What is the definition of natural numbers. Natural number

The nth root of the number x is a nonnegative number z that, when raised to the nth power, becomes x. The definition of the root is included in the list of basic arithmetic operations, which we get to know in childhood.

Mathematical notation

"Root" comes from the Latin word radix and today the word "radical" is used as a synonym for this mathematical term. Since the 13th century, mathematicians have denoted root extraction by the letter r with a horizontal bar above the radical expression. In the 16th century, the V designation was introduced, which gradually replaced the r sign, but the horizontal line remained. It is easy to type in a typography or write by hand, but the letter designation of the root - sqrt has spread in electronic publications and programming. This is how we will denote square roots in this article.

Square root

The square radical of a number x is a number z that, when multiplied by itself, becomes x. For example, if we multiply 2 by 2, we get 4. Two in this case is the square root of four. Multiply 5 by 5, we get 25, and now we already know the value of the expression sqrt (25). We can multiply and - 12 by -12 and get 144, and the radical 144 is both 12 and -12. Obviously, square roots can be both positive and negative numbers.

A kind of dualism of such roots is important for solving quadratic equations, therefore, when looking for answers in such problems, it is required to indicate both roots. When solving algebraic expressions, arithmetic square roots are used, that is, only their positive values.

Numbers whose square roots are whole are called perfect squares. There is a whole sequence of such numbers, the beginning of which looks like:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256…

The square roots of other numbers are irrational numbers. For example, sqrt (3) = 1.73205080757 ... and so on. This number is infinite and not periodic, which causes some difficulties in calculating such radicals.

High school mathematics states that you cannot extract square roots from negative numbers. As we learn in the university course of matanalysis, this can and should be done - for this, complex numbers are needed. However, our program is designed to extract the real values of the roots, so it does not calculate even radicals from negative numbers.

Cubic root

The cubic radical of a number x is a number z that, when multiplied by itself three times, gives the number x. For example, if we multiply 2 × 2 × 2, we get 8. Therefore, two is the cube root of eight. Multiplying the four by ourselves three times, we get 4 × 4 × 4 = 64. Obviously, the four is the cube root of 64. There is an infinite sequence of numbers whose cubic radicals are integers. Its beginning looks like:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744…

For the rest of the numbers, the cube roots are irrational numbers... Unlike square radicals, cubic roots, like any odd roots, can be extracted from negative numbers. It's all about the product of numbers less than zero. Minus for minus gives a plus - the known with school bench rule. And a minus for a plus - gives a minus. If we multiply negative numbers an odd number of times, then the result will also be negative, therefore, extract the odd radical from negative number nothing bothers us.

However, the calculator program works differently. Essentially, extracting a root is a reversed exponentiation. The square root is considered as exponentiation of 1/2, and the cubic root is considered as 1/3. The formula for exponentiation of 1/3 can be changed and expressed as 2/6. The result is the same, but you cannot extract such a root from a negative number. Thus, our calculator only calculates arithmetic roots from positive numbers.

Nth root

Such an ornate way of calculating radicals allows you to determine roots of any degree from any expression. You can extract the 5th root of the cube of a number or the 19th radical of a 12th power. All this is elegantly implemented in the form of raising to the power of 3/5 or 12/19, respectively.

Let's consider an example

Diagonal of a square

The irrationality of the diagonal of a square was known to the ancient Greeks. They faced the problem of calculating the diagonal of a flat square, since its length is always proportional to the root of two. The formula for determining the length of the diagonal is derived from and ultimately takes the form:

d = a × sqrt (2).

Let's find the square radical of two using our calculator. Let's enter in the cell "Number (x)" the value 2, and in the "Power (n)" also 2. As a result, we get the expression sqrt (2) = 1.4142. Thus, for a rough estimate of the diagonal of a square, it is sufficient to multiply its side by 1.4142.

Conclusion

The search for a radical is a standard arithmetic operation, without which scientific or design calculations are indispensable. Of course, we do not need to determine the roots to solve everyday problems, but our online calculator will definitely come in handy for schoolchildren or students to check homework assignments in algebra or mathematical analysis.

