How to take a root from a number. Research work on the topic: "Extracting square roots from large numbers without a calculator"

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Extracting a root from a large number. Dear friends!In this article, we will show you how to take the root of a large number without a calculator. This is necessary not only for solving certain types of USE problems (there are such - for movement), but also for the general mathematical development, it is desirable to know this analytical technique.

It would seem that everything is simple: factorize and extract. There is no problem. For example, the number 291600, when expanded, will give the product:

We calculate:

There is one BUT! The method is good if divisors 2, 3, 4 and so on are easily determined. But what if the number from which we extract the root is a product of prime numbers? For example, 152881 is the product of the numbers 17, 17, 23, 23. Try to find these divisors right away.

The essence of the method we are considering- this is pure analysis. The root with the accumulated skill is found quickly. If the skill is not worked out, but the approach is simply understood, then it is a little slower, but still determined.

Let's take the root of 190969.

First, let's determine between what numbers (multiples of a hundred) our result lies.

Obviously, the result of the root of a given number lies in the range from 400 to 500, because

400 2 =160000 and 500 2 =250000

Really:

in the middle, closer to 160,000 or 250,000?

The number 190969 is somewhere in the middle, but still closer to 160000. We can conclude that the result of our root will be less than 450. Let's check:

Indeed, it is less than 450, since 190,969< 202 500.

Now let's check the number 440:

So our result is less than 440, since 190 969 < 193 600.

Checking the number 430:

We have established that the result of this root lies in the range from 430 to 440.

The product of numbers ending in 1 or 9 gives a number ending in 1. For example, 21 times 21 equals 441.
The product of numbers ending in 2 or 8 gives a number ending in 4. For example, 18 times 18 equals 324.
The product of numbers ending in 5 gives a number ending in 5. For example, 25 times 25 equals 625.
The product of numbers ending in 4 or 6 gives a number ending in 6. For example, 26 times 26 equals 676.
The product of numbers ending in 3 or 7 gives a number ending in 9. For example, 17 times 17 equals 289.

Since the number 190969 ends with the number 9, then this product is either 433 or 437.

*Only they, when squared, can give 9 at the end.

We check:

So the result of the root will be 437.

That is, we kind of "felt" the right answer.

As you can see, the maximum that is required is to carry out 5 actions in a column. Perhaps you will immediately get to the point, or you will do just three actions. It all depends on how accurately you make the initial estimate of the number.

Extract your own root from 148996

Such a discriminant is obtained in the problem:

The motor ship passes along the river to the destination 336 km and after parking returns to the point of departure. Find the speed of the ship in still water, if the speed of the current is 5 km / h, the parking lasts 10 hours, and the ship returns to the point of departure 48 hours after leaving it. Give your answer in km/h.

View Solution

The result of the root is between the numbers 300 and 400:

300 2 =90000 400 2 =160000

Indeed, 90000<148996<160000.

The essence of further reasoning is to determine how the number 148996 is located (distanced) relative to these numbers.

Calculate the differences 148996 - 90000=58996 and 160000 - 148996=11004.

It turns out that 148996 is close (much closer) to 160000. Therefore, the result of the root will definitely be greater than 350 and even 360.

We can conclude that our result is greater than 370. Further, it is clear: since 148996 ends with the number 6, this means that you must square the number ending either in 4 or 6. *Only these numbers, when squared, give in end 6.

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell about the site in social networks.

root n th power of a natural number a the number is called n whose th power is equal to a. The root is denoted as follows: . The symbol √ is called root sign or sign of the radical, number a - root 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 the root n th degree

That root extraction- the action, the opposite of exponentiation, with the help of which, according to the given degree and according to the given exponent, the base of the degree is found.

Square root

The square root of a number a is the number whose square is a.

The operation by which the square root is calculated is called taking the square root.

Extracting the 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, it is required to find the number itself from it.

