Table formula decimal fractions. Decimal. Operations with decimals

In this tutorial, we'll look at each of these operations one by one.

Lesson content

Adding decimals

As we know, a decimal has an integer part and a fractional part. When adding decimals, the integer and fractional parts are added separately.

For example, let's add the decimals 3.2 and 5.3. It is more convenient to add decimal fractions in a column.

First, we write these two fractions in a column, while the integer parts must be under the integer parts, and the fractional ones under the fractional ones. In school, this requirement is called "comma under comma".

Let's write the fractions in a column so that the comma is under the comma:

We begin to add the fractional parts: 2 + 3 \u003d 5. We write down the five in the fractional part of our answer:

Now we add up the integer parts: 3 + 5 = 8. We write the eight in the integer part of our answer:

Now we separate the integer part from the fractional part with a comma. To do this, we again follow the rule "comma under comma":

Got the answer 8.5. So the expression 3.2 + 5.3 is equal to 8.5

In fact, not everything is as simple as it seems at first glance. Here, too, there are pitfalls, which we will now talk about.

Places in decimals

Decimals, like ordinary numbers, have their own digits. These are tenth places, hundredth places, thousandth places. In this case, the digits begin after the decimal point.

The first digit after the decimal point is responsible for the tenths place, the second digit after the decimal point for the hundredths place, the third digit after the decimal point for the thousandths place.

Decimal digits store some useful information. In particular, they report how many tenths, hundredths, and thousandths are in a decimal.

For example, consider the decimal 0.345

The position where the triple is located is called tenth place

The position where the four is located is called hundredths place

The position where the five is located is called thousandths

Let's look at this figure. We see that in the category of tenths there is a three. This suggests that there are three tenths in the decimal fraction 0.345.

If we add the fractions, and then we get the original decimal fraction 0.345

It can be seen that at first we got the answer, but converted it to a decimal fraction and got 0.345.

When adding decimal fractions, the same principles and rules are followed as when adding ordinary numbers. The addition of decimal fractions occurs by digits: tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.

Therefore, when adding decimal fractions, it is required to follow the rule "comma under comma". A comma under a comma provides the same order in which tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.

Example 1 Find the value of the expression 1.5 + 3.4

First of all, we add the fractional parts 5 + 4 = 9. We write the nine in the fractional part of our answer:

Now we add up the integer parts 1 + 3 = 4. We write down the four in the integer part of our answer:

Now we separate the integer part from the fractional part with a comma. To do this, we again observe the rule "comma under a comma":

Got the answer 4.9. So the value of the expression 1.5 + 3.4 is 4.9

Example 2 Find the value of the expression: 3.51 + 1.22

We write this expression in a column, observing the rule "comma under a comma"

First of all, add the fractional part, namely the hundredths 1+2=3. We write the triple in the hundredth part of our answer:

Now add tenths of 5+2=7. We write down the seven in the tenth part of our answer:

Now add the whole parts 3+1=4. We write down the four in the whole part of our answer:

We separate the integer part from the fractional part with a comma, observing the “comma under the comma” rule:

Got the answer 4.73. So the value of the expression 3.51 + 1.22 is 4.73

3,51 + 1,22 = 4,73

As with ordinary numbers, when adding decimal fractions, . In this case, one digit is written in the answer, and the rest are transferred to the next digit.

Example 3 Find the value of the expression 2.65 + 3.27

We write this expression in a column:

Add hundredths of 5+7=12. The number 12 will not fit in the hundredth part of our answer. Therefore, in the hundredth part, we write the number 2, and transfer the unit to the next bit:

Now we add the tenths of 6+2=8 plus the unit that we got from the previous operation, we get 9. We write the number 9 in the tenth of our answer:

Now add the whole parts 2+3=5. We write the number 5 in the integer part of our answer:

Got the answer 5.92. So the value of the expression 2.65 + 3.27 is 5.92

2,65 + 3,27 = 5,92

Example 4 Find the value of the expression 9.5 + 2.8

Write this expression in a column

We add the fractional parts 5 + 8 = 13. The number 13 will not fit in the fractional part of our answer, so we first write down the number 3, and transfer the unit to the next digit, or rather transfer it to the integer part:

Now we add the integer parts 9+2=11 plus the unit that we got from the previous operation, we get 12. We write the number 12 in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 12.3. So the value of the expression 9.5 + 2.8 is 12.3

9,5 + 2,8 = 12,3

When adding decimal fractions, the number of digits after the decimal point in both fractions must be the same. If there are not enough digits, then these places in the fractional part are filled with zeros.

Example 5. Find the value of the expression: 12.725 + 1.7

Before writing this expression in a column, let's make the number of digits after the decimal point in both fractions the same. The decimal fraction 12.725 has three digits after the decimal point, while the fraction 1.7 has only one. So in the fraction 1.7 at the end you need to add two zeros. Then we get the fraction 1,700. Now you can write this expression in a column and start calculating:

Add thousandths of 5+0=5. We write the number 5 in the thousandth part of our answer:

Add hundredths of 2+0=2. We write the number 2 in the hundredth part of our answer:

Add tenths of 7+7=14. The number 14 will not fit in a tenth of our answer. Therefore, we first write down the number 4, and transfer the unit to the next bit:

Now we add the integer parts 12+1=13 plus the unit that we got from the previous operation, we get 14. We write the number 14 in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 14,425. So the value of the expression 12.725+1.700 is 14.425

12,725+ 1,700 = 14,425

Subtraction of decimals

When subtracting decimal fractions, you must follow the same rules as when adding: “a comma under a comma” and “an equal number of digits after a decimal point”.

Example 1 Find the value of the expression 2.5 − 2.2

We write this expression in a column, observing the “comma under comma” rule:

We calculate the fractional part 5−2=3. We write the number 3 in the tenth part of our answer:

Calculate the integer part 2−2=0. We write zero in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

We got the answer 0.3. So the value of the expression 2.5 − 2.2 is equal to 0.3

2,5 − 2,2 = 0,3

Example 2 Find the value of the expression 7.353 - 3.1

This expression has a different number of digits after the decimal point. In the fraction 7.353 there are three digits after the decimal point, and in the fraction 3.1 there is only one. This means that in the fraction 3.1, two zeros must be added at the end to make the number of digits in both fractions the same. Then we get 3,100.

