What is the square root?

Among the many knowledge that is a sign of literacy, the alphabet is in the first place. The next, the same "sign" element, are the skills of addition and multiplication and, adjacent to them, but inverse in sense, arithmetic operations of subtraction-division. Assimilated in a distant school childhood, they serve faithfully and truthfully day and night: TV, newspaper, SMS, bills for payment. And everywhere we read, write, count, add, subtract, multiply. And, tell me, how often did you have to go through life, take roots, except at the dacha? For example, such an entertaining task, like, the square root of 12345 ... Is there still some powder in the flasks? Osilim? Nothing is easier! Where is my calculator ... And without it, hand-to-hand, weakly?

First, let's clarify what this 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 degree - here is the unity of opposites in the life application. Raising to a power, say, a square is a multiplication of a number by itself, that is, as taught in the school, X * X = A or in another record X2 = A, and the words - "X squared equals A". Then the inverse problem sounds like this: 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 known methods of calculation "in column", which help to perform any calculations using the first four arithmetic operations. Alas ... For square, and not only square, roots of such algorithms does not exist. And in that case, how to extract the square root without a calculator? Proceeding from the definition of the square root, the conclusion is one - it is necessary to select the result value by a sequential search of numbers, the square of which is close to the value of the radicand expression. Only that! Do not have time to pass an hour or two, as you can calculate, using a well-known method of multiplication in the "column", any square root. If you have the skills for this, just a couple of minutes. Even not quite an advanced calculator user or PC does it in one fell swoop - progress.

But seriously, the calculation of the square root is often carried out using the "artillery fork" method: first take a number whose square, approximately, corresponds to the root expression. It is better if "our square" is slightly less than this expression. Then correct the number according to one's own skill-understanding, for example, multiply by two, and ... again squared. If the result is greater than the number under the root, successively adjusting the original number, gradually approach 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-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 will calculate the square root of the previously mentioned number 12345. We do step by step:

1. Take, purely intuitive, X = 100. Let's calculate: Х * Х = 10000. Intuition at height - the result is less than 12345.

2. Let's try, too, purely intuitively, X = 120. Then: X * X = 14400. And again with intuition, the order is more than 12345.

3. The above "fork" 100 and 120. Choose the new numbers - 110 and 115. Obtain, respectively, 12100 and 13225 - the plug narrows.

4. Try on the "maybe" X = 111. We get X * X = 12321. This number is already close enough 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, pupils of the school and followers of Pythagoras, thought up to the use of square roots, for 800 years BC. And immediately, "ran into" new discoveries in the field of numbers. And where did that come from?

1. The solution of the problem with the extraction of 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 down 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 - here it is, the influence of the Pythagoras school. It turned out that in a triangle with a very specific single side dimension, the hypotenuse has a size that is expressed by a number that has "no end". So in mathematics irrational numbers appeared .

2. It is known that famine is a disaster. It turned out that this mathematical operation contains another trick - extracting the root, we do not know which square of positive, or negative, is the radicand. This uncertainty, a double result from one operation, is recorded.

The study of the problems connected with this phenomenon became a direction in mathematics called the theory of a complex variable, which has great practical significance in mathematical physics.

It is curious that the root designation - the radical - applied in his "Universal arithmetic" the same ubiquitous I. Newton, and exactly the modern form of the root record is known from 1690 from the book of the Frenchman Roll "The Guide of Algebra".