It's time to disassemble root extraction methods. They are based on the properties of the roots, in particular, on the equality, which is true for any non-negative number b.
Below we will consider in turn the main methods of extracting roots.
Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.
If the tables of squares, cubes, etc. is not at hand, it is logical to use the method of extracting the root, which involves decomposing the root number into simple factors.
Separately, it is worth dwelling on, which is possible for roots with odd exponents.
Finally, consider a method that allows you to sequentially find the digits of the value of the root.
Let's get started.
Using a table of squares, a table of cubes, etc.
In the simplest cases, tables of squares, cubes, etc. allow extracting roots. What are these tables?
The table of squares of integers from 0 to 99 inclusive (shown below) consists of two zones. The first zone of the table is located on a gray background; by selecting a certain row and a certain column, it allows you to make a number from 0 to 99. For example, let's select a row of 8 tens and a column of 3 units, with this we fixed the number 83. The second zone occupies the rest of the table. Each of its cells is located at the intersection of a certain row and a certain column, and contains the square of the corresponding number from 0 to 99 . At the intersection of our chosen row of 8 tens and column 3 of one, there is a cell with the number 6889, which is the square of the number 83.
Tables of cubes, tables of fourth powers of numbers from 0 to 99 and so on are similar to the table of squares, only they contain cubes, fourth powers, etc. in the second zone. corresponding numbers.
Tables of squares, cubes, fourth powers, etc. allow you to extract square roots, cube roots, fourth roots, etc. respectively from the numbers in these tables. Let us explain the principle of their application in extracting roots.
Let's say we need to extract the root of the nth degree from the number a, while the number a is contained in the table of nth degrees. According to this table, we find the number b such that a=b n . Then 
, therefore, the number b will be the desired root of the nth degree.
As an example, let's show how the cube root of 19683 is extracted using the cube table. We find the number 19 683 in the table of cubes, from it we find that this number is a cube of the number 27, therefore, 
.
It is clear that tables of n-th degrees are very convenient when extracting roots. However, they are often not at hand, and their compilation requires a certain amount of time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, one has to resort to other methods of extracting the roots.
Decomposition of the root number into prime factors
A fairly convenient way to extract the root from a natural number (if, of course, the root is extracted) is to decompose the root number into prime factors. His the essence is as follows: after it is quite easy to represent it as a degree with the desired indicator, which allows you to get the value of the root. Let's explain this point.
Let the root of the nth degree be extracted from a natural number a, and its value is equal to b. In this case, the equality a=b n is true. The number b as any natural number can be represented as a product of all its prime factors p 1 , p 2 , …, p m in the form p 1 p 2 … p m , and the root number a in this case is represented as (p 1 p 2 ... p m) n . Since the decomposition of the number into prime factors is unique, the decomposition of the root number a into prime factors will look like (p 1 ·p 2 ·…·p m) n , which makes it possible to calculate the value of the root as .
Note that if the factorization of the root number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n , then the root of the nth degree from such a number a is not completely extracted.
Let's deal with this when solving examples.
Example.
Take the square root of 144 .
Solution.
If we turn to the table of squares given in the previous paragraph, it is clearly seen that 144=12 2 , from which it is clear that the square root of 144 is 12 .
But in the light of this point, we are interested in how the root is extracted by decomposing the root number 144 into prime factors. Let's take a look at this solution.
Let's decompose 144 to prime factors:
That is, 144=2 2 2 2 3 3 . Based on the resulting decomposition, the following transformations can be carried out: 144=2 2 2 2 3 3=(2 2) 2 3 2 =(2 2 3) 2 =12 2. Hence, 
.
Using the properties of the degree and properties of the roots, the solution could be formulated a little differently: .
Answer:
To consolidate the material, consider the solutions of two more examples.
Example.
Calculate the root value.
Solution.
The prime factorization of the root number 243 is 243=3 5 . Thus, 
.
Answer:
Example.
Is the value of the root an integer?
Solution.
To answer this question, let's decompose the root number into prime factors and see if it can be represented as a cube of an integer.
We have 285 768=2 3 3 6 7 2 . The resulting decomposition is not represented as a cube of an integer, since the degree of the prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 is not taken completely.
Answer:
No.
Extracting roots from fractional numbers
It's time to figure out how the root is extracted from a fractional number. Let the fractional root number be written as p/q . According to the property of the root of the quotient, the following equality is true. From this equality it follows fraction root rule: The root of a fraction is equal to the quotient of dividing the root of the numerator by the root of the denominator.
Let's look at an example of extracting a root from a fraction.
Example.
What is the square root of the common fraction 25/169.
Solution.
According to the table of squares, we find that the square root of the numerator of the original fraction is 5, and the square root of the denominator is 13. Then 
. This completes the extraction of the root from the ordinary fraction 25/169.
Answer:
The root of a decimal fraction or a mixed number is extracted after replacing the root numbers with ordinary fractions.
Example.