Now you can write this expression in a column and calculate it:

Got the answer 4,253. So the value of the expression 7.353 − 3.1 is 4.253

7,353 — 3,1 = 4,253

As with ordinary numbers, sometimes you will have to borrow one from the adjacent bit if subtraction becomes impossible.

Example 3 Find the value of the expression 3.46 − 2.39

Subtract hundredths of 6−9. From the number 6 do not subtract the number 9. Therefore, you need to take a unit from the adjacent digit. Having borrowed one from the neighboring digit, the number 6 turns into the number 16. Now we can calculate the hundredths of 16−9=7. We write down the seven in the hundredth part of our answer:

Now subtract tenths. Since we took one unit in the category of tenths, the figure that was located there decreased by one unit. In other words, the tenth place is now not the number 4, but the number 3. Let's calculate the tenths of 3−3=0. We write zero in the tenth part of our answer:

Now subtract the integer parts 3−2=1. We write the unit in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 1.07. So the value of the expression 3.46−2.39 is equal to 1.07

3,46−2,39=1,07

Example 4. Find the value of the expression 3−1.2

This example subtracts a decimal from an integer. Let's write this expression in a column so that the integer part of the decimal fraction 1.23 is under the number 3

Now let's make the number of digits after the decimal point the same. To do this, after the number 3, put a comma and add one zero:

Now subtract tenths: 0−2. Do not subtract the number 2 from zero. Therefore, you need to take a unit from the adjacent digit. By borrowing one from the adjacent digit, 0 turns into the number 10. Now you can calculate the tenths of 10−2=8. We write down the eight in the tenth part of our answer:

Now subtract the whole parts. Previously, the number 3 was located in the integer, but we borrowed one unit from it. As a result, it turned into the number 2. Therefore, we subtract 1 from 2. 2−1=1. We write the unit in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 1.8. So the value of the expression 3−1.2 is 1.8

Decimal multiplication

Multiplying decimals is easy and even fun. To multiply decimals, you need to multiply them like regular numbers, ignoring the commas.

Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in both fractions, then count the same number of digits on the right in the answer and put a comma.

Example 1 Find the value of the expression 2.5 × 1.5

We multiply these decimal fractions as ordinary numbers, ignoring the commas. To ignore the commas, you can temporarily imagine that they are absent altogether:

We got 375. In this number, it is necessary to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in fractions of 2.5 and 1.5. In the first fraction there is one digit after the decimal point, in the second fraction there is also one. A total of two numbers.

We return to the number 375 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 3.75. So the value of the expression 2.5 × 1.5 is 3.75

2.5 x 1.5 = 3.75

Example 2 Find the value of the expression 12.85 × 2.7

Let's multiply these decimals, ignoring the commas:

We got 34695. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 12.85 and 2.7. In the fraction 12.85 there are two digits after the decimal point, in the fraction 2.7 there is one digit - a total of three digits.

We return to the number 34695 and begin to move from right to left. We need to count three digits from the right and put a comma:

Got the answer 34,695. So the value of the expression 12.85 × 2.7 is 34.695

12.85 x 2.7 = 34.695

Multiplying a decimal by a regular number

Sometimes there are situations when you need to multiply a decimal fraction by a regular number.

To multiply a decimal and an ordinary number, you need to multiply them, regardless of the comma in the decimal. Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the decimal fraction, then count the same number of digits to the right in the answer and put a comma.

For example, multiply 2.54 by 2

We multiply the decimal fraction 2.54 by the usual number 2, ignoring the comma:

We got the number 508. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.54. The fraction 2.54 has two digits after the decimal point.

We return to the number 508 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 5.08. So the value of the expression 2.54 × 2 is 5.08

2.54 x 2 = 5.08

Multiplying decimals by 10, 100, 1000

Multiplying decimals by 10, 100, or 1000 is done in the same way as multiplying decimals by regular numbers. It is necessary to perform the multiplication, ignoring the comma in the decimal fraction, then in the answer, separate the integer part from the fractional part, counting the same number of digits on the right as there were digits after the decimal point in the decimal fraction.

For example, multiply 2.88 by 10

Let's multiply the decimal fraction 2.88 by 10, ignoring the comma in the decimal fraction:

We got 2880. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.88. We see that in the fraction 2.88 there are two digits after the decimal point.

We return to the number 2880 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 28.80. We discard the last zero - we get 28.8. So the value of the expression 2.88 × 10 is 28.8

2.88 x 10 = 28.8

There is a second way to multiply decimal fractions by 10, 100, 1000. This method is much simpler and more convenient. It consists in the fact that the comma in the decimal fraction moves to the right by as many digits as there are zeros in the multiplier.

For example, let's solve the previous example 2.88×10 in this way. Without giving any calculations, we immediately look at the factor 10. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 2.88 we move the decimal point to the right by one digit, we get 28.8.

2.88 x 10 = 28.8

Let's try to multiply 2.88 by 100. We immediately look at the factor 100. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 2.88 we move the decimal point to the right by two digits, we get 288

2.88 x 100 = 288

Let's try to multiply 2.88 by 1000. We immediately look at the factor 1000. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 2.88 we move the decimal point to the right by three digits. The third digit is not there, so we add another zero. As a result, we get 2880.

2.88 x 1000 = 2880

Multiplying decimals by 0.1 0.01 and 0.001

Multiplying decimals by 0.1, 0.01, and 0.001 works in the same way as multiplying a decimal by a decimal. It is necessary to multiply fractions like ordinary numbers, and put a comma in the answer, counting as many digits on the right as there are digits after the decimal point in both fractions.

For example, multiply 3.25 by 0.1

We multiply these fractions like ordinary numbers, ignoring the commas:

We got 325. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 3.25 and 0.1. In the fraction 3.25 there are two digits after the decimal point, in the fraction 0.1 there is one digit. A total of three numbers.

