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Fraction. Multiplication of common, decimal, mixed fractions

As is known, multiplication of numbers comes down to the summation of partial products obtained by multiplying the current digit of the multiplier IN to the multiplicand L. For binary numbers, partial products are equal to the multiplicand or zero. Therefore, multiplication of binary numbers is reduced to sequential summation of partial products with a shift. For decimal numbers partial products can take 10 different meanings, including zero. Therefore, to obtain partial products, instead of multiplication, multiple sequential summation of the multiplicand L can be used. To illustrate the multiplication algorithm decimal numbers Let's use an example.

Example 2.26. Pa fig. 2.15, A The multiplication of integer decimal numbers A x b = 54 x 23 is given, starting from the least significant digit of the multiplier. The following algorithm is used for multiplication:

0 is taken as the initial state. The first sum is obtained by adding the multiplicand A = 54 to zero. Then the multiplicand is added to the first sum again A= 54. And finally, after the third summation, the first partial product is obtained, equal to 0 "+ 54 + 54 + 54 = 162;

Rice. 2.15. Algorithm for multiplying integer decimal numbers 54 x 23(A) and the principle of its implementation(b)

  • the first partial product is shifted one bit to the right (or the multiplicand to the left);
  • the multiplicand is added twice to the highest digits of the first partial product: 16 + 54 + 54 = 124;
  • after combining the resulting sum 124 with the least significant 2 of the first partial product, the product 1242 is found.

Let us consider, using an example, the possibility of a circuit implementation of an algorithm using the operations of summation, subtraction and shift.

Example 2.27. Let it be in the register R t the multiplicand is permanently stored A = 54. In the initial state to the register R 2 place the multiplier IN= 23, and register R 3 is loaded with zeros. To obtain the first partial product (162), we add the multiplicand three times to the contents of the register A = 54, decreasing the contents of the register each time by one R T After the least significant bit of the register R., becomes equal to zero, shift the contents of both registers /?. to the right by one bit, and R.,. Presence of 0 in the least significant digit R 2c indicates that the formation of the partial product is complete and a shift needs to be made. Then we perform two operations of adding the multiplicand A= 54 with the contents of the register and subtracting one from the contents of the register R 0. After the second operation, the least significant digit of the register R., will become equal to zero. Therefore, by shifting the contents of the registers to the right by one bit R 3 and R Y we obtain the required product P = 1242.

The implementation of the algorithm for multiplying decimal numbers in binary decimal codes (Fig. 2.16) has features associated with performing addition and subtraction operations

Rice. 2.16.

(see paragraph 2.3), as well as shifting the tetrad by four bits. Let's consider them under the conditions of Example 2.27.

Example 2.28. Multiplying floating point numbers. To obtain the product of numbers A and B c floating point must be defined M c = M l x M n, R With = P{ + R n. In this case, the rules of multiplication and algebraic addition of fixed-point numbers are used. The product is assigned a "+" sign if the multiplicand and the multiplier have the same signs, and a "-" sign if their signs are different. If necessary, the resulting mantissa is normalized with appropriate order correction.

Example 2.29. Multiplying binary normalized numbers:

When performing a multiplication operation, there may be special cases, which are processed by special processor commands. For example, if one of the factors is equal to zero, the multiplication operation is not performed (blocked) and a zero result is immediately generated.

In this article we will look at an action such as multiplication decimals. Let's start by stating the general principles, then show how to multiply one decimal fraction by another and consider the method of multiplication by a column. All definitions will be illustrated with examples. Then we will look at how to correctly multiply decimal fractions by ordinary, as well as mixed and natural numbers (including 100, 10, etc.)

In this material, we will only touch on the rules for multiplying positive fractions. Cases with negative numbers are dealt with separately in articles on multiplying rational and real numbers.

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Let's formulate general principles, which must be followed when solving problems on multiplying decimal fractions.

Let us first remember that decimal fractions are nothing more than special shape recordings of ordinary fractions, therefore, the process of multiplying them can be reduced to a similar one for ordinary fractions. This rule works for both finite and infinite fractions: after converting them to ordinary fractions, it is easy to multiply with them according to the rules we have already learned.

Let's see how such problems are solved.

