How to find the square root of a number manually. Square root. Actions with square roots. Module. Comparison of square roots

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 columnar multiplication, 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”.

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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:
  - = 7√3
= √(49 x 3)= √49 x √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 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.
- 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, 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.< 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. In our case, the first number on the left will be 7. Next, 4 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).
4. In our example, subtract 4 from 7 and get 3. 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 "_×_=".
5. 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.
  - 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. To master this method, imagine the number whose square root you need to find as the area of the square S. In this case, you will look for the length of the side L of such a square. We calculate the value of L such that L² = S.
    
    Give a letter for each number in the answer. Let us denote by A the first digit in the value of L (the desired square root). B will be the second digit, C the third and so on.
    
    Specify a letter for each pair of first digits. Let us denote by S a the first pair of digits in the value of S, by S b the second pair of digits, and so on.
    
    Understand the connection between this method and long division. Just like in the operation of division, where we are only interested in the next digit of the number we are dividing each time, when calculating a square root, we work with a pair of digits sequentially (to obtain the next one digit in the square root value).
  2. Consider the first pair of digits Sa of the number S (Sa = 7 in our example) and find its square root. In this case, the first digit A of the 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.
  3. 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.

Rational numbers

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

Complex numbers

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

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

where the modulus root is understood in the sense of an arithmetic value, and k can take the values k=0 and k=1, so the answer ends up with two different results.

Generalizations

Square roots are introduced as solutions to equations of the form for other objects: matrices, functions, operators, etc. Quite arbitrary multiplicative operations can be used as an operation, for example, superposition.

Square root in computer science

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

Algorithms for finding the square root

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

Taylor series expansion

at .

Arithmetic square root

For squares of numbers the following equalities are true:

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

3 steps are completed, the square root of 9 is 3.

The disadvantage of this method is that if the root being extracted is not an integer, then you can only find out its whole part, but not more precisely. At the same time, this method is quite accessible to children solving simple mathematical problems that require extracting the square root.

Rough estimate

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

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

Two and six are used because And

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

Geometric square root

To manually extract the root, a notation similar to long division is used.

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

A visual description of the algorithm:
Fact 1. \(\bullet\) Let's take some non-negative number \(a\) (that is, \(a\geqslant 0\) ). Then (arithmetic) square root from the number \(a\) is called such a non-negative number \(b\) , when squared we get the number \(a\) :\[\sqrt a=b\quad \text(same as )\quad a=b^2\] From the definition it follows that. \(a\geqslant 0, b\geqslant 0\)
These restrictions are an important condition for the existence of a square root and should be remembered!
Recall that any number when squared gives a non-negative result. That is, \(100^2=10000\geqslant 0\) and \((-100)^2=10000\geqslant 0\) .
\(\bullet\) What is \(\sqrt(25)\) equal to? We know that \(5^2=25\) and \((-5)^2=25\) . Since by definition we must find a non-negative number, then \(-5\) is not suitable, therefore, \(\sqrt(25)=5\) (since \(25=5^2\) ).
Finding the value of \(\sqrt a\) is called taking the square root of the number \(a\) , and the number \(a\) is called the radical expression.

\(\bullet\) Based on the definition, expression \(\sqrt(-25)\), \(\sqrt(-4)\), etc. don't make sense.
Fact 2. For quick calculations, it will be useful to learn the table of squares of natural numbers from \(1\) to \(20\) :