We return to the number 325 and begin to move from right to left. We need to count three digits on the right and put a comma. After counting three digits, we find that the numbers are over. In this case, you need to add one zero and put a comma:

We got the answer 0.325. So the value of the expression 3.25 × 0.1 is 0.325

3.25 x 0.1 = 0.325

There is a second way to multiply decimals by 0.1, 0.01 and 0.001. This method is much easier and more convenient. It consists in the fact that the comma in the decimal fraction moves to the left by as many digits as there are zeros in the multiplier.

For example, let's solve the previous example 3.25 × 0.1 in this way. Without giving any calculations, we immediately look at the factor 0.1. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 3.25 we move the decimal point to the left by one digit. Moving the comma one digit to the left, we see that there are no more digits before the three. In this case, add one zero and put a comma. As a result, we get 0.325

3.25 x 0.1 = 0.325

Let's try multiplying 3.25 by 0.01. Immediately look at the multiplier of 0.01. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 3.25 we move the comma to the left by two digits, we get 0.0325

3.25 x 0.01 = 0.0325

Let's try multiplying 3.25 by 0.001. Immediately look at the multiplier of 0.001. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 3.25 we move the decimal point to the left by three digits, we get 0.00325

3.25 × 0.001 = 0.00325

Do not confuse multiplying decimals by 0.1, 0.001 and 0.001 with multiplying by 10, 100, 1000. A common mistake most people make.

When multiplying by 10, 100, 1000, the comma is moved to the right by as many digits as there are zeros in the multiplier.

And when multiplying by 0.1, 0.01 and 0.001, the comma is moved to the left by as many digits as there are zeros in the multiplier.

If at first it is difficult to remember, you can use the first method, in which the multiplication is performed as with ordinary numbers. In the answer, you will need to separate the integer part from the fractional part by counting as many digits on the right as there are digits after the decimal point in both fractions.

Dividing a smaller number by a larger one. Advanced level.

In one of the previous lessons, we said that when dividing a smaller number by a larger one, a fraction is obtained, in the numerator of which is the dividend, and in the denominator is the divisor.

For example, to divide one apple into two, you need to write 1 (one apple) in the numerator, and write 2 (two friends) in the denominator. The result is a fraction. So each friend will get an apple. In other words, half an apple. A fraction is the answer to a problem how to split one apple between two

It turns out that you can solve this problem further if you divide 1 by 2. After all, a fractional bar in any fraction means division, which means that this division is also allowed in a fraction. But how? We are used to the fact that the dividend is always greater than the divisor. And here, on the contrary, the dividend is less than the divisor.

Everything will become clear if we remember that a fraction means crushing, dividing, dividing. This means that the unit can be split into as many parts as you like, and not just into two parts.

When dividing a smaller number by a larger one, a decimal fraction is obtained, in which the integer part will be 0 (zero). The fractional part can be anything.

So, let's divide 1 by 2. Let's solve this example with a corner:

One cannot be divided into two just like that. If you ask a question "how many twos are in one" , then the answer will be 0. Therefore, in private we write 0 and put a comma:

Now, as usual, we multiply the quotient by the divisor to pull out the remainder:

The moment has come when the unit can be split into two parts. To do this, add another zero to the right of the received one:

We got 10. We divide 10 by 2, we get 5. We write down the five in the fractional part of our answer:

Now we take out the last remainder to complete the calculation. Multiply 5 by 2, we get 10

We got the answer 0.5. So the fraction is 0.5

Half an apple can also be written using the decimal fraction 0.5. If we add these two halves (0.5 and 0.5), we again get the original one whole apple:

This point can also be understood if we imagine how 1 cm is divided into two parts. If you divide 1 centimeter into 2 parts, you get 0.5 cm

Example 2 Find the value of expression 4:5

How many fives are in four? Not at all. We write in private 0 and put a comma:

We multiply 0 by 5, we get 0. We write zero under the four. Immediately subtract this zero from the dividend:

Now let's start splitting (dividing) the four into 5 parts. To do this, to the right of 4, we add zero and divide 40 by 5, we get 8. We write the eight in private.

We complete the example by multiplying 8 by 5, and get 40:

We got the answer 0.8. So the value of the expression 4: 5 is 0.8

Example 3 Find the value of expression 5: 125

How many numbers 125 are in five? Not at all. We write 0 in private and put a comma:

We multiply 0 by 5, we get 0. We write 0 under the five. Immediately subtract from the five 0

Now let's start splitting (dividing) the five into 125 parts. To do this, to the right of this five, we write zero:

Divide 50 by 125. How many numbers 125 are in 50? Not at all. So in the quotient we again write 0

We multiply 0 by 125, we get 0. We write this zero under 50. Immediately subtract 0 from 50

Now we divide the number 50 into 125 parts. To do this, to the right of 50, we write another zero:

Divide 500 by 125. How many numbers are 125 in the number 500. In the number 500 there are four numbers 125. We write the four in private:

We complete the example by multiplying 4 by 125, and get 500

We got the answer 0.04. So the value of the expression 5: 125 is 0.04

Division of numbers without a remainder

So, let's put a comma in the quotient after the unit, thereby indicating that the division of integer parts is over and we proceed to the fractional part:

Add zero to the remainder 4

Now we divide 40 by 5, we get 8. We write the eight in private:

40−40=0. Received 0 in the remainder. So the division is completely completed. Dividing 9 by 5 results in a decimal of 1.8:

9: 5 = 1,8

Example 2. Divide 84 by 5 without a remainder

First we divide 84 by 5 as usual with a remainder:

Received in private 16 and 4 more in the balance. Now we divide this remainder by 5. We put a comma in the private, and add 0 to the remainder 4

Now we divide 40 by 5, we get 8. We write the eight in the quotient after the decimal point:

and complete the example by checking if there is still a remainder:

Dividing a decimal by a regular number

A decimal fraction, as we know, consists of an integer and a fractional part. When dividing a decimal fraction by a regular number, first of all you need:

divide the integer part of the decimal fraction by this number;
after the integer part is divided, you need to immediately put a comma in the private part and continue the calculation, as in ordinary division.

