Extracting the root is the reverse operation of raising a power. That is, taking the root of the number X, we get a number that squared will give the same number X.
Extracting the root is a fairly simple operation. A table of squares can make the extraction work easier. Because it is impossible to remember all the squares and roots by heart, but the numbers may be large.
Extracting the root of a number
Extraction square root from the number - simple. Moreover, this can be done not immediately, but gradually. For example, take the expression √256. Initially, it is difficult for an ignorant person to give an answer right away. Then we will do it step by step. First, we divide by just the number 4, from which we take the selected square as the root.
Let's represent: √(64 4), then it will be equivalent to 2√64. And as you know, according to the multiplication table 64 = 8 8. The answer will be 2*8=16.
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Extracting a complex root
The square root cannot be calculated from negative numbers, because any squared number is a positive number!
A complex number is the number i, which squared is equal to -1. That is, i2=-1.
In mathematics, there is a number that is obtained by taking the root of the number -1.
That is, it is possible to calculate the root of negative number, but this already applies to higher mathematics, not school mathematics.
Let's consider an example of such a root extraction: √(-49)=7*√(-1)=7i.
Online root calculator
Using our calculator, you can calculate the extraction of a number from the square root:
Converting Expressions Containing a Root Operation
The essence of transforming radical expressions is to decompose the radical number into simpler ones, from which the root can be extracted. Such as 4, 9, 25 and so on.
Let's give an example, √625. Let's divide the radical expression by the number 5. We get √(125 5), repeat the operation √(25 25), but we know that 25 is 52. Which means the answer will be 5*5=25.
But there are numbers for which the root cannot be calculated using this method and you just need to know the answer or have a table of squares at hand.
√289=√(17*17)=17
Bottom line
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The circle showed how you can extract square roots in a column. You can calculate the root with arbitrary precision, find any number of digits in its decimal notation, even if it turns out to be irrational. The algorithm was remembered, but questions remained. It was not clear where the method came from and why it gave the correct result. It wasn’t in the books, or maybe I was just looking in the wrong books. In the end, like much of what I know and can do today, I came up with it myself. I share my knowledge here. By the way, I still don’t know where the rationale for the algorithm is given)))
So, first I tell you “how the system works” with an example, and then I explain why it actually works.
Let’s take a number (the number was taken “out of thin air”, it just came to mind).
1. We divide its numbers into pairs: those to the left of decimal point, we group two from right to left, and those to the right - two from left to right. We get.
2. We extract the square root from the first group of numbers on the left - in our case this is (it is clear that the exact root may not be extracted, we take a number whose square is as close as possible to our number formed by the first group of numbers, but does not exceed it). In our case this will be a number. We write down the answer - this is the most significant digit of the root.
3. We square the number that is already in the answer - this - and subtract it from the first group of numbers on the left - from the number. In our case it remains .
4. We assign the following group of two numbers to the right: . We multiply the number that is already in the answer by , and we get .
5. Now watch carefully. We need to assign one digit to the number on the right, and multiply the number by, that is, by the same assigned digit. The result should be as close as possible to, but again no more than this number. In our case, this will be the number, we write it in the answer next to, on the right. This is the next digit in the decimal notation of our square root.
6. From subtract the product , we get .
7. Next, we repeat the familiar operations: we assign the following group of digits to the right, multiply by , to the resulting number > we assign one digit to the right, such that when multiplied by it we get a number smaller than , but closest to it - this is the next digit in decimal root notation.
The calculations will be written as follows:
And now the promised explanation. The algorithm is based on the formula
Comments: 50
2 Anton:
Too chaotic and confusing. Arrange everything point by point and number them. Plus: explain where we substitute in each action required values. I’ve never calculated a root root before – I found it difficult to figure it out.
5 Julia:
6 :
Yulia, 23 on this moment written on the right, these are the first two (on the left) already obtained digits of the root in the answer. Multiply by 2 according to the algorithm. We repeat the steps described in point 4.
7 zzz:
error in “6. From 167 we subtract the product 43 * 3 = 123 (129 nada), we get 38.”
I don’t understand how it turned out to be 08 after the decimal point...9 Fedotov Alexander:
And even in the pre-calculator era, we were taught at school not only the square root, but also the cube root in a column, but this was more tedious and painstaking work. It was easier to use Bradis tables or a slide rule, which we already studied in high school.
10 :
Alexander, you are right, you can extract roots of large powers into a column. I'm going to write just about how to find the cube root.
12 Sergei Valentinovich:
Dear Elizaveta Alexandrovna! In the late 70s, I developed a scheme for automatic (i.e., not by selection) calculation of quadra. root on the Felix adding machine. If you are interested, I can send you a description.
