Periodic decimal fractions. Fractions and Decimals and Actions on Them

Already in elementary school, students are faced with fractions. And then they appear in every topic. It is impossible to forget the actions with these numbers. Therefore, you need to know all the information about ordinary and decimals... These concepts are simple, the main thing is to understand everything in order.

What are fractions for?

The world around us consists of whole objects. Therefore, there is no need for shares. But everyday life constantly pushes people to work with parts of objects and things.

For example, chocolate has several slices. Consider a situation where its tile is formed by twelve rectangles. If you divide it in two, you get 6 parts. She will split well into three. But five will not be able to give a whole number of chocolate wedges.

By the way, these slices are already fractions. And their further division leads to the appearance of more complex numbers.

What is a fraction?

It is a number made up of the parts of one. Outwardly, it looks like two numbers separated by a horizontal or oblique line. This feature is called fractional. The number written at the top (left) is called the numerator. The bottom (right) is the denominator.

In fact, the fractional bar turns out to be a division sign. That is, the numerator can be called divisible, and the denominator can be called a divisor.

What fractions are there?

In mathematics, there are only two types of them: ordinary and decimal fractions. Schoolchildren are the first to meet in primary grades calling them simply "fractions". The second will recognize in the 5th grade. It is then that these names appear.

Ordinary fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the whole by a comma. For example, 4.7. Students need to be clear that the two examples given are completely different numbers.

Each fraction can be written as a decimal. This statement is almost always true in the opposite direction. There are rules that allow you to write a decimal fraction as an ordinary fraction.

What are the subspecies of these types of fractions?

Better to start in chronological order as they are being studied. First to go common fractions... Among them, 5 subspecies can be distinguished.

Correct. Its numerator is always less than the denominator.

Wrong. Its numerator is greater than or equal to the denominator.

Abbreviated / irreducible. It can be both right and wrong. What is important is whether the numerator with the denominator has common factors. If there are, then they are supposed to divide both parts of the fraction, that is, to reduce it.

Mixed. An integer is assigned to its usual correct (incorrect) fractional part. Moreover, it always stands on the left.

Composite. It is formed from two fractions separated by each other. That is, there are three fractional lines in it at once.

Decimal fractions have only two subspecies:

final, that is, the one in which the fractional part is limited (has an end);

infinite - a number whose digits after the decimal point do not end (they can be written endlessly).

How to convert a decimal to a fraction?

If it is a finite number, then the association based on the rule is applied - as I hear, so I write. That is, you need to correctly read it and write it down, but without a comma, but with a fractional line.

As a hint about the required denominator, you need to remember that it is always one and several zeros. The latter need to be written as many as there are digits in the fractional part of the number in question.

How to convert decimal fractions into ordinary fractions if their integer part is absent, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. It remains to write down only the fractional parts. The first number will have the denominator 10, the second - 100. That is, the given examples will have the numbers: 9/10, 5/100. Moreover, it turns out that the latter can be reduced by 5. Therefore, the result for it must be written 1/20.

How to make an ordinary fraction from a decimal if its integer part is nonzero? For example, 5.23 or 13.00108. In both examples, the integer part is read and its value written. In the first case, it is 5, in the second - 13. Then you need to go to the fractional part. The same operation is supposed to be carried out with them. The first number has 23/100, the second has 108/100000. The second value needs to be shortened again. The answer is the following mixed fractions: 5 23/100 and 13 27/25000.

How to convert an infinite decimal fraction to a fraction?

If it is non-periodic, then such an operation will fail. This fact is due to the fact that each decimal fraction is always translated into either a final or a periodic one.

The only thing you can do with such a fraction is to round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal - will never give an initial value. That is, infinite non-periodic fractions cannot be converted into ordinary ones. This must be remembered.

How to write an infinite periodic fraction as an ordinary fraction?

In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called a period. For example, 0.3 (3). Here "3" in the period. They are classified as rational, since they can be transformed into fractions.

Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period begins immediately from the comma. In the second, the fractional part begins with some numbers, and then the repetition begins.

