How to extract the square root of a fraction. Root extraction

Do you want to do well on the Unified State Examination in mathematics? Then you need to be able to count quickly, correctly and without a calculator. After all main reason loss of points on the Unified State Exam in mathematics - computational errors.

According to the rules conducting the Unified State Exam, it is prohibited to use a calculator during the mathematics exam. The price may be too high - removal from the exam.

In fact, you don’t need a calculator for the Unified State Examination in mathematics. All problems are solved without it. The main thing is attention, accuracy and some secret techniques, which we will tell you about.

Let's start with the main rule. If a calculation can be simplified, simplify it.

Here, for example, is the “devilish equation”:

Seventy percent of graduates solve it head-on. They calculate the discriminant using the formula, after which they say that the root cannot be extracted without a calculator. But you can divide the left and right sides of the equation by . It will work out

Which way is easier? :-)

Many schoolchildren do not like columnar multiplication. Nobody liked solving boring “examples” in fourth grade. However, in many cases it is possible to multiply numbers without a “column”, in a row. It's much faster.

Please note that we do not start with smaller digits, but with larger ones. It's convenient.

Now - division. It is not easy to divide “in a column” by . But remember that the division sign: and the fractional bar are the same thing. Let's write it as a fraction and reduce the fraction:

Another example.

How to square a two-digit number quickly and without any columns? We apply abbreviated multiplication formulas:

Sometimes it is convenient to use another formula:

Numbers ending in , are squared instantly.

Let's say we need to find the square of a number ( - not necessarily a number, any natural number). We multiply by and add to the result. All!

For example: (and attributed).

(and attributed).

This method is useful not only for squaring, but for taking the square root of numbers ending in .

How can you even extract the square root without a calculator? We'll show you two ways.

The first method is to factorize the radical expression.

For example, let's find
A number is divisible by (since the sum of its digits is divisible by ). Let's factorize:

Let's find it. This number is divisible by . It is also divided by. Let's factor it out.

Another example.

There is a second way. It is convenient if the number from which you need to extract the root cannot be factorized.

For example, you need to find . The number under the root is odd, it is not divisible by, is not divisible by, is not divisible by... You can continue to look for what it is divisible by, or you can do it easier - find this root by selection.

Obviously, a two-digit number was squared, which is between the numbers and , since , , and the number is between them. We already know the first digit in the answer, it is .

The last digit in the number is . Since , , the last digit in the answer is either , or . Let's check:
. It worked!

Let's find it.

This means that the first digit in the answer is five.

The last digit in the number is nine. , . This means that the last digit in the answer is either , or .

Let's check:

If the number from which you need to extract the square root ends in or, then the square root of it will be an irrational number. Because no integer square ends in or . Remember that in the tasks part Unified State Exam options in mathematics, the answer must be written as an integer or a finite decimal fraction, that is, it must be a rational number.

We encounter quadratic equations in problems and variants of the Unified State Examination, as well as in parts. They need to count the discriminant and then extract the root from it. And it is not at all necessary to look for roots from five digit numbers. In many cases, the discriminant can be factorized.

For example, in Eq.

Another situation in which the expression under the root can be factorized is taken from the problem.

Hypotenuse right triangle is equal to , one of the legs is equal to , find the second leg.

According to the Pythagorean theorem, it is equal to . You can count in a column for a long time, but it’s easier to use the abbreviated multiplication formula.

And now we’ll tell you the most interesting thing - why graduates lose precious points on the Unified State Exam. After all, errors in calculations do not just happen.

1 . The right way To lose points - sloppy calculations, in which something is corrected, crossed out, one number is written on top of another. Look at your drafts. Perhaps they look the same? :-)

Write legibly! Don't skimp on paper. If something is wrong, do not correct one number for another, it is better to write it again.

2. For some reason, many schoolchildren, when counting in a column, try to do it 1) very, very quickly, 2) in very small numbers, in the corner of their notebook, and 3) with a pencil. The result is this:

It's impossible to make anything out. So is it any surprise that the Unified State Exam score is lower than expected?

