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 the decimal point are grouped two from right to left, and those to the right are grouped 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 at the 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 square, but also cube root extract 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 it into a column and roots higher degrees. 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 automatically (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 this is a variation on the “Felix” of the Babylonian method of extracting a horse square method 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. Please explain.
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 the 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 decomposition 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 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 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. The 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 quantity 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!
Root n th degree natural number a this number is called n the th degree of which is equal to a. The root is designated as follows: . The symbol √ is called root sign or radical sign, number a - radical number, n - root exponent.
The action by which the root of a given degree is found is called root extraction.
Since, according to the definition of the concept of a root n th degree
That root extraction- an action inverse to raising to a power, with the help of which the base of the degree is found from a given degree and from a given exponent.
Square root
Square root of a number a is the number whose square is equal to a.
The action by which the square root is calculated is called square rooting.
Square Root- the opposite action of squaring (or raising a number to the second power). When squaring a number, you need to find its square. When extracting the square root, the square of the number is known; you need to use it to find the number itself.
Therefore, to check the correctness of the action, you can raise the found root to the second power and, if the degree is equal to the radical number, then the root was found correctly.
Let's look at extracting the square root and checking it using an example. Let's calculate or (the root exponent with a value of 2 is usually not written, since 2 is the smallest exponent and it should be remembered that if there is no exponent above the root sign, then the exponent 2 is implied), for this we need to find the number, when raised to the second the degree will be 49. Obviously, such a number is 7, since
7 7 = 7 2 = 49.
Calculating the square root
If given number is 100 or less, then the square root of it can be calculated using the multiplication table. For example, the square root of 25 is 5, because 5 5 = 25.
Now let's look at a way to find the square root of any number without using a calculator. For example, let's take the number 4489 and start calculating it step by step.
- Let us determine which digits the required root should consist of. Since 10 2 = 10 · 10 = 100, and 100 2 = 100 · 100 = 10000, it becomes clear that the desired root must be greater than 10 and less than 100, i.e. consist of tens and ones.
- Find the number of tens of the root. Multiplying tens yields hundreds, and there are 44 of them in our number, so the root must contain so many tens that the square of the tens gives approximately 44 hundreds. Therefore, the root must have 6 tens, because 60 2 = 3600, and 70 2 = 4900 (this is too much). Thus, we found out that our root contains 6 tens and several units, since it is in the range from 60 to 70.
- The multiplication table will help you determine the number of units in the root. Looking at the number 4489, we see that the last digit in it is 9. Now we look at the multiplication table and see that 9 units can only be obtained by squaring the numbers 3 and 7. This means the root of the number will be equal to 63 or 67.
- We check the numbers 63 and 67 we received by squaring them: 63 2 = 3969, 67 2 = 4489.
Do you have calculator addiction? Or do you think that it is very difficult to calculate, for example, except with a calculator or using a table of squares.
It happens that schoolchildren are tied to a calculator and even multiply 0.7 by 0.5 by pressing the treasured buttons. They say, well, I still know how to calculate, but now I’ll save time... When the exam comes... then I’ll strain myself...
So the fact is that there will already be plenty of “stressful moments” during the exam... As they say, water wears away stones. So in an exam, little things, if there are a lot of them, can ruin you...
Let's minimize the number of possible troubles.
Taking the square root of a large number
We will now only talk about the case when the result of extracting the square root is an integer.
Case 1.
So, let us at any cost (for example, when calculating the discriminant) need to calculate the square root of 86436.
We will expand the number 86436 into prime factors. Divide by 2, we get 43218; divide by 2 again, we get 21609. A number cannot be divisible by 2. But since the sum of the digits is divisible by 3, then the number itself is divisible by 3 (generally speaking, it is clear that it is also divisible by 9). . Divide by 3 again, and we get 2401. 2401 is not completely divisible by 3. Not divisible by five (does not end in 0 or 5).
We suspect divisibility by 7. Indeed, and ,
So, Complete order!
Case 2.
Let us need to calculate . It is inconvenient to act in the same way as described above. We are trying to factorize...
The number 1849 is not divisible by 2 (it is not even)…
It is not completely divisible by 3 (the sum of the digits is not a multiple of 3)...
It is not completely divisible by 5 (the last digit is neither 5 nor 0)…
It’s not completely divisible by 7, it’s not divisible by 11, it’s not divisible by 13... Well, how long will it take us to sort through all the prime numbers?
Let's think a little differently.
We understand that
We have narrowed our search. Now we go through the numbers from 41 to 49. Moreover, it is clear that since the last digit of the number is 9, then we should stop at options 43 or 47 - only these numbers, when squared, will give the last digit 9.
Well, here, of course, we stop at 43. Indeed,
P.S. How the hell do we multiply 0.7 by 0.5?
You should multiply 5 by 7, ignoring the zeros and signs, and then separate, going from right to left, two decimal places. We get 0.35.
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 - this is what underlay one of basic sciences our days. At first, these were particles of elementary mathematics, which made it possible to connect numbers with their physical expressions, later the conclusions began to be presented only theoretically (due to their abstraction), but after a while, as one scientist put it, “mathematics reached the ceiling of complexity when they disappeared from it.” all the numbers." The concept of “square root” appeared at a time when it could be easily supported by empirical data, going beyond the plane of calculations.
