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How to find the root of a large number. Square root

Chapter one.

Finding the largest integer square root from a given integer.

170. Preliminary remarks.

A) Since we will talk about extracting only the square root, to shorten the speech in this chapter, instead of “square” root we will say simply “root”.

b) If we square the numbers of the natural series: 1,2,3,4,5. . . , then we get the following table of squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,121,144. .,

Obviously, there are a lot of integers that are not in this table; Of course, it is impossible to extract the whole root from such numbers. Therefore, if you need to extract the root of any integer, for example. required to find √4082, then we agree to understand this requirement as follows: extract the whole root of 4082, if possible; if it is not possible, then we must find the largest integer whose square is 4082 (such a number is 63, since 63 2 = 3969, and 64 2 = 4090).

V) If this number is less than 100, then the root of it is found using the multiplication table; Thus, √60 would be 7, since seven 7 equals 49, which is less than 60, and eight 8 equals 64, which is greater than 60.

171. Extracting the root of a number less than 10,000 but greater than 100. Let's say we need to find √4082. Since this number is less than 10,000, its root is less than √l0,000 = 100. On the other hand, this number is greater than 100; this means that the root of it is greater than (or equal to 10). (If, for example, it was necessary to find √ 120 , then although the number 120 > 100, however √ 120 is equal to 10, because 11 2 = 121.) But every number that is greater than 10 but less than 100 has 2 digits; This means that the required root is the sum:

tens + ones,

and therefore its square must equal the sum:

This sum must be the greatest square of 4082.

Let's take the largest of them, 36, and assume that the square of the tens root will be equal to exactly this largest square. Then the number of tens in the root must be 6. Let us now check that this should always be the case, i.e., the number of tens in the root is always equal to the largest integer root of the number of hundreds of the radical.

Indeed, in our example, the number of tens of the root cannot be more than 6, since (7 dec.) 2 = 49 hundreds, which exceeds 4082. But it cannot be less than 6, since 5 dec. (with units) is less than 6 des., and meanwhile (6 des.) 2 = 36 hundreds, which is less than 4082. And since we are looking for the largest whole root, we should not take 5 des for the root, when even 6 tens are not many.

So, we have found the number of tens of the root, namely 6. We write this number to the right of the = sign, remembering that it means tens of the root. Raising it by the square, we get 36 hundreds. We subtract these 36 hundreds from the 40 hundreds of the radical number and subtract the remaining two digits of this number. The remainder 482 must contain 2 (6 dec.) (units) + (units)2. The product (6 dec.) (units) must be tens; therefore, the double product of tens by ones must be sought in the tens of the remainder, i.e., in 48 (we get their number by separating one digit on the right in the remainder of 48 "2). The doubled tens of the root make up 12. This means that if we multiply 12 by the units of the root ( which are still unknown), then we should get the number contained in 48. Therefore, we divide 48 by 12.

To do this, draw a vertical line to the left of the remainder and behind it (stepping back from the line one place to the left for the purpose that will now appear) we write double the first digit of the root, i.e. 12, and divide 48 by it. In the quotient we get 4.

However, we cannot guarantee in advance that the number 4 can be taken as units of the root, since we have now divided by 12 the entire number of tens of the remainder, while some of them may not belong to the double product of tens by units, but are part of the square of units. Therefore, the number 4 may be large. We need to try it out. It is obviously suitable if the sum 2 (6 dec.) 4 + 4 2 is no more than the remainder 482.

As a result, we get the sum of both at once. The resulting product turned out to be 496, which is greater than the remainder 482; That means number 4 is big. Then let's test the next smaller number 3 in the same way.

Examples.

In example 4, when dividing the 47 tens of the remainder by 4, we get 11 as a quotient. But since the number of units of the root cannot be a two-digit number 11 or 10, we must directly test the number 9.

In example 5, after subtracting 8 from the first face of the square, the remainder turns out to be 0, and the next face also consists of zeros. This shows that the desired root consists of only 8 tens, and therefore a zero must be put in place of the ones.

172. Extracting the root of a number greater than 10000. Let's say we need to find √35782. Since the radical number exceeds 10,000, the root of it is greater than √10000 = 100 and, therefore, it consists of 3 digits or more. No matter how many digits it consists of, we can always consider it as the sum of only tens and ones. If, for example, the root turns out to be 482, then we can count it as the amount of 48 des. + 2 units Then the square of the root will consist of 3 terms:

(dec.) 2 + 2 (dec.) (unit) + (unit) 2 .

