How to find the square root of a number manually. What is a square root? What is the square root of one hundred?

When solving various problems from a mathematics and physics course, pupils and students are often faced with the need to extract roots of the second, third or nth degree. Of course, in the century information technologies It won’t be difficult to solve this problem using a calculator. However, situations arise when it is impossible to use the electronic assistant.

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

Finding square roots using a table of squares

One of the simplest ways to calculate roots is to using a special table. What is it and how to use it correctly?

Using the table, you can find the square of any number from 10 to 99. The rows of the table contain the values ​​of tens, and the columns contain the values ​​of units. The cell at the intersection of a row and a column contains the square of a two-digit number. In order to calculate the square of 63, you need to find a row with a value of 6 and a column with a value of 3. At the intersection we will find a cell with the number 3969.

Since extracting the root is the inverse operation of squaring, to perform this action you must do the opposite: first find the cell with the number whose radical you want to calculate, then use the values ​​of the column and row to determine the answer. As an example, consider the calculation square root 169.

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

Similarly, you can calculate cube and nth roots using the appropriate tables.

The advantage of the method is its simplicity and the absence of additional calculations. The disadvantages are obvious: the method can only be used for a limited range of numbers (the number for which the root is found must be in the range from 100 to 9801). In addition, it will not work if the given number is not in the table.

Prime factorization

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

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

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

Heron's method

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

√R = √a + (R - a) / 2√a,

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

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

√111 = √121 + (111 - 121) / 2 ∙ √121 = 11 - 10 / 22 ≈ 10,55.

Now let's check the accuracy of the method:

10.55² = 111.3025.

The error of the method was approximately 0.3. If the accuracy of the method needs to be improved, you can repeat the previously described steps:

√111 = √111,3025 + (111 - 111,3025) / 2 ∙ √111,3025 = 10,55 - 0,3025 / 21,1 ≈ 10,536.

Let's check the accuracy of the calculation:

10.536² = 111.0073.

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

Calculating the root by long division

This method of finding the square root value is a little more complex than the previous ones. However, it is the most accurate among other calculation methods without a calculator.

Let's say that you need to find the square root accurate to 4 decimal places. Let's analyze the calculation algorithm using the example of an arbitrary number 1308.1912.

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

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

Bitwise square root calculation

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

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

1. Let's find out which digit of the square root value will be the most significant. To do this, let’s square 0, 10, 100, 1000, etc. and find out between which of them the radical number is located. We get that 10²< 781 < 100², т. е. старшим разрядом будут десятки.
2. Let's choose the value of tens. To do this, we will take turns raising to the power of 10, 20, ..., 90 until we get a number greater than 781. For our case, we get 10² = 100, 20² = 400, 30² = 900. The value of the result n will be within 20< n <30.
3. Similar to the previous step, the value of the units digit is selected. Let's square 21.22, ..., 29 one by one: 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784. We get that 27< n < 28.
4. Each subsequent digit (tenths, hundredths, etc.) is calculated in the same way as shown above. Calculations are carried out until the required accuracy is achieved.

Video

This video will show you how to find square roots without using a calculator.

Quite often, when solving problems, we are faced with large numbers from which we need to extract Square root. Many students decide that this is a mistake and begin to re-solve the entire example. Under no circumstances should you do this! There are two reasons for this:

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

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

So, the algorithm:

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

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

Root limitation

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

10 2 = 100;
20 2 = 400;
30 2 = 900;
40 2 = 1600;
...
90 2 = 8100;
100 2 = 10 000.

We get a series of numbers:

100; 400; 900; 1600; 2500; 3600; 4900; 6400; 8100; 10 000.

What do these numbers tell us? It's simple: we get boundaries. Take, for example, the number 1296. It lies between 900 and 1600. Therefore, its root cannot be less than 30 and greater than 40:

[Caption for the picture]

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

[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 - again without any complicated calculations! It is enough to know the special rule. Here it is:

The last digit of the square depends only on the last digit original number.

In other words, just look at the last digit of the square and we will immediately understand where the original number ends.

There are only 10 digits that can come in last place. Let's try to find out what they turn into when squared. Take a look at the table:

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 relative to the five. For example:

2 2 = 4;
8 2 = 64 → 4.

