Definition of square root. How to quickly extract square roots

Before the advent of calculators, students and teachers calculated square roots by hand. There are several ways to manually calculate the square root of a number. Some of them offer only an approximate solution, others give an exact answer.

Steps

Prime factorization

Factor the root number into factors that are square numbers. Depending on the root number, you will get an approximate or exact answer. Square numbers - numbers from which you can extract an integer Square root. Factors are numbers that, when multiplied, give the original number. For example, the factors of the number 8 are 2 and 4, since 2 x 4 = 8, the numbers 25, 36, 49 are square numbers, since √25 = 5, √36 = 6, √49 = 7. Square factors are factors , which are square numbers. First, try to factorize the root number into square factors.

• For example, calculate the square root of 400 (manually). First try factoring 400 into square factors. 400 is a multiple of 100, that is, divisible by 25 - this is a square number. Dividing 400 by 25 gives you 16. The number 16 is also a square number. Thus, 400 can be factored into square factors of 25 and 16, that is, 25 x 16 = 400.
• This can be written as follows: √400 = √(25 x 16).

1. Square root of the product of some terms is equal to the product square roots from each term, i.e. √(a x b) = √a x √b. Use this rule and take the square root of each square factor and multiply the results to find the answer.

   • In our example, take the square root of 25 and 16.
     • √(25 x 16)
     • √25 x √16
     • 5 x 4 = 20

2. If the radical number does not factor into two square factors (and it does in most cases), you will not be able to find the exact answer as an integer. But you can simplify the problem by decomposing the root number into a square factor and an ordinary factor (a number from which the whole square root cannot be taken). Then you will take the square root of the square factor and you will take the root of the ordinary factor.

   • For example, calculate the square root of the number 147. The number 147 cannot be factored into two square factors, but it can be factored into the following factors: 49 and 3. Solve the problem as follows:
     • = √(49 x 3)
     • = √49 x √3
     • = 7√3

3. If necessary, evaluate the value of the root. Now you can evaluate the value of the root (find an approximate value) by comparing it with the values ​​​​of the roots of square numbers that are closest (on both sides of the number line) to the root number. You will get the value of the root as a decimal fraction, which must be multiplied by the number behind the root sign.

   • Let's go back to our example. The root number is 3. The nearest square numbers to it are the numbers 1 (√1 = 1) and 4 (√4 = 2). Thus, the value of √3 lies between 1 and 2. Since the value of √3 is probably closer to 2 than to 1, our estimate is: √3 = 1.7. We multiply this value by the number at the root sign: 7 x 1.7 \u003d 11.9. If you do the calculations on a calculator, you get 12.13, which is pretty close to our answer.
     • This method also works with large numbers. For example, consider √35. The root number is 35. The nearest square numbers to it are the numbers 25 (√25 = 5) and 36 (√36 = 6). Thus, the value of √35 lies between 5 and 6. Since the value of √35 is much closer to 6 than it is to 5 (because 35 is only 1 less than 36), we can state that √35 is slightly less than 6. Checking with a calculator gives us the answer 5.92 - we were right.

4. Another way is to decompose the root number into prime factors. Prime factors are numbers that are only divisible by 1 and themselves. Write the prime factors in a row and find pairs of identical factors. Such factors can be taken out of the sign of the root.

   • For example, calculate the square root of 45. We decompose the root number into prime factors: 45 \u003d 9 x 5, and 9 \u003d 3 x 3. Thus, √45 \u003d √ (3 x 3 x 5). 3 can be taken out of the root sign: √45 = 3√5. Now we can estimate √5.
   • Consider another example: √88.
     • = √(2 x 44)
     • = √ (2 x 4 x 11)
     • = √ (2 x 2 x 2 x 11). You got three multiplier 2s; take a couple of them and take them out of the sign of the root.
     • = 2√(2 x 11) = 2√2 x √11. Now we can evaluate √2 and √11 and find an approximate answer.

   Calculating the square root manually

   Using column division

   1. This method involves a process similar to long division and gives an accurate answer. First, draw a vertical line dividing the sheet into two halves, and then to the right and a little lower top edge sheet to the vertical line, draw a horizontal line. Now divide the root number into pairs of numbers, starting with the fractional part after the decimal point. So, the number 79520789182.47897 is written as "7 95 20 78 91 82, 47 89 70".

      • For example, let's calculate the square root of the number 780.14. Draw two lines (as shown in the picture) and write the number in the top left as "7 80, 14". It is normal that the first digit from the left is an unpaired digit. Answer (root of given number) will be written on the top right.

   2. Given the first pair of numbers (or one number) from the left, find the largest integer n whose square is less than or equal to the pair of numbers (or one number) in question. In other words, find the square number that is closest to, but less than, the first pair of numbers (or single number) from the left, and take the square root of that square number; you will get the number n. Write the found n at the top right, and write down the square n at the bottom right.

