How to find out what the root is equal to. Square root. Actions with square roots. Module. Comparison of square roots
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Students always ask: “Why can’t I use a calculator in the 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 inverse to the action of squaring.
√81= 9 9 2 =81
If you take the square root of a positive number and square the result, you get the same number.
From small numbers that are exact squares of natural numbers, for example 1, 4, 9, 16, 25, ..., 100, square roots can be extracted orally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400 you can extract them using the selection method using some tips. Let's try to look at this method with an example.
Example: Extract the root of the number 676.
We notice that 20 2 = 400, and 30 2 = 900, which means 20< √676 < 900.
Exact squares of natural numbers end in 0; 1; 4; 5; 6; 9.
The number 6 is given by 4 2 and 6 2.
This means that 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 = 6400, and 90 2 = 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is equal to either 83 or 87.
Let's check: 83 2 = 6889.
Answer: √6889 = 83 .
If you find it difficult to solve using the selection method, you can factor the radical expression.
For example, find √893025.
Let's factor the number 893025, remember, you did this in the sixth grade.
We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.
More example: √20736. Let's factor the number 20736:
We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.
Of course, factorization requires knowledge of divisibility signs and factorization skills.
And finally, there is rule for extracting square roots. Let's get acquainted with this rule with examples.
Calculate √279841.
To extract the root of a multi-digit integer, we divide it from right to left into faces containing 2 digits (the leftmost edge may contain one digit). We write it like this: 27’98’41
To obtain the first digit of the root (5), we take the square root of the largest perfect square contained in the first face on the left (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is added to the difference (subtracted).
To the left of the resulting number 298, write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), test 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 edge (41) is added to the difference (94).
To the left of the resulting number 9441, write the double product of the digits of the root (52 ∙2 = 104), divide the number of all tens of the number 9441 (944/104 ≈ 9) by this product, test the quotient (1049 ∙9 = 9441) should be 9441 and write it down (9) after the second digit of the root.
We received the answer √279841 = 529.
Extract similarly roots of decimal fractions. Only the radical number must be divided into faces so that the comma is between the faces.
Example. Find the value √0.00956484.
Just remember that if a decimal fraction has an odd number of decimal places, the square root cannot be taken from it.
So now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn to solve problems, you need to solve them. And if you have any questions, .
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There are several methods for calculating the square root without a calculator.
How to find the root of a number - 1 way
- One method is to factor the number under the root. These components, when multiplied, form a radical value. The accuracy of the result depends on the number under the root.
- For example, if you take the number 1,600 and start factoring it, the reasoning will be structured as follows: this number is a multiple of 100, which means it can be divided by 25; since the root of the number 25 is taken, the number is square and suitable for further calculations; when dividing, we get another number - 64. This number is also square, so the root can be extracted well; After these calculations, under the root, you can write the number 1600 as the product of 25 and 64.
- One of the rules for extracting a root states that the root of the product of factors is equal to the number that is obtained by multiplying the roots of each factor. This means that: √(25*64) = √25 * √64. If we take the roots from 25 and 64, we get the following expression: 5 * 8 = 40. That is, the square root of the number 1600 is 40.
- But it happens that the number under the root cannot be decomposed into two factors, from which the whole root is extracted. Typically this can only be done for one of the multipliers. Therefore, most often it is not possible to find an absolutely exact answer in such an equation.
- In this case, only an approximate value can be calculated. Therefore, you need to take the root of the multiplier, which is a square number. This value is then multiplied by the root of the second number that is not the square term of the equation.
- It looks like this, for example, let’s take the number 320. It can be decomposed into 64 and 5. You can extract the whole root from 64, but not from 5. Therefore, the expression will look like this: √320 = √(64*5) = √64*√5 = 8√5.
- If necessary, you can find the approximate value of this result by calculating
√5 ≈ 2.236, therefore √320 = 8 * 2.236 = 17.88 ≈ 18. - Also, the number under the root can be decomposed into several prime factors, and the same ones can be taken out from under it. Example: √75 = √(5*5*3) = 5√3 ≈ 8.66 ≈ 9.
How to find the root of a number - method 2
- Another way is to do long division. Division occurs in a similar way, but you just need to look for square numbers, from which you can then extract the root.
- In this case, we write the square number on top and subtract it on the left side, and the extracted root from below.