The rule by which you need to write an infinite decimal in the form of an ordinary fraction will be different for the indicated two types of numbers. It is quite easy to write down pure periodic fractions with ordinary ones. As with the final ones, they need to be converted: write the period into the numerator, and the denominator will be the number 9, repeated as many times as the period contains.

For example, 0, (5). The number does not have an integer part, so you need to immediately start with the fractional part. In the numerator write 5, and in the denominator one 9. That is, the answer will be the fraction 5/9.

Rule on how to write an ordinary decimal periodic fraction that is mixed.

Look at the length of the period. So many 9 will have the denominator.

Write down the denominator: first nines, then zeros.

To determine the numerator, you need to write down the difference of two numbers. All digits after the decimal point, together with the period, will be decremented. Subtracted - it is without a period.

For example, 0.5 (8) - write down the periodic decimal fraction in the form of an ordinary one. There is one digit in the fractional part before the period. So zero will be one. There is also only one number in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.

To determine the numerator from 58, you need to subtract 5. It turns out 53. The answer, for example, will have to write 53/90.

How are common fractions converted to decimals?

The simplest option turns out to be a number, the denominator of which is 10, 100, and so on. Then the denominator is simply discarded, and a comma is placed between the fractional and integer parts.

There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. Only the denominator is supposed to multiply, but also the numerator by the same number.

For all other cases, a simple rule comes in handy: divide the numerator by the denominator. In this case, you can get two options for answers: a final or a periodic decimal fraction.

Actions with common fractions

Addition and subtraction

Students get to know them earlier than others. Moreover, first the fractions have the same denominators, and then they are different. General rules can be summarized in such a plan.

Find the least common multiple of the denominators.

Write down additional factors to all common fractions.

Multiply the numerators and denominators by the factors defined for them.

Add (subtract) the numerators of the fractions, and leave the common denominator unchanged.

If the numerator of the reduced number is less than the subtracted one, then you need to find out if we have a mixed number or a regular fraction.

In the first case, you need to take one unit from the whole part. Add the denominator to the numerator of the fraction. And then do the subtraction.

In the second, it is necessary to apply the rule of subtracting the larger from the smaller number. That is, subtract the modulus of the decreasing from the modulus of the subtracted, and put the sign "-" in response.

Look carefully at the result of addition (subtraction). If you get an incorrect fraction, then it is supposed to select the whole part. That is, divide the numerator by the denominator.

Multiplication and division

To fulfill them, the fractions do not need to be reduced to common denominator... This makes it easier to follow through. But they still have to follow the rules.

When multiplying ordinary fractions, you need to consider the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be canceled.

Multiply the numerators.

Multiply the denominators.

If you get a cancellable fraction, then it is supposed to be simplified again.

When dividing, you must first replace division with multiplication, and the divisor (second fraction) with the reciprocal (swap the numerator and denominator).

Then proceed as in multiplication (starting from point 1).

In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written in the form wrong fraction... That is, with the denominator 1. Then proceed as described above.

Decimal actions

Addition and subtraction

Of course, you can always turn a decimal into a fraction. And to act according to the already described plan. But sometimes it is more convenient to act without this translation. Then the rules for adding and subtracting them will be exactly the same.

Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Add the missing number of zeros to it.

Write fractions so that the comma is below the comma.

Add (subtract) as natural numbers.

Remove the comma.

Multiplication and division

It is important that you do not need to add zeros here. Fractions are supposed to be left as they are given in the example. And then go according to plan.

For multiplication, you need to write fractions one below the other, ignoring the commas.

Multiply as natural numbers.

Put a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.

To divide, you first need to transform the divisor: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.

Multiply the dividend by the same number.

Divide a decimal by a natural number.

Put a comma in the answer at the moment when the division of the whole part ends.

What if there are both types of fractions in one example?

Yes, in mathematics, there are often examples in which you need to perform actions on ordinary and decimal fractions. In such tasks, two solutions are possible. You need to objectively weigh the numbers and choose the best one.