      • In our case, the first number on the left will be the number 7. Next, 4< 7, то есть 2 2 < 7 и n = 2. Напишите 2 сверху справа - это первая цифра в искомом квадратном корне. Напишите 2×2=4 справа снизу; вам понадобится это число для последующих вычислений.

   3. Subtract the square of the number n you just found from the first pair of numbers (or one number) from the left. Write the result of the calculation under the subtrahend (the square of the number n).

      • In our example, subtract 4 from 7 to get 3.

   4. Take down the second pair of numbers and write it down next to the value obtained in the previous step. Then double the number at the top right and write the result at the bottom right with "_×_=" appended.

      • In our example, the second pair of numbers is "80". Write "80" after the 3. Then, doubling the number from the top right gives 4. Write "4_×_=" from the bottom right.

   5. Fill in the blanks on the right.

      • In our case, if instead of dashes we put the number 8, then 48 x 8 \u003d 384, which is more than 380. Therefore, 8 is too large a number, but 7 is fine. Write 7 instead of dashes and get: 47 x 7 \u003d 329. Write 7 from the top right - this is the second digit in the desired square root of the number 780.14.

   6. Subtract the resulting number from the current number on the left. Write the result from the previous step below the current number on the left, find the difference and write it below the subtracted one.

      • In our example, subtract 329 from 380, which equals 51.

   7. Repeat step 4. If the demolished pair of numbers is the fractional part of the original number, then put the separator (comma) of the integer and fractional parts in the desired square root from the top right. On the left, carry down the next pair of numbers. Double the number at the top right and write the result at the bottom right with "_×_=" appended.

      • In our example, the next pair of numbers to be demolished will be the fractional part of the number 780.14, so put the separator of the integer and fractional parts in the desired square root from the top right. Demolish 14 and write down at the bottom left. Double the top right (27) is 54, so write "54_×_=" at the bottom right.

   8. Repeat steps 5 and 6. Find it largest number in place of dashes on the right (instead of dashes, you need to substitute the same number) so that the multiplication result is less than or equal to the current number on the left.

      • In our example, 549 x 9 = 4941, which is less than the current number on the left (5114). Write 9 on the top right and subtract the result of the multiplication from the current number on the left: 5114 - 4941 = 173.

   9. If you need to find more decimal places for the square root, write a pair of zeros next to the current number on the left and repeat steps 4, 5 and 6. Repeat steps until you get the accuracy of the answer you need (number of decimal places).

      Understanding the process

      1. To master this method, imagine the number whose square root you need to find as the area of ​​​​the square S. In this case, you will look for the length of the side L of such a square. Calculate the value of L for which L² = S.

         Enter a letter for each digit in your answer. Denote by A the first digit in the value of L (the desired square root). B will be the second digit, C the third and so on.

         Specify a letter for each pair of leading digits. Denote by S a the first pair of digits in the value S, by S b the second pair of digits, and so on.

         Explain the connection of this method with long division. As in the division operation, where each time we are only interested in one next digit of the divisible number, when calculating the square root, we work with a pair of digits in sequence (to obtain the next one digit in the square root value).

      2. Consider the first pair of digits Sa of the number S (Sa = 7 in our example) and find its square root. In this case, the first digit A of the sought value of the square root will be such a digit, the square of which is less than or equal to S a (that is, we are looking for such an A that satisfies the inequality A² ≤ Sa< (A+1)²). В нашем примере, S1 = 7, и 2² ≤ 7 < 3²; таким образом A = 2.

         • Let's say we need to divide 88962 by 7; here the first step will be similar: we consider the first digit of the divisible number 88962 (8) and select the largest number that, when multiplied by 7, gives a value less than or equal to 8. That is, we are looking for a number d for which the inequality is true: 7 × d ≤ 8< 7×(d+1). В этом случае d будет равно 1.

      3. Mentally imagine the square whose area you need to calculate. You are looking for L, that is, the length of the side of a square whose area is S. A, B, C are numbers in the number L. You can write it differently: 10A + B \u003d L (for a two-digit number) or 100A + 10B + C \u003d L (for three-digit number) and so on.

         • Let (10A+B)² = L² = S = 100A² + 2×10A×B + B². Remember that 10A+B is a number whose B stands for ones and A stands for tens. For example, if A=1 and B=2, then 10A+B equals the number 12. (10A+B)² is the area of ​​the whole square, 100A² is the area of ​​the large inner square, B² is the area of ​​the small inner square, 10A×B is the area of ​​each of the two rectangles. Adding the areas of the figures described, you will find the area of ​​the original square.