For example, let's divide 4.8 by 2

Let's write this example as a corner:

Now let's divide the whole part by 2. Four divided by two is two. We write the deuce in private and immediately put a comma:

Now we multiply the quotient by the divisor and see if there is a remainder from the division:

4−4=0. The remainder is zero. We do not write zero yet, since the solution is not completed. Then we continue to calculate, as in ordinary division. Take down 8 and divide it by 2

8: 2 = 4. We write the four in the quotient and immediately multiply it by the divisor:

Got the answer 2.4. Expression value 4.8: 2 equals 2.4

Example 2 Find the value of the expression 8.43:3

We divide 8 by 3, we get 2. Immediately put a comma after the two:

Now we multiply the quotient by the divisor 2 × 3 = 6. We write the six under the eight and find the remainder:

We divide 24 by 3, we get 8. We write the eight in private. We immediately multiply it by the divisor to find the remainder of the division:

24−24=0. The remainder is zero. Zero is not recorded yet. Take the last three of the dividend and divide by 3, we get 1. Immediately multiply 1 by 3 to complete this example:

Got the answer 2.81. So the value of the expression 8.43: 3 is equal to 2.81

Dividing a decimal by a decimal

To divide a decimal fraction into a decimal fraction, in the dividend and in the divisor, move the comma to the right by the same number of digits as there are after the decimal point in the divisor, and then divide by a regular number.

For example, divide 5.95 by 1.7

Let's write this expression as a corner

Now, in the dividend and in the divisor, we move the comma to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we must move the comma to the right by one digit in the dividend and in the divisor. Transferring:

After moving the decimal point to the right by one digit, the decimal fraction 5.95 turned into a fraction 59.5. And the decimal fraction 1.7, after moving the decimal point to the right by one digit, turned into the usual number 17. And we already know how to divide the decimal fraction by the usual number. Further calculation is not difficult:

The comma is moved to the right to facilitate division. This is allowed due to the fact that when multiplying or dividing the dividend and the divisor by the same number, the quotient does not change. What does it mean?

This is one of the interesting features of division. It is called the private property. Consider expression 9: 3 = 3. If in this expression the dividend and the divisor are multiplied or divided by the same number, then the quotient 3 will not change.

Let's multiply the dividend and divisor by 2 and see what happens:

(9 × 2) : (3 × 2) = 18: 6 = 3

As can be seen from the example, the quotient has not changed.

The same thing happens when we carry a comma in the dividend and in the divisor. In the previous example, where we divided 5.91 by 1.7, we moved the comma one digit to the right in the dividend and divisor. After moving the comma, the fraction 5.91 was converted to the fraction 59.1 and the fraction 1.7 was converted to the usual number 17.

In fact, inside this process, multiplication by 10 took place. Here's what it looked like:

5.91 × 10 = 59.1

Therefore, the number of digits after the decimal point in the divisor depends on what the dividend and divisor will be multiplied by. In other words, the number of digits after the decimal point in the divisor will determine how many digits in the dividend and in the divisor the comma will be moved to the right.

Decimal division by 10, 100, 1000

Dividing a decimal by 10, 100, or 1000 is done in the same way as . For example, let's divide 2.1 by 10. Let's solve this example with a corner:

But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is moved to the left by as many digits as there are zeros in the divisor.

Let's solve the previous example in this way. 2.1: 10. We look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 2.1, you need to move the comma to the left by one digit. We move the comma to the left by one digit and see that there are no more digits left. In this case, we add one more zero before the number. As a result, we get 0.21

Let's try to divide 2.1 by 100. There are two zeros in the number 100. So in the divisible 2.1, you need to move the comma to the left by two digits:

2,1: 100 = 0,021

Let's try to divide 2.1 by 1000. There are three zeros in the number 1000. So in the divisible 2.1, you need to move the comma to the left by three digits:

2,1: 1000 = 0,0021

Decimal division by 0.1, 0.01 and 0.001

Dividing a decimal by 0.1, 0.01, and 0.001 is done in the same way as . In the dividend and in the divisor, you need to move the comma to the right by as many digits as there are after the decimal point in the divisor.

For example, let's divide 6.3 by 0.1. First of all, we move the commas in the dividend and in the divisor to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we move the commas in the dividend and in the divisor to the right by one digit.

After moving the decimal point to the right by one digit, the decimal fraction 6.3 turns into the usual number 63, and the decimal fraction 0.1, after moving the decimal point to the right by one digit, turns into one. And dividing 63 by 1 is very simple:

So the value of the expression 6.3: 0.1 is equal to 63

But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is transferred to the right by as many digits as there are zeros in the divisor.

Let's solve the previous example in this way. 6.3:0.1. Let's look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 6.3, you need to move the comma to the right by one digit. We move the comma to the right by one digit and get 63

Let's try to divide 6.3 by 0.01. Divisor 0.01 has two zeros. So in the divisible 6.3, you need to move the comma to the right by two digits. But in the dividend there is only one digit after the decimal point. In this case, one more zero must be added at the end. As a result, we get 630

Let's try dividing 6.3 by 0.001. The divisor of 0.001 has three zeros. So in the divisible 6.3, you need to move the comma to the right by three digits:

6,3: 0,001 = 6300

Tasks for independent solution

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Of the many fractions found in arithmetic, those with 10, 100, 1000 in the denominator deserve special attention - in general, any power of ten. These fractions have a special name and notation.

A decimal is any number whose denominator is a power of ten.

Decimal examples:

Why was it necessary to isolate such fractions at all? Why do they need their own entry form? There are at least three reasons for this:

Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, bring the fractions to a common denominator. In decimal fractions, none of this is required;
Reduction of calculations. Decimals add and multiply according to their own rules, and after a little practice you will work with them much faster than with ordinary ones;
Ease of recording. Unlike ordinary fractions, decimals are written in one line without loss of clarity.

Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you demand change in the amount of 2/3 rubles in a store :)

Rules for writing decimal fractions

The main advantage of decimal fractions is a convenient and visual notation. Namely:

Decimal notation is a form of decimal notation where the integer part is separated from the fractional part using a regular dot or comma. In this case, the separator itself (dot or comma) is called the decimal point.

For example, 0.3 (read: “zero integer, 3 tenths”); 7.25 (7 integers, 25 hundredths); 3.049 (3 integers, 49 thousandths). All examples are taken from the previous definition.