The first way: represent ordinary decimal

It is suitable if, when dividing or translating, finite fractions are obtained. If at least one number gives the periodic part, then this technique is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to count them.

The second way: write down decimal fractions with ordinary

This technique turns out to be convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, a very large ordinary fraction can turn out and decimal notations will allow you to count the task faster and easier. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.

This article is about decimals... Here we will deal with decimal notation of fractional numbers, introduce the concept of a decimal fraction and give examples of decimal fractions. Next, let's talk about the decimal places and 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 basic actions with decimal fractions. Finally, we will set the position of the 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 regular ordinary fractions, are read in the same way as these ordinary fractions, only "zero integers" is added beforehand. For example, the decimal fraction 0.12 corresponds to the ordinary fraction 12/100 (read "twelve hundredths"), therefore, 0.12 reads 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, decimal 56.002 is a mixed number, so decimal 56.002 reads "fifty-six point two thousandths."

Decimal places

In the notation of decimal fractions, as well as in the notation natural numbers, the meaning of each digit depends on its position. Indeed, the number 3 in the decimal fraction 0.3 means three tenths, in the decimal fraction 0.0003 - three ten thousandths, and in the decimal fraction 30,000,152 - three tens of thousands. So we can talk about decimal places, as well as about the digits in natural numbers.

The names of the digits in decimal fractions to the decimal point completely coincide with the names of 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 decimal 37.051, the number 3 is in the tens place, 7 is in the ones place, 0 is in the tenth place, 5 is in the hundredth place, 1 is in the thousandth place.

The decimal places also differ in order of precedence. If we move from digit to digit from left to right in the decimal notation, then we will move from senior To least significant digits... For example, the hundredth place is older than the tenth place, and the millionth place is less than the hundredth place. In this final decimal fraction, we can talk about the most significant and least significant digits. For example, in decimal fraction 604.9387 senior (higher) the rank is the rank of hundreds, and junior (inferior)- the ten-thousandth category.

For decimal fractions, there is a digit expansion. It is similar to the expansion in terms of the digits of natural numbers. For example, the decimal expansion of 45.6072 is as follows: 45.6072 = 40 + 5 + 0.6 + 0.007 + 0.0002. And the properties of addition from the expansion of a decimal fraction by digits allow you to switch 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.

Final decimals

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

Definition.

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

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

However, not every common fraction can be represented as a final 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 into a final decimal fraction. We will talk more about this in the section of the theory of converting ordinary fractions to decimal fractions.

Infinite decimals: periodic fractions and non-periodic fractions

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

Definition.

Infinite decimal fractions- these are decimal fractions, in the recording of which there are an infinite number of digits.

It is clear that we cannot write infinite decimal fractions in full, therefore, they are limited to only some finite number digits after the decimal point and put an ellipsis, indicating an endless sequence of numbers. 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 infinite 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 decimal fractions(or simply periodic fractions) Are infinite decimal fractions, in the notation of which, starting from some decimal place, some digit or group of digits is repeated infinitely, 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 of the form 152.

For infinite periodic decimal fractions, a special notation is 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 different periods can be specified for the same periodic decimal fraction. For example, the periodic decimal fraction 0.73333 ... can be viewed 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 so 0.73 (333), etc. Here, in order to avoid ambiguity and discrepancies, we agree to consider the shortest of all possible sequences of repeating digits, and starting from the closest position to the decimal point, as the decimal fraction period. That is, the period of the decimal fraction 0.73333 ... will be considered a sequence of one digit 3, and the frequency 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 frequency starts from the third digit after the decimal point, that is, 4.7412121212 ... = 4.74 (12).

Infinite decimal periodic fractions are obtained by converting ordinary fractions into decimal fractions, the denominators of which contain prime factors other than 2 and 5.

Here it is worth mentioning about 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 a period of 0, and it is customary to replace them with periodic fractions with a period of 0. For this, period 9 is replaced with a period of 0, and the value of the next highest rank is increased by one. For example, a fraction with a period of 9 like 7.24 (9) is replaced by a periodic fraction with a period of 0 like 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 the 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 decimal fractions, which do not contain an infinitely repeating sequence of numbers. They are called non-periodic.