Engineering calculator online

We are in a hurry to present a free engineering calculator to everyone. With its help, any student can quickly and, most importantly, easily perform various kinds of mathematical calculations online.

Calculator taken from the site - web 2.0 scientific calculator

A simple and easy-to-use engineering calculator with an unobtrusive and understandable interface will truly be useful to the widest circle of Internet users. Now, when you need a calculator, visit our website and use a free engineering calculator.

An engineering calculator can do as simple arithmetic operations, and rather complex mathematical calculations.

Web20calc is an engineering calculator that has great amount functions, for example, how to calculate all elementary functions. The calculator also supports trigonometric functions, matrices, logarithms and even graphing.

Undoubtedly, Web20calc will be of interest to that group of people who, in search of simple solutions, types a query in search engines: mathematical online calculator... A free web application will help you instantly calculate the result of some mathematical expression, for example, subtract, add, divide, extract a root, raise to a power, etc.

In the expression, you can use the operations of exponentiation, addition, subtraction, multiplication, division, percentage, constant PI. For complex calculations, use parentheses.

Engineering calculator features:

1. basic arithmetic operations;
2. work with numbers in a standard form;
3. calculation of trigonometric roots, functions, logarithms, exponentiation;
4. statistical calculations: addition, arithmetic mean or standard deviation;
5. application of a memory cell and user-defined functions of 2 variables;
6. work with angles in radian and degree measures.

The engineering calculator allows you to use a variety of mathematical functions:

Extraction of roots (square root, cubic, and n-th root);
ex (e to the x power), exponent;
trigonometric functions: sine - sin, cosine - cos, tangent - tan;
inverse trigonometric functions: arcsine - sin-1, arccosine - cos-1, arctangent - tan-1;
hyperbolic functions: sine - sinh, cosine - cosh, tangent - tanh;
logarithms: binary logarithm base two - log2x, decimal logarithm base ten - log, natural logarithm - ln.

This engineering calculator also includes a quantity calculator with the ability to convert physical quantities for various measurement systems - computer units, distance, weight, time, etc. With this function, you can instantly convert miles to kilometers, pounds to kilograms, seconds to hours, etc.

To make mathematical calculations, first enter a sequence of mathematical expressions in the appropriate field, then click on the equal sign and see the result. You can enter values directly from the keyboard (for this, the calculator area must be active, therefore, it will not be superfluous to put the cursor in the input field). Among other things, data can be entered using the buttons on the calculator itself.

To build graphs in the input field, write the function as indicated in the field with examples or use the specially designed toolbar (to go to it, click on the button with the icon in the form of a graph). To convert values press Unit, to work with matrices - Matrix.

Instructions

To raise a number to the 1/3 power, enter that number, then click on the exponentiation button and type in the approximate value of the number 1/3 - 0.333. This accuracy is sufficient for most calculations. However, the accuracy of the calculations is very easy to improve - just add as many triples as it will fit on the calculator indicator (for example, 0.3333333333333333). Then press the "=" button.

To calculate the root of the third power using your computer, start the Windows calculator program. The procedure for calculating the root of the third degree is completely similar to that described above. The only difference is in the design of the exponentiation button. It is labeled “x ^ y” on the calculator's virtual keyboard.

The root of the third degree can also be calculated in MS Excel. To do this, enter "=" in any cell and select the "insert" (fx) icon. Select the "DEGREE" function in the window that appears and press the "OK" button. In the window that appears, enter the value of the number for which you want to calculate the root of the third degree. In "Degree" enter the number "1/3". Dial the number 1/3 in this form - as usual. After that, click the "OK" button. In the cell of the table where it was created, the cube root of per this number.

If you have to calculate the root of the third degree all the time, then slightly improve the method described above. As the number from which you want to extract the root, specify not the number itself, but the cell of the table. After that, just each time enter the original number into this cell - its cube root will appear in the cell with the formula.