In writing, a comma is usually used as a decimal point. Here and below, the comma will also be used throughout the site.

To write an arbitrary decimal fraction in the specified form, you need to follow three simple steps:

Write out the numerator separately;
Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
If the decimal point has shifted, and after it there are zeros at the end of the record, they must be crossed out.

It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, any number of zeros can be assigned to the left of any number without harm to health. It's ugly, but sometimes useful.

At first glance, this algorithm may seem rather complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:

Task. For each fraction, indicate its decimal notation:

The numerator of the first fraction: 73. We shift the decimal point by one sign (because the denominator is 10) - we get 7.3.

The numerator of the second fraction: 9. We shift the decimal point by two digits (because the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange notation like “.09”.

The numerator of the third fraction: 10029. We shift the decimal point by three digits (because the denominator is 1000) - we get 10.029.

The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10.500. There are extra zeros at the end of the number. We cross them out - we get 10.5.

Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as is done in the last example. However, in no case should you do this with zeros that are inside the number (which are surrounded by other digits). That is why we got 10.029 and 10.5, and not 1.29 and 1.5.

So, we figured out the definition and form of recording decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.

Change from fractions to decimals

Consider a simple numerical fraction of the form a / b . You can use the basic property of a fraction and multiply the numerator and denominator by such a number that you get a power of ten below. But before doing so, please read the following:

There are denominators that are not reduced to the power of ten. Learn to recognize such fractions, because they cannot be worked with according to the algorithm described below.

That's it. Well, how to understand whether the denominator is reduced to the power of ten or not?

The answer is simple: factorize the denominator into prime factors. If only factors 2 and 5 are present in the expansion, this number can be reduced to the power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the degree of ten.

Task. Check if the specified fractions can be represented as decimals:

We write out and factorize the denominators of these fractions:

20 \u003d 4 5 \u003d 2 2 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.

12 \u003d 4 3 \u003d 2 2 3 - there is a "forbidden" factor 3. The fraction cannot be represented as a decimal.

640 \u003d 8 8 10 \u003d 2 3 2 3 2 5 \u003d 2 7 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction is represented as a decimal.

48 \u003d 6 8 \u003d 2 3 2 3 \u003d 2 4 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.

So, we figured out the denominator - now we will consider the entire algorithm for switching to decimal fractions:

Factorize the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
Count how many twos and fives are present in the decomposition (there will be no other numbers there, remember?). Choose such an additional multiplier so that the number of twos and fives is equal.
Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.

Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest such factor from all possible ones.

And one more thing: if there is an integer part in the original fraction, be sure to convert this fraction to an improper one - and only then apply the described algorithm.

Task. Convert these numbers to decimals:

Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, a fraction can be represented as a decimal. There are two twos and no fives in the expansion, so the additional factor is 5 2 = 25. The number of twos and fives will be equal to it. We have:

Now let's deal with the second fraction. To do this, note that 24 \u003d 3 8 \u003d 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.

The last two fractions have denominators 5 (a prime number) and 20 = 4 5 = 2 2 5 respectively - only twos and fives are present everywhere. At the same time, in the first case, “for complete happiness”, there is not enough multiplier 2, and in the second - 5. We get:

Switching from decimals to ordinary

The reverse conversion - from decimal notation to normal - is much easier. There are no restrictions and special checks here, so you can always convert a decimal fraction into a classic “two-story” one.

The translation algorithm is as follows:

Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing - do not overdo it and do not cross out the internal zeros surrounded by other numbers;
Calculate how many digits are in the original decimal fraction after the decimal point. Take the number 1 and add as many zeros to the right as you counted the characters. This will be the denominator;
Actually, write down the fraction whose numerator and denominator we just found. Reduce if possible. If there was an integer part in the original fraction, now we will get an improper fraction, which is very convenient for further calculations.

Task. Convert decimals to ordinary: 0.008; 3.107; 2.25; 7,2008.

We cross out the zeros on the left and the commas - we get the following numbers (these will be numerators): 8; 3107; 225; 72008.

In the first and second fractions after the decimal point there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.

Finally, let's combine the numerators and denominators into ordinary fractions:

As can be seen from the examples, the resulting fraction can very often be reduced. Once again, I note that any decimal fraction can be represented as an ordinary one. The reverse transformation is not always possible.

In this article, we will understand what a decimal fraction is, what features and properties it has. Go! 🙂

The decimal fraction is a special case of ordinary fractions (in which the denominator is a multiple of 10).

Definition

Decimals are fractions whose denominators are numbers consisting of one and a certain number of zeros following it. That is, these are fractions with a denominator of 10, 100, 1000, etc. Otherwise, a decimal fraction can be characterized as a fraction with a denominator of 10 or one of the powers of ten.

Fraction examples:

, ,

A decimal fraction is written differently than a common fraction. Operations with these fractions are also different from operations with ordinary ones. The rules for operations on them are to a large extent close to the rules for operations on integers. This, in particular, determines their relevance in solving practical problems.

Representation of a fraction in decimal notation

There is no denominator in the decimal notation, it displays the number of the numerator. In general, decimal fractions are written as follows:

where X is the integer part of the fraction, Y is its fractional part, "," is the decimal point.

For the correct representation of an ordinary fraction as a decimal, it is required that it be correct, that is, with a highlighted integer part (if possible) and a numerator that is less than the denominator. Then, in decimal notation, the integer part is written before the decimal point (X), and the numerator of the ordinary fraction is written after the decimal point (Y).

If the numerator represents a number with a number of digits less than the number of zeros in the denominator, then in the Y part the missing number of digits in the decimal notation is filled with zeros in front of the numerator digits.

Example:

If the ordinary fraction is less than 1, i.e. does not have an integer part, then 0 is written in decimal form for X.

In the fractional part (Y), after the last significant (other than zero) digit, an arbitrary number of zeros can be entered. It does not affect the value of the fraction. And vice versa: all zeros at the end of the fractional part of the decimal fraction can be omitted.

Reading decimals

Part X is read in the general case as follows: "X integers."