Now we can reason in exactly the same way as when finding √4082 (in the previous paragraph). The only difference will be that to find the tens of the root of 4082 we had to extract the root of 40, and this could be done using the multiplication table; now, to obtain tens√35782, we will have to take the root of 357, which cannot be done using the multiplication table. But we can find √357 using the technique that was described in the previous paragraph, since the number 357< 10 000. Наибольший целый корень из 357 оказывается 18. Значит, в √3"57"82 должно быть 18 десятков. Чтобы найти единицы, надо из 3"57"82 вычесть квадрат 18 десятков, для чего достаточно вычесть квадрат 18 из 357 сотен и к остатку снести 2 последние цифры подкоренного числа. Остаток от вычитания квадpaта 18 из 357 у нас уже есть: это 33. Значит, для получения остатка от вычитания квадрата 18 дес. из 3"57"82, достаточно к 33 приписать справа цифры 82.

Next, we proceed as we did when finding √4082, namely: to the left of the remainder 3382 we draw a vertical line and behind it we write (stepping back one space from the line) twice the number of tens of the root found, i.e. 36 (twice 18). In the remainder, we separate one digit on the right and divide the number of tens of the remainder, i.e. 338, by 36. In the quotient we get 9. We test this number, for which we assign it to 36 on the right and multiply by it. The product turned out to be 3321, which is less than the remainder. This means that the number 9 is suitable, we write it at the root.

In general, to extract the square root of any integer, you must first extract the root of its hundreds; if this number is more than 100, then you will have to look for the root of the number of hundreds of these hundreds, that is, of the tens of thousands of this number; if this number is more than 100, you will have to take the root from the number of hundreds of tens of thousands, that is, from the millions of a given number, etc.

Examples.

IN last example, having found the first digit and subtracted its square, we get a remainder of 0. We take down the next 2 digits 51. Separating the tens, we get 5 des, while the double found digit of the root is 6. This means that from dividing 5 by 6 we get 0. We put in root 0 is in second place and add the next 2 digits to the remainder; we get 5110. Then we continue as usual.

In this example, the required root consists of only 9 hundreds, and therefore zeros must be placed in the places of tens and in places of ones.

Rule. To extract the square root of a given integer, divide it from right hand to the left, on the edge, there are 2 digits in each, except for the last one, which may contain one digit.
To find the first digit of the root, take the square root of the first face.
To find the second digit, the square of the first digit of the root is subtracted from the first face, the second face is taken to the remainder, and the number of tens of the resulting number is divided by double the first digit of the root; the resulting integer is tested.
This test is carried out like this: behind the vertical line (to the left of the remainder) write twice the previously found number of the root and to it, with right side, the tested digit is assigned, the resulting number is multiplied by the tested digit after this addition. If after multiplication the result is a number greater than the remainder, then the tested digit is not suitable and the next smaller digit must be tested.
The next digits of the root are found using the same technique.
If, after removing a face, the number of tens of the resulting number turns out to be less than the divisor, that is, less than twice the found part of the root, then they put 0 at the root, remove the next face and continue the action further.

173. Number of digits of the root. From the consideration of the process of finding the root, it follows that the root has as many digits as the radical number contains faces of 2 digits each (the left face may have one digit).

Chapter two.

Extracting confidants square roots from whole and fractional numbers .

For extracting the square root of polynomials, see the additions to the 2nd part of § 399 et seq.

174. Signs of an exact square root. The exact square root of a given number is a number whose square is exactly equal to the given number. Let us indicate some signs by which one can judge whether an exact root can be extracted from a given number or not:

A) If the exact whole root is not extracted from a given whole number (the remainder is obtained when extracting), then the fractional exact root cannot be found from such a number, since any fraction that is not equal to a whole number, when multiplied by itself, also produces a fraction in the product, not an integer.

b) Since the root of a fraction is equal to the root of the numerator divided by the root of the denominator, the exact root of an irreducible fraction cannot be found if it cannot be extracted from the numerator or the denominator. For example, it is impossible to extract the exact root from the fractions 4/5, 8/9 and 11/15, since in the first fraction it cannot be extracted from the denominator, in the second - from the numerator, and in the third - neither from the numerator nor from the denominator.