Example 1

Calculate the product of 1.5 and 0.75.

Solution: First, let's replace decimal fractions with ordinary ones. We know that 0.75 is 75/100, and 1.5 is 15/10. We can reduce the fraction and select the whole part. We will write the resulting result 125 1000 as 1, 125.

Answer: 1 , 125 .

We can use the column counting method, just like for natural numbers.

Example 2

Multiply one periodic fraction 0, (3) by another 2, (36).

First, let's reduce the original fractions to ordinary ones. We will get:

0 , (3) = 0 , 3 + 0 , 03 + 0 , 003 + 0 , 003 + . . . = 0 , 3 1 - 0 , 1 = 0 , 3 9 = 3 9 = 1 3 2 , (36) = 2 + 0 , 36 + 0 , 0036 + . . . = 2 + 0 , 36 1 - 0 , 01 = 2 + 36 99 = 2 + 4 11 = 2 4 11 = 26 11

Therefore, 0, (3) · 2, (36) = 1 3 · 26 11 = 26 33.

The resulting common fraction can lead to decimal form, dividing the numerator by the denominator in a column:

Answer: 0 , (3) · 2 , (36) = 0 , (78) .

If we have infinite non-periodic fractions in the problem statement, then we need to perform preliminary rounding (see the article on rounding numbers if you have forgotten how to do this). After this, you can perform the multiplication action with already rounded decimal fractions. Let's give an example.

Example 3

Calculate the product of 5, 382... and 0, 2.

Solution

In our problem we have an infinite fraction that must first be rounded to hundredths. It turns out that 5.382... ≈ 5.38. It makes no sense to round the second factor to hundredths. Now you can calculate the required product and write down the answer: 5.38 0.2 = 538 100 2 10 = 1 076 1000 = 1.076.

Answer: 5.382…·0.2 ≈ 1.076.

The column counting method can be used not only for natural numbers. If we have decimals, we can multiply them in exactly the same way. Let's derive the rule:

Definition 1

Multiplying decimal fractions by column is performed in 2 steps:

1. Perform column multiplication, not paying attention to commas.

2. Bet in the final number decimal point, separating it as many digits on the right side as both factors contain decimal places together. If the result is not enough numbers for this, add zeros to the left.

Let's look at examples of such calculations in practice.

Example 4

Multiply the decimals 63, 37 and 0, 12 by columns.

Solution

First, let's multiply numbers, ignoring decimal points.

Now we need to put the comma in the right place. It will separate the four digits on the right side because the sum of the decimals in both factors is 4. There is no need to add zeros, because enough signs:

Answer: 3.37 0.12 = 7.6044.

Example 5

Calculate how much 3.2601 times 0.0254 is.

Solution

We count without commas. We get the following number:

We will put a comma separating 8 digits on the right side, because the original fractions together have 8 decimal places. But our result has only seven digits, and we cannot do without additional zeros:

Answer: 3.2601 0.0254 = 0.08280654.

How to multiply a decimal by 0.001, 0.01, 01, etc.

Multiplying decimals by such numbers is common, so it is important to be able to do it quickly and accurately. Let's write down a special rule that we will use for this multiplication:

Definition 2

If we multiply a decimal by 0, 1, 0, 01, etc., we end up with a number similar to the original fraction, with the decimal point moved to the left by required quantity signs. If there are not enough numbers to transfer, you need to add zeros to the left.

So, to multiply 45, 34 by 0, 1, you need to move the decimal point in the original decimal fraction by one place. We will end up with 4, 534.

Example 6

Multiply 9.4 by 0.0001.

Solution

We will have to move the decimal point four places according to the number of zeros in the second factor, but the numbers in the first factor are not enough for this. We assign the necessary zeros and find that 9.4 · 0.0001 = 0.00094.

Answer: 0 , 00094 .

For infinite decimals we use the same rule. So, for example, 0, (18) · 0, 01 = 0, 00 (18) or 94, 938... · 0, 1 = 9, 4938.... etc.

The process of such multiplication is no different from the action of multiplying two decimal fractions. It is convenient to use the column multiplication method if the problem statement contains a final decimal fraction. In this case, it is necessary to take into account all the rules that we talked about in the previous paragraph.