- Now the second value needs to be doubled and written from the bottom right in the form: number_x_=. The gaps must be filled in with a number that is less than or equal to the required value on the left - just like in normal division.
- If necessary, this result is again subtracted from the left. Such calculations continue until the result is achieved. You can also add zeros until you reach the desired number of decimal places.
It's time to sort it out root extraction methods. They are based on the properties of roots, in particular, on the equality, which is true for any non-negative number b.
Below we will look at the main methods of extracting roots one by one.
Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.
If tables of squares, cubes, etc. If you don’t have it at hand, it’s logical to use the method of extracting the root, which involves decomposing the radical number into prime factors.
It is worth special mentioning what is possible for roots with odd exponents.
Finally, let's consider a method that allows us to sequentially find the digits of the root value.
Let's get started.
Using a table of squares, a table of cubes, etc.
In the simplest cases, tables of squares, cubes, etc. allow you to extract 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 specific row and a specific column, it allows you to compose 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 cell 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 ones there is a cell with the number 6,889, 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. accordingly from the numbers in these tables. Let us explain the principle of their use when extracting roots.
Let's say we need to extract the nth root of the number a, while the number a is contained in the table of nth powers. Using 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 to use a cube table to extract the cube root of 19,683. We find the number 19,683 in the table of cubes, from it we find that this number is the cube of the number 27, therefore, 
.
It is clear that tables of nth powers are very convenient for extracting roots. However, they are often not at hand, and compiling them requires some time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, you have to resort to other methods of root extraction.
Factoring a radical number into prime factors
A fairly convenient way to extract the root of a natural number (if, of course, the root is extracted) is to decompose the radical number into prime factors. His the point is this: after that it is quite easy to represent it as a power with the desired exponent, which allows you to obtain the value of the root. Let's clarify this point.
Let the nth root of a natural number a be taken and its value equal b. In this case, the equality a=b n is true. The number b, like any natural number, can be represented as the product of all its prime factors p 1 , p 2 , …, p m in the form p 1 ·p 2 ·…·p m , and the radical number a in this case is represented as (p 1 ·p 2 ·…·p m) n . Since the decomposition of a number into prime factors is unique, the decomposition of the radical number a into prime factors will have the form (p 1 ·p 2 ·…·p m) n, which makes it possible to calculate the value of the root as.
Note that if the decomposition into prime factors of a radical number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n, then the nth root of such a number a is not completely extracted.
Let's figure this out when solving examples.
Example.
Take the square root of 144.
Solution.
If you look at the table of squares given in the previous paragraph, you can clearly see that 144 = 12 2, from which it is clear that the square root of 144 is 12.
But in light of this point, we are interested in how the root is extracted by decomposing the radical number 144 into prime factors. Let's 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 the properties of the roots, the solution could be formulated a little differently: .
Answer:
To consolidate the material, consider the solutions to two more examples.
Example.
Calculate the value of the root.
Solution.
The prime factorization of the radical number 243 has the form 243=3 5 . Thus, 
.
Answer:
Example.
Is the root value an integer?
Solution.
To answer this question, let's factor the radical 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 expansion cannot be represented as a cube of an integer, since the power of the prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 cannot be extracted completely.
Answer:
No.
Extracting roots from fractional numbers
It's time to figure out how to extract the root of a fractional number. Let the fractional radical number be written as p/q. According to the property of the root of a quotient, the following equality is true. From this equality it follows rule for extracting the root of a fraction: The root of a fraction is equal to the quotient of the root of the numerator divided 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.
Using the table of squares, we find that the square root of the numerator of the original fraction is equal to 5, and the square root of the denominator is equal to 13. Then 
. This completes the extraction of the root of the common fraction 25/169.
Answer:
The root of a decimal fraction or mixed number is extracted after replacing the radical numbers with ordinary fractions.
Example.
Take the cube root of the decimal fraction 474.552.
Solution.
Let's imagine the original decimal fraction as an ordinary 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 
. All that remains is to complete the calculations 
.
Answer:
.
Taking the root of a negative number
It is worthwhile to dwell on extracting roots from negative numbers. When studying roots, we said that when the root exponent is an odd number, then there can be a negative number under the root sign. We gave these entries the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, 
. This equality gives rule for extracting odd roots from negative numbers: to extract the root of a negative number, you need to take the root of the opposite positive number, and put a minus sign in front of the result.
Let's look at the example solution.
Example.