\[\begin(array)(|ll|) \hline 1^2=1 & \quad11^2=121 \\ 2^2=4 & \quad12^2=144\\ 3^2=9 & \quad13 ^2=169\\ 4^2=16 & \quad14^2=196\\ 5^2=25 & \quad15^2=225\\ 6^2=36 & \quad16^2=256\\ 7^ 2=49 & \quad17^2=289\\ 8^2=64 & \quad18^2=324\\ 9^2=81 & \quad19^2=361\\ 10^2=100& \quad20^2= 400\\ \hline \end(array)\]
Fact 3.
What operations can you do with square roots? \(\bullet\) The sum or difference of square roots is NOT EQUAL to the square root of the sum or difference, that is Thus, if you need to calculate, for example, \(\sqrt(25)+\sqrt(49)\) , then initially you must find the values of \(\sqrt(25)\) and \(\sqrt(49)\ ) and then fold them. Hence, \[\sqrt(25)+\sqrt(49)=5+7=12\] If the values \(\sqrt a\) or \(\sqrt b\) cannot be found when adding \(\sqrt a+\sqrt b\), then such an expression is not transformed further and remains as it is. For example, in the sum \(\sqrt 2+ \sqrt (49)\) we can find \(\sqrt(49)\) is \(7\) , but \(\sqrt 2\) cannot be transformed in any way, That's why \(\sqrt 2+\sqrt(49)=\sqrt 2+7\). Unfortunately, this expression cannot be simplified further\(\bullet\) The product/quotient of square roots is equal to the square root of the product/quotient, that is \[\sqrt a\cdot \sqrt b=\sqrt(ab)\quad \text(s)\quad \sqrt a:\sqrt b=\sqrt(a:b)\] (provided that both sides of the equalities make sense)
Example: \(\sqrt(32)\cdot \sqrt 2=\sqrt(32\cdot 2)=\sqrt(64)=8\); \(\sqrt(768):\sqrt3=\sqrt(768:3)=\sqrt(256)=16\); \(\sqrt((-25)\cdot (-64))=\sqrt(25\cdot 64)=\sqrt(25)\cdot \sqrt(64)= 5\cdot 8=40\).
\(\bullet\) Using these properties, it is convenient to find square roots of large numbers by factoring them.
Let's look at an example. Let's find \(\sqrt(44100)\) . Since \(44100:100=441\) , then \(44100=100\cdot 441\) . According to the criterion of divisibility, the number \(441\) is divisible by \(9\) (since the sum of its digits is 9 and is divisible by 9), therefore, \(441:9=49\), that is, \(441=9\ cdot 49\) . Thus we got:\[\sqrt(44100)=\sqrt(9\cdot 49\cdot 100)= \sqrt9\cdot \sqrt(49)\cdot \sqrt(100)=3\cdot 7\cdot 10=210\] Let's look at another example:
\[\sqrt(\dfrac(32\cdot 294)(27))= \sqrt(\dfrac(16\cdot 2\cdot 3\cdot 49\cdot 2)(9\cdot 3))= \sqrt( \ dfrac(16\cdot4\cdot49)(9))=\dfrac(\sqrt(16)\cdot \sqrt4 \cdot \sqrt(49))(\sqrt9)=\dfrac(4\cdot 2\cdot 7)3 =\dfrac(56)3\] \ \(\bullet\) Let's show how to enter numbers under the square root sign using the example of the expression \(5\sqrt2\) (short notation for the expression \(5\cdot \sqrt2\)). Since \(5=\sqrt(25)\) , then
Note also that, for example,
1) \(\sqrt2+3\sqrt2=4\sqrt2\) ,
2) \(5\sqrt3-\sqrt3=4\sqrt3\)

Why is that? Let's explain using example 1). As you already understand, we cannot somehow transform the number \(\sqrt2\). Let's imagine that \(\sqrt2\) is some number \(a\) . Accordingly, the expression \(\sqrt2+3\sqrt2\) is nothing more than \(a+3a\) (one number \(a\) plus three more of the same numbers \(a\)). And we know that this is equal to four such numbers \(a\) , that is, \(4\sqrt2\) .