Take the cube root of the decimal 474.552.
Solution.
Let's represent the original decimal as a common fraction: 474.552=474552/1000 . Then 
. It remains to extract the cube roots that are in the numerator and denominator of the resulting fraction. Because 474 552=2 2 2 3 3 3 13 13 13=(2 3 13) 3 =78 3 and 1 000=10 3 , then 
And 
. It remains only to complete the calculations 
.
Answer:
.
Extracting the root of a negative number
Separately, it is worth dwelling on extracting roots from negative numbers. When studying roots, we said that when the exponent of the root is an odd number, then a negative number can be under the sign of the root. We gave such notations the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, we have 
. This equality gives rule for extracting odd roots from negative numbers: to extract the root of a negative number, you need to extract the root of the opposite positive number, and put a minus sign in front of the result.
Let's consider an example solution.
Example.
Find the root value.
Solution.
Let's transform the original expression so that a positive number appears under the root sign: 
. Now we replace the mixed number with an ordinary fraction: 
. We apply the rule of extracting the root from an ordinary fraction: 
. It remains to calculate the roots in the numerator and denominator of the resulting fraction: 
.
Here is a summary of the solution: 
.
Answer:
.
Bitwise Finding the Root Value
In the general case, under the root there is a number that, using the techniques discussed above, cannot be represented as the nth power of any number. But at the same time, there is a need to know the value of a given root, at least up to a certain sign. In this case, to extract the root, you can use an algorithm that allows you to consistently obtain a sufficient number of values of the digits of the desired number.
The first step of this algorithm is to find out what is the most significant bit of the root value. To do this, the numbers 0, 10, 100, ... are successively raised to the power n until a number exceeding the root number is obtained. Then the number that we raised to the power of n in the previous step will indicate the corresponding high order.
For example, consider this step of the algorithm when extracting the square root of five. We take the numbers 0, 10, 100, ... and square them until we get a number greater than 5 . We have 0 2 =0<5 , 10 2 =100>5 , which means that the most significant digit will be the units digit. The value of this bit, as well as lower ones, will be found in the next steps of the root extraction algorithm.
All the following steps of the algorithm are aimed at successive refinement of the value of the root due to the fact that the values of the next digits of the desired value of the root are found, starting from the highest and moving to the lowest. For example, the value of the root in the first step is 2 , in the second - 2.2 , in the third - 2.23 , and so on 2.236067977 ... . Let us describe how the values of the bits are found.
Finding bits is carried out by enumeration of their possible values 0, 1, 2, ..., 9 . In this case, the nth powers of the corresponding numbers are calculated in parallel, and they are compared with the root number. If at some stage the value of the degree exceeds the radical number, then the value of the digit corresponding to the previous value is considered found, and the transition to the next step of the root extraction algorithm is made, if this does not happen, then the value of this digit is 9 .
Let us explain all these points using the same example of extracting the square root of five.
First, find the value of the units digit. We will iterate over the values 0, 1, 2, …, 9 , calculating respectively 0 2 , 1 2 , …, 9 2 until we get a value greater than the radical number 5 . All these calculations are conveniently presented in the form of a table: 
So the value of the units digit is 2 (because 2 2<5
, а 2 3 >5 ). Let's move on to finding the value of the tenth place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the obtained values \u200b\u200bwith the root number 5: 
Since 2.2 2<5
, а 2,3 2 >5 , then the value of the tenth place is 2 . You can proceed to finding the value of the hundredths place: 
So the next value of the root of five is found, it is equal to 2.23. And so you can continue to find values further: 2,236, 2,2360, 2,23606, 2,236067, … .
To consolidate the material, we will analyze the extraction of the root with an accuracy of hundredths using the considered algorithm.
First, we define the senior digit. To do this, we cube the numbers 0, 10, 100, etc. until we get a number greater than 2,151.186 . We have 0 3 =0<2 151,186 , 10 3 =1 000<2151,186 , 100 3 =1 000 000>2 151.186 , so the most significant digit is the tens digit.
Let's define its value. 
Since 10 3<2 151,186
, а 20 3 >2,151.186 , then the value of the tens digit is 1 . Let's move on to units. 
Thus, the value of the ones place is 2 . Let's move on to ten. 
Since even 12.9 3 is less than the radical number 2 151.186 , the value of the tenth place is 9 . It remains to perform the last step of the algorithm, it will give us the value of the root with the required accuracy. 
At this stage, the value of the root is found up to hundredths: 
.
In conclusion of this article, I would like to say that there are many other ways to extract roots. But for most tasks, those that we studied above are sufficient.
Bibliography.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).
Bibliographic description: Pryamostanov S. M., Lysogorova L. V. Methods for extracting a square root // Young scientist. 2017. №2.2. S. 76-77..02.2019).
Keywords : square root, square root extraction.