From numbers from which the exact root cannot be extracted, only approximate roots can be extracted.

175. Approximate root accurate to 1. An approximate square root, accurate to within 1, of a given number (integer or fractional, it doesn’t matter) is an integer that satisfies the following two requirements:

1) the square of this number is not greater than the given number; 2) but the square of this number increased by 1 is greater than this number. In other words, an approximate square root accurate to 1 is the largest integer square root of a given number, that is, the root that we learned to find in the previous chapter. This root is called approximate to within 1, because to obtain an exact root, we would have to add some fraction less than 1 to this approximate root, so if instead of the unknown exact root we take this approximate one, we will make an error less than 1.

Rule. To extract an approximate square root accurate to within 1, you need to extract the largest integer root of the integer part of the given number.

The number found by this rule is an approximate root with a disadvantage , since it lacks the exact root of a certain fraction (less than 1). If we increase this root by 1, we get another number in which there is some excess over the exact root, and this excess is less than 1. This root increased by 1 can also be called an approximate root with an accuracy of 1, but with an excess. (The names: “with deficiency” or “with excess” in some mathematical books are replaced by other equivalent ones: “by deficiency” or “by excess.”)

176. Approximate root with an accuracy of 1/10. Let's say we need to find √2.35104 with an accuracy of 1/10. This means that you need to find a decimal fraction that would consist of whole units and tenths and that would satisfy two following requirements:

1) the square of this fraction does not exceed 2.35104, but 2) if we increase it by 1/10, then the square of this increased fraction exceeds 2.35104.

To find such a fraction, we first find an approximate root accurate to 1, that is, we extract the root only from the integer 2. We get 1 (and the remainder is 1). We write the number 1 at the root and put a comma after it. Now we will look for the number of tenths. To do this, we take down to remainder 1 the digits 35 to the right of the decimal point, and continue the extraction as if we were extracting the root of the integer 235. We write the resulting number 5 in the root in the place of tenths. We don't need the remaining digits of the radical number (104). That the resulting number 1.5 will actually be an approximate root with an accuracy of 1/10 can be seen from the following. If we were to find the largest integer root of 235 with an accuracy of 1, we would get 15. So:

15 2 < 235, but 16 2 >235.

Dividing all these numbers by 100, we get:

This means that the number 1.5 is the decimal fraction that we called an approximate root with an accuracy of 1/10.

Using this technique, we can also find the following approximate roots with an accuracy of 0.1:

177. Approximate square root to within 1/100 to 1/1000, etc.

Suppose we need to find an approximate √248 with an accuracy of 1/100. This means: find a decimal fraction that would consist of whole, tenths and hundredths parts and that would satisfy two requirements:

1) its square does not exceed 248, but 2) if we increase this fraction by 1/100 then the square of this increased fraction exceeds 248.

We will find such a fraction in the following sequence: first we will find the whole number, then the tenths figure, then the hundredths figure. The root of an integer is 15 integers. To get the tenths figure, as we have seen, you need to add to the remainder 23 2 more digits to the right of the decimal point. In our example, these numbers are not present at all; we put zeros in their place. By adding them to the remainder and continuing as if we were finding the root of the integer 24,800, we will find the tenths figure 7. It remains to find the hundredths figure. To do this, we add 2 more zeros to the remainder 151 and continue extraction, as if we were finding the root of the integer 2,480,000. We get 15.74. That this number is really an approximate root of 248 with an accuracy of 1/100 can be seen from the following. If we were to find the largest integer square root of the integer 2,480,000, we would get 1574; Means:

1574 2 < 2,480,000, but 1575 2 > 2,480,000.

Dividing all numbers by 10,000 (= 100 2), we get:

This means that 15.74 is that decimal fraction that we called an approximate root with an accuracy of 1/100 of 248.

Applying this technique to finding an approximate root with an accuracy of 1/1000 to 1/10000, etc., we find the following.

Rule. To extract from this whole numbers or from this decimal approximate root with an accuracy of 1/10 to 1/100 to 1/100, etc., first find an approximate root with an accuracy of 1, extracting the root from an integer (if it is not there, write about the root of 0 integers).