Find the value of the root.
Solution.
Let's transform the original expression so that there is a positive number under the root sign: 
. Now replace the mixed number with an ordinary fraction: 
. We apply the rule for extracting the root of an ordinary fraction: 
. It remains to calculate the roots in the numerator and denominator of the resulting fraction: 
.
Here is a short summary of the solution: 
.
Answer:
.
Bitwise determination of 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 in this case there is a need to know the meaning 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 sequentially obtain a sufficient number of digit values of the desired number.
The first step of this algorithm is to find out what the most significant bit of the root value is. To do this, the numbers 0, 10, 100, ... are sequentially raised to the power n until the moment when a number exceeds the radical number is obtained. Then the number that we raised to the power n at the previous stage will indicate the corresponding most significant digit.
For example, consider this step of the algorithm when extracting the square root of five. 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 the most significant digit will be the ones digit. The value of this bit, as well as the lower ones, will be found in the next steps of the root extraction algorithm.
All the following steps of the algorithm are aimed at sequentially clarifying the value of the root by finding the values of the next bits of the desired value of the root, starting with the highest one and moving to the lowest ones. For example, the value of the root at the first step turns out to be 2, at the second – 2.2, at the third – 2.23, and so on 2.236067977…. Let us describe how the values of the digits are found.
The digits are found by searching through 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 radical 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 is made to the next step of the root extraction algorithm; if this does not happen, then the value of this digit is equal to 9.
Let us explain these points using the same example of extracting the square root of five.
First we find the value of the units digit. We will go through the values 0, 1, 2, ..., 9, calculating 0 2, 1 2, ..., 9 2, respectively, until we get a value greater than the radical number 5. It is convenient to present all these calculations in the form of a table: 
So the value of the units digit is 2 (since 2 2<5
, а 2 3 >5). Let's move on to finding the value of the tenths place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the resulting values with the radical number 5: 
Since 2.2 2<5
, а 2,3 2 >5, then the value of the tenths place is 2. You can proceed to finding the value of the hundredths place: 
This is how the next value of the root of five was found, it is equal to 2.23. And so you can continue to find values: 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 determine the most significant 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 determine its value. 
Since 10 3<2 151,186
, а 20 3 >2 151.186, then the value of the tens place is 1. Let's move on to units. 
Thus, the value of the ones digit is 2. Let's move on to tenths. 
Since even 12.9 3 is less than the radical number 2 151.186, then the value of the tenths 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 accurate 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, the ones we studied above are sufficient.
References.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).
Sokolov Lev Vladimirovich, 8th grade student of the Municipal Educational Institution “Tugulymskaya V(S)OSH”
Purpose of the work: find and show those methods of extracting square roots that can be used without having a calculator at hand.
Download:
Preview:
Regional scientific and practical conference
students of Tugulym urban district
Finding square roots of large numbers without a calculator
Performer: Lev Sokolov,
MCOU "Tugulymskaya V(S)OSH",
8th grade
Head: Sidorova Tatyana
Nikolaevna
r.p. Tugulym, 2016
Introduction 3
Chapter 1. Method of factorization 4
Chapter 2. Extracting square roots with corner 4
Chapter 3. Method of using the table of squares of two-digit numbers 6
Chapter 4. Formula of Ancient Babylon 6
Chapter 6. Canadian method 7
Chapter 7. Guessing selection method 8
Chapter 8. Method of deductions for odd number 8
Conclusion 10
References 11
Appendix 12
Introduction
Relevance of the studyWhen I studied the topic of square roots this school year, I became interested in the question of how you can take the square root of large numbers without a calculator.
I became interested and decided to study this issue deeper than it is stated in the school curriculum, and also to prepare a mini-book with the simplest ways to extract square roots from large numbers without a calculator.
Purpose of the work: find and show those methods of extracting square roots that can be used without having a calculator at hand.
Tasks:
- Study the literature on this issue.
- Consider the features of each method found and its algorithm.
- Show practical application of acquired knowledge and evaluate
The degree of complexity in using various methods and algorithms.
- Create a mini-book on the most interesting algorithms.
Object of study:mathematical symbols are square roots.
Subject of research:Features of methods for extracting square roots without a calculator.
Research methods:
- Finding methods and algorithms for extracting square roots from large numbers without a calculator.
- Comparison of the found methods.
- Analysis of the obtained methods.