Example 7

Calculate how much 15 · 2.27 is.

Solution

Multiply by column original numbers and separate the two commas.

Answer: 15 · 2.27 = 34.05.

If we perform periodic decimal multiplication by natural number, you must first change the decimal fraction to an ordinary fraction.

Example 8

Calculate the product of 0 , (42) and 22 .

Let us reduce the periodic fraction to ordinary form.

0 , (42) = 0 , 42 + 0 , 0042 + 0 , 000042 + . . . = 0 , 42 1 - 0 , 01 = 0 , 42 0 , 99 = 42 99 = 14 33

0, 42 22 = 14 33 22 = 14 22 3 = 28 3 = 9 1 3

We can write the final result in the form of a periodic decimal fraction as 9, (3).

Answer: 0 , (42) 22 = 9 , (3) .

Infinite fractions You must first round before making calculations.

Example 9

Calculate how much 4 · 2, 145... will be.

Solution

Let's round the original infinite decimal fraction to hundredths. After this we come to multiplying a natural number and a final decimal fraction:

4 2.145… ≈ 4 2.15 = 8.60.

Answer: 4 · 2, 145… ≈ 8, 60.

How to multiply a decimal by 1000, 100, 10, etc.

Multiplying a decimal fraction by 10, 100, etc. is often encountered in problems, so we will analyze this case separately. The basic rule of multiplication is:

Definition 3

To multiply a decimal fraction by 1000, 100, 10, etc., you need to move its comma to 3, 2, 1 digits depending on the multiplier and discard the extra zeros on the left. If there are not enough numbers to move the comma, add as many zeros to the right as we need.

Let's show with an example exactly how to do this.

Example 10

Multiply 100 and 0.0783.

Solution

To do this, we need to move the comma in the decimal fraction to 2 digits in right side. We will end up with 007, 83 The zeros on the left can be discarded and the result written as 7, 38.

Answer: 0.0783 100 = 7.83.

Example 11

Multiply 0.02 by 10 thousand.

Solution: We will move the comma four digits to the right. We don’t have enough signs for this in the original decimal fraction, so we’ll have to add zeros. In this case, three 0 will be enough. The result is 0, 02000, move the comma and get 00200, 0. Ignoring the zeros on the left, we can write the answer as 200.

Answer: 0.02 · 10,000 = 200.

The rule we have given will work the same in the case of infinite decimal fractions, but here you should be very careful about the period of the final fraction, since it is easy to make a mistake in it.

Example 12

Calculate the product of 5.32 (672) times 1,000.

Solution: first of all, we will write the periodic fraction as 5, 32672672672 ..., so the probability of making a mistake will be less. After this, we can move the comma to the required number of characters (three). The result will be 5326, 726726... Let's enclose the period in brackets and write the answer as 5,326, (726).

Answer: 5, 32 (672) · 1,000 = 5,326, (726) .

If the problem conditions contain infinite non-periodic fractions that must be multiplied by ten, one hundred, a thousand, etc., do not forget to round them before multiplying.

To perform multiplication of this type, you need to represent the decimal fraction as an ordinary fraction and then proceed according to the already familiar rules.

Example 13

Multiply 0, 4 by 3 5 6

Solution

​First, let's convert the decimal fraction to an ordinary fraction. We have: 0, 4 = 4 10 = 2 5.

We received a response in the form mixed number. You can write it as a periodic fraction 1, 5 (3).

Answer: 1 , 5 (3) .

If an infinite non-periodic fraction is involved in the calculation, you need to round it to a certain number and then multiply it.

Example 14

Calculate the product 3, 5678. . . · 2 3

Solution

We can represent the second factor as 2 3 = 0, 6666…. Next, round both factors to the thousandth place. After this, we will need to calculate the product of two final decimal fractions 3.568 and 0.667. Let's count with a column and get the answer:

The final result must be rounded to thousandths, since it was to this digit that we rounded the original numbers. It turns out that 2.379856 ≈ 2.380.

Answer: 3, 5678. . . · 2 3 ≈ 2, 380

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In the last lesson we learned how to add and subtract decimals (see lesson “Adding and subtracting decimals”). At the same time, we assessed how much calculations are simplified compared to ordinary “two-story” fractions.