Definition.

Non-periodic decimals(or simply non-periodic fractions) Are infinite decimal fractions without a period.

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

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

Decimal actions

One of the actions with decimal fractions is comparison, four basic arithmetic are also defined decimal actions: addition, subtraction, multiplication and division. Let's consider separately each of the actions with decimal fractions.

Comparison of decimals is essentially based on comparing common fractions that correspond to compared decimal fractions. However, converting decimal fractions into ordinary fractions is a rather laborious operation, and infinite non-periodic fractions cannot be represented as an ordinary fraction, so it is convenient to use a bitwise comparison of decimal fractions. Bitwise comparison of decimal fractions 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 - decimal multiplication... The multiplication of final decimal fractions is carried out in the same way as subtraction of decimal fractions, rules, examples, solutions to multiplication with a column of natural numbers. In the case of periodic fractions, multiplication can be reduced to multiplication of ordinary fractions. In turn, the multiplication of infinite non-periodic decimal fractions after they are rounded is reduced to the multiplication of finite decimal fractions. We recommend for further study the material of the article multiplication of decimal fractions, rules, examples, solutions.

Decimal fractions on the coordinate ray

There is a one-to-one correspondence between dots and decimal fractions.

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

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, the decimal fraction 1.4 corresponds to the ordinary fraction 14/10, therefore the point with the coordinate 1.4 is removed from the origin in the positive direction by 14 segments equal to a tenth of a unit segment.

Decimal fractions can be marked on the coordinate ray, starting from the decomposition of this decimal fraction into digits. For example, suppose we need to build a point with a coordinate of 16.3007, since 16.3007 = 16 + 0.3 + 0.0007, then you can get to this point by sequentially postponing 16 unit segments from the origin, 3 segments, the length of which equal to a tenth of a unit, and 7 segments, the length of which is equal to ten thousandths of a unit segment.

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

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

The reverse process of obtaining a decimal fraction corresponding to a given point on the coordinate ray is the so-called decimal segment measurement... Let's figure out how it is carried out.

Let our task be to get from the origin to a given point of the coordinate line (or infinitely approach it if it is impossible to get into it). In 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 unit, then segments whose length is equal to a hundredth of a unit, etc. Writing down the number of deferred segments of each length, we get a 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 postpone 1 unit segment and 4 segments, the length of which is equal to a tenth of a unit. Thus, point M corresponds to the decimal fraction 1.4.

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

Bibliography.

Maths: textbook. for 5 cl. general education. institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., Erased. - M .: Mnemozina, 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 .: Mnemosina, 2008 .-- 288 p .: ill. ISBN 978-5-346-00897-2.
Algebra: study. for 8 cl. 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 (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.

As you know, the set of rational numbers (Q) includes the set of integers (Z), which in turn includes the set of natural numbers (N). In addition to integers, rational numbers include fractions.

Why, then, the whole set of rational numbers is sometimes considered as infinite decimal periodic fractions? Indeed, in addition to fractions, they also include whole numbers, as well as non-periodic fractions.

The fact is that all integers, as well as any fraction, can be represented as an infinite periodic decimal fraction. That is, for all rational numbers, you can use the same notation.

How does an infinite periodic decimal fraction appear? In it, a repeating group of digits after the decimal point is placed in brackets. For example, 1.56 (12) is a fraction in which the group of numbers 12 is repeated, that is, the fraction has the value 1.561212121212 ... and so on without end. A repeating group of numbers is called a period.

However, in a similar form, we can represent any number, if we consider it a period of the digit 0, which is also repeated endlessly. For example, the number 2 is the same as 2.00000 .... Therefore, it can be written as an infinite periodic fraction, that is, 2, (0).

