What is a square root definition. Square root. Actions with square roots. Module. Comparison of square roots
Among the many knowledge that is a sign of literacy, the alphabet comes first. The next, equally “sign” element is the skills of addition-multiplication and, adjacent to them, but opposite in meaning, arithmetic operations of subtraction-division. The skills learned in distant school childhood serve faithfully day and night: TV, newspaper, SMS, and everywhere we read, write, count, add, subtract, multiply. And, tell me, have you often had to extract roots in your life, except at the dacha? For example, such an entertaining problem, like the square root of the number 12345... Is there still gunpowder in the flasks? Can we handle it? Nothing could be simpler! Where is my calculator... And without it, hand-to-hand combat is weak?
First, let's clarify what it is - the square root of a number. Generally speaking, “taking the root of a number” means performing the arithmetic operation opposite to raising it to a power - here you have the unity of opposites in life application. Let's say a square is the multiplication of a number by itself, i.e., as taught at school, X * X = A or in another notation X2 = A, and in words - “X squared equals A.” Then the inverse problem sounds like this: the square root of the number A is the number X, which, when squared, equals A.
Taking the square root
From the school arithmetic course, methods of calculations “in a column” are known, which help to perform any calculations using the first four arithmetic operations. Alas... For square, and not only square, roots, such algorithms do not exist. And in this case, how to extract the square root without a calculator? Based on the definition of the square root, there is only one conclusion - it is necessary to select the value of the result by sequentially enumerating numbers whose square approaches the value of the radical expression. That's all! Before an hour or two has passed, you can calculate using well famous trick column multiplication, any square root. If you have the skills, this will only take a couple of minutes. Even a not-so-advanced user of a calculator or PC can do this in one fell swoop - progress.
But seriously, the calculation of the square root is often performed using the “artillery fork” technique: first take a number whose square approximately corresponds to the radical expression. It is better if “our square” is slightly smaller than this expression. Then they adjust the number according to their own skill and understanding, for example, multiply by two, and... square it again. If the result more number under the root, successively adjusting the original number, gradually approaching its “colleague” under the root. As you can see, there is no calculator, only the ability to count “in a column.” Of course, there are many scientifically proven and optimized algorithms for calculating square roots, but for “ home use"The above technique gives 100% confidence in the result.
Yes, I almost forgot, to confirm our increased literacy, let’s calculate the square root earlier the specified number 12345. Let’s do it step by step:
1. Let's take, purely intuitively, X=100. Let's calculate: X * X = 10000. Intuition is at its best - the result is less than 12345.
2. Let’s try, also purely intuitively, X = 120. Then: X * X = 14400. And again, intuition is in order - the result is more than 12345.
3. Above we got a “fork” of 100 and 120. Let’s choose new numbers - 110 and 115. We get, respectively, 12100 and 13225 - the fork narrows.
4. Let’s try “maybe” X=111. We get X * X = 12321. This number is already quite close to 12345. In accordance with the required accuracy, the “fit” can be continued or stopped at the result obtained. That's all. As promised - everything is very simple and without a calculator.
Just a little history...
Thought of using it square roots also Pythagoreans, students of the school and followers of Pythagoras, 800 BC. and right there, we “ran into” new discoveries in the field of numbers. And where did that come from?
1. Solving the problem with extracting the root gives the result in the form of numbers of a new class. They were called irrational, in other words, “unreasonable”, because. they are not written as a complete number. The most classic example of this kind is the square root of 2. This case corresponds to calculating the diagonal of a square with a side equal to 1 - this is the influence of the Pythagorean school. It turned out that in a triangle with a very specific unit size of sides, the hypotenuse has a size that is expressed by a number that “has no end.” This is how they appeared in mathematics
2. It is known that it turned out that this mathematical operation contains another catch - when extracting the root, we do not know which number, positive or negative, is the square of the radical expression. This uncertainty, a double result from one operation, is written down.
The study of problems related to this phenomenon has become a direction in mathematics called the theory of complex variables, which has a large practical significance in mathematical physics.
It is curious that the same omnipresent I. Newton used the designation of the root - radical - in his “Universal Arithmetic”, and exactly modern look notation of the root has been known since 1690 from the book of the Frenchman Rolle “Manual of Algebra”.