Unfortunately, this effect does not occur with multiplying and dividing decimals. In some cases, decimal notation even complicates these operations.

First, let's introduce a new definition. We'll see him quite often, and not just in this lesson.

The significant part of a number is everything between the first and last non-zero digit, including the ends. It's about about numbers only, the decimal point is not taken into account.

The digits included in the significant part of a number are called significant digits. They can be repeated and even be equal to zero.

For example, consider several decimal fractions and write out the corresponding significant parts:

  1. 91.25 → 9125 (significant figures: 9; 1; 2; 5);
  2. 0.008241 → 8241 (significant figures: 8; 2; 4; 1);
  3. 15.0075 → 150075 (significant figures: 1; 5; 0; 0; 7; 5);
  4. 0.0304 → 304 (significant figures: 3; 0; 4);
  5. 3000 → 3 (there is only one significant figure: 3).

Please note: the zeros inside the significant part of the number do not go anywhere. We have already encountered something similar when we learned to convert decimal fractions to ordinary fractions (see lesson “ Decimals”).

This point is so important, and mistakes are made here so often, that I will publish a test on this topic in the near future. Be sure to practice! And we, armed with the concept of the significant part, will proceed, in fact, to the topic of the lesson.

Multiplying Decimals

The multiplication operation consists of three successive steps:

  1. For each fraction, write down the significant part. You will get two ordinary integers - without any denominators and decimal points;
  2. Multiply these numbers in any convenient way. Directly, if the numbers are small, or in a column. We obtain the significant part of the desired fraction;
  3. Find out where and by how many digits the decimal point in the original fractions is shifted to obtain the corresponding significant part. Perform reverse shifts for the significant part obtained in the previous step.

Let me remind you once again that zeros on the sides of the significant part are never taken into account. Ignoring this rule leads to errors.

  1. 0.28 12.5;
  2. 6.3 · 1.08;
  3. 132.5 · 0.0034;
  4. 0.0108 1600.5;
  5. 5.25 · 10,000.

We work with the first expression: 0.28 · 12.5.

  1. Let's write out the significant parts for the numbers from this expression: 28 and 125;
  2. Their product: 28 · 125 = 3500;
  3. In the first factor the decimal point is shifted 2 digits to the right (0.28 → 28), and in the second it is shifted by 1 more digit. In total, you need a shift to the left by three digits: 3500 → 3,500 = 3.5.

Now let's look at the expression 6.3 · 1.08.

  1. Let's write down the significant parts: 63 and 108;
  2. Their product: 63 · 108 = 6804;
  3. Again, two shifts to the right: by 2 and 1 digit, respectively. Total - again 3 digits to the right, so the reverse shift will be 3 digits to the left: 6804 → 6.804. This time there are no trailing zeros.

We reached the third expression: 132.5 · 0.0034.

  1. Significant parts: 1325 and 34;
  2. Their product: 1325 · 34 = 45,050;
  3. In the first fraction, the decimal point moves to the right by 1 digit, and in the second - by as many as 4. Total: 5 to the right. We shift by 5 to the left: 45,050 → .45050 = 0.4505. The zero was removed at the end, and added at the front so as not to leave a “naked” decimal point.

The following expression is: 0.0108 · 1600.5.

  1. We write the significant parts: 108 and 16 005;
  2. We multiply them: 108 · 16,005 = 1,728,540;
  3. We count the numbers after the decimal point: in the first number there are 4, in the second there are 1. The total is again 5. We have: 1,728,540 → 17.28540 = 17.2854. At the end, the “extra” zero was removed.

Finally, the last expression: 5.25 10,000.

  1. Significant parts: 525 and 1;
  2. We multiply them: 525 · 1 = 525;
  3. The first fraction is shifted 2 digits to the right, and the second fraction is shifted 4 digits to the left (10,000 → 1.0000 = 1). Total 4 − 2 = 2 digits to the left. We perform a reverse shift by 2 digits to the right: 525, → 52,500 (we had to add zeros).