14 Vlad aus Engelsstadt:
(((Extracting the square root of the column)))
The algorithm is simplified if you use the 2nd number system, which is studied in computer science, but is also useful in mathematics. A.N. Kolmogorov presented this algorithm in popular lectures for schoolchildren. His article can be found in the “Chebyshev Collection” (Mathematical Journal, look for a link to it on the Internet)
By the way, say:
G. Leibniz at one time toyed with the idea of transitioning from the 10th number system to the binary one because of its simplicity and accessibility for beginners (primary schoolchildren). But breaking established traditions is like breaking a fortress gate with your forehead: it’s possible, but it’s useless. So it turns out, as according to the most quoted bearded philosopher in the old days: the traditions of all dead generations suppress the consciousness of the living.Until next time.
15 Vlad aus Engelsstadt:
))Sergey Valentinovich, yes, I’m interested...((
I bet that this is a variation on the “Felix” of the Babylonian method of extracting the square knight using the method of successive approximations. This algorithm was covered by Newton's method (tangent method)
I wonder if I was wrong in my forecast?
18 :
2Vlad aus Engelsstadt
Yes, the algorithm in binary should be simpler, that's pretty obvious.
About Newton's method. Maybe that's true, but it's still interesting
20 Kirill:
Thanks a lot. But there is still no algorithm, no one knows where it came from, but the result is correct. THANKS A LOT! I've been looking for this for a long time)
21 Alexander:
How will you extract the root from a number where the second group from left to right is very small? for example, everyone's favorite number is 4,398,046,511,104. After the first subtraction, it is not possible to continue everything according to the algorithm. Can you explain please.
22 Alexey:
Yes, I know this method. I remember reading it in the book “Algebra” of some old edition. Then, by analogy, he himself deduced how to extract the cube root in a column. But there it’s already more complicated: each digit is determined not by one (as for a square), but by two subtractions, and even there you have to multiply long numbers every time.
23 Artem:
There are typos in the example of extracting the square root of 56789.321. The group of numbers 32 is assigned twice to the numbers 145 and 243, in the number 2388025 the second 8 must be replaced by 3. Then the last subtraction should be written as follows: 2431000 – 2383025 = 47975.
Additionally, when dividing the remainder by the doubled value of the answer (without taking into account the comma), we obtain an additional number of significant digits (47975/(2*238305) = 0.100658819...), which should be added to the answer (√56789.321 = 238.305... = 238.305100659).24 Sergey:
Apparently the algorithm came from Isaac Newton’s book “General Arithmetic or a book on arithmetic synthesis and analysis.” Here is an excerpt from it:
ABOUT EXTRACTING ROOTS
To extract the square root of a number, you must first place a dot above its digits, starting from the ones. Then you should write in the quotient or radical the number whose square is equal to or closest in disadvantage to the numbers or number preceding the first point. After subtracting this square, the remaining digits of the root will be sequentially found by dividing the remainder by twice the value of the already extracted part of the root and subtracting each time from the remainder of the square the last found digit and its tenfold product by the named divisor.
25 Sergey:
Please also correct the title of the book “General Arithmetic or a book about arithmetic synthesis and analysis”
26 Alexander:
thanks for interesting material. But this method seems to me somewhat more complicated than what is needed, for example, for a schoolchild. I use a simpler method based on expanding a quadratic function using the first two derivatives. Its formula is:
sqrt(x)= A1+A2-A3, where
A1 is the integer whose square is closest to x;
A2 is a fraction, the numerator is x-A1, the denominator is 2*A1.
For most numbers found in school course, this is enough to get the result accurate to the hundredth.
If you need more exact result, take
A3 is a fraction, the numerator is A2 squared, the denominator is 2*A1+1.
Of course, to use it you need a table of squares of integers, but this is not a problem at school. Remembering this formula is quite simple.
However, it confuses me that I obtained A3 empirically as a result of experiments with a spreadsheet and I do not quite understand why this member has this appearance. Maybe you can give me some advice?27 Alexander:
Yes, I've considered these considerations too, but the devil is in the details. You write:
“since a2 and b differ quite little.” The question is exactly how little.
This formula works well on numbers in the second ten and much worse (not up to hundredths, only up to tenths) on numbers in the first ten. Why this happens is difficult to understand without the use of derivatives.28 Alexander:
I will clarify what I see as the advantage of the formula I propose. It does not require the not entirely natural division of numbers into pairs of digits, which, as experience shows, is often performed with errors. Its meaning is obvious, but for a person familiar with analysis, it is trivial. Works well on numbers from 100 to 1000, which are the most common numbers encountered in school.