Rational numbersThe non-negative square root of a positive number is called arithmetic square root and is denoted using the radical sign.
Complex numbers
Over the field of complex numbers there are always two solutions, differing only in sign (with the exception of the square root of zero). The root of a complex number is often denoted as , but this notation must be used carefully. Common mistake:
To extract the square root of a complex number, it is convenient to use the exponential form of writing a complex number: if
,
,
where the modulus root is understood in the sense of an arithmetic value, and k can take the values k=0 and k=1, so the answer ends up with two different results.
Generalizations
Square roots are introduced as solutions to equations of the form for other objects: matrices, functions, operators, etc. Quite arbitrary multiplicative operations can be used as an operation, for example, superposition.
Square root in computer science
In many function-level programming languages (as well as markup languages like LaTeX), the square root function is written as sqrt(from English square root"Square root").
Algorithms for finding the square root
Finding or calculating the square root of a given number is called extraction(square) root.
Taylor series expansion
at .Arithmetic square root
For squares of numbers the following equalities are true:
That is, 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, like this:
3 steps are completed, the square root of 9 is 3.
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 can solve simple problems. math problems, requiring square root extraction.
Rough estimate
Many algorithms for calculating square roots of a positive real number S require some initial value. If the initial value is too far from the real value of the root, the calculations become slower. Therefore, it is useful to have a rough estimate, which may be very imprecise, but is easy to calculate. If S≥ 1, let D will be the number of digits S to the left of the decimal point. If S < 1, пусть D will be the number of consecutive zeros to the right of the decimal point, taken with a minus sign. Then the rough estimate looks like this:
If D odd, D = 2n+ 1, then use 
If D even, D = 2n+ 2, then use 
Two and six are used because 
And
When working in a binary system (as inside computers), a different evaluation should be used (here D is the number of binary digits).
Geometric square root
To manually extract the root, a notation similar to long division is used.
The number whose root we are looking for is written down. To the right of it we will gradually obtain the numbers of the desired root. Let's take the root of a number with a finite number of decimal places. To begin, mentally or with marks, we divide the number N into groups of two digits to the left and to the right of the decimal point. If necessary, groups are padded with zeros - the integer part is padded on the left, the fractional part on the right. So 31234.567 can be represented as 03 12 34. 56 70. Unlike division, demolition is carried out in such groups of 2 digits.
Visual description of the algorithm:
What is a square root?
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”) This concept is very simple. Natural, I would say. Mathematicians try to find a reaction for every action. There is addition - there is also subtraction. There is multiplication - there is also division. There is squaring... So there is also taking the square root! That's all. This action ( square root
) in mathematics is indicated by this icon: The icon itself is called "a beautiful word".
radical How to extract the root? It's better to look at.
examples
But what is the square root of zero? No problem! What number squared does zero make? Yes, it gives zero! Means:
Got it, what is square root? Then we consider examples:
Answers (in disarray): 6; 1; 4; 9; 5.
Decided? Really, how much easier is that?!
But... What does a person do when he sees some task with roots?
A person begins to feel sad... He does not believe in the simplicity and lightness of his roots. Although he seems to know what is square root...
This is because the person ignored several important points when studying the roots. Then these fads take cruel revenge on tests and exams...
Point one. You need to recognize the roots by sight!
What is the square root of 49? Seven? Right! How did you know it was seven? Squared seven and got 49? Right! Please note that extract the root out of 49 we had to do the reverse operation - square 7! And make sure we don't miss. Or they could have missed...
This is the difficulty root extraction. Square You can use any number without any problems. Multiply a number by itself with a column - that's all. But for root extraction There is no such simple and fail-safe technology. We have to pick up answer and check if it is correct by squaring it.
This complex creative process - choosing an answer - is greatly simplified if you remember squares of popular numbers. Like a multiplication table. If, say, you need to multiply 4 by 6, you don’t add four 6 times, do you? The answer 24 immediately comes up. Although, not everyone gets it, yes...
To work freely and successfully with roots, it is enough to know the squares of numbers from 1 to 20. Moreover there And back. Those. you should be able to easily recite both, say, 11 squared and the square root of 121. To achieve this memorization, there are two ways. The first is to learn the table of squares. This will be a great help in solving examples. The second is to solve more examples. This will greatly help you remember the table of squares.