Everyone knows that taking the square root without a calculator is very difficult.
task. When we don’t have a calculator at hand, we start by using the selection method to try to remember the data from the table of squares of integers, but this does not always help. For example, a table of squares of integers does not answer questions such as, for example, extracting the root of 75, 37,885,108,18061 and others, even approximately.
Also, the use of a calculator is often prohibited during the OGE and Unified State Examinations.
tables of squares of integers, but you need to extract the root of 3136 or 7056, etc.
But while studying the literature on this topic, I learned that taking roots from such numbers
Perhaps without a table and a calculator, people learned long before the invention of the microcalculator. While researching this topic, I found several ways to solve this problem.
Chapter 1. Method of factorization into prime factors
To extract the square root, you can factor the number into its prime factors and take the square root of the product.
This method is usually used when solving problems with roots at school.
3136│2 7056│2
1568│2 3528│2
784│2 1764│2
392│2 882│2
196│2 441│3
98│2 147│3
49│7 49│7
7│7 7│7
√3136 = √2²∙2²∙2²∙7² = 2∙2∙2∙7 = 56 √3136 = √2²∙2²∙3²∙7² = 2∙2∙3∙7 = 84
Many people use it successfully and consider it the only one. Extracting the root by factorization is a time-consuming task, which also does not always lead to the desired result. Try taking the square root of 209764? Factoring into prime factors gives the product 2∙2∙52441. What to do next? Everyone faces this problem, and in their answer they calmly write down the remainder of the decomposition under the sign of the root. Of course, you can do the decomposition using trial and error and selection if you are sure that you will get a beautiful answer, but practice shows that very rarely tasks with complete decomposition are offered. More often than not, we see that the root cannot be completely extracted.
Therefore, this method only partially solves the problem of extraction without a calculator.
Chapter 2. Extracting square roots with a corner
To extract the square root using a corner andLet's look at the algorithm:
1st step. The number 8649 is divided into edges from right to left; each of which must contain two digits. We get two faces:.
2nd step. Taking the square root of the first face of 86, we getwith a disadvantage. The number 9 is the first digit of the root.
3rd step. The number 9 is squared (9 2
= 81) and subtract the number 81 from the first face, we get 86-81=5. The number 5 is the first remainder.
4th step. To the remainder 5 we add the second side 49, we get the number 549.
5th step . We double the first digit of the root 9 and, writing from the left, we get -18
We need to assign the largest digit to the number so that the product of the number we get by this digit would be either equal to the number 549 or less than 549. This is the number 3. It is found by selection: the number of tens of the number 549, that is, the number 54 divided by 18, we get 3, since 183 ∙ 3 = 549. The number 3 is the second digit of the root.
6th step. We find the remainder 549 – 549 = 0. Since the remainder is zero, we got the exact value of the root – 93.
Let me give you another example: extract √212521
Algorithm steps | Example | Comments |
|
Divide the number into groups of 2 digits each from right to left | 21’ 25’ 21 | The total number of groups formed determines the number of digits in the answer |
|
For the first group of numbers, select a number whose square will be the largest, but not exceed the numbers of the first group | 1 group – 21 4 2 =16 number - 4 | The number found is written in the first place in the answer. |
|
From the first group of numbers, subtract the square of the first digit of the answer found in step 2 | 21’ 25’ 21 | ||
To the remainder found in step 3, add the second group of numbers to the right (remove) | 21’ 25’ 21 16__ | ||
To the doubled first digit of the answer, add a digit to the right such that the product of the resulting number by this digit is the largest, but does not exceed the number found in step 4 | 4*2=8 number - 6 86*6=516 | The found number is written in the answer in second place |
|
From the number obtained in step 4, subtract the number obtained in step 5. Take the third group to the remainder | 21’ 25’ 21 | ||
To the doubled number consisting of the first two digits of the answer, add a digit to the right such that the product of the resulting number by this digit is the largest, but does not exceed the number obtained in step 6 | 46*2=92 number 1 921*1=921 | The found number is written in third place in the answer. |
|
Write down answer | √212521=461 |
Chapter 3. How to use the table of squares of two-digit numbers
I learned about this method from the Internet. The method is very simple and allows you to instantly extract the square root of any integer from 1 to 100 with an accuracy of tenths without a calculator. One condition for this method is the presence of a table of squares of numbers up to 99.
(It is in all 8th grade algebra textbooks, and is offered as reference material in the OGE exam.)