29 Alexander:
By the way, I did some digging and found the exact expression for A3 in my formula:
A3= A22 /2(A1+A2)30 vasil stryzhak:
In our time, with the widespread use of computer technology, the question of extracting the square knight from a number is not worth it from a practical point of view. But for mathematics lovers, they are undoubtedly of interest various options solutions to this problem. IN school curriculum a method for this calculation without involving additional funds should take place on a par with multiplication and long division. The calculation algorithm must not only be memorized, but also understandable. Classic method, provided in this material for discussion with disclosure of the essence, fully complies with the above criteria.
A significant drawback of the method proposed by Alexander is the use of a table of squares of integers. The author is silent about the majority of numbers encountered in the school course. As for the formula, in general it appeals to me in terms of high accuracy calculations.31 Alexander:
for 30 vasil stryzhak
I didn't keep anything quiet. The table of squares is supposed to be up to 1000. In my time at school they simply learned it by heart and it was in all mathematics textbooks. I explicitly named this interval.
As for computer technology, it is not used mainly in mathematics lessons, unless the topic of using a calculator is specifically discussed. Calculators are now built into devices that are prohibited for use on the Unified State Exam.32 vasil stryzhak:
Alexander, thanks for the clarification! I thought that for the proposed method it is theoretically necessary to remember or use a table of squares of all two-digit numbers. Then for radical numbers not included in the interval from 100 to 10000, you can use the technique of increasing or decreasing them by required amount orders of comma transfer.
33 vasil stryzhak:
39 ALEXANDER:
MY FIRST PROGRAM IN IAMB LANGUAGE ON THE SOVIET MACHINE “ISKRA 555″ WAS WRITTEN TO EXTRACT THE SQUARE ROOT OF A NUMBER USING THE COLUMN EXTRACTION ALGORITHM! and now I forgot how to extract it manually!
Sokolov Lev Vladimirovich, 8th grade student of the Municipal Educational Institution “Tugulymskaya V(S)OSH”
Goal of the work: find and show those extraction methods square roots, which can be used without having a calculator at hand.
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Regional scientific and practical conference
students of Tugulym urban district
Extracting square roots from 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 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
The relevance of research,when I was studying the topic of square roots in this academic year, then I was interested in the question of how you can extract the square root of large numbers without a calculator.
I became interested and decided to study this issue deeper than it is set out in the school curriculum, and also to prepare a mini-book with the most in simple ways extracting square roots of large numbers without a calculator.
Goal of the work: find and show those methods of extracting square roots that can be used without having a calculator at hand.
Tasks:
- Study literature on this issue.
- Consider the features of each method found and its algorithm.
- Show practical use acquired knowledge and evaluate
Difficulty to use in various ways and algorithms.
- Create a mini-book on the most interesting algorithms.
Object of study:mathematical symbols are square roots.
Subject of study: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 number found 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 the answer in third place |
|
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 in the following way finding 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 to the number x natural number a (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 we see, the Babylonian method gives a good approximation to exact value 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 size 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 suggested English students Mathematical College of London, but everyone in his life at least once involuntarily used this method. It is based on selection different meanings squares of similar 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 subtraction = 6, so the 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 who solve 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 physical and mathematical literature, 1979
- Tkacheva M.V. Home math. Book for 8th grade students educational institutions. – 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 study: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 There are 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!
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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 square roots: 1. Method of factoring into prime factors 2. Extracting square roots 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 computational 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, you 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:
When solving some problems, you will need to take the square root of large number. How to do it?
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 quick 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, you 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 a 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.
Preferably an engineering one - one that has a button with a root sign: “√”. Usually, to extract the root, it is enough to type the number itself, and then press the button: “√”.
In most modern mobile phones There is a “calculator” application with a root extraction function. The procedure for finding the root of a number using a telephone calculator is similar to the above.
Example.
Find from 2.
Turn on the calculator (if it is turned off) and successively press the buttons with the image of two and root (“2” “√”). As a rule, you do not need to press the “=” key. As a result, we get a number like 1.4142 (the number of digits and “roundness” depends on the bit depth and calculator settings).
Note: When trying to find the root, the calculator usually gives an error.
If you have access to a computer, then finding the root of a number is very easy.
1. You can use the Calculator application, available on almost any computer. For Windows XP, this program can be launched as follows:
“Start” - “All Programs” - “Accessories” - “Calculator”.
It is better to set the view to “normal”. By the way, unlike a real calculator, the button for extracting the root is marked “sqrt” and not “√”.
If you can’t get to the calculator using the indicated method, you can run the standard calculator “manually”:
“Start” - “Run” - “calc”.
