Mathematics was born when a person became aware of himself and began to position himself as an autonomous unit of the world. The desire to measure, compare, calculate what surrounds you is what underlay one of fundamental sciences our days. At first, these were pieces of elementary mathematics, which made it possible to associate numbers with their physical expressions, later the conclusions began to be presented only theoretically (due to their abstractness), but after a while, as one scientist put it, "mathematics reached the ceiling of complexity when all numbers." The concept of "square root" appeared at a time when it could be easily supported by empirical data, going beyond the plane of calculations.

How it all started

The first mention of the root, which on this moment denoted as √, was recorded in the writings of the Babylonian mathematicians, who laid the foundation for modern arithmetic. Of course, they looked a little like the current form - the scientists of those years first used bulky tablets. But in the second millennium BC. e. they came up with an approximate calculation formula that showed how to take the square root. The photo below shows a stone on which Babylonian scientists carved the output process √2, and it turned out to be so correct that the discrepancy in the answer was found only in the tenth decimal place.

In addition, the root was used if it was necessary to find the side of a triangle, provided that the other two were known. Well, when solving quadratic equations, there is no escape from extracting the root.

Along with the Babylonian works, the object of the article was also studied in the Chinese work "Mathematics in Nine Books", and the ancient Greeks came to the conclusion that any number from which the root is not extracted without a remainder gives an irrational result.

Origin this term associated with the Arabic representation of the number: ancient scientists believed that the square of an arbitrary number grows from the root, like a plant. In Latin, this word sounds like radix (you can trace a pattern - everything that has a "root" semantic load, consonantly, whether it be radish or sciatica).

Scientists of subsequent generations picked up this idea, designating it as Rx. For example, in the 15th century, in order to indicate that the square root is taken from an arbitrary number a, they wrote R 2 a. Habitual modern look"tick" √ appeared only in the 17th century thanks to Rene Descartes.

Our days

Mathematically, the square root of y is the number z whose square is y. In other words, z 2 =y is equivalent to √y=z. However this definition relevant only for arithmetic root, since it implies a non-negative value of the expression. In other words, √y=z, where z is greater than or equal to 0.

AT general case, which acts to determine algebraic root, the value of the expression can be either positive or negative. Thus, due to the fact that z 2 =y and (-z) 2 =y, we have: √y=±z or √y=|z|.

Due to the fact that love for mathematics has only increased with the development of science, there are various manifestations of attachment to it, not expressed in dry calculations. For example, along with such interesting events as the day of Pi, the holidays of the square root are also celebrated. They are celebrated nine times in a hundred years, and are determined according to the following principle: the numbers that denote the day and month in order must be the square root of the year. So, next time this holiday will be celebrated on April 4, 2016.

Properties of the square root on the field R

Almost all mathematical expressions have a geometric basis, this fate did not pass and √y, which is defined as the side of a square with area y.

How to find the root of a number?

There are several calculation algorithms. The simplest, but at the same time quite cumbersome, is the usual arithmetic calculation, which is as follows:

1) from the number whose root we need, odd numbers are subtracted in turn - until the remainder of the output is less than the subtracted one or even zero. The number of moves will eventually become the desired number. For example, the calculation square root out of 25:

Following odd number is 11, we have the following remainder: 1<11. Количество ходов - 5, так что корень из 25 равен 5. Вроде все легко и просто, но представьте, что придется вычислять из 18769?

For such cases, there is a Taylor series expansion:

√(1+y)=∑((-1) n (2n)!/(1-2n)(n!) 2 (4 n))y n , where n takes values ​​from 0 to

+∞, and |y|≤1.

Graphic representation of the function z=√y

Consider an elementary function z=√y on the field of real numbers R, where y is greater than or equal to zero. Her chart looks like this:

The curve grows from the origin and necessarily crosses the point (1; 1).

Properties of the function z=√y on the field of real numbers R

1. The domain of definition of the considered function is the interval from zero to plus infinity (zero is included).

2. The range of values ​​of the considered function is the interval from zero to plus infinity (zero is again included).

3. The function takes the minimum value (0) only at the point (0; 0). There is no maximum value.

4. The function z=√y is neither even nor odd.

5. The function z=√y is not periodic.

6. There is only one point of intersection of the graph of the function z=√y with the coordinate axes: (0; 0).

7. The intersection point of the graph of the function z=√y is also the zero of this function.

8. The function z=√y is continuously growing.

9. The function z=√y takes only positive values, therefore, its graph occupies the first coordinate angle.

Options for displaying the function z=√y

In mathematics, to facilitate the calculation of complex expressions, the power form of writing the square root is sometimes used: √y=y 1/2. This option is convenient, for example, in raising a function to a power: (√y) 4 =(y 1/2) 4 =y 2 . This method is also a good representation for differentiation with integration, since thanks to it the square root is represented by an ordinary power function.