The Y part is read according to the number in the denominator. For the denominator 10, you should read: "Y tenths", for the denominator 100: "Y hundredths", for the denominator 1000: "Y thousandths" and so on ... 😉

Another approach to reading is considered more correct, based on counting the number of digits of the fractional part. To do this, you need to understand that the fractional digits are located in a mirror image with respect to the digits of the integer part of the fraction.

Names for correct reading are given in the table:

Based on this, the reading should be based on the correspondence to the name of the category of the last digit of the fractional part.

3.5 reads "three point five"
0.016 reads like "zero point sixteen thousandths"

Converting an arbitrary ordinary fraction to a decimal

If the denominator of an ordinary fraction is 10 or some power of ten, then the fraction is converted as described above. In other situations, additional transformations are needed.

There are 2 ways to translate.

The first way of translation

The numerator and denominator must be multiplied by such an integer that the denominator is 10 or one of the powers of ten. And then the fraction is represented in decimal notation.

This method is applicable for fractions, the denominator of which is decomposed only into 2 and 5. So, in the previous example . If there are other prime factors in the expansion (for example, ), then you will have to resort to the 2nd method.

The second way of translation

The 2nd method is to divide the numerator by the denominator in a column or on a calculator. The integer part, if any, is not involved in the transformation.

The long division rule that results in a decimal fraction is described below (see Dividing Decimals).

Convert decimal to ordinary

To do this, its fractional part (to the right of the comma) should be written as a numerator, and the result of reading the fractional part should be written as the corresponding number in the denominator. Further, if possible, you need to reduce the resulting fraction.

End and Infinite Decimal

The decimal fraction is called final, the fractional part of which consists of a finite number of digits.

All the above examples contain exactly the final decimal fractions. However, not every ordinary fraction can be represented as a final decimal. If the 1st translation method for a given fraction is not applicable, and the 2nd method demonstrates that the division cannot be completed, then only an infinite decimal fraction can be obtained.

It is impossible to write an infinite fraction in its full form. In an incomplete form, such fractions can be represented:

as a result of reduction to the desired number of decimal places;
in the form of a periodic fraction.

A fraction is called periodic, in which, after the decimal point, an infinitely repeating sequence of digits can be distinguished.

The remaining fractions are called non-periodic. For non-periodic fractions, only the 1st representation method (rounding) is allowed.

An example of a periodic fraction: 0.8888888 ... There is a repeating figure 8 here, which, obviously, will be repeated indefinitely, since there is no reason to assume otherwise. This number is called fraction period.

Periodic fractions are pure and mixed. A decimal fraction is pure, in which the period begins immediately after the decimal point. A mixed fraction has 1 or more digits before the decimal point.

54.33333 ... - periodic pure decimal fraction

2.5621212121 ... - periodic mixed fraction

Examples of writing infinite decimals:

The 2nd example shows how to properly form a period in a periodic fraction.

Converting periodic decimals to ordinary

To convert a pure periodic fraction into an ordinary period, write it in the numerator, and write in the denominator a number consisting of nines in an amount equal to the number of digits in the period.

A mixed recurring decimal is translated as follows:

you need to form a number consisting of the number after the decimal point before the period, and the first period;
from the resulting number subtract the number after the decimal point before the period. The result will be the numerator of an ordinary fraction;
in the denominator, you need to enter a number consisting of the number of nines equal to the number of digits of the period, followed by zeros, the number of which is equal to the number of digits of the number after the decimal point before the 1st period.

Decimal Comparison

Decimal fractions are compared initially by their whole parts. The larger is the fraction that has the larger integer part.

If the integer parts are the same, then the digits of the corresponding digits of the fractional part are compared, starting from the first (from the tenths). The same principle applies here: the larger of the fractions, which has a larger rank of tenths; if the tenths digits are equal, the hundredths digits are compared, and so on.

Insofar as

, since with equal integer parts and equal tenths in the fractional part, the 2nd fraction has more hundredths.

Adding and subtracting decimals

Decimals are added and subtracted in the same way as whole numbers, writing the corresponding digits one under the other. To do this, you need to have decimal points under each other. Then the units (tens, etc.) of the integer part, as well as the tenths (hundredths, etc.) of the fractional part will match. The missing digits of the fractional part are filled with zeros. Directly The process of addition and subtraction is carried out in the same way as for integers.

Decimal multiplication

To multiply decimal fractions, you need to write them one under the other, aligned with the last digit and not paying attention to the location of the decimal points. Then you need to multiply the numbers in the same way as when multiplying integers. After receiving the result, you should recalculate the number of digits after the decimal point in both fractions and separate the total number of fractional digits in the resulting number with a comma. If there are not enough digits, they are replaced by zeros.

Multiplying and dividing decimals by 10 n

These actions are simple and come down to moving the decimal point. P in multiplication, the comma is moved to the right (the fraction increases) by the number of digits equal to the number of zeros in 10 n, where n is an arbitrary integer power. That is, a certain number of digits are transferred from the fractional part to the integer. When dividing, respectively, the comma is transferred to the left (the number decreases), and some of the digits are transferred from the integer part to the fractional part. If there are not enough digits to transfer, then the missing digits are filled with zeros.

Dividing a decimal and an integer by an integer and a decimal

Dividing a decimal by an integer is the same as dividing two integers. Additionally, only the position of the decimal point must be taken into account: when demolishing the digit of the digit followed by a comma, it is necessary to put a comma after the current digit of the generated answer. Then you need to keep dividing until you get zero. If there are not enough signs in the dividend for complete division, zeros should be used as them.

Similarly, 2 integers are divided into a column if all the digits of the dividend have been demolished, and the full division has not yet been completed. In this case, after the demolition of the last digit of the dividend, a decimal point is placed in the resulting answer, and zeros are used as the demolished digits. Those. the dividend here, in fact, is represented as a decimal fraction with a zero fractional part.

To divide a decimal fraction (or an integer) by a decimal number, it is necessary to multiply the dividend and the divisor by the number 10 n, in which the number of zeros is equal to the number of digits after the decimal point in the divisor. In this way, they get rid of the decimal point in the fraction by which you want to divide. Further, the division process is the same as described above.