The same can be done with any finite fraction. For example:

0,125 = 0,1250000... = 0,125(0)

However, in practice, they do not use the transformation of a finite fraction into an infinite periodic fraction. Therefore, they separate finite fractions and infinite periodic ones. Thus, it is more correct to say that rational numbers include

all integers,
finite fractions,
infinite periodic fractions.

At the same time, they just remember that whole numbers and finite fractions are representable in theory in the form of infinite periodic fractions.

On the other hand, the concepts of finite and infinite fractions are applicable to decimal fractions. If we talk about ordinary fractions, then both a finite and an infinite decimal fraction can be uniquely represented as an ordinary fraction. So, from the point of view of ordinary fractions, periodic and finite fractions are one and the same. In addition, whole numbers can also be represented as a fraction if we imagine that we are dividing this number by 1.

How to represent a decimal infinite periodic fraction in the form of an ordinary fraction? More often they use something like this algorithm:

The fraction is brought to the form so that only the period appears after the decimal point.
The infinite periodic fraction is multiplied by 10 or 100, or ... so that the comma moves to the right by one period (that is, one period is in the integer part).
Equate the original fraction (a) to the variable x, and the fraction (b) obtained by multiplying by the number N - to Nx.
Subtract x from Nx. Subtract a from b. That is, they make up the equation Nx - x = b - a.
When solving the equation, an ordinary fraction is obtained.

An example of converting an infinite periodic decimal fraction to an ordinary fraction:
x = 1.13333 ...
10x = 11.3333 ...
10x * 10 = 11.33333 ... * 10
100x = 113.3333 ...
100x - 10x = 113.3333 ... - 11.3333 ...
90x = 102
x =

Remember how in the very first lesson about decimal fractions I said that there are number fractions that cannot be represented as decimals (see the lesson "Decimal fractions")? We also learned how to factor the denominators of fractions to check if there are numbers other than 2 and 5.

So: I lied. And today we will learn how to convert absolutely any number fraction into decimal. At the same time, we will get acquainted with a whole class of fractions with an infinite significant part.

A periodic decimal fraction is any decimal fraction that has:

The significant part consists of an infinite number of digits;

At certain intervals, the numbers in the significant part are repeated.

The set of repeating numbers that make up significant part, is called the periodic part of the fraction, and the number of digits in this set is called the period of the fraction. The rest of the segment of the significant part, which is not repeated, is called the non-periodic part.

Since there are many definitions, it is worth considering in detail several such fractions:

This fraction occurs most often in problems. Non-periodic part: 0; periodic part: 3; period length: 1.

Non-periodic part: 0.58; periodic part: 3; period length: again 1.

Non-periodic part: 1; periodic part: 54; period length: 2.

Non-periodic part: 0; periodic part: 641025; period length: 6. For convenience, repeating parts are separated from each other by a space - this is not necessary in this solution.

Non-periodic part: 3066; periodic part: 6; period length: 1.

As you can see, the definition of a periodic fraction is based on the concept significant part of the number... Therefore, if you have forgotten what it is, I recommend repeating it - see the lesson "".

Go to periodic decimal

Consider an ordinary fraction of the form a / b. Let's expand its denominator into prime factors. There are two options:

In the expansion there are only factors 2 and 5. These fractions are easily reduced to decimal - see the lesson "Decimal fractions". We are not interested in such;
There is something else in the expansion besides 2 and 5. In this case, the fraction is not representable as a decimal, but you can make a periodic decimal from it.

To set a periodic decimal fraction, you need to find its periodic and non-periodic part. How? Convert the fraction to an incorrect fraction, and then divide the numerator by the denominator "with an angle".

In this case, the following will occur:

Split first whole part if there is one;
There may be several numbers after the decimal point;
After a while, the numbers will start repeat.

That's all! The repeating numbers after the decimal point are designated by the periodic part, and the one in front - by the non-periodic part.

Task. Convert common fractions to periodic decimal:

All fractions are without an integer part, so we just divide the numerator by the denominator with a "corner":

As you can see, the remnants are repeated. Let's write the fraction in the "correct" form: 1.733 ... = 1.7 (3).