You will need

calculator or computer

Instructions

To count the root third degree, use the calculator. It is desirable that this was not an ordinary calculator, but a calculator used for engineering calculations. However, even on such a calculator, you will not find a special button for extracting the root. third degree... Therefore, use a function to raise a number to a power. Extracting the root third degree corresponds to raising to the power 1/3 (one third).
To raise a number to the 1/3 power, type the number itself on the calculator's keyboard. Then press the "Exponentiation" button. Such a button, depending on the type of calculator, may look like xy (y - as a superscript). Since most calculators do not have the ability to work with ordinary (non-decimal) fractions, instead of the number 1/3, type its approximate value: 0.33. To get more accurate calculations, you need to increase the number of "triplets", for example, dial 0.33333333333333. Then, click the "=" button.
To count the root third degree on a computer, use a standard Windows calculator. The procedure is completely similar to that described in the previous paragraph of the instructions. The only difference is the symbol for the exponentiation button. On a "computer" calculator, it looks like x ^ y.
If the root third degree have to be considered systematically, then use MS Excel. To count the root third degree in "Excel", enter in any cell the sign "=", and then, select the icon "fx" - insert a function. In the window that appears in the "Select a function" list, select the line "DEGREE". Click the "Ok" button. In the newly appeared window, enter in the "Number" line the value of the number from which you want to extract the root. In the line "Degree" enter the number "1/3" and click "OK". The required value of the cube root from the original number will appear in the cell of the table.

From a large number we have already disassembled without a calculator. In this article, we'll look at how to extract the cube root (root of the third power). I'll make a reservation that we are talking about natural numbers. How long do you think it takes to orally calculate roots such as:

Quite a bit, and if you train two or three times for 20 minutes, then you can extract any such root in 5 seconds orally.

* It should be noted that we are talking about such numbers under the root, which are the result of cubing natural numbers from 0 to 100.

We know that:

So, the number a, which we will find is a natural number from 0 to 100. Look at the table of cubes of these numbers (the results of raising to the third power):

You can easily extract the cube root of any number in this table. What do you need to know?

1. These are cubes of multiples of ten:

I would even say that these are "beautiful" numbers, they are easy to remember. It's easy to learn.

2. This is a property of numbers in the product.

Its essence lies in the fact that when raising to the third power of a certain number, the result will have a singularity. Which one?

For example, let's cube 1, 11, 21, 31, 41, etc. You can look at the table.

1 3 = 1, 11 3 = 1331, 21 3 = 9261, 31 3 = 26791, 41 3 = 68921 …

That is, when we cube a number with one at the end, the result will always be a number with one at the end.

When cubing a number with a 2 at the end, the result will always be a number with an 8 at the end.

Let's show the correspondence in the plate for all numbers:

The knowledge of the presented two points is quite enough.

Let's consider some examples:

Extract the cube root of 21952.

This number is in the range from 8000 to 27000. This means that the result of the root lies in the range from 20 to 30. The number 29952 ends with 2. This option is possible only when a number with an eight at the end is raised to a cube. So the root result is 28.

Extract the cube root of 54852.

This number is in the range from 27000 to 64000. This means that the result of the root lies in the range from 30 to 40. The number 54852 ends with 2. This option is possible only when a number with an eight at the end is raised to a cube. So the root result is 38.

Extract the cube root of 571787.

This number is in the range from 512000 to 729000. This means that the result of the root lies in the range from 80 to 90. The number 571787 ends with 7. This option is possible only when a number with a three at the end is raised to a cube. So the root result is 83.

Extract the cube root of 614125.

This number is in the range from 512000 to 729000. This means that the result of the root lies in the range from 80 to 90. The number 614125 ends with 5. This option is possible only when a number with a five at the end is raised to a cube. So the root result is 85.

I think you can now easily extract the cube root of the number 681472.

Of course, it takes a little practice to extract such roots orally. But having restored the two indicated tablets on paper, you can easily extract such a root within a minute, in any case.

After you have found the result, be sure to check (raise it to the third degree). * Nobody canceled multiplication by a column 😉

On the exam itself, there are no problems with such "ugly" roots. For example, in you want to extract the cube root of 1728. I think that this is not a problem for you now.

If you know any interesting calculation techniques without a calculator, send me, I will publish it over time.That's all. Success to you!

Best regards, Alexander Krutitskikh.

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