3. Many schoolchildren are accustomed to ignoring parentheses in expressions. Sometimes this happens:

Remember that the equal sign is not placed just anywhere, but only between equal amounts. Write competently, even in draft form.

4 . Huge number computational errors associated with fractions. If you are dividing a fraction by a fraction, use what
A “hamburger” is drawn here, that is multi-story fraction. It is extremely difficult to get the correct answer using this method.

Let's summarize.

Checking the tasks of the first part profile Unified State Examination in mathematics - automatic. There is no “almost right” answer here. Either he is correct or he is not. One computational error - and hello, the task does not count. Therefore, it is in your interests to learn to count quickly, correctly and without a calculator.

The tasks of the second part of the profile Unified State Examination in mathematics are checked by an expert. Take care of him! Let him understand both your handwriting and the logic of the decision.

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

Steps

Prime factorization

Factor the radical number into factors that are square numbers. Depending on the radical number, you will get an approximate or exact answer. Square numbers are numbers from which the whole square root can be taken. 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 factor the radical number into square factors.

For example, calculate the square root of 400 (by hand). 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 the square factors of 25 and 16, that is, 25 x 16 = 400.
This can be written as follows: √400 = √(25 x 16).

Square root of the product of some terms equal to the product square roots from each term, that is, √(a x b) = √a x √b. Use this rule to take the square root of each square factor and multiply the results to find the answer.
- In our example, take the root of 25 and 16.
  - √(25 x 16)
  - √25 x √16
  - 5 x 4 = 20
If the radical number does not decompose into two square factor(and this happens in most cases), you will not be able to find the exact answer in the form of a whole number. But you can simplify the problem by decomposing the radical 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 will take the root of the common 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 factorized into the following factors: 49 and 3. Solve the problem as follows:
  - = √(49 x 3)
  - = √49 x √3
  - = 7√3
If necessary, estimate the value of the root. Now you can estimate the value of the root (find approximate value), comparing it with the values of the roots of square numbers that are closest (on both sides on the number line) to the radical number. You will receive the root value as a decimal fraction, which must be multiplied by the number behind the root sign.
- Let's return to our example. The radical number is 3. The square numbers closest to it will be the numbers 1 (√1 = 1) and 4 (√4 = 2). Thus, the value of √3 is located 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 = 11.9. If you do the math on a calculator, you'll get 12.13, which is pretty close to our answer.
  - This method also works with large numbers. For example, consider √35. The radical number is 35. The closest square numbers to it will be the numbers 25 (√25 = 5) and 36 (√36 = 6). Thus, the value of √35 is located between 5 and 6. Since the value of √35 is much closer to 6 than to 5 (because 35 is only 1 less than 36), we can say that √35 is slightly less than 6. Check on the calculator gives us the answer 5.92 - we were right.
Another way is to factor the radical number into prime factors. Prime factors are numbers that are divisible only by 1 and themselves. Write it down prime factors in a row and find pairs of identical factors. Such factors can be taken out of the root sign.
- For example, calculate the square root of 45. We factor the radical number into prime factors: 45 = 9 x 5, and 9 = 3 x 3. Thus, √45 = √(3 x 3 x 5). 3 can be taken out as a root sign: √45 = 3√5. Now we can estimate √5.
- Let's look at another example: √88.
  - = √(2 x 44)
  - = √ (2 x 4 x 11)
  - = √ (2 x 2 x 2 x 11). You received three multipliers of 2; take a couple of them and move them beyond the root sign.
  - = 2√(2 x 11) = 2√2 x √11. Now you can evaluate √2 and √11 and find an approximate answer.
Calculating square root manually