The result is a fraction: 0.5833 ... = 0.58 (3).

We write in the normal form: 4.0909 ... = 4, (09).

We get the fraction: 0.4141 ... = 0, (41).

The transition from periodic decimal to a common fraction

Consider the periodic decimal fraction X = abc (a 1 b 1 c 1). It is required to transfer it to the classic "two-story" one. To do this, we will follow four simple steps:

Find the period of the fraction, i.e. count how many digits are in the periodic part. Let it be the number k;
Find the value of the expression X · 10 k. This is equivalent to shifting the decimal point a full period to the right - see the lesson “Multiplying and dividing decimal fractions”;
Subtract the original expression from the resulting number. In this case, the periodic part is "burned", and remains regular fraction;
Find X in the resulting equation. We convert all decimal fractions into ordinary ones.

Task. Reduce the numbers to an improper fraction:

9,(6);
32,(39);
0,30(5);
0,(2475).

We work with the first fraction: X = 9, (6) = 9.666 ...

The brackets contain only one digit, so the period is k = 1. Then we multiply this fraction by 10 k = 10 1 = 10. We have:

10X = 10 9.6666 ... = 96.666 ...

Subtract the original fraction and solve the equation:

10X - X = 96.666 ... - 9.666 ... = 96 - 9 = 87;
9X = 87;
X = 87/9 = 29/3.

Now let's deal with the second fraction. So X = 32, (39) = 32.393939 ...

Period k = 2, so we multiply everything by 10 k = 10 2 = 100:

100X = 100 32.393939 ... = 3239.3939 ...

Subtract the original fraction again and solve the equation:

100X - X = 3239.3939 ... - 32.3939 ... = 3239 - 32 = 3207;
99X = 3207;
X = 3207/99 = 1069/33.

We proceed to the third fraction: X = 0.30 (5) = 0.30555 ... The scheme is the same, so I'll just give the calculations:

Period k = 1 ⇒ multiply everything by 10 k = 10 1 = 10;

10X = 10 0.30555 ... = 3.05555 ...
10X - X = 3.0555 ... - 0.305555 ... = 2.75 = 11/4;
9X = 11/4;
X = (11/4): 9 = 11/36.

Finally, the last fraction: X = 0, (2475) = 0.2475 2475 ... Again, for convenience, the periodic parts are separated from each other by spaces. We have:

k = 4 ⇒ 10 k = 10 4 = 10,000;
10,000X = 10,000 0.2475 2475 = 2475.2475 ...
10,000X - X = 2475.2475 ... - 0.2475 2475 ... = 2475;
9999X = 2475;
X = 2475: 9999 = 25/101.

That if they know the theory of series, then no metamatic concepts can be introduced without it. Moreover, these people believe that those who do not use it everywhere are ignorant. Let's leave the views of these people on their conscience. Let's better understand what an infinite periodic fraction is and how to deal with it for us, uneducated people who know no limits.

Divide 237 by 5. No, you don't need to launch Calculator. Let's better remember middle (or even elementary?) School and just divide by a column:

Well, remember? Then you can get down to business.

The concept of "fraction" in mathematics has two meanings:

Non-integer number.
Non-integer notation.

There are two types of fractions - in the sense, two forms of writing non-integers:

Simple (or vertical) fractions like 1/2 or 237/5.
Decimal fractions such as 0.5 or 47.4.

Note that, in general, the very use of the fraction-notation does not mean that what is written is a fraction-number, for example 3/3 or 7.0 - not fractions in the first sense of the word, but in the second, of course, fractions.

In mathematics, in general, from time immemorial, decimal counting has been adopted, and therefore decimal fractions are more convenient than simple ones, that is, a fraction with a decimal denominator (Vladimir Dal. Explanatory dictionary living Great Russian language. "Ten").

And if so, then I want to make any vertical fraction decimal ("horizontal"). And for this you just need to divide the numerator by the denominator. Take, for example, the fraction 1/3 and try to make a decimal out of it.