As you can see, the last digit is the same in both cases. This means that, for example, the root of 3364 must end in 2 or 8. On the other hand, we remember the restriction from the previous paragraph. We get:

[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 all! 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 all! 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:

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

The problem of finding a root in mathematics is the inverse problem of raising a number to a power. There are different roots: roots of the second degree, roots of the third degree, roots of the fourth degree, and so on. It depends on what power the number was originally raised to. The root is indicated by the symbol: √ is a square root, that is, the root of the second degree; if the root has a degree greater than the second, then the corresponding degree is assigned above the root sign. The number that is under the root sign is a radical expression. When finding a root, there are several rules that will help you not make a mistake in finding the root:

• An even root (if the degree is 2, 4, 6, 8, etc.) of a negative number does NOT exist. If the radical expression is negative, but the root of an odd degree is sought (3, 5, 7, and so on), then the result will be negative.
• The root of any power of one is always one: √1 = 1.
• The root of zero is zero: √0 = 0.

How to find the root of 100

If the problem does not say what root of the degree needs to be found, then it usually means that it is necessary to find the root of the second degree (square).
Let's find √100 = ? We need to find a number that, when raised to the second power, gives the number 100. Obviously, such a number is the number 10, since: 10 2 = 100. Therefore, √100 = 10: the square root of 100 is 10.

What is a square root?

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

This concept is very simple. Natural, I would say. Mathematicians try to find a reaction for every action. There is addition - there is also subtraction. There is multiplication - there is also division. There is squaring... So there is also taking the square root! That's all. This action ( square root) in mathematics is indicated by this icon:

The icon itself is called a beautiful word " radical".

How to extract the root? It's better to look at examples.

What is the square root of 9? What number squared will give us 9? 3 squared gives us 9! Those:

But what is the square root of zero? No problem! What number squared does zero make? Yes, it gives zero! Means:

Got it, what is square root? Then we consider examples:

Answers (in disarray): 6; 1; 4; 9; 5.

Decided? Really, how much easier is that?!

But... What does a person do when he sees some task with roots?

A person begins to feel sad... He does not believe in the simplicity and lightness of his roots. Although he seems to know what is square root...

This is because the person ignored several important points when studying the roots. Then these fads take cruel revenge on tests and exams...

Point one. You need to recognize the roots by sight!

What is the square root of 49? Seven? Right! How did you know it was seven? Squared seven and got 49? Right! Please note that extract the root out of 49 we had to do the reverse operation - square 7! And make sure we don't miss. Or they could have missed...

This is the difficulty root extraction. Square You can use any number without any problems. Multiply a number by itself with a column - that's all. But for root extraction There is no such simple and fail-safe technology. We have to pick up answer and check if it is correct by squaring it.

This complex creative process - choosing an answer - is greatly simplified if you remember squares of popular numbers. Like a multiplication table. If, say, you need to multiply 4 by 6, you don’t add four 6 times, do you? The answer 24 immediately comes up. Although, not everyone gets it, yes...

To work freely and successfully with roots, it is enough to know the squares of numbers from 1 to 20. Moreover there And back. Those. you should be able to easily recite both, say, 11 squared and the square root of 121. To achieve this memorization, there are two ways. The first is to learn the table of squares. This will be a great help in solving examples. The second is to solve more examples. This will greatly help you remember the table of squares.

And no calculators! For testing purposes only. Otherwise, you will slow down mercilessly during the exam...

So, what is square root And How extract roots- I think it’s clear. Now let's find out WHAT we can extract them from.

Point two. Root, I don't know you!

What numbers can you take square roots from? Yes, almost any of them. It's easier to understand what it's from it is forbidden extract them.

Let's try to calculate this root:

To do this, we need to choose a number that squared will give us -4. We select.

What, it doesn't fit? 2 2 gives +4. (-2) 2 gives again +4! That's it... There are no numbers that, when squared, will give us a negative number! Although I know these numbers. But I won’t tell you). Go to college and you will find out for yourself.

The same story will happen with any negative number. Hence the conclusion:

An expression in which there is a negative number under the square root sign - doesn't make sense! This is a forbidden operation. It is as forbidden as dividing by zero. Remember this fact firmly! Or in other words:

You cannot extract square roots from negative numbers!

But of all the others, it’s possible. For example, it is quite possible to calculate

At first glance, this is very difficult. Selecting fractions and squaring them... Don't worry. When we understand the properties of roots, such examples will be reduced to the same table of squares. Life will become easier!

Okay, fractions. But we still come across expressions like:

It's OK. All the same. The square root of two is the number that, when squared, gives us two. Only this number is completely uneven... Here it is:

What’s interesting is that this fraction never ends... Such numbers are called irrational. In square roots this is the most common thing. By the way, this is why expressions with roots are called irrational. It is clear that writing such an infinite fraction all the time is inconvenient. Therefore, instead of an infinite fraction, they leave it like this:

If, when solving an example, you end up with something that cannot be extracted, like:

then we leave it like that. This will be the answer.