Fact 4.
\(\bullet\) They often say “you can’t extract the root” when you can’t get rid of the sign \(\sqrt () \ \) of the root (radical) when finding the value of a number. For example, you can take the root of the number \(16\) because \(16=4^2\) , therefore \(\sqrt(16)=4\) . But it is impossible to extract the root of the number \(3\), that is, to find \(\sqrt3\), because there is no number that squared will give \(3\) .
Such numbers (or expressions with such numbers) are irrational. For example, numbers \(\sqrt3, \ 1+\sqrt2, \ \sqrt(15)\) and so on. are irrational.
Also irrational are the numbers \(\pi\) (the number “pi”, approximately equal to \(3.14\)), \(e\) (this number is called the Euler number, it is approximately equal to \(2.7\)) etc.
\(\bullet\) Please note that any number will be either rational or irrational. And together all rational and all irrational numbers form a set called a set of real numbers. This set is denoted by the letter \(\mathbb(R)\) .
This means that all the numbers that we currently know are called real numbers.

Fact 5.
\(\bullet\) The modulus of a real number \(a\) is a non-negative number \(|a|\) equal to the distance from the point \(a\) to \(0\) on the real line. For example, \(|3|\) and \(|-3|\) are equal to 3, since the distances from the points \(3\) and \(-3\) to \(0\) are the same and equal to \(3 \) .
\(\bullet\) If \(a\) is a non-negative number, then \(|a|=a\) .
Example: \(|5|=5\) ; \(\qquad |\sqrt2|=\sqrt2\) .
\(\bullet\) If \(a\) is a negative number, then \(|a|=-a\) . Example: \(|-5|=-(-5)=5\) ;.
\(\qquad |-\sqrt3|=-(-\sqrt3)=\sqrt3\)
They say that for negative numbers the modulus “eats” the minus, while positive numbers, as well as the number \(0\), are left unchanged by the modulus. This rule only applies to numbers. If under your modulus sign there is an unknown \(x\) (or some other unknown), for example, \(|x|\) , about which we do not know whether it is positive, zero or negative, then get rid of the modulus we can not. In this case, this expression remains the same: \(|x|\) . \(\bullet\) The following formulas hold: \[(\large(\sqrt(a^2)=|a|))\]\[(\large((\sqrt(a))^2=a)), \text( provided ) a\geqslant 0\]
Very often the following mistake is made: they say that \(\sqrt(a^2)\) and \((\sqrt a)^2\) are one and the same. This is only true if \(a\) is a positive number or zero. But if \(a\) is a negative number, then this is false. It is enough to consider this example. Let's take instead of \(a\) the number \(-1\) . Then \(\sqrt((-1)^2)=\sqrt(1)=1\) , but the expression \((\sqrt (-1))^2\) does not exist at all (after all, it is impossible to use the root sign put negative numbers!). Therefore, we draw your attention to the fact that \(\sqrt(a^2)\) is not equal to \((\sqrt a)^2\) ! Example: 1)\(\sqrt(\left(-\sqrt2\right)^2)=|-\sqrt2|=\sqrt2\)<0\) ;

, because \(-\sqrt2 \(\phantom(00000)\) 2) \((\sqrt(2))^2=2\) .
\(\bullet\) Since \(\sqrt(a^2)=|a|\) , then \[\sqrt(a^(2n))=|a^n|\]
(the expression \(2n\) denotes an even number)
That is, when extracting the root of a number that is to some degree, this degree is halved.
Example:
1) \(\sqrt(4^6)=|4^3|=4^3=64\)