And no calculators! For testing purposes only. Otherwise, you will slow down mercilessly during the exam...
So, what is square root And How extract roots- I think it’s clear. Now let's find out WHAT we can extract them from.
Point two. Root, I don't know you!
What numbers can you take square roots from? Yes, almost any of them. It's easier to understand what it's from it is forbidden extract them.
Let's try to calculate this root:
To do this, we need to choose a number that squared will give us -4. We select.
What, it doesn't fit? 2 2 gives +4. (-2) 2 gives again +4! That's it... There are no numbers that, when squared, will give us a negative number! Although I know these numbers. But I won’t tell you). Go to college and you will find out for yourself.
The same story will happen with any negative number. Hence the conclusion:
An expression in which there is a negative number under the square root sign - doesn't make sense! This is a forbidden operation. It is as forbidden as dividing by zero. Remember this fact firmly! Or in other words:
Square roots of negative numbers can't be removed!
But of all the others, it’s possible. For example, it is quite possible to calculate
At first glance, this is very difficult. Selecting fractions and squaring them... Don't worry. When we understand the properties of roots, such examples will be reduced to the same table of squares. Life will become easier!
Okay, fractions. But we still come across expressions like:
It's OK. All the same. The square root of two is the number that, when squared, gives us two. Only this number is completely uneven... Here it is:
What’s interesting is that this fraction never ends... Such numbers are called irrational. In square roots this is the most common thing. By the way, this is why expressions with roots are called irrational. It is clear that writing such an infinite fraction all the time is inconvenient. Therefore, instead of an infinite fraction, they leave it like this:
If, when solving an example, you end up with something that cannot be extracted, like:
then we leave it like that. This will be the answer.
You need to clearly understand what the icons mean
Of course, if the root of the number is taken smooth, you must do this. The answer to the task is in the form, for example
Quite a complete answer.
And, of course, you need to know the approximate values from memory:
This knowledge greatly helps to assess the situation in complex tasks.
Point three. The most cunning.
The main confusion in working with roots is caused by this point. It is he who gives confidence in his own abilities... Let's deal with this point properly!
First, let's take the square root of four of them again. Have I already bothered you with this root?) Never mind, now it will be interesting!
What number does 4 square? Well, two, two - I hear dissatisfied answers...
Right. Two. But also minus two will give 4 squared... Meanwhile, the answer
correct and the answer
gross mistake. Like this.
So what's the deal?
Indeed, (-2) 2 = 4. And under the definition of the square root of four minus two quite suitable... This is also the square root of four.
But! IN school course Mathematicians usually consider square roots only non-negative numbers! That is, zero and all positive. Even a special term was invented: from the number A- This non-negative number whose square is A. Negative results when extracting an arithmetic square root are simply discarded. At school, everything is square roots - arithmetic. Although this is not particularly mentioned.
Okay, that's understandable. It's even better not to bother with negative results... This is not yet confusion.
Confusion begins when solving quadratic equations. For example, you need to solve the following equation.
The equation is simple, we write the answer (as taught):
This answer (absolutely correct, by the way) is just an abbreviated version two answers:
Stop, stop! Just above I wrote that the square root is a number Always non-negative! And here is one of the answers - negative! Disorder. This is the first (but not the last) problem that causes distrust of the roots... Let's solve this problem. Let's write down the answers (purely for understanding!) like this:
The parentheses do not change the essence of the answer. I just separated it with brackets signs from root. Now you can clearly see that the root itself (in brackets) is still a non-negative number! And the signs are result of solving the equation. After all, when solving any equation we must write All Xs that, when substituted into the original equation, will give the correct result. The root of five (positive!) with both a plus and a minus fits into our equation.
Like this. If you just take the square root from anything, you Always you get one non-negative result. For example:
Because it - arithmetic square root.
But if you decide something quadratic equation, type:
That Always it turns out two answer (with plus and minus):
Because this is the solution to the equation.
Hope, what is square root You've got your points clear. Now it remains to find out what can be done with the roots, what their properties are. And what are the points and pitfalls... sorry, stones!)
All this is in the following lessons.
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