Then they find the number of tenths. To do this, add to the remainder the 2 digits of the radical number to the right of the decimal point (if they are not there, add two zeros to the remainder), and continue extraction as is done when extracting the root of an integer. The resulting number is written at the root in the place of tenths.

Then find the hundredths number. To do this, two numbers to the right of those that were just removed are added to the remainder, etc.

Thus, when extracting the root of an integer with a decimal fraction, it is necessary to divide into faces 2 digits each, starting from the decimal point, both to the left (in the integer part of the number) and to the right (in the fractional part).

Examples.

1) Find up to 1/100 roots: a) √2; b) √0.3;

In the last example, we converted the fraction 3/7 to a decimal by calculating 8 decimal places to form the 4 faces needed to find the 4 decimal places of the root.

178. Description of the table of square roots. At the end of this book is a table of square roots calculated with four digits. Using this table, you can quickly find the square root of a whole number (or decimal fraction) that is expressed in no more than four digits. Before explaining how this table is structured, we note that we can always find the first significant digit of the desired root without the help of tables by just looking at the radical number; we can also easily determine which decimal place the first digit of the root means and, therefore, where in the root, when we find its digits, we must put a comma. Here are some examples:

1) √5"27,3 . The first digit will be 2, since the left side of the radical number is 5; and the root of 5 is equal to 2. In addition, since in the integer part of the radical there are only 2 faces, then in the integer part of the desired root there must be 2 digits and, therefore, its first digit 2 must mean tens.

2) √9.041. Obviously, in this root the first digit will be 3 prime units.

3) √0.00"83"4. The first significant digit is 9, since the face from which the root would have to be taken to obtain the first significant digit is 83, and the root of 83 is 9. Since the required number will not contain either whole numbers or tenths, the first digit 9 must mean hundredths.

4) √0.73"85. The first significant figure is 8 tenths.

5) √0.00"00"35"7. The first significant figure will be 5 thousandths.

Let's make one more remark. Let us assume that we need to extract the root of a number which, after discarding the occupied word in it, is represented by a series of numbers like this: 5681. This root can be one of the following:

If we take the roots that we underline with one line, then they will all be expressed by the same series of numbers, precisely those numbers that are obtained when extracting the root from 5681 (these will be the numbers 7, 5, 3, 7). The reason for this is that the faces into which the radical number has to be divided when finding the digits of the root will be the same in all these examples, therefore the digits for each root will be the same (only the position of the decimal point will, of course, be different). In the same way, in all the roots underlined by us with two lines, the same numbers should be obtained, exactly those that are used to express √568.1 (these numbers will be 2, 3, 8, 3), and for the same reason. Thus, the digits of the roots of the numbers represented (by dropping the comma) by the same row of numbers 5681 will be of two (and only two) kind: either this is the row 7, 5, 3, 7, or the row 2, 3, 8, 3. The same, obviously, can be said about any other series of numbers. Therefore, as we will now see, in the table, each row of digits of the radical number corresponds to 2 rows of digits for the roots.

Now we can explain the structure of the table and how to use it. For clarity of explanation, we have shown the beginning of the first page of the table here.

This table is located on several pages. On each of them, in the first column on the left, the numbers 10, 11, 12... (up to 99) are placed. These numbers express the first 2 digits of the number from which the square root is sought. In the top horizontal line (as well as in the bottom) are the numbers: 0, 1, 2, 3... 9, representing the 3rd digit of this number, and then further to the right are the numbers 1, 2, 3. . . 9, representing the 4th digit of this number. All other horizontal lines contain 2 four-digit numbers expressing the square roots of the corresponding numbers.

Suppose you need to find the square root of some number, either an integer or expressed as a decimal fraction. First of all, we find, without the help of tables, the first digit of the root and its digit. Then we will discard the comma in this number, if there is one. Let us first assume that after discarding the comma, only 3 digits will remain, for example. 114. We find in the tables in the leftmost column the first 2 digits, i.e. 11, and move from them to the right along the horizontal line until we reach the vertical column, at the top (and bottom) of which is the 3rd digit of the number , i.e. 4. In this place we find two four-digit numbers: 1068 and 3376. Which of these two numbers should be taken and where to place the comma in it, this is determined by the first digit of the root and its digit, which we found earlier. So, if we need to find √0.11"4, then the first digit of the root is 3 tenths, and therefore we must take 0.3376 for the root. If we needed to find √1.14, then the first digit of the root would be 1, and we Then we would take 1.068.