Open the table and check the speed of finding the answer. But first, a few recommendations: the leftmost column will be integers in the answer, the topmost line will be tenths in the answer. And then everything is simple: close the last two digits of the number in the table and find the one you need, not exceeding the radical number, and then follow the rules of this table.
Let's look at an example. Let's find the value √87.
We close the last two digits of all numbers in the table and find close ones for 87 - there are only two of them 86 49 and 88 37. But 88 is already a lot.
So, there is only one thing left - 8649.
The left column gives the answer 9 (these are integers), and the top line 3 (these are tenths). This means √87≈ 9.3. Let's check on MK √87 ≈ 9.327379.
Fast, simple, accessible during the exam. But it is immediately clear that roots larger than 100 cannot be extracted using this method. The method is convenient for tasks with small roots and in the presence of a table.
Chapter 4. Formula of Ancient Babylon
The ancient Babylonians used the following method to find the approximate value of the square root of their number x. They represented the number x as the sum of a 2 +b, where a 2 the closest exact square of the natural number a to the number x (a 2 . (1)
Using formula (1), we extract the square root, for example, from the number 28:
The result of extracting the root of 28 using MK is 5.2915026.
As you can see, the Babylonian method gives a good approximation to the exact value of the root.
Chapter 5. Method of discarding a complete square
(only for four-digit numbers)
It’s worth clarifying right away that this method is applicable only to extracting the square root of an exact square, and the finding algorithm depends on the value of the radical number.
- Extracting roots up to number 75 2 = 5625
For example: √¯3844 = √¯ 37 00 + 144 = 37 + 25 = 62.
We present the number 3844 as a sum by selecting the square 144 from this number, then discarding the selected square, tonumber of hundreds of the first term(37) we always add 25 . We get the answer 62.
This way you can only extract square roots up to 75 2 =5625!
2) Extracting roots after number 75 2 = 5625
How to verbally extract square roots from numbers greater than 75 2 =5625?
For example: √7225 = √ 70 00 + 225 = 70 + √225 = 70 + 15 = 85.
Let us explain, we will present 7225 as the sum of 7000 and the selected square 225. Thenadd the square root to the number of hundreds out of 225, equal to 15.
We get the answer 85.
This method of finding is very interesting and to some extent original, but during my research I encountered it only once in the work of a Perm teacher.
Perhaps it has been little studied or has some exceptions.
It is quite difficult to remember due to the duality of the algorithm and is applicable only for four-digit numbers of exact roots, but I worked through many examples and became convinced of its correctness. In addition, this method is available to those who have already memorized the squares of numbers from 11 to 29, because without their knowledge it will be useless.
Chapter 6. Canadian method
√ X = √ S + (X - S) / (2 √ S), where X is the number to be square rooted and S is the number of the nearest exact square.
Let's try to take the square root of 75
√ 75 = 9 + (- 6/18) = 9 - 0,333 = 8,667
With a detailed study of this method, one can easily prove its similarity to the Babylonian one and argue for the copyright of the invention of this formula, if there is one in reality. The method is simple and convenient.
Chapter 7. Guessing selection method
This method is offered by English students at the London College of Mathematics, but everyone has involuntarily used this method at least once in their lives. It is based on selecting different values of the squares of close numbers by narrowing the search area. Anyone can master this method, but it is unlikely to be used, because it requires repeated calculation of the product of a column of not always correctly guessed numbers. This method loses both in the beauty of the solution and in time. The algorithm is simple:
Let's say you want to take the square root of 75.
Since 8 2 = 64 and 9 2 = 81, you know the answer is somewhere in between.
Try building 8.5 2 and you will get 72.25 (too little)
Now try 8.6 2 and you get 73.96 (too small, but getting closer)
Now try 8.7 2 and you will get 75.69 (too big)
Now you know the answer is between 8.6 and 8.7
Try building 8.65 2 and you will get 74.8225 (too small)
Now try 8.66 2... and so on.
Continue until you get an answer that is accurate enough for you.
Chapter 8. Odd number deduction method
Many people know the method of extracting the square root by factoring a number into prime factors. In my work I will present another way by which you can find out the integer part of the square root of a number. The method is very simple. Note that the following equalities are true for squares of numbers:
1=1 2
1+3=2 2
1+3+5=3 2
1+3+5+7=4 2 etc.