Please note last example: as the decimal point moves to different directions, the total shift is found through the difference. This is very important point! Here's another example:

Consider the numbers 1.5 and 12,500. We have: 1.5 → 15 (shift by 1 to the right); 12,500 → 125 (shift 2 to the left). We “step” 1 digit to the right, and then 2 to the left. As a result, we stepped 2 − 1 = 1 digit to the left.

Decimal division

Division is perhaps the most complex operation. Of course, here you can act by analogy with multiplication: divide the significant parts, and then “move” the decimal point. But in this case, many subtleties arise that negate the potential savings.

So let's take a look universal algorithm, which is a little longer, but much more reliable:

  1. Convert all decimal fractions to ordinary fractions. With a little practice, this step will take you a matter of seconds;
  2. Divide the resulting fractions in the classic way. In other words, multiply the first fraction by the “inverted” second (see lesson “Multiplying and dividing numerical fractions");
  3. If possible, present the result again as a decimal fraction. This step is also quick, since the denominator is often already a power of ten.

Task. Find the meaning of the expression:

  1. 3,51: 3,9;
  2. 1,47: 2,1;
  3. 6,4: 25,6:
  4. 0,0425: 2,5;
  5. 0,25: 0,002.

Let's consider the first expression. First, let's convert fractions to decimals:

Let's do the same with the second expression. The numerator of the first fraction will again be factorized:

There is an important point in the third and fourth examples: after getting rid of decimal notation reducible fractions arise. However, we will not perform this reduction.

The last example is interesting because the numerator of the second fraction contains a prime number. There is simply nothing to factorize here, so we consider it straight ahead:

Sometimes division results in an integer (I'm talking about the last example). In this case, the third step is not performed at all.

In addition, when dividing, “ugly” fractions often arise that cannot be converted to decimals. This distinguishes division from multiplication, where the results are always represented in decimal form. Of course, in this case the last step is again not performed.

Pay also attention to the 3rd and 4th examples. In them we do not intentionally shorten ordinary fractions, derived from decimals. Otherwise, this will complicate the inverse task - representing the final answer again in decimal form.

Remember: the basic property of a fraction (like any other rule in mathematics) in itself does not mean that it must be applied everywhere and always, at every opportunity.

Purpose of the service. The online calculator is designed for multiplying binary numbers.

Number #1

Number No. 2


Example No. 1. Multiply binary numbers 111 and 101.
Solution.
1 1 1
1 0 1
= = = = =
1 1 1
0 0 0
1 1 1
= = = = =
0 0 0 1 1

During summation, an overflow occurred in bits 2, 3, 4. Moreover, the overflow also occurred in the most significant digit, so we write 1 in front of the resulting number, and we get: 100011
In the decimal number system, this number has the following form:
To translate, you need to multiply the digit of a number by the corresponding degree of digit.
100011 = 2 5 *1 + 2 4 *0 + 2 3 *0 + 2 2 *0 + 2 1 *1 + 2 0 *1 = 32 + 0 + 0 + 0 + 2 + 1 = 35
Let's check the result of multiplication in the decimal number system. To do this, we convert the numbers 111 and 101 into decimal notation.
111 2 = 2 2 *1 + 2 1 *1 + 2 0 *1 = 4 + 2 + 1 = 7
101 2 = 2 2 *1 + 2 1 *0 + 2 0 *1 = 4 + 0 + 1 = 5
7 x 5 = 35

Example No. 2. Find the binary product 11011*1100. Convert the answer to decimal system.
Solution. We start multiplication from the lowest digits: if the current digit of the second number is 0, then we write zeros everywhere, if 1, then we rewrite the first number.