2) \(\sqrt((-25)^2)=|-25|=25\) (note that if the module is not supplied, it turns out that the root of the number is equal to \(-25\) ; but we remember , that by definition of a root this cannot happen: when extracting a root, we should always get a positive number or zero)
3) \(\sqrt(x^(16))=|x^8|=x^8\) (since any number to an even power is non-negative)
Fact 6.<\sqrt b\) , то \(a(the expression \(2n\) denotes an even number)
How to compare two square roots? \(\bullet\) For square roots it is true: if \(\sqrt a 1) compare \(\sqrt(50)\) and \(6\sqrt2\) . First, let's transform the second expression into<72\) , то и \(\sqrt{50}<\sqrt{72}\) . Следовательно, \(\sqrt{50}<6\sqrt2\) .
\(\sqrt(36)\cdot \sqrt2=\sqrt(36\cdot 2)=\sqrt(72)\)
. Thus, since \(50<50<64\) , то \(7<\sqrt{50}<8\) , то есть число \(\sqrt{50}\) находится между числами \(7\) и \(8\) .
2) Between what integers is \(\sqrt(50)\) located? Since \(\sqrt(49)=7\) , \(\sqrt(64)=8\) , and \(49 3) Let's compare \(\sqrt 2-1\) and \(0.5\) . Let's assume that \(\sqrt2-1>0.5\) :<0,5\) .
Note that adding a certain number to both sides of the inequality does not affect its sign. Multiplying/dividing both sides of an inequality by a positive number also does not affect its sign, but multiplying/dividing by a negative number reverses the sign of the inequality!
You can square both sides of an equation/inequality ONLY IF both sides are non-negative. For example, in the inequality from the previous example you can square both sides, in the inequality \(-3<\sqrt2\) нельзя (убедитесь в этом сами)! \(\bullet\) It should be remembered that \[\begin(aligned) &\sqrt 2\approx 1.4\\ &\sqrt 3\approx 1.7 \end(aligned)\] Knowing the approximate meaning of these numbers will help you when comparing numbers!
\(\bullet\) In order to extract the root (if it can be extracted) from some large number that is not in the table of squares, you must first determine between which “hundreds” it is located, then – between which “tens”, and then determine the last digit of this number. Let's show how this works with an example.
Let's take \(\sqrt(28224)\) . We know that \(100^2=10\,000\), \(200^2=40\,000\), etc. Note that \(28224\) is between \(10\,000\) and \(40\,000\) . Therefore, \(\sqrt(28224)\) is between \(100\) and \(200\) .
Now let’s determine between which “tens” our number is located (that is, for example, between \(120\) and \(130\)). Also from the table of squares we know that \(11^2=121\) , \(12^2=144\) etc., then \(110^2=12100\) , \(120^2=14400 \) , \(130^2=16900\) , \(140^2=19600\) , \(150^2=22500\) , \(160^2=25600\) , \(170^2=28900 \) . So we see that \(28224\) is between \(160^2\) and \(170^2\) . Therefore, the number \(\sqrt(28224)\) is between \(160\) and \(170\) .
Let's try to determine the last digit. Let's remember what single-digit numbers, when squared, give \(4\) at the end? These are \(2^2\) and \(8^2\) . Therefore, \(\sqrt(28224)\) will end in either 2 or 8. Let's check this. Let's find \(162^2\) and \(168^2\) :
\(162^2=162\cdot 162=26224\)
\(168^2=168\cdot 168=28224\) .

In order to adequately solve the Unified State Examination in mathematics, you first need to study theoretical material, which introduces you to numerous theorems, formulas, algorithms, etc. At first glance, it may seem that this is quite simple. However, finding a source in which the theory for the Unified State Exam in mathematics is presented in an easy and understandable way for students with any level of training is, in fact, a rather difficult task. School textbooks cannot always be kept at hand. And finding basic formulas for the Unified State Exam in mathematics can be difficult even on the Internet.

Why is it so important to study theory in mathematics not only for those taking the Unified State Exam?

Because it broadens your horizons. Studying theoretical material in mathematics is useful for anyone who wants to get answers to a wide range of questions related to knowledge of the world around them. Everything in nature is ordered and has a clear logic. This is precisely what is reflected in science, through which it is possible to understand the world.
Because it develops intelligence. By studying reference materials for the Unified State Exam in mathematics, as well as solving various problems, a person learns to think and reason logically, to formulate thoughts competently and clearly. He develops the ability to analyze, generalize, and draw conclusions.

We invite you to personally evaluate all the advantages of our approach to systematization and presentation of educational materials.