Rule: 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, to get the square root of 36 and 121 this is:
Total number of subtractions = 6, so square root of 36 = 6.
Total number of subtractions = 11, so √121 = 11.
Another example: let's find √529
Solution: 1)_529
2)_528
3)_525
4)_520
5)_513
6)_504
7)_493
8)_480
9)_465
10)_448
11)_429
12)_408
13)_385
14)_360
15)_333
16)_304
17)_273
18)_240
19)_205
20)_168
21)_129
22)_88
23)_45
Answer: √529 = 23
Scientists call this method arithmetic square root extraction, and behind the scenes the “turtle method” because of its slowness.
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. Try to extract the square root of a number, for example, 5963364 in this way and you will understand that it “works”, of course, without errors for exact roots, but it is very, very long in the solution.
Conclusion
The root extraction methods described in this work are found in many sources. However, understanding them turned out to be a difficult task for me, which aroused considerable interest. The presented algorithms will allow everyone who is interested in this topic to quickly master the skills of calculating the square root; they can be used when checking their solution and do not depend on a calculator.
As a result of the research, I came to the conclusion: various methods of extracting the square root without a calculator are necessary in a school mathematics course in order to develop calculation skills.
The theoretical significance of the study - the main methods for extracting square roots are systematized.
Practical significance:in creating a mini-book containing a reference diagram for extracting square roots in various ways (Appendix 1).
Literature and Internet sites:
- I.N. Sergeev, S.N. Olehnik, S.B. Gashkov “Apply mathematics.” – M.: Nauka, 1990
- Kerimov Z., “How to find a whole root?” Popular scientific and mathematical magazine "Kvant" No. 2, 1980
- Petrakov I.S. “mathematics clubs in grades 8-10”; Book for teachers.
–M.: Education, 1987
- Tikhonov A.N., Kostomarov D.P. “Stories about applied mathematics.” - M.: Nauka. Main editorial office of physical and mathematical literature, 1979
- Tkacheva M.V. Home math. A book for 8th grade students. – Moscow, Enlightenment, 1994.
- Zhokhov V.I., Pogodin V.N. Reference tables in mathematics.-M.: LLC Publishing House “ROSMEN-PRESS”, 2004.-120 p.
- http://translate.google.ru/translate
- http://www.murderousmaths.co.uk/books/sqroot.htm
- http://ru.wikipedia.ord /wiki /teorema/
Good afternoon, dear guests!
My name is Lev Sokolov, I study in the 8th grade at evening school.
I present to your attention a work on the topic: “Finding square roots of large numbers without a calculator."
When studying a topicsquare roots this school year, I was interested in the question of how to extract the square root of large numbers without a calculator and I decided to study it more deeply, since next year I have to take an exam in mathematics.
The purpose of my work:find and show ways to extract square roots without a calculator
To achieve the goal I decided the following tasks:
1. Study the literature on this issue.
2. Consider the features of each method found and its algorithm.
3. Show the practical application of the acquired knowledge and assess the degree of complexity in using various methods and algorithms.
4.Create a mini-book according to the most interesting algorithms.
The object of my research wassquare roots.
Subject of research:ways to extract square roots without a calculator.
Research methods:
1. Search for methods and algorithms for extracting square roots from large numbers without a calculator.
2. Comparison and analysis of the methods found.
I found and studied 8 ways to find square roots without a calculator and put them into practice. The names of the methods found are shown on the slide.
I will focus on those that I liked.
I will show with an example how you can extract the square root of the number 3025 using prime factorization.
The main disadvantage of this method- it takes a lot of time.
Using the formula of Ancient Babylon, I will extract the square root of the same number 3025.
The method is convenient only for small numbers.
From the same number 3025 we extract the square root using a corner.
In my opinion, this is the most universal method; it can be applied to any numbers.
IN Modern science knows many ways to extract the square root without a calculator, but I have not studied all of them.
Practical significance of my work:in creating a mini-book containing a reference diagram for extracting square roots in various ways.
The results of my work can be successfully used in mathematics, physics and other subjects where extracting roots without a calculator is required.
Thank you for your attention!