Students always ask: “Why can't I use a calculator on a math exam? How to extract the square root of a number without a calculator? Let's try to answer this question.

How to extract the square root of a number without the help of a calculator?

Action square root extraction the opposite of squaring.

√81= 9 9 2 =81

If we take the square root of a positive number and square the result, we get the same number.

From small numbers that are perfect squares natural numbers, for example 1, 4, 9, 16, 25, ..., 100 square roots can be extracted verbally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract the square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400, you can extract using the selection method using some tips. Let's try an example to consider this method.

Example: Extract the root of the number 676.

We notice that 20 2 \u003d 400, and 30 2 \u003d 900, which means 20< √676 < 900.

Exact squares of natural numbers end in 0; 1; 4; 5; 6; 9.
The number 6 is given by 4 2 and 6 2 .
So, if the root is taken from 676, then it is either 24 or 26.

It remains to check: 24 2 = 576, 26 2 = 676.

Answer: √676 = 26 .

More example: √6889 .

Since 80 2 \u003d 6400, and 90 2 \u003d 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is either 83 or 87.

Check: 83 2 = 6889.

Answer: √6889 = 83 .

If you find it difficult to solve by the selection method, then you can factorize the root expression.

For example, find √893025.

Let's factorize the number 893025, remember, you did it in the sixth grade.

We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.

More example: √20736. Let's factorize the number 20736:

We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.

Of course, factoring requires knowledge of divisibility criteria and factoring skills.

And finally, there is square root rule. Let's look at this rule with an example.

Calculate √279841.

To extract the root of a multi-digit integer, we split it from right to left into faces containing 2 digits each (there may be one digit in the left extreme face). Write like this 27'98'41

To get the first digit of the root (5), we extract the square root of the largest exact square contained in the first left face (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is attributed (demolished) to the difference.
To the left of the resulting number 298, they write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), experience the quotient (102 ∙ 2 = 204 should not be more than 298) and write (2) after the first digit of the root.
Then the resulting quotient 204 is subtracted from 298, and the next facet (41) is attributed (demolished) to the difference (94).
To the left of the resulting number 9441, they write the double product of the digits of the root (52 ∙ 2 = 104), divide by this product the number of all tens of the number 9441 (944/104 ≈ 9), experience the quotient (1049 ∙ 9 = 9441) should be 9441 and write it down (9) after the second digit of the root.

We got the answer √279841 = 529.

Similarly extract roots of decimals. Only the radical number must be divided into faces so that the comma is between the faces.

Example. Find the value √0.00956484.

You just have to remember that if decimal has an odd number of decimal places, it does not take exactly the square root.

So, now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn how to solve problems, you need to solve them. And if you have any questions, sign up for my lessons.

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Among the many knowledge that is a sign of literacy, the alphabet is in the first place. The next, the same "sign" element, are the skills of addition-multiplication and, adjacent to them, but reverse in meaning, arithmetic operations of subtraction-division. The skills learned in distant school childhood serve faithfully day and night: TV, newspaper, SMS, And everywhere we read, write, count, add, subtract, multiply. And, tell me, have you often had to take roots in life, except in the country? For example, such an entertaining problem, like, the square root of the number 12345 ... Is there still gunpowder in the powder flasks? Can we do it? Yes, there is nothing easier! Where is my calculator ... And without it, hand-to-hand, weak?

First, let's clarify what it is - the square root of a number. Generally speaking, "to extract a root from a number" means to perform the arithmetic operation opposite to raising to a power - here you have the unity of opposites in life application. let's say a square is a multiplication of a number by itself, i.e., as they taught at school, X * X = A or in another notation X2 = A, and in words - “X squared equals A”. Then the inverse problem sounds like this: the square root of the number A, is the number X, which, when squared, is equal to A.

Extracting the square root

From school course arithmetic knows methods of calculating "in a column", which help to perform any calculations using the first four arithmetic operations. Alas ... For square, and not only square, roots of such algorithms do not exist. And in this case, how to extract the square root without a calculator? Based on the definition of the square root, there is only one conclusion - it is necessary to select the value of the result by sequential enumeration of numbers, the square of which approaches the value of the root expression. Only and everything! Before an hour or two has passed, how can you calculate using well well-known trick multiplication into a "column", any square root. If you have the skills, a couple of minutes is enough for this. Even a not quite advanced calculator or PC user does it in one fell swoop - progress.

But seriously, the calculation of the square root is often performed using the “artillery fork” technique: first, they take a number whose square approximately corresponds to the root expression. It is better if "our square" is slightly less than this expression. Then they correct the number according to their own skill-understanding, for example, multiply by two, and ... square it again. If the result more number under the root, sequentially adjusting the original number, gradually approaching its "colleague" under the root. As you can see - no calculator, only the ability to count "in a column". Of course, there are many scientifically reasoned and optimized square root algorithms, but for " home use» The above technique gives 100% confidence in the result.