2. To find the root of a number, you can also use some programs installed on your computer. In addition, the program has its own built-in calculator.
For example, for the MS Excel application, you can do the following sequence of actions:
Launch MS Excel.
We write down in any cell the number from which we need to extract the root.
Move the cell pointer to a different location
Press the function selection button (fx)
Select the “ROOT” function
We specify a cell with a number as an argument to the function
Click “OK” or “Enter”
Advantage this method is that now it is enough to enter any value into the cell with the number, as in the function, .
Note.
There are several other, more exotic ways to find the root of a number. For example, in a “corner”, using a slide rule or Bradis tables. However, these methods are not discussed in this article due to their complexity and practical uselessness.
Video on the topic
Sources:
- how to find the root of a number
Sometimes situations arise when you have to perform some kind of mathematical calculations, including extracting square roots and to a greater extent from the number. The "n" root of "a" is the number nth degree which is the number "a".
Instructions
To find the root "n" of , do the following.
On your computer, click “Start” - “All Programs” - “Accessories”. Then go to the “Service” subsection and select “Calculator”. You can do this manually: Click Start, type "calk" in the Run box, and press Enter. Will open. To extract the square root of a number, enter it into the calculator and press the button labeled "sqrt". The calculator will extract the second degree root, called the square root, from the entered number.
In order to extract a root whose degree is higher than the second, you need to use another type of calculator. To do this, in the calculator interface, click the “View” button and select the “Engineering” or “Scientific” line from the menu. This type of calculator has the necessary to calculate nth root degree function.
To extract the root of the third degree (), on an “engineering” calculator, enter the desired number and press the “3√” button. To obtain a root whose degree is higher than 3, enter the desired number, press the button with the “y√x” icon and then enter the number - the exponent. After this, press the equal sign (the “=” button) and you will get the desired root.
If your calculator does not have the "y√x" function, the following.
To extract the cube root, enter the radical expression, then put a check mark in the check box, which is located next to the inscription “Inv”. With this action, you will reverse the functions of the calculator buttons, i.e., by clicking on the cube button, you will extract the cube root. On the button that you
Mathematics originated when man became aware of himself and began to position himself as an autonomous unit of the world. The desire to measure, compare, count what surrounds you is what underlay one of the fundamental sciences of our days. At first it was particles 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 disappeared from it.” The concept of “square root” appeared at a time when it could be easily supported by empirical data, going beyond the plane of calculations.
Where it all began
The first mention of the root, which is currently denoted as √, was recorded in the works of Babylonian mathematicians, who laid the foundation for modern arithmetic. Of course, they bore little resemblance to the current form - scientists of those years first used bulky tablets. But in the second millennium BC. e. They derived an approximate calculation formula that showed how to extract the square root. The photo below shows a stone on which Babylonian scientists carved the process for deducing √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 a 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 cannot be extracted without a remainder gives an irrational result.
The origin of this term is associated with the Arabic representation of number: ancient scientists believed that a square any number grows from a root like a plant. In Latin, this word sounds like radix (you can trace a pattern - everything that has a “root” meaning is consonant, be it radish or radiculitis).
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 of an arbitrary number a was taken, they wrote R 2 a. Habitual modern view"tick" √ appeared only in the 17th century thanks to Rene Descartes.
Our days
In mathematical terms, the square root of a number y is the number z whose square is equal to y. In other words, z 2 =y is equivalent to √y=z. However this definition 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 applies to determining an 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 the love for mathematics has only increased with the development of science, there are various manifestations of affection for it that are not expressed in dry calculations. For example, along with such interesting phenomena as Pi Day, square root holidays are also celebrated. They are celebrated nine times every hundred years, and are determined according to the following principle: the numbers that indicate in order the day and month must be the square root of the year. So, in 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, and √y, which is defined as the side of a square with area y, has not escaped this fate.
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 at the output is less than the subtracted one or even equal to zero. The number of moves will ultimately 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
Let's consider the elementary function z=√y on the field of real numbers R, where y is greater than or equal to zero. Its schedule looks like this:
The curve grows from the origin and necessarily intersects the point (1; 1).
Properties of the function z=√y on the field of real numbers R
1. The domain of definition of the function under consideration 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 its 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 as an ordinary power function.
And in programming, replacing the symbol √ is the combination of letters sqrt.
It is worth noting that in this area the square root is in great demand, since it is part of most geometric formulas necessary for calculations. The counting algorithm itself is quite complex and is based on recursion (a function that calls itself).
Square root in 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 root of 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 were solved even with a negative discriminant. In C, the same properties are relevant for the square root as in R, the only thing is that the restrictions on the radical expression are removed.