Graphical representation of decimals

Graphically, decimal fractions are represented by means of a coordinate line. For this, single segments are additionally divided into 10 equal parts, just as centimeters and millimeters are deposited on a ruler at the same time. This ensures that decimals are displayed accurately and can be compared objectively.

In order for the longitudinal divisions on single segments to be the same, one should carefully consider the length of the single segment itself. It should be such that the convenience of additional division can be ensured.

DECIMAL FRACTIONS. ACTIONS ON DECIMAL FRACTIONS

(lesson summary)

Tumysheva Zamira Tansykbaevna, teacher of mathematics, school-gymnasium No. 2

Khromtau, Aktobe region, Republic of Kazakhstan

This development of the lesson is intended as a lesson-generalization of the chapter "Actions on decimal fractions". It can be used in both 5th grade and 6th grade. The lesson is conducted in the form of a game.

Decimals. Operations on decimals.(lesson summary)

Target:

Practicing the skills and abilities of adding, subtracting, multiplying and dividing decimal fractions into natural numbers and decimal fractions

Creating conditions for the development of independent work skills, self-control and self-esteem, the development of intellectual qualities: attention, imagination, memory, the ability to analyze and generalize

To instill cognitive interest in the subject and develop self-confidence

LESSON PLAN:

1. Organizational part.

3. The theme and purpose of our lesson.

4. The game "To the treasured flag!"

5. The game "Number mill".

6. Lyrical digression.

7. Verification work.

8. The game "Encryption" (work in pairs)

9. Summing up.

10. Homework.

1. Organizational part. Hello. Have a seat.

2. An overview of the rules for performing arithmetic operations with decimal fractions.

Rule for adding and subtracting decimals:

1) equalize the number of decimal places in these fractions;

2) write down one under the other so that the comma is under the comma;

3) without noticing the comma, perform the action (addition or subtraction), and put a comma under the commas as a result.

3,455 + 0,45 = 3,905 3,5 + 4 = 7,5 15 – 7,88 = 7,12 4,57 - 3,2 = 1,37

3,455 + 3,5 _15,00 _ 4,57

0,450 4,0 7,88 3,20

3,905 7,5 7,12 1,37

When adding and subtracting, natural numbers are written as a decimal fraction with decimal places equal to zero.

Rule for multiplying decimals:

1) ignoring the comma, multiply the numbers;

2) in the resulting product, separate with a comma as many digits from right to left as they are separated by a comma in decimal fractions.

When multiplying a decimal fraction by bit units (10, 100, 1000, etc.), the comma is moved to the right by as many numbers as there are zeros in the bit unit

17.25 4 = 69

x 1 7.2 5

6 9,0 0

15.256 100 = 1525.6

.5 0.52 = 2.35

X 0.5 2

4,5

2 7 0

2 0 8__

2,3 5 0

When multiplying, natural numbers are written as natural numbers.

The rule for dividing decimal fractions by a natural number:

1) divide the whole part of the dividend, put a comma in the private;

2) continue dividing.

When dividing to the remainder, we take down only one number from the dividend.

If in the process of dividing a decimal fraction there remains a remainder, then by assigning the required number of zeros to it, we continue the division until the remainder is zero.

15,256: 100 = 0,15256

0,25: 1000 = 0,00025

When dividing a decimal fraction into bit units (10, 100, 1000, etc.), the comma is moved to the left by as many numbers as there are zeros in the bit unit.

18,4: 8 = 2,3

_ 18,4 І_8_

16 2,3

2 4

22,2: 25 = 0,88

22,2 І_25_

0 0,888

22 2

20 0

2 20

2 00

200

3,56: 4 = 0,89

3,56 І_4_

0 0,89

3 5

3 2

When dividing, natural numbers are written as natural numbers.

Rule for dividing decimals by decimals:

1) we move the comma in the divisor to the right so that we get a natural number;

2) move the comma in the dividend to the right of as many numbers as it was moved in the divisor;

3) we divide the decimal fraction by a natural number.

3,76: 0,4 = 9, 4

_ 3,7,6 I_0,4,_

3 6 9, 4

1 6

The game "To the cherished flag!"

Rules of the game: From each team, one student is called to the board, who perform an oral count from the bottom step. The solver of one example marks the answer in the table. Then he is replaced by another member of the team. There is a movement up - to the coveted flag. Students in the field verbally check the results of their players. If the answer is incorrect, another member of the team comes to the board to continue solving the problems. Team captains call the students to work at the board. The first team to reach the flag with the fewest students wins.

Game "Number Mill"

Rules of the game: Numbers are written in the circles of the mill. The arrows connecting the circles indicate the actions. The task is to perform sequential actions, moving along the arrow from the center to the outer circle. Performing sequential actions along the indicated route, you will find the answer in one of the circles below. The result of performing actions for each arrow is written in the oval next to it.

Lyrical digression.

Lifshitz's poem "Three tenths"

Who is this

From portfolio

Throws in annoyance

hateful puzzler,

Pencil case and notebooks

And sticks his diary.

Without blushing,

Under an oak sideboard.

To lie under the sideboard? ..

Please get to know:

Kostya Zhigalin.

The victim of eternal nit-picking, -

He failed again.

And hisses

To disheveled

Looking problem book:

I'm just not lucky!

I'm just a loser!

What is the reason

His resentment and annoyance?

That the answer didn't fit

Only three tenths.

This is a real waste!

And to him, of course,

find fault

Strict

Maria Petrovna.

Three tenths...

Tell me about this error

And, perhaps, on the faces

You will see a smile.

Three tenths...

And yet about this error

I beg you

listen to me

No smile.

If b, building your house.

The one you live in.

Architect

a little

Wrong

In counting, -

What would happen.

Do you know Kostya Zhigalin?

This house

would have turned

In a heap of ruins!

You enter the bridge.

He is reliable and durable.

Don't be an engineer

Accurate in his drawings, -

Would you, Kostya,

Falling down

into the cold river

Wouldn't say thank you

That person!

Here is the turbine.

It has a shaft

Bored by turners.

If the turner

In work

Wasn't very accurate.

It would be done, Kostya,

Great misfortune:

It would destroy the turbine

into small pieces!

Three tenths -

And the walls

Are being erected

Koso!