Therefore, to check the correctness of the action taken, you can raise the found root to the second degree, and if the degree is equal to the root number, then the root was found correctly.

Consider extracting the square root and its verification with an example. We calculate or (the root exponent with the value 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, this number is 7, since

7 7 = 7 2 = 49.

Calculating the square root

If the 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 x 5 = 25.

Now consider 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 step by step.

Let us determine which digits the desired root should consist of. Since 10 2 \u003d 10 10 \u003d 100, and 100 2 \u003d 100 100 \u003d 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 produces hundreds, our number is 44, so the root must contain so many tens that the square of tens gives approximately 44 hundreds. Therefore, there should be 6 tens at the root, because 60 2 \u003d 3600, and 70 2 \u003d 4900 (this is too much). Thus, we found out that our root contains 6 tens and several ones, since it is in the range from 60 to 70.
The multiplication table will help determine the number of units at 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. So the root of the number will be 63 or 67.
We check the numbers we got 63 and 67 by squaring them: 63 2 \u003d 3969, 67 2 \u003d 4489.

Preferably engineering - one in which there is a button with a root sign: "√". Usually, to extract the root, it is enough to type the number itself, and then press the button: “√”.

Most modern mobile phones have a "calculator" application with a root extraction function. The procedure for finding the root of a number using a telephone calculator is similar to the above.
Example.
Find from 2.
We turn on the calculator (if it is turned off) and successively press the buttons with the image of two and the root (“2”, “√”). Pressing the "=" key is usually not necessary. As a result, we get a number like 1.4142 (the number of characters and "roundness" depends on the bit depth and calculator settings).
Note: when trying to find the root, the calculator usually gives an error.

If you have access to a computer, then finding the root of a number is very simple.
1. You can use the Calculator application available on almost any computer. For Windows XP, this program can be run as follows:
"Start" - "All Programs" - "Accessories" - "Calculator".
It is better to set the view to "normal". By the way, unlike a real calculator, the button for extracting the root is marked as "sqrt", not "√".

If you do not get to the calculator in the specified way, then you can start the standard calculator “manually”:
"Start" - "Run" - "calc".
2. To find the root of a number, you can also use some programs installed on your computer. In addition, the program has its own built-in calculator.

For example, for the MS Excel application, you can do the following sequence of actions:
We start MS Excel.

We write in any cell the number from which you want to extract the root.

Move the cell pointer to a different location

Press the function selection button (fx)

Select the "ROOT" function

As a function argument, specify a cell with a number

Press "OK" or "Enter"
The advantage of this method is that now it is enough to enter any value into the cell with a number, as in with the function immediately appears.
Note.
There are several other, more exotic ways to find the root of a number. For example, a "corner", using a slide rule or Bradis tables. However, these methods are not considered in this article due to their complexity and practical uselessness.

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 by all means (for example, when calculating the discriminant) need to calculate the square root of 86436.

We will decompose the number 86436 into prime factors. We divide by 2, we get 43218; again we divide by 2, - we get 21609. The number is not divisible by 2 more. But since the sum of the digits is divisible by 3, then the number itself is divisible by 3 (generally speaking, it can be seen that it is also divisible by 9). . Once again we divide by 3, we get 2401. 2401 is not completely divisible by 3. Not divisible by five (does not end with 0 or 5).

We suspect divisibility by 7. Indeed, a ,

So, full order!

Case 2

Let us need to calculate . It is inconvenient to act in the same way as described above. Trying to factorize...

The number 1849 is not completely 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 not 5 or 0) ...

It is not completely divisible by 7, it is not divisible by 11, it is not divisible by 13 ... Well, how long will it take us to go through all the prime numbers like this?

Let's argue a little differently.

We understand that

We narrowed down the search. Now we sort through the numbers from 41 to 49. Moreover, it is clear that since the last digit of the number is 9, then it is worth stopping at options 43 or 47 - only these numbers, when squared, will give the last digit 9.

Well, here already, 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.