Among the many knowledge that is a sign of literacy, alphabet is in the first place. The next, the same “symbolic” element, is the addition – multiplication skills and the arithmetic operations of subtraction – division adjacent to them, but inverse in meaning. The skills acquired in distant school childhood serve faithfully day and night: TV, newspaper, SMS, payment bills. And everywhere we read, write, count, add, subtract, multiply. And, tell me, have you often had to take roots in life, except in the country house? For example, such an entertaining puzzle, such as the square root of 12345 ... Is there still gunpowder in the flocks? Can we handle it? Yes, there is nothing easier! Where is my calculator ... And without it, melee, weak?
First, let's clarify what it is - the square root of a number. Generally speaking, “extracting a root from a number” means performing an arithmetic operation opposite to raising to a power - here you have the unity of opposites in a life application. Raising, say, a square, is the multiplication of a number by itself, that is, as taught at school, X * X = A or in another record X2 = A, and with the words “X squared equals A”. Then the inverse problem is: the square root of the number A, is the number X, which, being squared is equal to A.
Extract the square root
From the school course of arithmetic, there are known methods of computing "in a column" that help to perform any calculations using the first four arithmetic operations. Alas ... For square, and not only square, the roots of 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 a sequential search of numbers whose square approaches the value of the radical expression. That's all! Before an hour or two has passed, any square root can be calculated using the well-known method of multiplication in a “column”. If you have the skills, a couple of minutes is enough for this. Even a not-so-advanced calculator or PC user does it in one fell swoop - progress.
But seriously, the square root calculation is often performed using the “artillery fork” technique: first, take a number whose square roughly corresponds to the radical expression. Better if “our square” is slightly smaller than this expression. Then the number is adjusted according to one’s own skill-understanding, for example, they are multiplied by two, and ... they are again squared. If the result is greater than the number under the root, sequentially adjusting the original number, gradually approach its “colleague” under the root. As you can see - no calculator, only the ability to count "in a column." Of course, there are many scientifically-reasoned and optimized algorithms for calculating the square root, but for "home use" the above technique gives 100% confidence in the result.
Yes, I almost forgot to confirm my increased literacy, we calculate the square root of the previously indicated number 12345. We do it step by step:
1. Take, purely intuitively, X = 100. We calculate: X * X = 10000. Intuition at altitude - the result is less than 12345.
2. Let's try, also purely intuitively, X = 120. Then: X * X = 14400. And again, with intuition, the order is - the result is more than 12345.
3. The “fork” 100 and 120 is obtained above. We choose new numbers - 110 and 115. We get, respectively, 12100 and 13225 - the fork narrows.
4. We try to "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 on the result obtained. That's all. As promised - everything is very simple and without a calculator.
Quite a bit of history ...
The Pythagoreans, school students and followers of Pythagoras, 800 years BC thought of using square roots. and right there, “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 finished 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 - here it is, the influence of the Pythagorean school. It turned out that in a triangle with a very specific unitary size of the sides, the hypotenuse has a size that is expressed by a number that has “no end”. So in mathematics appeared irrational numbers.
2. It is known that dashing trouble began. It turned out that this mathematical operation contains another catch: when we extract the root, we do not know the square of which number, positive or negative, is the radical expression. This uncertainty, the double result of one operation, is recorded.
The study of the problems associated with this phenomenon has become a direction in mathematics called the theory of a complex variable, which is of great practical importance in mathematical physics.
It is curious that the radical notation used the same ubiquitous I. Newton in his Universal Arithmetic, and the modern form of the root notation has been known since 1690 from the French book of Rolle's Guide to Algebra.