This way we can easily find:

√5.30 = 2.302; √7"18 = 26.80; √0.91"6 = 0.9571, etc.

Let us now assume that we need to find the root of a number expressed (by dropping the decimal point) in 4 digits, for example, √7"45.6. Noting that the first digit of the root is 2 tens, we find for the number 745, as has now been explained, the digits 2729 (we only notice this number with our finger, but do not write it down). Then we move further to the right from this number until on the right side of the table (behind the last bold line) we meet the vertical column that is marked at the top (and bottom) 4. the th digit of this number, i.e. the number 6, and find the number 1 there. This will be an amendment that must be applied (in the mind) to the previously found number 2729; we get 2730. We write this number down and put a comma in it in the proper place. : 27.30.

In this way we find, for example:

√44.37 = 6.661; √4.437 = 2.107; √0.04"437 =0.2107, etc.

If the radical number is expressed by only one or two digits, then we can assume that there are one or two zeros after these digits, and then proceed as explained for a three-digit number. For example, √2.7 =√2.70 =1.643; √0.13 = √0.13"0 = 0.3606, etc..

Finally, if the radical number is expressed by more than 4 digits, then we will take only the first 4 of them, and discard the rest, and to reduce the error, if the first of the discarded digits is 5 or more than 5, then we will increase by l the fourth of the retained digits . So:

√357,8| 3 | = 18,91; √0,49"35|7 | = 0.7025; etc.

Comment. The tables indicate the approximate square root, sometimes with a deficiency, sometimes with an excess, namely the one of these approximate roots that comes closer to the exact root.

179. Extracting square roots from ordinary fractions. The exact square root of an irreducible fraction can be extracted only when both terms of the fraction are exact squares. In this case, it is enough to extract the root of the numerator and denominator separately, for example:

The approximate square root of an ordinary fraction with some decimal precision can most easily be found if we first reverse common fraction to a decimal, calculating in this fraction the number of decimal places after the decimal point that would be twice the number of decimal places in the desired root.

However, you can do it differently. Let's explain this at following example:

Find approximate √ 5 / 24

Let's make the denominator an exact square. To do this, it would be enough to multiply both terms of the fraction by the denominator 24; but in this example you can do it differently. Let's decompose 24 into prime factors: 24 = 2 2 2 3. From this expansion it is clear that if 24 is multiplied by 2 and another 3, then in the product each prime factor will be repeated an even number of times, and, therefore, the denominator will become a square:

It remains to calculate √30 with some accuracy and divide the result by 12. It must be borne in mind that dividing by 12 will also reduce the fraction indicating the degree of accuracy. So, if we find √30 with an accuracy of 1/10 and divide the result by 12, we will obtain an approximate root of the fraction 5/24 with an accuracy of 1/120 (namely 54/120 and 55/120)

Chapter three.

Graph of a functionx = √y .

180. Inverse function. Let some equation be given that determines at as a function of X , for example, like this: y = x 2 . We can say that it determines not only at as a function of X , but also, conversely, determines X as a function of at , albeit in an implicit way. To make this function explicit, we need to solve this equation for X , taking at for a known number; So, from the equation we took we find: y = x 2 .

Algebraic expression, obtained for x after solving the equation that determines y as a function of x, is called the inverse function of the one that determines y.

So, the function x = √y inverse function y = x 2 . If, as is customary, we denote the independent variable X , and the dependent at , then the inverse function obtained now can be expressed as follows: y = √ x . Thus, in order to obtain the inverse function of a given (direct) one, from the equation that determines this this function, output X depending on y and in the resulting expression replace y on x , A X on y .

181. Graph of a function y = √ x . This function is not possible with negative value X , but it is possible to calculate it (with any accuracy) for any positive value x , and for each such value the function receives two different meanings with the same absolute value, but with opposite signs. If you are familiar √ If we denote only the arithmetic value of the square root, then these two values ​​of the function can be expressed as follows: y= ± √x To plot a graph of this function, you must first compile a table of its values. The easiest way to create this table is from the table of direct function values:

y = x 2 .

x

y

if the values at take as values X , and vice versa:

y= ± √x

By plotting all these values ​​on the drawing, we get the following graph.