And in programming, the replacement for the symbol √ is the combination of letters sqrt.

It is worth noting that in this area the square root is in great demand, as it is part of most of the geometric formulas necessary for calculations. The counting algorithm itself is quite complicated and is based on recursion (a function that calls itself).

The square root in the complex field C

By and large, it was the subject of this article that stimulated the discovery of the field of complex numbers C, since mathematicians were haunted by the question of obtaining an even degree root from a negative number. This is how the imaginary unit i appeared, which is characterized by a very interesting property: its square is -1. Thanks to this, quadratic equations and with a negative discriminant got a solution. In C, for the square root, the same properties are relevant as in R, the only thing is that the restrictions on the root expression are removed.

Exponentiation implies that a given number must be multiplied by itself a certain number of times. For example, raising the number 2 to the fifth power would look like this:

The number that needs to be multiplied by itself is called the base of the degree, and the number of multiplications is its exponent. Raising to a power corresponds to two opposite actions: finding the exponent and finding the base.

root extraction

Finding the base of an exponent is called root extraction. This means that you need to find the number that needs to be raised to the power of n to get the given one.

For example, it is necessary to extract the 4th root of the number 16, i.e. to determine, you need to multiply by itself 4 times to get 16 in the end. This number is 2.

Such an arithmetic operation is written using a special sign - the radical: √, above which the exponent is indicated on the left.

arithmetic root

If the exponent is an even number, then the root can be two numbers with the same modulus, but c is positive and negative. So, in the given example it can be numbers 2 and -2.

The expression must be unambiguous, i.e. have one result. For this, the concept of an arithmetic root was introduced, which can only be a positive number. An arithmetic root cannot be less than zero.

Thus, in the example considered above, only the number 2 will be the arithmetic root, and the second answer - -2 - is excluded by definition.

Square root

For some degrees that are used more often than others, there are special names that are originally associated with geometry. We are talking about raising to the second and third degrees.

To the second power, the length of the side of the square when you need to calculate its area. If you need to find the volume of a cube, the length of its edge is raised to the third power. Therefore, it is called the square of the number, and the third is called the cube.

Accordingly, the root of the second degree is called the square, and the root of the third degree is called the cubic. The square root is the only one of the roots that does not have an exponent above the radical when written:

So, the arithmetic square root of a given number is a positive number that must be raised to the second power to get the given number.

The area of ​​a square plot of land is 81 dm². Find his side. Suppose the length of the side of the square is X decimetres. Then the area of ​​the plot is X² square decimetres. Since, according to the condition, this area is 81 dm², then X² = 81. The length of the side of a square is a positive number. A positive number whose square is 81 is the number 9. When solving the problem, it was required to find the number x, the square of which is 81, i.e. solve the equation X² = 81. This equation has two roots: x 1 = 9 and x 2 \u003d - 9, since 9² \u003d 81 and (- 9)² \u003d 81. Both numbers 9 and - 9 are called the square roots of the number 81.

Note that one of square roots X= 9 is a positive number. It is called the arithmetic square root of 81 and is denoted √81, so √81 = 9.

Arithmetic square root of a number a is a non-negative number whose square is equal to a.

For example, the numbers 6 and -6 are the square roots of 36. The number 6 is the arithmetic square root of 36, since 6 is a non-negative number and 6² = 36. The number -6 is not an arithmetic root.

Arithmetic square root of a number a denoted as follows: √ a.

The sign is called the arithmetic square root sign; a is called a root expression. Expression √ a read like this: the arithmetic square root of a number a. For example, √36 = 6, √0 = 0, √0.49 = 0.7. In cases where it is clear that we are talking about an arithmetic root, they briefly say: "the square root of a«.

The act of finding the square root of a number is called taking the square root. This action is the reverse of squaring.

Any number can be squared, but not every number can be square roots. For example, it is impossible to extract the square root of the number - 4. If such a root existed, then, denoting it with the letter X, we would get the wrong equality x² \u003d - 4, since there is a non-negative number on the left, and a negative one on the right.