Preview:
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Slide captions:
Extracting square roots from large numbers without a calculator Performer: Lev Sokolov, MKOU "Tugulymskaya V(S)OSH", 8th grade Leader: Sidorova Tatyana Nikolaevna I category, mathematics teacher r.p. Tugulym
The correct application of methods can be learned through application and a variety of examples. G. Zeiten Purpose of the work: to find and show those methods of extracting square roots that can be used without having a calculator at hand. Objectives: - Study the literature on this issue. - Consider the features of each method found and its algorithm. - Show the practical application of the acquired knowledge and assess the degree of complexity in using various methods and algorithms. - Create a mini-book on the most interesting algorithms.
Object of study: square roots Subject of study: methods of extracting square roots without a calculator. Research methods: Search for methods and algorithms for extracting square roots from large numbers without a calculator. Comparison of the found methods. Analysis of the obtained methods.
Methods for extracting a square root: 1. Method of factoring into prime factors 2. Extracting a square root using a corner 3. Method of using a table of squares of two-digit numbers 4. Formula of Ancient Babylon 5. Method of discarding a perfect square 6. Canadian method 7. Method of guessing 8. Method of deductions odd number
Method of factoring into prime factors To extract a square root, you can factor a number into prime factors and extract the square root of the product. 3136│2 7056│2 209764│2 1568│2 3528│2 104882│2 784│2 1764│2 52441│229 392│2 882│2 229│229 196│2 4 41│3 98│2 147│3 √209764 = √2∙2∙52441 = 49│7 49│7 = √2²∙229² = 458. 7│7 7│7 √3136 = √ 2²∙2²∙2²∙7² = 2∙2∙2∙7 = 56. √7056 = √2²∙2²∙3²∙7² = 2∙2∙3∙7 = 84. It is not always easy to decompose, more often it is not completely removed, it takes a lot of time.
Formula of Ancient Babylon (Babylonian method) Algorithm for extracting the square root using the ancient Babylonian method. 1. Present the number c as the sum a² + b, where a² is the exact square of the natural number a closest to the number c (a² ≈ c); 2. The approximate value of the root is calculated using the formula: The result of extracting the root using a calculator is 5.292.
Extracting a square root with a corner The method is almost universal, since it is applicable to any numbers, but composing a rebus (guessing the number at the end of a number) requires logic and good computing skills with a column.
Algorithm for extracting a square root using a corner 1. Divide the number (5963364) into pairs from right to left (5`96`33`64) 2. Extract the square root from the first group on the left (- number 2). This is how we get the first digit of the number. 3. Find the square of the first digit (2 2 =4). 4. Find the difference between the first group and the square of the first digit (5-4=1). 5. We take down the next two digits (we get the number 196). 6. Double the first digit we found and write it on the left behind the line (2*2=4). 7. Now we need to find the second digit of the number: double the first digit we found becomes the tens digit of the number, when multiplied by the number of units, we need to get a number less than 196 (this is the number 4, 44*4=176). 4 is the second digit of &. 8. Find the difference (196-176=20). 9. We demolish the next group (we get the number 2033). 10. Double the number 24, we get 48. 11. 48 tens in the number, when multiplied by the number of ones, we should get a number less than 2033 (484*4=1936). The units digit we found (4) is the third digit of the number. Then the process is repeated.
Odd number subtraction method (arithmetic method) Square root algorithm: Subtract odd numbers in order until the remainder is less than the next number to be subtracted or equal to zero. Count the number of actions performed - this number is the integer part of the number of the square root being extracted. Example 1: calculate 1. 9 − 1 = 8; 8 − 3 = 5; 5 − 5 = 0. 2. 3 actions completed
36 - 1 = 35 - 3 = 32 - 5 = 27 - 7 = 20 - 9 = 11 - 11 = 0 total number of subtractions = 6, so square root of 36 = 6. 121 – 1 = 120 - 3 = 117- 5 = 112 - 7 = 105 - 9 = 96 - 11 = 85 – 13 = 72 - 15 = 57 – 17 = 40 - 19 = 21 - 21 = 0 Total number of subtractions = 11, so square root of 121 = 11. 5963364 = ??? Russian scientists behind the scenes call it the “turtle method” because of its slowness. It is inconvenient for large numbers.
The theoretical significance of the study - the main methods for extracting square roots are systematized. Practical significance: in creating a mini-book containing a reference diagram for extracting square roots in various ways.
Thank you for your attention!
Preview:
Some problems require taking the square root of a large number. How to do this?
Odd number deduction method.
The method is very simple. Note that the following equalities are true for squares of numbers:
1=1 2
1+3=2 2
1+3+5=3 2
1+3+5+7=4 2 etc.
