3364 must end in 2 or 8. On the other hand, we remember the restriction from the previous paragraph. We get:

Red squares indicate that we do not yet know this figure. But the root lies in the range from 50 to 60, on which there are only two numbers ending in 2 and 8:

That's it! 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 there will be only one candidate for the roots!

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 gives the original number will be the root.

For example, for the number 3364 we found two candidate numbers: 52 and 58. Let's square them:

52 2 = (50 +2) 2 = 2500 + 2 50 2 + 4 = 2704;
58 2 = (60 − 2) 2 = 3600 − 2 60 2 + 4 = 3364.

That's it! It turned out that the root is 58! At the same time, to simplify the calculations, I used the formula for the squares of the sum and difference. Thanks to this, I didn’t even have to multiply the numbers into a column! This is another level of optimization of calculations, but, of course, completely optional :)

Examples of calculating roots

Theory is, of course, good. But let's check it in practice.

[Caption for the picture]

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:

All that remains is to square each number and compare it 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:

[Caption for the picture]

900 < 1369 < 1600;
30 2 < 1369 < 40 2;

Let's look at the last digit:

1369 → 9;
33; 37.

Square it:

33 2 = (30 + 3) 2 = 900 + 2 30 3 + 9 = 1089 ≠ 1369;
37 2 = (40 − 3) 2 = 1600 − 2 40 3 + 9 = 1369.

Here is the answer: 37.

Task. Calculate the square root:

[Caption for the picture]

We limit the number:

2500 < 2704 < 3600;
50 2 < 2704 < 60 2;

Let's look at the last digit:

2704 → 4;
52; 58.

Square it:

52 2 = (50 + 2) 2 = 2500 + 2 50 2 + 4 = 2704;

We received the answer: 52. The second number will no longer need to be squared.

Task. Calculate the square root:

[Caption for the picture]

We limit the number:

3600 < 4225 < 4900;
60 2 < 4225 < 70 2;

Let's look at the last digit:

4225 → 5;
65.

As you can see, after the second step there is only one option left: 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 look at the reasons. There are two of them:

• In any normal mathematics exam, be it the State Examination or the Unified State Exam, the use of calculators is prohibited. And if you bring a calculator into class, you can easily be kicked out of the exam.
• Don't be like stupid Americans. Which are not like roots - they cannot add two prime numbers. And when they see fractions, they generally become hysterical.

How to extract the root from the number. In this article we will learn how to take the square root of four and five digit numbers.

Let's take the square root of 1936 as an example.

Hence, .

The last digit in the number 1936 is the number 6. The square of the number 4 and the number 6 ends at 6. Therefore, 1936 can be the square of the number 44 or the number 46. It remains to check using multiplication.

Means,

Let's take the square root of the number 15129.

Hence, .

The last digit in the number 15129 is the number 9. The square of the number 3 and the number 7 ends at 9. Therefore, 15129 can be the square of the number 123 or the number 127. Let's check using multiplication.

Means,

How to extract the root - video

And now I suggest you watch Anna Denisova’s video - "How to extract the root ", author of the site" Simple physics", in which she explains how to find square and cube roots without a calculator.

The video discusses several ways to extract roots:

1. The easiest way to extract the square root.

2. By selection using the square of the sum.

3. Babylonian method.

4. Method of extracting the square root of a column.

5. A quick way to extract the cube root.

6. Method of extracting the cube root in a column.