Expression √ a only makes sense when a ≥ 0. The definition of the square root can be briefly written as: √ a ≥ 0, (√a)² = a. Equality (√ a)² = a valid for a ≥ 0. Thus, to make sure that the square root of a non-negative number a equals b, i.e., that √ a =b, you need to check that the following two conditions are met: b ≥ 0, b² = a.

The square root of a fraction

Let's calculate . Note that √25 = 5, √36 = 6, and check if the equality holds.

As and , then the equality is true. So, .

Theorem: If a a≥ 0 and b> 0, that is, the root of the fraction equal to the root from the numerator divided by the root of the denominator. It is required to prove that: and .

Since √ a≥0 and √ b> 0, then .

By the property of raising a fraction to a power and determining the square root the theorem is proven. Let's look at a few examples.

Calculate , according to the proven theorem .

Second example: Prove that , if a ≤ 0, b < 0. .

Another example: Calculate .

.

Square root transformation

Taking the multiplier out from under the sign of the root. Let an expression be given. If a a≥ 0 and b≥ 0, then by the theorem on the root of the product, we can write:

Such a transformation is called factoring out the root sign. Consider an example;

Calculate at X= 2. Direct substitution X= 2 in radical expression leads to complex calculations. These calculations can be simplified if we first remove the factors from under the root sign: . Now substituting x = 2, we get:.

So, when taking out the factor from under the root sign, they represent the root expression in the form of a product in which one or more factors are squares non-negative numbers. The root product theorem is then applied and the root of each factor is taken. Consider an example: Simplify the expression A = √8 + √18 - 4√2 by taking out the factors from under the root sign in the first two terms, we get:. We emphasize that the equality valid only when a≥ 0 and b≥ 0. if a < 0, то .

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Students always ask: “Why can't I use a calculator on a math exam? How to extract the square root of a number without a calculator? Let's try to answer this question.

How to extract the square root of a number without the help of a calculator?

Action square root extraction the opposite of squaring.

√81= 9 9 2 =81

If from positive number take the square root and square the result, we get the same number.

From small numbers that are perfect squares natural numbers, for example 1, 4, 9, 16, 25, ..., 100 square roots can be extracted verbally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract the square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400, you can extract using the selection method using some tips. Let's try an example to consider this method.

Example: Extract the root of the number 676.

We notice that 20 2 \u003d 400, and 30 2 \u003d 900, which means 20< √676 < 900.

Exact squares of natural numbers end in 0; one; 4; 5; 6; nine.
The number 6 is given by 4 2 and 6 2 .
So, if the root is taken from 676, then it is either 24 or 26.

It remains to check: 24 2 = 576, 26 2 = 676.

Answer: √676 = 26 .

More example: √6889 .

Since 80 2 \u003d 6400, and 90 2 \u003d 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is either 83 or 87.

Check: 83 2 = 6889.

Answer: √6889 = 83 .

If you find it difficult to solve by the selection method, then you can factorize the root expression.

For example, find √893025.

Let's factorize the number 893025, remember, you did it in the sixth grade.

We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.

More example: √20736. Let's factorize the number 20736:

We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.

Of course, factoring requires knowledge of divisibility criteria and factoring skills.

And finally, there is square root rule. Let's look at this rule with an example.

Calculate √279841.

To extract the root of a multi-digit integer, we split it from right to left into faces containing 2 digits each (there may be one digit in the left extreme face). Write like this 27'98'41

To get the first digit of the root (5), we extract the square root of the largest exact square contained in the first left face (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is attributed (demolished) to the difference.
To the left of the received number 298, they write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), experience the quotient (102 ∙ 2 = 204 should be no more than 298) and write (2) after the first digit of the root.
Then the resulting quotient 204 is subtracted from 298, and the next facet (41) is attributed (demolished) to the difference (94).
To the left of the resulting number 9441, they write the double product of the digits of the root (52 ∙ 2 = 104), divide by this product the number of all tens of the number 9441 (944/104 ≈ 9), experience the quotient (1049 ∙ 9 = 9441) should be 9441 and write it down (9) after the second digit of the root.

We got the answer √279841 = 529.

Similarly extract roots of decimals. Only the radical number must be divided into faces so that the comma is between the faces.

Example. Find the value √0.00956484.

You just have to remember that if decimal has an odd number of decimal places, it does not take exactly the square root.

So, now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn how to solve problems, you need to solve them. And if you have any questions, sign up for my lessons.

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