In the same drawing we depicted (with a broken line) the graph of the direct function y = x 2 . Let's compare these two graphs with each other.

182. The relationship between the graphs of direct and inverse functions. To compile a table of values ​​of the inverse function y= ± √x we took for X those numbers that are in the table of the direct function y = x 2 served as values ​​for at , and for at took those numbers; which in this table were the values ​​for x . It follows from this that both graphs are the same, only the graph of the direct function is so located relative to the axis at - how the graph of the inverse function is located relative to the axis X - ov. As a result, if we bend the drawing around a straight line OA bisecting a right angle xOy , so that the part of the drawing containing the semi-axis Oh , fell on the part that contains the axle shaft Oh , That Oh compatible with Oh , all divisions Oh will coincide with divisions Oh , and parabola points y = x 2 will align with the corresponding points on the graph y= ± √x . For example, points M And N , whose ordinate 4 , and the abscissas 2 And - 2 , will coincide with the points M" And N" , for which the abscissa 4 , and the ordinates 2 And - 2 . If these points coincide, this means that the straight lines MM" And NN" perpendicular to OA and divide this straight line in half. The same can be said for all other corresponding points in both graphs.

Thus, the graph of the inverse function should be the same as the graph of the direct function, but these graphs are located differently, namely symmetrically with each other relative to the bisector of the angle xOy . We can say that the graph of the inverse function is a reflection (as in a mirror) of the graph of the direct function relative to the bisector of the angle xOy .

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. Why this is so will be explained 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.

The power of the root is always a natural number, that is, it is impossible to solve the following 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 a cube root or a root of the third degree. 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 in practice, by solving many examples and memorizing tables of squares and cubes large 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 large number that finding the correct squared number would be very difficult, if possible at all. 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 have a 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. More exact result Special engineering calculators can give you, but the full result simply does not fit on the display of ordinary ones. 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 of a simple calculator, then you need to double-click 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.

    Calculating (or extracting) the square root can be done in several ways, but all of them are not very simple. It’s easier, of course, to use a calculator. But if this is not possible (or you want to understand the essence of the square root), I can advise you to go the following way, its algorithm is:

    If you don’t have the strength, desire or patience for such lengthy calculations, you can resort to rough selection; its advantage is that it is incredibly fast and, with proper ingenuity, accurate. Example:

    When I was in school (early 60s), we were taught to take the square root of any number. The technique is simple, outwardly similar to long division, but to present it here will require half an hour of time and 4-5 thousand characters of text. But why do you need this? You have a phone or other gadget, nm has a calculator. There is a calculator on any computer. Personally, I prefer to do these types of calculations in Excel.

    Often in school you need to find square roots different numbers. But if we are used to constantly using a calculator for this, then in exams this will not be possible, so we need to learn to look for the root without the help of a calculator. And it is in principle possible to do this.

    The algorithm is as follows:

    Look first at last digit your number:

    For example,

    Now we need to determine approximately the value for the root of the leftmost group

    In the case when a number has more than two groups, then you need to find the root like this:

    But the next number should be the largest, you need to choose it like this:

    Now we need to form a new number A by adding the following group to the remainder that was obtained above.

    In our examples:

  • The column is higher, and when more than fifteen characters are needed, then computers and phones with calculators most often rest. It remains to check whether the description of the technique will take 4-5 thousand characters.

    Take any number, count pairs of digits to the right and left from the decimal point

    For example, 1234567890.098765432100

    A pair of digits is like a two-digit number. The root of a two-digit is one-digit. We select a single digit whose square is less than the first pair of digits. In our case it is 3.

    As when dividing by a column, we write out this square under the first pair and subtract it from the first pair. The result is underlined. 12 - 9 = 3. Add the second pair of numbers to this difference (it will be 334). To the left of the number of berms, the double value of that part of the result that has already been found is supplemented with a number (we have 2 * 6 = 6), such that when multiplied by the not obtained number, it does not exceed the number with the second pair of digits. We get that the found figure is five. Again we find the difference (9), add the next pair of digits to get 956, again write out the doubled part of the result (70), add it again the right number and so on until it stops. Or to the required accuracy of calculations.