Rule: 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, to get the square root of 36 and 121 is:
36 - 1 = 35 - 3 = 32 - 5 = 27 - 7 = 20 - 9 = 11 - 11 = 0
Total number of subtractions = 6, so square root of 36 = 6.
121 - 1 = 120 - 3 = 117- 5 = 112 - 7 = 105 - 9 = 96 - 11 = 85 – 13 = 72 - 15 = 57 – 17 = 40 - 19 = 21 - 21 = 0
Total number of subtractions = 11, so√121 = 11.
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 to three decimal places. Here is their formula:
√ X = √ S + (X - S) / (2 √ S), where X is the number to be square rooted and S is the number of the nearest exact square.
Example. Take the square root of 75.
X = 75, S = 81. This means that √ S = 9.
Let's calculate √75 using this formula: √ 75 = 9 + (75 - 81) / (2∙9)
√ 75 = 9 + (- 6/18) = 9 - 0,333 =
8,667
A method for extracting square roots using a corner.
1. Divide the number (5963364) into pairs from right to left (5`96`33`64)
2. Take the square root of the first group on the left (- number 2). This is how we get the first digit of the number.
3. Find the square of the first digit (2 2 =4).
4. Find the difference between the first group and the square of the first digit (5-4=1).
5. We take down the next two digits (we get the number 196).
6. Double the first digit we found and write it on the left behind the line (2*2=4).
7. Now we need to find the second digit of the number: double the first digit we found becomes the tens digit of the number, when multiplied by the number of units, we need to get a number less than 196 (this is the number 4, 44*4=176). 4 is the second digit of &.
8. Find the difference (196-176=20).
9. We demolish the next group (we get the number 2033).
10. Double the number 24, we get 48.
There are 11.48 tens in a number, when multiplied by the number of ones, we should get a number less than 2033 (484*4=1936). The units digit we found (4) is the third digit of the number.
Action square rootinverse to the action of squaring.
√81= 9 9 2 =81.
Selection method.
Example: Extract the root of the number 676.
We notice that 20 2 = 400, and 30 2 = 900, which means 20
Exact squares of natural numbers end in 0; 1; 4; 5; 6; 9.
The number 6 gives 4 2 and 6 2 .
This means that if the root is taken from 676, then it is either 24 or 26.
Remaining to check: 24 2 = 576, 26 2 = 676.
Answer: √ 676 = 26.
Another example: √6889.
Since 80 2 = 6400, and 90 2 = 8100, then 80 The number 9 gives 3 2 and 7 2 , then √6889 is equal to either 83 or 87.
Let's check: 83 2 = 6889.
Answer: √6889 = 83.
If you find it difficult to solve using the selection method, you can factor the radical expression.
For example, find √893025.
Let's factor the number 893025, remember, you did this in the sixth grade.
We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.
Babylonian method.
Step #1. Present the number x as a sum: x=a 2 + b, where a 2 the closest exact square of the natural number a to the number x.
Step #2. Use formula:
Example. Calculate.
Arithmetic method.
We subtract all odd numbers from the number in order until the remainder is less than the next number to be subtracted or equal to zero. Having counted the number of actions performed, we determine the integer part of the square root of the number.
Example. Calculate the integer part of a number.
Solution. 12 - 1 = 11; 11 - 3 = 8; 8 - 5 = 3; 3 3 - integer part of the number. So, .
Method (known as Newton's method)is as follows.
Let a 1 - first approximation of the number(as a 1 you can take the values of the square root of a natural number - an exact square not exceeding .
This method allows you to extract the square root of a large number with any accuracy, although with a significant drawback: the cumbersomeness of the calculations.
Evaluation method.
Step #1. Find out the range in which the original root lies (100; 400; 900; 1600; 2500; 3600; 4900; 6400; 8100; 10,000).
Step #2. Using the last digit, determine which digit the desired number ends with.
Units digit of x | ||||||||||
Units digit of x 2 |
Step #3. Square the expected numbers and determine the desired number from them.
Example 1. Calculate .
Solution. 2500 50 2 2 50
= *2 or = *8.
52
2
= (50 +2)
2
= 2500 + 2 50 2 + 4 = 2704;
58
2
= (60 − 2)
2
= 3600 − 2 60 2 + 4 = 3364.
Therefore = 58.