Three tenths -

And collapse

wagons

Off the slope!

make a mistake

Only three tenths

Pharmacy, -

Medicine becomes poison

Will kill a man!

We smashed and drove

Fascist gang.

Your father gave

Battery command.

Make a mistake on arrival

At least three tenths

The shells would not overtake

Damned fascists.

You think about it

My friend, in cold blood

And say.

Was it not right

Maria Petrovna?

To be honest

Think about it, Kostya.

It's not long to lie

Diary under the buffet!

Test work on the topic "Decimal fractions" (mathematics -5)

9 slides will appear on the screen in sequence. Students write down the number of the option and the answers to the question in their notebooks. For example, Option 2

1. C; 2. A; etc.

QUESTION 1

Option 1

When multiplying a decimal fraction by 100, you need to move the comma in this fraction:

A. to the left by 2 digits; B. to the right by 2 digits; C. do not change the place of the comma.

Option 2

When multiplying a decimal fraction by 10, you need to move the comma in this fraction:

A. right 1 digit; B. to the left by 1 digit; C. do not change the place of the comma.

QUESTION 2

Option 1

The sum 6.27 + 6.27 + 6.27 + 6.27 + 6.27 as a product is written as follows:

A. 6.27 5; B. 6.27 6.27; S. 6.27 4.

Option 2

The sum 9.43 + 9.43 + 9.43 + 9.43 as a product is written as follows:

A. 9.43 9.43; B. 6 9.43; S. 9.43 4.

QUESTION 3

Option 1

In the product 72.43 18 after the decimal point will be:

Option 2

In the product of 12.453 35 after the decimal point will be:

A. 2 digits; B. 0 digits; C. 3 digits.

QUESTION 4

Option 1

In quotient 76.4:2 after the decimal point will be:

A. 2 digits; B. 0 digits; C. 1 digit.

Option 2

In private 95.4:6 after the decimal point will be:

A. 1 digit; B. 3 digits; C. 2 digits.

QUESTION 5

Option 1

Find the value of the expression 34.5: x + 0.65 y, at x=10 y=100:

A. 35.15; B. 68.45; S. 9.95.

Option 2

Find the value of the expression 4.9 x +525:y, at x=100 y=1000:

A. 4905.25; B. 529.9; pp. 490,525.

QUESTION 6

Option 1

The area of a rectangle with sides 0.25 and 12 cm is

A. 3; B. 0.3; S. 30.

Option 2

The area of a rectangle with sides 0.5 and 36 cm is

A. 1.8; V. 18; C. 0.18.

QUESTION 7

Option 1

Two students left the school at the same time in opposite directions. The speed of the first student is 3.6 km/h, the speed of the second student is 2.56 km/h. After 3 hours the distance between them will be:

A. 6.84 km; V. 18.48 km; S. 3.12 km

Option 2

Two cyclists left the school at the same time in opposite directions. The speed of the first is 11.6 km/h, the speed of the second is 13.06 km/h. After 4 hours the distance between them will be:

A. 5.84 km; V. 100.8 km; S. 98.64 km

Option 1

Option 2

Check your answers. Put a "+" for a correct answer and a "-" for an incorrect answer.

Game "Encryption"

Rules of the game: Each desk is given a card with a task that has a code-letter. After completing the steps and getting the result, write down the code-letter of your card under the number corresponding to your answer.

As a result, we get the proposal:

6,8

420

21,6

420

306

65,8

21,6

Summing up the lesson.

Scores for test work are announced.

Homework #1301, 1308, 1309

Thank you for your attention!!!

To write a rational number m / n as a decimal fraction, you need to divide the numerator by the denominator. In this case, the quotient is written as a finite or infinite decimal fraction.

Write the given number as a decimal.

Decision. Divide the numerator of each fraction by its denominator: a) divide 6 by 25; b) divide 2 by 3; in) divide 1 by 2, and then add the resulting fraction to unity - the integer part of this mixed number.

Irreducible ordinary fractions whose denominators contain no prime divisors other than 2 and 5 , are written as a final decimal fraction.

AT example 1 when a) denominator 25=5 5; when in) the denominator is 2, so we got the final decimals 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a final decimal.

Is it possible, without dividing into a column, to convert such an ordinary fraction into a decimal fraction, the denominator of which does not contain other divisors, except 2 and 5? Let's figure it out! What fraction is called decimal and is written without a fractional line? Answer: a fraction with a denominator of 10; 100; 1000 etc. And each of these numbers is a product equal number of twos and fives. Actually: 10=2 5 ; 100=2 5 2 5 ; 1000=2 5 2 5 2 5 etc.

Therefore, the denominator of an irreducible ordinary fraction will need to be represented as a product of twos and fives, and then multiplied by 2 and (or) 5 so that the twos and fives become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. So that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which the denominator was multiplied.

Express the following fractions as a decimal:

Decision. Each of these fractions is irreducible. Let us decompose the denominator of each fraction into prime factors.

20=2 2 5. Conclusion: one "five" is missing.

8=2 2 2. Conclusion: there are not enough three "fives".

25=5 5. Conclusion: two "twos" are missing.

Comment. In practice, they often do not use the factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is a unit with zeros (10 or 100 or 1000, etc.). And then the numerator is multiplied by the same number.

So, in case a)(example 2) from the number 20 you can get 100 by multiplying by 5, therefore, you need to multiply the numerator and denominator by 5.

When b)(example 2) from the number 8, the number 100 will not work, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.

When in)(example 2) out of 25 you get 100 when multiplied by 4. This means that the numerator 8 must also be multiplied by 4.

An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodical decimal fraction. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosing it in parentheses.

When b)(example 1 ) the repeated digit is one and equals 6. Therefore, our result 0.66... will be written like this: 0,(6) . They read: zero integers, six in the period.

If there is one or more non-recurring digits between the comma and the first period, then such a periodic fraction is called a mixed periodic fraction.

An irreducible common fraction whose denominator together with others multiplier contains multiplier 2 or 5 , becomes mixed periodic fraction.

Write the number as a decimal:

Any rational number can be written as an infinite periodic decimal fraction.

Write the number as an infinite periodic fraction.