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 the square of an arbitrary 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. The “tick”, familiar to modern eyes, 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, the next time we will celebrate this holiday is April 4, 2016.
Properties of the square root on the field R
Almost all mathematical expressions are based on geometric basis, this fate did not escape √y, which is defined as the side of a square with area y.
How to find the root of a number?
There are several calculation algorithms. The simplest, but at the same time quite cumbersome, is the usual arithmetic calculation, which is as follows:
1) from the number whose root we need, odd numbers are subtracted in turn - until the remainder 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:
Following odd number- this is 11, the remainder is as follows: 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, as 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.
Do you want to do well on the Unified State Examination in mathematics? Then you need to be able to count quickly, correctly and without a calculator. After all, the main reason for losing points on the Unified State Exam in mathematics is computational errors.
According to the rules of the Unified State Exam, it is prohibited to use a calculator during the mathematics exam. The price may be too high - removal from the exam.
In fact, you don’t need a calculator for the Unified State Examination in mathematics. All problems are solved without it. The main thing is attention, accuracy and some secret techniques, which we will tell you about.
Let's start with the main rule. If a calculation can be simplified, simplify it.
Here, for example, is the “devilish equation”:
Seventy percent of graduates solve it head-on. They calculate the discriminant using the formula, after which they say that the root cannot be extracted without a calculator. But you can divide the left and right sides of the equation by . It will work out
Which way is easier? :-)
Many schoolchildren do not like columnar multiplication. Nobody liked solving boring “examples” in fourth grade. However, in many cases it is possible to multiply numbers without a “column”, in a row. It's much faster.
Please note that we do not start with smaller digits, but with larger ones. It's convenient.
Now - division. It is not easy to divide “in a column” by . But remember that the division sign: and the fractional bar are the same thing. Let's write it as a fraction and reduce the fraction:
Another example.
How to square a two-digit number quickly and without any columns? We apply abbreviated multiplication formulas:
Sometimes it is convenient to use another formula:
Numbers ending in , are squared instantly.
Let's say we need to find the square of a number ( - not necessarily a number, but any natural number). We multiply by and add to the result. All!
For example: (and attributed).
(and attributed).
(and attributed).
This method is useful not only for squaring, but for taking the square root of numbers ending in .
How can you even extract the square root without a calculator? We'll show you two ways.
The first method is to factorize the radical expression.
For example, let's find
A number is divisible by (since the sum of its digits is divisible by ). Let's factorize:
Let's find it. This number is divisible by . It is also divided by. Let's factor it out.
Another example.
There is a second way. It is convenient if the number from which you need to extract the root cannot be factorized.
For example, you need to find . The number under the root is odd, it is not divisible by, is not divisible by, is not divisible by... You can continue to look for what it is divisible by, or you can do it easier - find this root by selection.
Obviously, a two-digit number was squared, which is between the numbers and , since , , and the number is between them. We already know the first digit in the answer, it is .
The last digit in the number is . Since , , the last digit in the answer is either , or . Let's check:
. It worked!
Let's find it.
This means that the first digit in the answer is five.
The last digit in the number is nine. , . This means that the last digit in the answer is either , or .
Let's check:
If the number from which you need to extract the square root ends in or, then the square root of it will be an irrational number. Because no integer square ends in or . Remember that in some of the problems on the Unified State Exam in mathematics, the answer must be written as an integer or a final decimal fraction, that is, it must be a rational number.
We encounter quadratic equations in problems and variants of the Unified State Exam, as well as in parts. They need to count the discriminant and then extract the root from it. And it is not at all necessary to look for roots from five-digit numbers. In many cases, the discriminant can be factorized.
For example, in Eq.
Another situation in which the expression under the root can be factorized is taken from the problem.
The hypotenuse of a right triangle is equal to , one of the legs is equal to , find the second leg.
According to the Pythagorean theorem, it is equal to . You can count in a column for a long time, but it’s easier to use the abbreviated multiplication formula.
And now we’ll tell you the most interesting thing - why graduates lose precious points on the Unified State Exam. After all, errors in calculations do not just happen.
1. A sure way to lose points is sloppy calculations in which something is corrected, crossed out, or one number is written on top of another. Look at your drafts. Perhaps they look the same? :-)
Write legibly! Don't save paper. If something is wrong, do not correct one number for another, it is better to write it again.
2. For some reason, many schoolchildren, when counting in a column, try to do it 1) very, very quickly, 2) in very small numbers, in the corner of their notebook, and 3) with a pencil. The result is this:
It's impossible to take anything apart. So is it surprising that the Unified State Exam score is lower than expected?
3. Many schoolchildren are accustomed to ignoring parentheses in expressions. Sometimes this happens:
Remember that the equal sign is not placed just anywhere, but only between equal values. Write competently, even in draft form.
4. A huge number of computational errors involve fractions. If you are dividing a fraction by a fraction, use what
A “hamburger” is drawn here, that is, a multi-story fraction. It is extremely difficult to get the correct answer using this method.
Let's summarize.
Checking the tasks of the first part of the profile Unified State Examination in mathematics is automatic. There is no “almost right” answer here. Either he is correct or he is not. One computational error - and hello, the task does not count. Therefore, it is in your interests to learn to count quickly, correctly and without a calculator.
The tasks of the second part of the profile Unified State Examination in mathematics are checked by an expert. Take care of him! Let him understand both your handwriting and the logic of the decision.