Using long division
1. This method involves a process similar to long division and provides an accurate answer. First, draw a vertical line dividing the sheet into two halves, and then to the right and slightly below the top edge of the sheet to vertical line draw horizontal line. Now divide the radical 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 given number in the form “7 80, 14” at the top left. It is normal that the first digit from the left is an unpaired digit. Answer (root of given number) you will write down on the top right.
2. For the first pair of numbers (or single number) from the left, find the largest integer n whose square is less than or equal to the pair of numbers (or single number) in question. In other words, find the square number that is closest to, but smaller 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 n you found at the top right, and write the square of n at the bottom right.
  - In our case, the first number on the left will be 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 single number) on the left. Write the result of the calculation under the subtrahend (the square of the number n).
  - In our example, subtract 4 from 7 and 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 the addition of "_×_=".
  - In our example, the second pair of numbers is "80". Write "80" after the 3. Then, double the number on the top right gives 4. Write "4_×_=" on the bottom right.
5. Fill in the blanks on the right.
  - In our case, if we put the number 8 instead of dashes, then 48 x 8 = 384, which is more than 380. Therefore, 8 is too large a number, but 7 will do. Write 7 instead of dashes and get: 47 x 7 = 329. Write 7 at 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 under the current number on the left, find the difference and write it under the subtrahend.
  - In our example, subtract 329 from 380, which equals 51.
7. Repeat step 4. If the pair of numbers to be carried is the fractional part of the original number, then put the separator (comma) with an integer and fractional parts in the desired square root from the top right. On the left, bring down the next pair of numbers. Double the number at the top right and write the result at the bottom right with the addition of "_×_=".
  - In our example, the next pair of numbers to be removed will be the fractional part of the number 780.14, so place the separator of the integer and fractional parts in the desired square root in the upper right. Take down 14 and write it down at the bottom left. Double the number on the top right (27) is 54, so write "54_×_=" on the bottom right.
8. Repeat steps 5 and 6. Find one greatest number in place of the dashes on the right (instead of the dashes you need to substitute the same number) so that the result of the multiplication 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 couple of zeros to the left of the current number and repeat steps 4, 5, and 6. Repeat steps until you get the answer precision (number of decimal places) you need.
Understanding the Process
1. 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 desired value of the square root will be a digit whose square is less than or equal to S a (that is, we are looking for an A such that 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.
2. Mentally imagine a 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 equal to S. A, B, C are the numbers in the number L. You can write it differently: 10A + B = L (for a two-digit number) or 100A + 10B + C = 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 in which the digit B stands for units and the digit A stands for tens. For example, if A=1 and B=2, then 10A+B is equal to the number 12. (10A+B)²- this is the area of the entire square, 100A²- area of the large inner square, B²- area of the small inner square, 10A×B- the area of each of the two rectangles. By adding up the areas of the described figures, you will find the area of the original square.

Preferably an engineering one - one that has a button with a root sign: “√”. Usually, to extract the root, it is enough to type the number itself, and then press the button: “√”.

In most modern mobile phones There is a “calculator” application with a root extraction function. The procedure for finding the root of a number using a telephone calculator is similar to the above.
Example.
Find from 2.
Turn on the calculator (if it is turned off) and successively press the buttons with the image of two and root (“2” “√”). As a rule, you do not need to press the “=” key. As a result, we get a number like 1.4142 (the number of digits and “roundness” depends on the bit depth and calculator settings).
Note: When trying to find the root, the calculator usually gives an error.

If you have access to a computer, then finding the root of a number is very simple.
1. You can use the Calculator application, available on almost any computer. For Windows XP, this program can be launched as follows:
“Start” - “All Programs” - “Accessories” - “Calculator”.
It is better to set the view to “normal”. By the way, unlike a real calculator, the button for extracting the root is marked “sqrt” and not “√”.

If you can’t get to the calculator using the indicated method, you can run the standard calculator “manually”:
“Start” - “Run” - “calc”.
2. To find the root of a number, you can also use some programs installed on your computer. In addition, the program has its own built-in calculator.

For example, for the MS Excel application, you can do the following sequence of actions:
Launch MS Excel.

We write down in any cell the number from which we need to extract the root.