You need to clearly understand what the icons mean

Of course, if the root of the number is taken smooth, you must do this. The answer to the task is in the form, for example

Quite a complete answer.

And, of course, you need to know the approximate values ​​from memory:

This knowledge greatly helps to assess the situation in complex tasks.

Point three. The most cunning.

The main confusion in working with roots is caused by this point. It is he who gives confidence in his own abilities... Let's deal with this point properly!

First, let's take the square root of four of them again. Have I already bothered you with this root?) Never mind, now it will be interesting!

What number does 4 square? Well, two, two - I hear dissatisfied answers...

Right. Two. But also minus two will give 4 squared... Meanwhile, the answer

correct and the answer

gross mistake. Like this.

So what's the deal?

Indeed, (-2) 2 = 4. And under the definition of the square root of four minus two quite suitable... This is also the square root of four.

But! In the school mathematics course, it is customary to consider square roots only non-negative numbers! That is, zero and all are positive. Even a special term was invented: from the number A- This non-negative number whose square is A. Negative results when extracting an arithmetic square root are simply discarded. At school, everything is square roots - arithmetic. Although this is not particularly mentioned.

Okay, that's understandable. It's even better not to bother with negative results... This is not yet confusion.

Confusion begins when solving quadratic equations. For example, you need to solve the following equation.

The equation is simple, we write the answer (as taught):

This answer (absolutely correct, by the way) is just an abbreviated version two answers:

Stop, stop! Just above I wrote that the square root is a number Always non-negative! And here is one of the answers - negative! Disorder. This is the first (but not the last) problem that causes distrust of the roots... Let's solve this problem. Let's write down the answers (just for understanding!) like this:

The parentheses do not change the essence of the answer. I just separated it with brackets signs from root. Now you can clearly see that the root itself (in brackets) is still a non-negative number! And the signs are result of solving the equation. After all, when solving any equation we must write All Xs that, when substituted into the original equation, will give the correct result. The root of five (positive!) with both a plus and a minus fits into our equation.

Like this. If you just take the square root from anything, you Always you get one non-negative result. For example:

Because it - arithmetic square root.

But if you are solving some quadratic equation, like:

That Always it turns out two answer (with plus and minus):

Because this is the solution to the equation.

Hope, what is square root You've got your points clear. Now it remains to find out what can be done with the roots, what their properties are. And what are the points and pitfalls... sorry, stones!)

All this is in the following lessons.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Among the many knowledge that is a sign of literacy, the alphabet comes first. The next, equally “sign” element is the skills of addition-multiplication and, adjacent to them, but opposite in meaning, arithmetic operations of subtraction-division. The skills learned in distant school childhood serve faithfully day and night: TV, newspaper, SMS, and everywhere we read, write, count, add, subtract, multiply. And, tell me, have you often had to extract roots in your life, except at the dacha? For example, such an entertaining problem, like the square root of the number 12345... Is there still gunpowder in the flasks? Can we handle it? Nothing could be simpler! Where is my calculator... And without it, hand-to-hand combat is weak?

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

Taking the square root

From the school arithmetic course, methods of calculations “in a column” are known, which help to perform any calculations using the first four arithmetic operations. Alas... For square, and not only square, roots, such algorithms do not exist. And in this case, how to extract the square root without a calculator? Based on the definition of the square root, there is only one conclusion - it is necessary to select the value of the result by sequentially enumerating numbers whose square approaches the value of the radical expression. That's all! Before an hour or two has passed, you can calculate, using the well-known method of multiplication in a “column”, any square root. If you have the skills, this will only take a couple of minutes. Even a not-so-advanced user of a calculator or PC can do this in one fell swoop - progress.

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

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

1. Let's take, purely intuitively, X=100. Let's calculate: X * X = 10000. Intuition is at its best - the result is less than 12345.

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

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

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

Just a little history...

The Pythagoreans, students of the school and followers of Pythagoras, came up with the idea of ​​​​using square roots, 800 years BC. and then we “ran into” new discoveries in the field of numbers. And where did that come from?

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

2. It is known that it turned out that this mathematical operation contains another catch - when extracting the root, we do not know which number, positive or negative, is the square of the radical expression. This uncertainty, the double result from one operation, is recorded in this way.

The study of problems related to this phenomenon has become a direction in mathematics called the theory of complex variables, which has great practical importance in mathematical physics.

It is curious that the same ubiquitous I. Newton used the designation of the root - radical - in his “Universal Arithmetic”, and exactly the modern form of notation of the root has been known since 1690 from the book of the Frenchman Rolle “Manual of Algebra”.