1 1 0 1 1
1 1 0 0
= = = = = = = =
0 0 0 0 0
0 0 0 0 0
1 1 0 1 1
1 1 0 1 1
= = = = = = = =
0 1 0 0 0 1 0 0

During summation, an overflow occurred in bits 3, 4, 5, 6, 7. Moreover, the overflow also occurred in the most significant digit, so we write 1 in front of the resulting number, and we get: 101000100

101000100 = 2 8 *1 + 2 7 *0 + 2 6 *1 + 2 5 *0 + 2 4 *0 + 2 3 *0 + 2 2 *1 + 2 1 *0 + 2 0 *0 = 256 + 0 + 64 + 0 + 0 + 0 + 4 + 0 + 0 = 324
Let's check the result of multiplication in the decimal number system. To do this, we convert the numbers 11011 and 1100 into decimal notation.
11011 = 2 4 *1 + 2 3 *1 + 2 2 *0 + 2 1 *1 + 2 0 *1 = 16 + 8 + 0 + 2 + 1 = 27
1100 = 2 3 *1 + 2 2 *1 + 2 1 *0 + 2 0 *0 = 8 + 4 + 0 + 0 = 12
27 x 12 = 324

Example No. 3. 1101.11*101
We will multiply numbers without taking into account floating point: 110111 x 101
We start multiplication from the lowest digits: if the current digit of the second number is 0, then we write zeros everywhere, if 1, then we rewrite the first number.

1 1 0 1 1 1
1 0 1
= = = = = = = =
1 1 0 1 1 1
0 0 0 0 0 0
1 1 0 1 1 1
= = = = = = = =
0 0 0 1 0 0 1 1

During summation, an overflow occurred in bits 2, 3, 4, 5, 6, 7. Moreover, the overflow also occurred in the most significant digit, so we write 1 in front of the resulting number, and we get: 100010011
Since we multiplied without taking into account floating point, we write the final result as: 1000100.11
In the decimal number system, this number has the following form:
1000100 = 2 6 *1 + 2 5 *0 + 2 4 *0 + 2 3 *0 + 2 2 *1 + 2 1 *0 + 2 0 *0 = 64 + 0 + 0 + 0 + 4 + 0 + 0 = 68
To convert the fractional part, you need to divide the digit of the number by the corresponding degree of digit.
11 = 2 -1 *1 + 2 -2 *1 = 0.75
As a result, we get the number 68.75
Let's check the result of multiplication in the decimal number system. To do this, we convert the numbers 1101.11 and 101 into decimal notation.
1101 = 2 3 *1 + 2 2 *1 + 2 1 *0 + 2 0 *1 = 8 + 4 + 0 + 1 = 13
11 = 2 -1 *1 + 2 -2 *1 = 0.75
As a result, we get the number 13.75
Convert the number: 101 2 = 2 2 *1 + 2 1 *0 + 2 0 *1 = 4 + 0 + 1 = 5
13.75 x 5 = 68.75

1. An ordinary fraction whose denominator is 10, 100, 1000, etc. is called a decimal fraction.

2. Fractions with a denominator of 10 n can be written as a decimal.

3. If you add one or more zeros to the decimal fraction on the right, you get a fraction equal to the given one.

4. If in a decimal fraction one or more zeros are removed from the right, you will get a fraction equal to the given one.

5. The integer part from the fractional part in the decimal notation of a number is separated by a comma.

6. The fractional part from the integer part in the decimal notation of a number is separated by a comma.

7. A decimal fraction that has a finite number of digits after the decimal point is called a finite decimal fraction.

8. A decimal fraction that has an infinite number of digits after the decimal point is called an infinite decimal fraction.

9. Infinite decimal fractions are divided into periodic and non-periodic decimal fractions

10. A consecutively repeated digit or minimal group of digits in the notation of an infinite decimal fraction after the decimal point is called the period of this infinite decimal fraction.

11. Irreducible ordinary fractions whose denominators do not contain prime factors other than 2 and 5 are written as a final decimal fraction.

12. Irreducible ordinary fractions, in the denominator of which, in addition to 2 and 5, there are others prime factors, are written as an infinite decimal fraction.

13. The rule for converting a decimal fraction into an ordinary fraction.

To write a decimal fraction as a fraction, you need to:

1) leave the whole part unchanged;

2) write the number after the decimal point in the numerator, and in the denominator - one and as many zeros as there are digits after the decimal point in the decimal fraction.

14. The rule for converting a fraction to a decimal.

1) (1 method) In order to write an irreducible ordinary fraction, the denominator of which does not contain other prime factors other than 2 and 5, as a decimal, you need to present it as a fraction with the denominator 10,100,1000, etc.

(2nd method) – divide the numerator by the denominator.

2) In order to write an irreducible ordinary fraction, in the denominator of which, in addition to 2 and 5, there are other prime factors as a decimal, you need to divide the numerator by the denominator.

15. Decimal places –…hundreds, tens, units, tenths, hundredths, thousandths…ten-thousandths….

16. The numbers in the decimal fraction to the right of the decimal point are called decimals.

17. Comparison of decimals:

1) (1st method) On a coordinate ray, the smaller decimal fraction is located to the left, and the larger decimal fraction is located to the right. Equal decimal fractions are represented on the coordinate ray by the same point.


2) (2nd method) Decimal fractions are compared place by digit, starting with the highest digit.

1) If the integer parts of decimal fractions are different, then the greater is the decimal fraction whose integer part is larger, and the lesser is the decimal fraction whose integer part is smaller.

2) if the integer parts of decimal fractions are the same, then the greater is the decimal fraction whose first of the non-matching digits written after the decimal point is greater.

18. Rules for rounding the whole part of a decimal fraction. To round a decimal fraction to the decimal place tens, hundreds, etc., you can discard its fractional part and apply the rule of rounding natural numbers to the learned number.

19. Rules for rounding the fractional part of a decimal. To round a decimal to the units, tenths, hundredths, etc. place, you can:

1) discard all digits following this digit;

2) if the first discarded digit is 5, 6, 7, 8, 9, then increase the resulting number by one digit to which we round;

3) if the first discarded digit is 0,1,2,3,4. then leave the resulting number unchanged.

20. The rule for adding (subtracting) decimal fractions. To add (subtract) decimal fractions, you need to:

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

2) write them down one after the other so that the comma is under the comma, and the numbers of the same digits are one under the other;

3) perform addition (subtraction) bit by bit;

4) place a comma in the resulting value of the sum (difference) under the commas of the terms (minued and subtracted).

21. The rule for multiplying a decimal fraction by a natural number. To multiply a decimal fraction by a natural number, you need to:

1) multiply it by this number, ignoring the comma;

2) in the resulting product, separate as many digits on the right with a comma as there are in the decimal fraction separated by a comma.

22. The rule for multiplying a decimal fraction by the numbers 10,100,1000, etc. To multiply a decimal fraction by 10,100,1000, etc., you need to move the decimal point to the right by as many digits as there are zeros in the digit unit.

23. The rule for multiplying a decimal fraction by the numbers 0.1; 0.01; 0.01, etc. To multiply a decimal by 0.1; 0.01; 0.01, etc., you need to move the decimal point to the left by as many digits as there are decimal places in the divisor.

24. Rule for multiplying decimals. To multiply decimal fractions:

1) multiply them without paying attention to the comma;

2) in the resulting product, separate with a comma as many digits on the right as there are separated by a comma in two factors together.

25. The rule for dividing a decimal fraction by numbers 10,100,1000, etc. To divide a decimal fraction by 10,100,1000, etc., you need to move the decimal point to the left by as many digits as there are zeros in the digit unit.

26. The rule for dividing a decimal fraction by numbers 0.1; 0.01; 0.01, etc. To divide a decimal by 0.1; 0.01; 0.01, etc., you need to move the decimal point to the right by as many digits as there are decimal places in the divisor.

27. The rule for dividing a decimal fraction by a natural number. To divide a decimal fraction by a natural number, you need to:

1) divide it by this number, ignoring the comma; 2) in the resulting quotient, separate as many digits on the right with a comma as there are separated by a comma in the decimal fraction.

28. Dividing a decimal by a decimal. To divide a number by a decimal fraction:

1) in the dividend and divisor, move the comma to the right by as many digits as there are after the decimal point in the divisor;

2) perform division by a natural number.

Comment:

For example, 0.333...=0,(3). They read: “About as many as three in a period.” If in infinite decimal periodic fraction the period begins immediately after the decimal point, it is called a pure decimal periodic fraction. If a periodic decimal fraction has other decimal places between the decimal point and the period, it is called a mixed periodic decimal fraction. Integers can be written as a pure periodic decimal fraction with a period equal to the number zero. Infinite decimal non-periodic fractions are called irrational numbers. Irrational numbers are written only in the form of an infinite decimal non-periodic fraction.