Move the cell pointer to a different location

Press the function selection button (fx)

Select the “ROOT” function

We specify a cell with a number as an argument to the function

Click “OK” or “Enter”
Advantage this method is that now it is enough to enter any value into the cell with the number, as in the function, .
Note.
There are several other, more exotic ways to find the root of a number. For example, “corner”, using slide rule or Bradis tables. However, these methods are not discussed in this article due to their complexity and practical uselessness.

Video on the topic

Sources:

how to find the root of a number

Sometimes situations arise when you have to perform some mathematical calculations, including extracting square roots and roots to a greater extent from the number. The "n" root of "a" is the number nth degree which is the number "a".

Instructions

To find the root "n" of , do the following.

On your computer, click “Start” - “All Programs” - “Accessories”. Then go to the “Service” subsection and select “Calculator”. You can do this manually: click Start, type "calk" in the Run box, and press Enter. Will open. To extract the square root of a number, enter it into the calculator and press the button labeled "sqrt". The calculator will extract the second degree root, called the square root, from the entered number.

In order to extract a root whose degree is higher than the second, you need to use another type of calculator. To do this, in the calculator interface, click the “View” button and select the “Engineering” or “Scientific” line from the menu. This type of calculator has the necessary to calculate the root nth degree function.

To extract the root of the third degree (), on an “engineering” calculator, enter the desired number and press the “3√” button. To obtain a root whose degree is higher than 3, enter the desired number, press the button with the “y√x” icon and then enter the number - the exponent. After that, press the equal sign (the “=” button) and you will get the desired root.

If your calculator does not have the "y√x" function, the following.

To extract cube root enter the radical expression, then put a mark in the check box, which is located next to the inscription “Inv”. With this action, you will reverse the functions of the calculator buttons, i.e., by clicking on the cube button, you will extract the cube root. On the button that you

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:

Roots of large numbers do appear in problems. Especially in text ones;
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 get most powerful weapon against square roots.

So, the algorithm:

Limit the required root above and below to numbers that are 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. 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:

[Caption for the picture]

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 - and again without any complex calculations! Enough to know 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 appear on last place. Let's try to find out what they turn into when 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. 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:

[Caption for the picture]

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:

[Caption for the picture]

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 calculation optimization, but, of course, it is 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:

At any normal exam in mathematics, 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 just roots - they are two prime numbers They can’t fold it. And when they see fractions, they generally become hysterical.

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; from such numbers, of course, it is impossible to extract whole root. 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 47 tens of the remainder by 4, we get 11 in the quotient. But since the number of units of the root cannot be double digit number 11 or 10, then you need to 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 last digits radical number. We already have the remainder from subtracting the square 18 from 357: this is 33. This means that to obtain the remainder from subtracting the square 18 des. from 3"57"82, it is enough to add the numbers 82 to 33 on the right.

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 a given 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, 2 digits each, except 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, more balance, 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 there are as many digits in the root as there are faces of 2 digits each in the radical number (the left face may have one digit).

Chapter two.

Extracting approximate square roots of integers and fractions .

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 integer (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 the fraction equal to the root from the numerator divided by the root of the denominator, then the exact root of irreducible fraction cannot be found if it cannot be extracted from the numerator or 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 with an accuracy of 1 is the largest integer square root of a given number, i.e. the root that we learned to find in 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. (Names: “with a deficiency” or “with an excess” in some math books 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 the following two 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 number of tenths, 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 a given decimal fraction an 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 the integer (if it is not there, write about the root 0 whole).

Then they find the number of tenths. To do this, add two digits of the radical number to the right of the decimal point to the remainder (if they are not there, add two zeros to the remainder), and continue extraction as is done when extracting a root from 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 means the first digit of the root 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. First significant figure 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 being 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, integer or expressed decimal. 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 decomposition it is clear that if 24 is multiplied by 2 and another 3, then each prime factor will be repeated in the product even number times, and therefore the denominator becomes 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 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, nose 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 .

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 create a table of values 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 .