How to extract the root of a fraction. Square root
On the mug she showed how you can extract it into a column square roots. You can calculate the root with arbitrary precision, find any number of digits in it 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 had a hard time figuring 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, but also the cube root in a column, but this is 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 ( junior 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 (ignoring the comma), we get the additional quantity significant figures(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 a more accurate 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 experimentally as a result of experiments with spreadsheet and I don’t quite understand why this member looks like this. 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:
Nowadays, widespread use computer technology, the question of extracting a square knight from a number is not worth it from a practical point of view. But for mathematics lovers, various options for solving this problem will undoubtedly be of interest. IN school curriculum the method of this calculation without the involvement of additional funds should take place on a par with multiplication and division into a column. 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 I like it due to the relatively high accuracy of the calculation.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!
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In the preface to his first edition, “In the Kingdom of Ingenuity” (1908), E. I. Ignatiev writes: “... intellectual initiative, quick wit and “ingenuity” cannot be “drilled into” or “put into” anyone’s head. The results are reliable only when the introduction to the field of mathematical knowledge is made in an easy and pleasant way, using objects and examples from ordinary and everyday situations, selected with appropriate wit and entertainment.”
In the preface to the 1911 edition “The Role of Memory in Mathematics” E.I. Ignatiev writes “... in mathematics it is not the formulas that should be remembered, but the process of thinking.”
To extract the square root, there are tables of squares for two-digit numbers; you can decompose the number into prime factors and take the square root of the product. A table of squares is sometimes not enough; extracting the root by factoring 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. By trial and error, selection - this, of course, can be done if you are sure that this is an integer. The method I want to propose allows you to take the square root in any case.
Once upon a time at the institute (Perm State Pedagogical Institute) we were introduced to this method, which I now want to talk about. I never wondered whether this method had a proof, so now I had to deduce some of the proof myself.
The basis of this method is the composition of the number =.
=&, i.e. & 2 =596334.
1. Divide the number (5963364) into pairs from right to left (5`96`33`64)
2. Extract the square root of the first group on the left ( - number 2). This is how we get the first digit of &.
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, which 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 ones digit we found (4) is the third digit of the number &.
I have given the proof for the following cases:
1. Extracting the square root of a three-digit number;
2. Extracting the square root of a four-digit number.
Approximate methods for extracting square roots (without using a calculator).
1. 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 a 2 + b, where a 2 is the exact square of the natural number a (a 2 ? x) closest to the number x, and used the formula 
. (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
2. Isaac Newton developed a method for taking square roots that dates back to Heron of Alexandria (circa 100 AD). This method (known as Newton's method) is as follows.
Let a 1- the first approximation of a number (as a 1 you can take the values of the square root of a natural number - an exact square not exceeding X) .
Next, more accurate approximation a 2 numbers
found by the formula 
.
Instructions
Select a multiplier for the radical number, the removal of which from under root is really an expression - otherwise the operation will lose . For example, if under the sign root with an exponent equal to three (cube root), it costs number 128, then from under the sign you can take out, for example, number 5. At the same time, the radical number 128 will have to be divided by 5 cubed: ³√128 = 5∗³√(128/5³) = 5∗³√(128/125) = 5∗³√1.024. If the presence of a fractional number under the sign root does not contradict the conditions of the problem, then it is possible in this form. If you need a simpler option, then first break the radical expression into such integer factors, the cube root of one of which will be an integer number m. For example: ³√128 = ³√(64∗2) = ³√(4³∗2) = 4∗³√2.
Use to select factors of a radical number if it is not possible to calculate the powers of a number in your head. This is especially true for root m with an exponent greater than two. If you have access to the Internet, you can perform calculations built into search engines Google and Nigma computing. For example, if you need to find the largest integer factor that can be taken out from under the cubic sign root for the number 250, then go to the Google website and enter the query “6^3” to check if it is possible to remove it from under the sign root six. The search engine will show a result equal to 216. Alas, 250 cannot be divided without a remainder by this number. Then enter the query 5^3. The result will be 125, and this allows you to divide 250 into factors of 125 and 2, which means taking it out of the sign root number 5, leaving there number 2.
Sources:
- how to get it out from under the roots
- Square root of the product
Take it out from under root one of the factors is necessary in situations where you need to simplify mathematical expression. There are times when it is impossible to perform the necessary calculations using a calculator. For example, if letter designations for variables are used instead of numbers.
Instructions
Break down the radical expression into simple factors. See which of the factors is repeated the same number of times, indicated in the indicators root, or more. For example, you need to take the fourth root of a. In this case, the number can be represented as a*a*a*a = a*(a*a*a)=a*a3. Indicator root in this case it will correspond with factor a3. It needs to be taken out of the sign.
Extract the root of the resulting radicals separately where possible. Extraction root is the algebraic operation inverse to exponentiation. Extraction root to an arbitrary power from a number, find a number that, when raised to this arbitrary power, will result in given number. If extraction root cannot be produced, leave the radical expression under the sign root just the way it is. As a result of the above actions, you will be removed from under sign root.
Video on the topic
note
Be careful when writing radical expressions in the form of factors - an error at this stage will lead to incorrect results.
When extracting roots, it is convenient to use special tables or tables of logarithmic roots - this will significantly reduce the time it takes to find the right decision.
Sources:
- root extraction sign in 2019
Simplification of algebraic expressions is required in many areas of mathematics, including when solving equations higher degrees, differentiation and integration. Several methods are used, including factorization. To apply this method, you need to find and make a general factor behind brackets.
Instructions
Carrying out the total multiplier brackets- one of the most common methods of decomposition. This technique is used to simplify the structure of long algebraic expressions, i.e. polynomials. The general number can be a number, a monomial or a binomial, and to find it, the distributive property of multiplication is used.
Number. Look carefully at the coefficients of each polynomial to see if they can be divided by the same number. For example, in the expression 12 z³ + 16 z² – 4 it is obvious factor 4. After the transformation, you get 4 (3 z³ + 4 z² - 1). In other words, this number is the least common integer divisor of all coefficients.
Monomial. Determine whether the same variable is in each of the terms of the polynomial. Assuming this is the case, now look at the coefficients as in the previous case. Example: 9 z^4 – 6 z³ + 15 z² – 3 z.
Each element of this polynomial contains a variable z. In addition, all coefficients are numbers that are multiples of 3. Therefore, the common factor will be the monomial 3 z:3 z (3 z³ – 2 z² + 5 z - 1).
Binomial.For brackets general factor of two, a variable and a number, which is a common polynomial. Therefore, if factor-the binomial is not obvious, then you need to find at least one root. Select the free term of the polynomial; this is a coefficient without a variable. Now apply the method of substitution into the general expression of all integer divisors of the free term.
Consider: z^4 – 2 z³ + z² - 4 z + 4. Check to see if any of the integer factors of 4 are z^4 – 2 z³ + z² - 4 z + 4 = 0. By simple substitution, find z1 = 1 and z2 = 2, which means for brackets we can remove the binomials (z - 1) and (z - 2). To find the remaining expression, use sequential long division.
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
Taking the square root of a number is easy. 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.
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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From the course you will not only learn dozens of techniques for simplified and quick multiplication, addition, multiplication, division, and calculating percentages, but you will also practice them in special assignments and educational games! Mental arithmetic also requires a lot of attention and concentration, which are actively trained when solving interesting problems.
