12.10.2019

How to calculate the root of a number. Research paper on the topic: "Extracting square roots of large numbers without a calculator"

Extracting the root is the reverse operation of raising a power. That is, taking the root of the number X, we get a number that squared will give the same number X.

Extracting the root is a fairly simple operation. A table of squares can make the extraction work easier. Because it is impossible to remember all the squares and roots by heart, but the numbers may be large.

Extracting the root of a number

Extraction square root from the number - simple. Moreover, this can be done not immediately, but gradually. For example, take the expression √256. Initially, it is difficult for an ignorant person to give an answer right away. Then we will do it step by step. First, we divide by just the number 4, from which we take the selected square as the root.

Let's represent: √(64 4), then it will be equivalent to 2√64. And as you know, according to the multiplication table 64 = 8 8. The answer will be 2*8=16.

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Extracting a complex root

The square root cannot be calculated from negative numbers, because any squared number is a positive number!

A complex number is the number i, which squared is equal to -1. That is, i2=-1.

In mathematics, there is a number that is obtained by taking the root of the number -1.

That is, it is possible to calculate the root of negative number, but this already applies to higher mathematics, not school.

Let's consider an example of such a root extraction: √(-49)=7*√(-1)=7i.

Online root calculator

Using our calculator, you can calculate the extraction of a number from the square root:

Converting Expressions Containing a Root Operation

The essence of transforming radical expressions is to decompose the radical number into simpler ones, from which the root can be extracted. Such as 4, 9, 25 and so on.

Let's give an example, √625. Let's divide the radical expression by the number 5. We get √(125 5), repeat the operation √(25 25), but we know that 25 is 52. Which means the answer will be 5*5=25.

But there are numbers for which the root cannot be calculated using this method and you just need to know the answer or have a table of squares at hand.

√289=√(17*17)=17

Bottom line

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From the course you will not only learn dozens of techniques for simplified and quick multiplication, addition, multiplication, division, and calculating percentages, but you will also practice them in special tasks and educational games! Mental arithmetic also requires a lot of attention and concentration, which are actively trained when solving interesting problems.

And do you have calculator addiction? Or do you think that it is very difficult to calculate, for example, except with a calculator or using a table of squares.

It happens that schoolchildren are tied to a calculator and even multiply 0.7 by 0.5 by pressing the treasured buttons. They say, well, I still know how to calculate, but now I’ll save time... When the exam comes... then I’ll strain myself...

So the fact is that there will already be plenty of “stressful moments” during the exam... As they say, water wears away stones. So in an exam, little things, if there are a lot of them, can ruin you...

Let's minimize the number of possible troubles.

Taking the square root of a large number

We will now only talk about the case when the result of extracting the square root is an integer.

Case 1.

So, let us at any cost (for example, when calculating the discriminant) need to calculate the square root of 86436.

We will factor the number 86436 into prime factors. Divide by 2, we get 43218; divide by 2 again, we get 21609. A number cannot be divisible by 2. But since the sum of the digits is divisible by 3, then the number itself is divisible by 3 (generally speaking, it is clear that it is also divisible by 9). . Divide by 3 again, and we get 2401. 2401 is not completely divisible by 3. Not divisible by five (does not end in 0 or 5).

We suspect divisibility by 7. Indeed, and ,

So, Complete order!

Case 2.

Let us need to calculate . It is inconvenient to act in the same way as described above. We are trying to factorize...

The number 1849 is not divisible by 2 (it is not even)…

It is not completely divisible by 3 (the sum of the digits is not a multiple of 3)...

It is not completely divisible by 5 (the last digit is neither 5 nor 0)…

It’s not completely divisible by 7, it’s not divisible by 11, it’s not divisible by 13... Well, how long will it take us to sort through all the prime numbers?

Let's think a little differently.

We understand that

We have narrowed our search. Now we go through the numbers from 41 to 49. Moreover, it is clear that since the last digit of the number is 9, then we should stop at options 43 or 47 - only these numbers, when squared, will give the last digit 9.

Well, here, of course, we stop at 43. Indeed,

P.S. How the hell do we multiply 0.7 by 0.5?

You should multiply 5 by 7, ignoring the zeros and signs, and then separate, going from right to left, two decimal places. We get 0.35.

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).

The square root of the product of some terms is equal to the product of the square roots of 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 factor into two square factors (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 an approximate value) by comparing it with the values of the roots of the square numbers that are closest (on both sides of the number line) to the radical number. You will get the value of the root as decimal, 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 the prime factors in a series 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 gives 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, draw a horizontal line to the vertical 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. You will write the answer (the root of this number) at 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 being transferred is the fractional part of the original number, then put a separator (comma) between the integer and fractional parts in the required square root at 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 in 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 the largest 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 square root value 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 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 easy.
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, in a “corner”, using a 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 kind of mathematical calculations, including extracting square roots and 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 this, 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

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 as follows:

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 it is required to find the square roots of 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 this is, in principle, possible to do.

The algorithm is as follows:

Look at the last digit of your number first:

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.

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

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. We again find the difference (9), add the next pair of digits to get 956, again write out the doubled part of the result (70), again supplement it with the desired digit, 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 for large numbers will not work, 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 8th grade.

To do this you need to break multi-digit number from right to left on the edge there are 2 digits :

The first digit of the root is the whole root of the left side, 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.