    Firstly, in order to calculate the square root, you need to know the multiplication table well. The most simple examples- this is 25 (5 by 5 = 25) and so on. If you take more complex numbers, you can use this table, where the horizontal line is units and the vertical line is tens.

    Eat good way how to find the root of a number without the help of calculators. To do this you will need a ruler and a compass. The point is that you find on the ruler the value that is under your root. For example, put a mark next to 9. Your task is to divide this number into an equal number of segments, that is, into two lines of 4.5 cm each, and into an even segment. It is easy to guess that in the end you will get 3 segments of 3 centimeters each.

    The method is not easy and is not suitable for large numbers, but it can be calculated without a calculator.

    Without the help of a calculator, the method of extracting the square root was taught in Soviet times at school in the 8th grade.

    To do this, you need to break a multi-digit number from right to left into edges of 2 digits :

    The first digit of the root is the whole root of the left side, in in this case, 5.

    We subtract 5 squared from 31, 31-25 = 6 and add the next side to the six, we have 678.

    The next digit x is matched to the double five so that

    10x*x was the maximum, but less than 678.

    x=6, since 106*6 = 636,

    Now we calculate 678 - 636 = 42 and add the next edge 92, we have 4292.

    Again we are looking for the maximum x such that 112x*x lt; 4292.

    Answer: the root is 563

    You can continue this way as long as necessary.

    In some cases, you can try to decompose the radical number into two or more square factors.

    It is also useful to remember the table (or at least some part of it) - squares natural numbers from 10 to 99.

    I propose a version I invented for extracting the square root of a column. It differs from the generally known one, with the exception of the selection of numbers. But as I found out later, this method already existed many years before I was born. The great Isaac Newton described it in his book General Arithmetic or a book about arithmetic synthesis and analysis. So here I present my vision and rationale for the algorithm of the Newton method. There is no need to memorize the algorithm. You can simply use the diagram in the figure as a visual aid if necessary.

    With the help of tables, you can not calculate, but find the square roots of the numbers that are in the tables. The easiest way to calculate not only square roots, but also other degrees, is by the method of successive approximations. For example, we calculate the square root of 10739, replace the last three digits with zeros and extract the root of 10000, we get 100 with a disadvantage, so we take the number 102, square it, we get 10404, which is also less than the given one, we take 103*103=10609 again with a disadvantage, we take 103.5*103.5=10712.25, take even more 103.6*103.6=10732, take 103.7*103.7=10753.69, which is already in excess. You can take the root of 10739 to be approximately equal to 103.6. More precisely 10739=103.629... . . Similarly, we calculate the cube root, first from 10000 we get approximately 25*25*25=15625, which is in excess, we take 22*22*22=10.648, we take a little more than 22.06*22.06*22.06=10735, which is very close to the given one.

Let's look at this algorithm using an example. We'll find

1st step. We divide the number under the root into two-digit faces (from right to left):

2nd step. We take the square root of the first face, i.e. from the number 65, we get the number 8. Under the first face we write the square of the number 8 and subtract. We assign the second face (59) to the remainder:

(number 159 is the first remainder).

3rd step. We double the found root and write the result on the left:

4th step. We separate one digit on the right in the remainder (159), and on the left we get the number of tens (it is equal to 15). Then we divide 15 by double the first digit of the root, i.e. by 16, since 15 is not divisible by 16, then the quotient results in zero, which we write as the second digit of the root. So, in the quotient we got the number 80, which we double again, and remove the next edge

(the number 15,901 is the second remainder).

5th step. In the second remainder we separate one digit from the right and divide the resulting number 1590 by 160. We write the result (number 9) as the third digit of the root and add it to the number 160. We multiply the resulting number 1609 by 9 and find the next remainder (1420):

IN further action are performed in the sequence specified in the algorithm (the root can be extracted with the required degree of accuracy).

Comment. If the radical expression is a decimal fraction, then its whole part is divided into edges of two digits from right to left, the fractional part - two digits from left to right, and the root is extracted according to the specified algorithm.

DIDACTIC MATERIAL

1. Take the square root of the number: a) 32; b) 32.45; c) 249.5; d) 0.9511.