At the lessons of mathematics, I got acquainted with the concept of a square root, and the operation of extracting a square root. I became interested in extracting the square root is only possible using a table of squares, using a calculator, or is there a way to extract it manually. I have found several ways: the Ancient Babylon formula, through the solution of equations, the method of discarding the full square, Newton's method, the geometric method, the graphic method (, ), the guessing method, the odd number subtraction method.
Consider the following methods:
Let's decompose into prime factors using the signs of divisibility 27225=5*5*3*3*11*11. Thus
- TO Canadian method. This fast method was discovered by young scientists at one of Canada's leading universities in the 20th century. Its accuracy is no more than two or three decimal places.
where x is the number to take the root from, c is the number of the nearest square), for example:
=5,92
- column. This method allows you to find the approximate value of the root of any real number with any predetermined accuracy. The disadvantages of the method include the increasing complexity of the calculation with an increase in the number of digits found. To manually extract the root, a notation similar to division by a column is used.
Square Root Algorithm
1. From a comma separately fractional and separately whole parts are divided on the edge of two numbers in each face ( kiss part - from right to left; fractional- from left to right). It is possible that the integer part may contain one digit, and the fractional part may contain zeros.
2. Extraction starts from left to right, and we select a number whose square does not exceed the number in the first face. We square this number and write it under the number in the first face.
3. We find the difference between the number in the first face and the square of the selected first number.
4. To the resulting difference we demolish the next face, the resulting number will be divisible. We form divider. We double the first selected digit of the answer (multiply by 2), we get the number of tens of the divisor, and the number of units should be such that its product by the whole divisor does not exceed the dividend. We write down the selected number in the answer.
5. To the resulting difference, we demolish the next face and perform actions according to the algorithm. If this face turns out to be the face of the fractional part, then put a comma in the answer. (Fig. 1.)
|
|
|
In this way, you can extract numbers with different accuracy, for example, with an accuracy of thousandths. (Fig.2)
Considering the various methods of extracting the square root, we can conclude: in each case, you need to decide on the choice of the most effective one in order to spend less time on solving
Literature:
- Kiselev A. Elements of Algebra and Analysis. Part one.-M.-1928
Keywords: square root, square root.
Annotation: The article describes methods for extracting a square root, and provides examples of extracting roots.
Root formulas. properties of square roots.
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
In the previous lesson, we figured out what a square root is. It's time to figure out what are formulas for roots, what are root properties and what can be done about it all.
Root Formulas, Root Properties, and Rules for Actions with Roots- it's essentially the same thing. There are surprisingly few formulas for square roots. Which, of course, pleases! Rather, you can write a lot of all sorts of formulas, but only three are enough for practical and confident work with roots. Everything else flows from these three. Although many stray in the three formulas of the roots, yes ...
Let's start with the simplest. Here she is:
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. Learning - with interest!)
you can get acquainted with functions and derivatives.
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 the fundamental sciences of our days. At first, these were pieces of elementary mathematics, which made it possible to connect 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 is currently 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 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.
The origin of this term is 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 (one can trace a pattern - everything that has a "root" semantic load is consonant, be it 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. The “tick” √, familiar to the modern look, 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 is relevant only for the 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.
In general, which is valid for determining an algebraic root, the value of an 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 affection for 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 equals zero. The number of moves will eventually become the desired number. For example, calculating the square root of 25:
The next odd number is 11, the remainder is: 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 function under consideration 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.
And do you have dependency on the calculator? Or do you think that, except with a calculator or using a table of squares, it is very difficult to calculate, for example,.
It happens that schoolchildren are tied to a calculator and even multiply 0.7 by 0.5 by pressing the cherished buttons. They say, well, I still know how to calculate, but now I’ll save time ... There will be an exam ... then I’ll tense up ...
So the fact is that there will be plenty of “tense moments” at the exam anyway ... As they say, water wears away a stone. So on the exam, little things, if there are a lot of them, can knock you down ...
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 by all means (for example, when calculating the discriminant) need to calculate the square root of 86436.
We will decompose the number 86436 into prime factors. We divide by 2, we get 43218; again we divide by 2, - we get 21609. The number is not divisible by 2 more. But since the sum of the digits is divisible by 3, then the number itself is divisible by 3 (generally speaking, it can be seen that it is also divisible by 9). . Once again we divide by 3, we get 2401. 2401 is not completely divisible by 3. Not divisible by five (does not end with 0 or 5).
We suspect divisibility by 7. Indeed, a ,
So, full order!
Case 2
Let us need to calculate . It is inconvenient to act in the same way as described above. Trying to factorize...
The number 1849 is not completely 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 not 5 or 0) ...
It is not completely divisible by 7, it is not divisible by 11, it is not divisible by 13 ... Well, how long will it take us to go through all the prime numbers like this?
Let's argue a little differently.
We understand that
We narrowed down the search. Now we sort through the numbers from 41 to 49. Moreover, it is clear that since the last digit of the number is 9, then it is worth stopping at options 43 or 47 - only these numbers, when squared, will give the last digit 9.
Well, here already, 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.