Yes, I almost forgot, in order to confirm our increased literacy, we calculate the square root of the previously indicated number 12345. We do it step by step:

1. Take, purely intuitively, X=100. Let's calculate: X * X = 10000. Intuition is on top - the result is less than 12345.

2. Let's try, also purely intuitively, X = 120. Then: X * X = 14400. And again, with intuition, the order - the result is more than 12345.

3. Above, a “fork” of 100 and 120 is obtained. Let's choose new numbers - 110 and 115. We get, respectively, 12100 and 13225 - the fork narrows.

4. We try on "maybe" X = 111. We get X * X = 12321. This number is already quite close to 12345. In accordance with the required accuracy, the “fitting” can be continued or stopped at the result obtained. That's all. As promised - everything is very simple and without a calculator.

Quite a bit of history...

Even the Pythagoreans, students of the school and followers of Pythagoras, thought of using square roots, 800 BC. and right there, "ran into" new discoveries in the field of numbers. And where did it come from?

1. The solution of the problem with the extraction of the root, gives the result in the form of numbers of a new class. They were called irrational, in other words, "unreasonable", because. they are not written as a complete number. The most classic example of this kind is the square root of 2. This case corresponds to the calculation of the diagonal of a square with a side equal to 1 - here it is, the influence of the Pythagorean school. It turned out that in a triangle with a very specific unit size of the sides, the hypotenuse has a size that is expressed by a number that "has no end." So in mathematics appeared

2. It is known that It turned out that this mathematical operation contains one more catch - extracting the root, we do not know what square of which number, positive or negative, is the root expression. This uncertainty, the double result from one operation, is written down.

The study of the problems associated with this phenomenon has become a direction in mathematics called the theory of a complex variable, which has a large practical value in mathematical physics.

It is curious that the root designation - radical - was used in his "Universal Arithmetic" by the same ubiquitous I. Newton, but exactly modern look The root record has been known since 1690 from the book of the Frenchman Roll "Guide to Algebra".

The non-negative square root of a positive number is called arithmetic square root and is denoted using the radical sign.

Complex numbers

Over the field of complex numbers there are always two solutions, differing only in sign (except for the square root of zero). The root of a complex number is often denoted as , but this notation must be used with care. Common mistake:

To extract the square root of a complex number, it is convenient to use the exponential notation of a complex number: if

where the root of the modulo is understood in the sense of an arithmetic value, and k can take on the values ​​k=0 and k=1, so the result is two different results in the answer.

Generalizations

Square roots are introduced as solutions to equations of the form and for other objects: matrices, functions, operators, etc. In this case, quite arbitrary multiplicative operations can be used, for example, superposition.

Square root in computer science

In many functional-level programming languages ​​(as well as markup languages ​​like LaTeX), the square root function is denoted as sqrt(from English. square root"Square root").

Algorithms for finding the square root

Finding or calculating the square root of a given number is called extraction(square) root.

Taylor series expansion

Arithmetic square root

For squares of numbers, the following equalities are true:

That is, you can find out the integer part of the square root of a number by subtracting from it all odd numbers in order, until the remainder is less than the next subtracted number or equal to zero, and counting the number of actions performed. For example, like this:

Performed 3 steps, the square root of 9 is 3.

The disadvantage of this method is that if the extracted root is not an integer, then you can find out only its integer part, but not more accurately. At the same time, this method is quite accessible to children who solve the simplest problems. math problems, requiring the extraction of the square root.

Rough estimate

Many algorithms for calculating the square roots of a positive real number S require some initial value. If the initial value is too far from the real value of the root, the calculations slow down. Therefore, it is useful to have a rough estimate that can be very inaccurate but is easy to calculate. If S≥ 1, let D will be the number of digits S to the left of the decimal point. If S < 1, пусть D will be the number of consecutive zeros to the right of the decimal point, taken with a minus sign. Then a rough estimate looks like this:

If D odd, D = 2n+ 1, then we use If D even, D = 2n+ 2, then we use

Two and six are used because And

When working in a binary system (like inside computers), a different estimate should be used (here D is the number of binary digits).

Geometric square root

To manually extract the root, a notation similar to column division is used. The number whose root we are looking for is written out. To the right of it, we will gradually get the numbers of the desired root. Let the root be extracted from a number with a finite number of decimal places. To begin with, mentally or with labels, we divide the number N into groups of two digits to the left and right of the decimal point. If necessary, groups are padded with zeros - the integer part is padded on the left, the fractional on the right. So 31234.567 can be represented as 03 12 34 . 56 70. Unlike division, demolition is carried out in such groups of 2 digits.

Visual description of the algorithm: