Ordinary and decimal fractions and operations on them. Decimals

This article is about decimals. Here we will deal with the decimal notation of fractional numbers, introduce the concept of a decimal fraction and give examples of decimal fractions. Next, let's talk about the digits of decimal fractions, give the names of the digits. After that, we will focus on infinite decimal fractions, say about periodic and non-periodic fractions. Next, we list the main actions with decimal fractions. In conclusion, we establish the position of decimal fractions on the coordinate ray.

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Decimal notation of a fractional number

Reading decimals

Let's say a few words about the rules for reading decimal fractions.

Decimal fractions, which correspond to the correct ordinary fractions, are read in the same way as these ordinary fractions, only “zero whole” is added beforehand. For example, the decimal fraction 0.12 corresponds to the ordinary fraction 12/100 (it reads “twelve hundredths”), therefore, 0.12 is read as “zero point twelve hundredths”.

Decimal fractions, which correspond to mixed numbers, are read in exactly the same way as these mixed numbers. For example, the decimal fraction 56.002 corresponds to a mixed number, therefore, the decimal fraction 56.002 is read as "fifty-six point two thousandths."

Places in decimals

In the notation of decimal fractions, as well as in the notation of natural numbers, the value of each digit depends on its position. Indeed, the number 3 in decimal 0.3 means three tenths, in decimal 0.0003 - three ten thousandths, and in decimal 30,000.152 - three tens of thousands. Thus, we can talk about digits in decimals, as well as about digits in natural numbers.

The names of the digits in the decimal fraction to the decimal point completely coincide with the names of the digits in natural numbers. And the names of the digits in the decimal fraction after the decimal point are visible from the following table.

For example, in the decimal fraction 37.051, the number 3 is in the tens place, 7 is in the units place, 0 is in the tenth place, 5 is in the hundredth place, 1 is in the thousandth place.

The digits in the decimal fraction also differ in seniority. If we move from digit to digit from left to right in the decimal notation, then we will move from senior to junior ranks. For example, the hundreds digit is older than the tenths digit, and the millionths digit is younger than the hundredths digit. In this final decimal fraction, we can talk about the most significant and least significant digits. For example, in decimal 604.9387 senior (highest) the digit is the hundreds digit, and junior (lowest)- ten-thousandth place.

For decimal fractions, expansion into digits takes place. It is analogous to the expansion in digits of natural numbers. For example, the decimal expansion of 45.6072 is: 45.6072=40+5+0.6+0.007+0.0002 . And the properties of addition from the expansion of a decimal fraction into digits allow you to go to other representations of this decimal fraction, for example, 45.6072=45+0.6072 , or 45.6072=40.6+5.007+0.0002 , or 45.6072= 45.0072+0.6 .

End decimals

Up to this point, we have only talked about decimal fractions, in the record of which there is a finite number of digits after the decimal point. Such fractions are called final decimal fractions.

Definition.

End decimals- These are decimal fractions, the records of which contain a finite number of characters (digits).

Here are some examples of final decimals: 0.317 , 3.5 , 51.1020304958 , 230 032.45 .

However, not every common fraction can be represented as a finite decimal fraction. For example, the fraction 5/13 cannot be replaced by an equal fraction with one of the denominators 10, 100, ..., therefore, it cannot be converted to a final decimal fraction. We'll talk more about this in the theory section of converting ordinary fractions to decimal fractions.

Infinite decimals: periodic fractions and non-periodic fractions

In writing a decimal fraction after a decimal point, you can allow the possibility of an infinite number of digits. In this case, we will come to the consideration of the so-called infinite decimal fractions.

Definition.

Endless decimals- These are decimal fractions, in the record of which there is an infinite number of digits.

It is clear that we cannot write the infinite decimal fractions in full, therefore, in their recording they are limited to only a certain finite number of digits after the decimal point and put an ellipsis indicating an infinitely continuing sequence of digits. Here are some examples of infinite decimal fractions: 0.143940932…, 3.1415935432…, 153.02003004005…, 2.111111111…, 69.74152152152….

If you look closely at the last two endless decimal fractions, then in the fraction 2.111111111 ... the infinitely repeating number 1 is clearly visible, and in the fraction 69.74152152152 ..., starting from the third decimal place, the repeating group of numbers 1, 5 and 2 is clearly visible. Such infinite decimal fractions are called periodic.

Definition.

Periodic decimals(or simply periodic fractions) are infinite decimal fractions, in the record of which, starting from a certain decimal place, some digit or group of digits, which is called fraction period.

For example, the period of the periodic fraction 2.111111111… is the number 1, and the period of the fraction 69.74152152152… is a group of numbers like 152.

For infinite periodic decimal fractions, a special notation has been adopted. For brevity, we agreed to write the period once, enclosing it in parentheses. For example, the periodic fraction 2.111111111… is written as 2,(1) , and the periodic fraction 69.74152152152… is written as 69.74(152) .

It is worth noting that for the same periodic decimal fraction, you can specify different periods. For example, the periodic decimal 0.73333… can be considered as a fraction 0.7(3) with a period of 3, as well as a fraction 0.7(33) with a period of 33, and so on 0.7(333), 0.7 (3333), ... You can also look at the periodic fraction 0.73333 ... like this: 0.733(3), or like this 0.73(333), etc. Here, in order to avoid ambiguity and inconsistencies, we agree to consider as the period of a decimal fraction the shortest of all possible sequences of repeating digits, and starting from the closest position to the decimal point. That is, the period of the decimal fraction 0.73333… will be considered a sequence of one digit 3, and the periodicity starts from the second position after the decimal point, that is, 0.73333…=0.7(3) . Another example: the periodic fraction 4.7412121212… has a period of 12, the periodicity starts from the third digit after the decimal point, that is, 4.7412121212…=4.74(12) .

Infinite decimal periodic fractions are obtained by converting to decimal fractions of ordinary fractions whose denominators contain prime factors other than 2 and 5.

Here it is worth mentioning periodic fractions with a period of 9. Here are examples of such fractions: 6.43(9) , 27,(9) . These fractions are another notation for periodic fractions with period 0, and it is customary to replace them with periodic fractions with period 0. To do this, period 9 is replaced by period 0, and the value of the next highest digit is increased by one. For example, a fraction with period 9 of the form 7.24(9) is replaced by a periodic fraction with period 0 of the form 7.25(0) or an equal final decimal fraction of 7.25. Another example: 4,(9)=5,(0)=5 . The equality of a fraction with a period of 9 and its corresponding fraction with a period of 0 is easily established after replacing these decimal fractions with their equal ordinary fractions.

Finally, let's take a closer look at infinite decimals, which do not have an infinitely repeating sequence of digits. They are called non-periodic.

Definition.

Non-recurring decimals(or simply non-periodic fractions) are infinite decimals with no period.

Sometimes non-periodic fractions have a form similar to that of periodic fractions, for example, 8.02002000200002 ... is a non-periodic fraction. In these cases, you should be especially careful to notice the difference.

Note that non-periodic fractions are not converted to ordinary fractions, infinite non-periodic decimal fractions represent irrational numbers.

Operations with decimals

One of the actions with decimals is comparison, and four basic arithmetic are also defined operations with decimals: addition, subtraction, multiplication and division. Consider separately each of the actions with decimal fractions.

Decimal Comparison essentially based on a comparison of ordinary fractions corresponding to the compared decimal fractions. However, converting decimal fractions to ordinary ones is a rather laborious operation, and infinite non-repeating fractions cannot be represented as an ordinary fraction, so it is convenient to use a bitwise comparison of decimal fractions. Bitwise comparison of decimals is similar to comparison of natural numbers. For more detailed information, we recommend that you study the article material comparison of decimal fractions, rules, examples, solutions.

Let's move on to the next step - multiplying decimals. Multiplication of final decimal fractions is carried out similarly to the subtraction of decimal fractions, rules, examples, solutions to multiplication by a column of natural numbers. In the case of periodic fractions, multiplication can be reduced to the multiplication of ordinary fractions. In turn, the multiplication of infinite non-periodic decimal fractions after their rounding is reduced to the multiplication of finite decimal fractions. We recommend further study of the material of the article multiplication of decimal fractions, rules, examples, solutions.

Decimals on the coordinate beam

There is a one-to-one correspondence between dots and decimals.

Let's figure out how points are constructed on the coordinate ray corresponding to a given decimal fraction.

We can replace finite decimal fractions and infinite periodic decimal fractions with ordinary fractions equal to them, and then construct the corresponding ordinary fractions on the coordinate ray. For example, a decimal fraction 1.4 corresponds to an ordinary fraction 14/10, therefore, the point with coordinate 1.4 is removed from the origin in the positive direction by 14 segments equal to a tenth of a single segment.

Decimal fractions can be marked on the coordinate beam, starting from the expansion of this decimal fraction into digits. For example, let's say we need to build a point with a coordinate of 16.3007 , since 16.3007=16+0.3+0.0007 , then we can get to this point by sequentially laying 16 unit segments from the origin of coordinates, 3 segments, the length of which equal to a tenth of a unit, and 7 segments, the length of which is equal to a ten thousandth of a unit segment.

This method of constructing decimal numbers on the coordinate beam allows you to get as close as you like to the point corresponding to an infinite decimal fraction.

It is sometimes possible to accurately plot a point corresponding to an infinite decimal. For example, , then this infinite decimal fraction 1.41421... corresponds to the point of the coordinate ray, remote from the origin by the length of the diagonal of a square with a side of 1 unit segment.

The reverse process of obtaining a decimal fraction corresponding to a given point on the coordinate beam is the so-called decimal measurement of a segment. Let's see how it is done.

Let our task be to get from the origin to a given point on the coordinate line (or infinitely approach it if it is impossible to get to it). With a decimal measurement of a segment, we can sequentially postpone any number of unit segments from the origin, then segments whose length is equal to a tenth of a single segment, then segments whose length is equal to a hundredth of a single segment, etc. By writing down the number of plotted segments of each length, we get the decimal fraction corresponding to a given point on the coordinate ray.

For example, to get to point M in the above figure, you need to set aside 1 unit segment and 4 segments, the length of which is equal to the tenth of the unit. Thus, the point M corresponds to the decimal fraction 1.4.

It is clear that the points of the coordinate beam, which cannot be reached during the decimal measurement, correspond to infinite decimal fractions.

Bibliography.

• Maths: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
• Maths. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Subject: Decimals. Adding and subtracting decimals

Lesson: Decimal notation of fractional numbers

The denominator of a fraction can be expressed as any natural number. Fractional numbers in which the denominator is expressed by the number 10; 100; 1000;…, where n , agreed to write without a denominator. Any fractional number whose denominator is 10; 100; 1000 etc. (that is, a one with several zeros) can be represented as a decimal notation (as a decimal fraction). First, write the integer part, then the numerator of the fractional part, and separate the integer part from the fractional part with a comma.

For example,

If the whole part is missing, i.e. the fraction is correct, then the integer part is written as 0.

To write a decimal correctly, the numerator of the fractional part must have as many digits as there are zeros in the fractional part.

1. Write as a decimal.

2. Represent the decimal as a fraction or mixed number.

3. Read the decimals.

12.4 - 12 whole 4 tenths;

0.3 - 0 whole 3 tenths;

1.14 - 1 whole 14 hundredths;

2.07 - 2 whole 7 hundredths;

0.06 - 0 point 6;

0.25 - 0 whole 25 hundredths;

1.234 - 1 whole 234 thousandths;

1.230 - 1 whole 230 thousandths;

1.034 - 1 whole 34 thousandths;

1.004 - 1 whole 4 thousandths;

1.030 - 1 whole 30 thousandths;

0.010101 - 0 point 10101 ppm.

4. Move the comma in each digit 1 digit to the left and read the numbers.

34,1; 310,2; 11,01; 10,507; 2,7; 3,41; 31,02; 1,101; 1,0507; 0,27.

5. Move the comma in each of the numbers 1 digit to the right and read the resulting number.

1,37; 0,1401; 3,017; 1,7; 350,4; 13,7; 1,401; 30,17; 17; 3504.

6. Express in meters and centimeters.

3.28 m = 3 m + .

7. Express in tons and kilograms.

24.030 t = 24 t.

8. Write down the quotient as a decimal fraction.

1710: 100 = ;

64: 10000 =

803: 100 =

407: 10 =

9. Express in dm.

5 dm 6 cm = 5 dm + ;

9 mm =

Decimals are divided into three following classes: finite decimals, infinite periodic decimals, and infinite non-periodic decimals.

End decimals

Definition . Ending decimal (decimal) call a fraction or a mixed number having a denominator of 10, 100, 1000, 10000, etc.

For example,

Decimal fractions also include such fractions that can be reduced to fractions having a denominator 10, 100, 1000, 10000, etc., using the basic property of fractions.

For example,

Statement . An irreducible simple fraction or an irreducible mixed non-integer is a finite decimal fraction if and only if the decomposition of their denominators into prime factors contains only the numbers 2 and 5 as factors, and in arbitrary powers.

For decimals there is special recording method A that uses a comma. The integer part of the fraction is written to the left of the decimal point, and the numerator of the fractional part is written to the right, before which such a number of zeros are added so that the number of digits after the decimal point is equal to the number of zeros in the denominator of the decimal fraction.

For example,

Note that the decimal fraction will not change if you assign several zeros to the right or left of it.

For example,

3,14 = 3,140 =
= 3,1400 = 003,14 .

The numbers before the comma (to the left of the comma) in decimal notation of the final decimal fraction, form a number called integer part of decimal.

The digits after the decimal point (to the right of the decimal point) in the decimal notation of the final decimal fraction are called decimal places.

There is a finite number of decimal places in the final decimal. Decimals form fractional part of a decimal.

Multiplying and dividing decimals by 10, 100, 1000, etc.

To multiply a decimal by 10, 100, 1000, 10000, etc., enough move comma to the right for 1, 2, 3, 4, etc. decimal places respectively.

fractional number.

Decimal notation of a fractional number is a set of two or more digits from $0$ to $9$, between which is the so-called \textit (decimal point).

Example 1

For example, $35.02; $100.7; $123 \ $456.5; $54.89.

The leftmost digit in the decimal representation of a number cannot be zero, except when the decimal point is immediately after the first digit $0$.

Example 2

For example, $0.357; $0.064.

Often the decimal point is replaced by a decimal point. For example, $35.02$; $100.7$; $123 \ 456.5$; $54.89.

Decimal definition

Definition 1

Decimals are fractional numbers that are represented in decimal notation.

For example, $121.05; $67.9; $345.6700.

Decimals are used for a more compact representation of regular fractions whose denominators are the numbers $10$, $100$, $1\000$, etc. and mixed numbers whose denominators are $10$, $100$, $1\000$, etc.

For example, the common fraction $\frac(8)(10)$ can be written as the decimal $0.8$, and the mixed number $405\frac(8)(100)$ as the decimal $405.08$.

Reading decimals

Decimals that correspond to regular fractions are read the same as ordinary fractions, only the phrase "zero integers" is added in front. For example, the common fraction $\frac(25)(100)$ (read "twenty-five hundredths") corresponds to the decimal fraction $0.25$ (read "zero point twenty-five hundredths").

Decimals that correspond to mixed numbers are read the same way as mixed numbers. For example, the mixed number $43\frac(15)(1000)$ corresponds to the decimal fraction $43.015$ (read "forty-three point fifteen thousandths").

Places in decimals

In decimal notation, the value of each digit depends on its position. Those. in decimal fractions, the concept also takes place discharge.

The digits in decimal fractions up to the decimal point are called the same as the digits in natural numbers. The digits in decimal fractions after the decimal point are listed in the table:

Picture 1.

Example 3

For example, in the decimal fraction $56,328$, $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenth place, $2$ is in the hundredth place, $8$ is in the thousandth place.

The digits in decimal fractions are distinguished by seniority. When reading a decimal fraction, they move from left to right - from senior discharge to junior.

Example 4

For example, in decimal $56.328$, the most significant (highest) digit is the tens digit, and the least significant (lowest) digit is the thousandths digit.

A decimal fraction can be expanded into digits in the same way as expansion into digits of a natural number.

Example 5

For example, let's expand the decimal fraction $37,851$ into digits:

$37,851=30+7+0,8+0,05+0,001$

End decimals

Definition 2

End decimals are called decimal fractions, the records of which contain a finite number of characters (digits).

For example, $0.138; $5.34; $56.123456; $350,972.54.

Any final decimal fraction can be converted to a common fraction or a mixed number.

Example 6

For example, the final decimal fraction $7.39$ corresponds to the fractional number $7\frac(39)(100)$, and the final decimal fraction $0.5$ corresponds to the proper fraction $\frac(5)(10)$ (or any fraction, which is equal to it, for example, $\frac(1)(2)$ or $\frac(10)(20)$.

Converting an ordinary fraction to a decimal fraction

Convert common fractions with denominators $10, 100, \dots$ to decimals

Before converting some proper ordinary fractions to decimals, they must first be “prepared”. The result of such preparation should be the same number of digits in the numerator and the number of zeros in the denominator.

The essence of the “preliminary preparation” of correct ordinary fractions for conversion to decimal fractions is to add on the left in the numerator such a number of zeros that the total number of digits becomes equal to the number of zeros in the denominator.

Example 7

For example, let's prepare the common fraction $\frac(43)(1000)$ for conversion to decimal and get $\frac(043)(1000)$. And the ordinary fraction $\frac(83)(100)$ does not need to be prepared.

Let's formulate rule for converting a proper common fraction with denominator $10$, or $100$, or $1\000$, $\dots$ to a decimal fraction:

write $0$;

put a decimal point after it;

write down the number from the numerator (together with added zeros after preparation, if necessary).

Example 8

Convert proper fraction $\frac(23)(100)$ to decimal.

Solution.

The denominator is the number $100$, which contains $2$ two zeros. The numerator contains the number $23$, which contains $2$.digits. this means that preparation for this fraction for conversion to decimal is not necessary.

Let's write $0$, put a decimal point and write the number $23$ from the numerator. We get the decimal fraction $0.23$.

Answer: $0,23$.

Example 9

Write the proper fraction $\frac(351)(100000)$ as a decimal.

Solution.

The numerator of this fraction has $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction needs to be prepared for conversion to decimal. To do this, add $5-3=2$ zeros to the left in the numerator: $\frac(00351)(100000)$.

Now we can form the desired decimal fraction. To do this, write $0$, then put a comma and write the number from the numerator. We get the decimal fraction $0.00351$.

Answer: $0,00351$.

Let's formulate rule for converting improper common fractions with denominators $10$, $100$, $\dots$ to decimals:

write a number from the numerator;

separate with a decimal point as many digits on the right as there are zeros in the denominator of the original fraction.

Example 10

Convert improper common fraction $\frac(12756)(100)$ to decimal.

Solution.

Let's write the number from the numerator $12756$, then separate the digits on the right with a decimal point $2$, because the denominator of the original fraction $2$ is zero. We get the decimal fraction $127.56$.

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 digits of the numerator.

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:

1. as a result of reduction to the desired number of decimal places;
2. 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:

1. you need to form a number consisting of the number after the decimal point before the period, and the first period;
2. from the resulting number subtract the number after the decimal point before the period. The result will be the numerator of an ordinary fraction;
3. 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.

Because the

, 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 When multiplying, 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.