Even a completely uneducated person will notice: no matter how much they divide, they will not split: so triplets will appear indefinitely. So we will write down: 0.33 ... We mean here "the number that is obtained when you divide 1 by 3", or, in short, "one third." Naturally, one third is a fraction in the first sense of the word, and "1/3" and "0.33 ..." are fractions in the second sense of the word, that is recording forms a number that is on the number line at such a distance from zero that if you postpone it three times, you get one.

Now let's try to divide 5 by 6:

Again, write down: 0.833 ... We mean "the number that is obtained when you divide 5 by 6," or, in short, "five sixths." However, confusion arises here: do I mean 0.83333 (and then the triplets are repeated), or 0.833833 (and then 833 is repeated). Therefore, the notation with ellipses does not suit us: it is not clear where the repeated part starts from (it is called the "period"). Therefore, we will take the period in brackets, like this: 0, (3); 0.8 (3).

0, (3) is not easy equals one third is there is one third, because we specially invented this notation to represent this number as a decimal fraction.

This entry is called infinite periodic fraction, or just a periodic fraction.

Whenever we divide one number by another, if a finite fraction is not obtained, then an infinite periodic fraction is obtained, that is, one day the sequences of numbers will definitely start repeating. Why this is so can be understood purely speculatively by looking carefully at the long division algorithm:

In the places marked with checkmarks, different pairs of numbers cannot be obtained all the time (because there are, in principle, a finite set of such pairs). And as soon as such a pair appears there, which already existed, the difference will also be the same - and then the whole process will begin to repeat itself. There is no need to check this, as it is quite obvious that if you repeat the same steps, the results will be the same.

Now that we understand well the essence periodic fraction, let's try to multiply one third by three. Yes, we get, of course, one, but let's write this fraction in decimal form and multiply it in a column (there is no ambiguity because of the ellipsis here, since all the digits after the decimal point are the same):

And again we notice that nines, nines and nines will appear after the decimal point all the time. That is, using, inversely, parenthesis, we get 0, (9). Since we know that the product of one third and three is one, then 0, (9) is such a bizarre notation for one. However, it is impractical to use this form of notation, because the unit is perfectly written without using a period, like this: 1.

As you can see, 0, (9) is one of those cases when an integer is written in the form of a fraction, like 3/3 or 7.0. That is, 0, (9) is a fraction only in the second sense of the word, but not in the first.

So, without any limits and series, we figured out what 0, (9) is and how to deal with it.

But still, let's remember that in fact we are smart and studied analysis. Indeed, it is difficult to deny that:

But, perhaps, no one will argue with the fact that:

All this is, of course, true. Indeed, 0, (9) is both the sum of the reduced series, and the doubled sine of the indicated angle, and natural logarithm Euler's numbers.

But neither one nor the other, nor the third is a definition.

Asserting that 0, (9) is the sum of an infinite series 9 / (10 n), for n from unity, is the same as claiming that the sine is the sum of an infinite Taylor series:

it quite right and this is the most important fact for computational mathematics, but this is not a definition, and, most importantly, it does not bring a person closer to understanding essence sinus. The essence of the sine of a certain angle is that it is just the ratio of the opposite leg angle to the hypotenuse.

Duck, a periodic fraction is just decimal fraction, which is obtained when long division the same set of numbers is repeated. There is no trace of analysis here.

And here the question arises: where generally did we take the number 0, (9)? What do we divide by a column to get it? Indeed, there are no such numbers, when dividing by each other by a column, we would have endlessly appearing nines. But we managed to get this number by multiplying the column 0, (3) by 3? Not really. After all, you need to multiply from right to left in order to correctly take into account the transfers of the digits, and we did it from left to right, cleverly taking advantage of the fact that transfers do not appear anywhere anyway. Therefore, the legality of writing 0, (9) depends on whether we recognize the legality of such a multiplication in a column or not.

Therefore, we can generally say that the notation 0, (9) is incorrect - and to a certain extent be right. However, since the notation a, (b) is accepted, it is simply ugly to abandon it when b = 9; it is better to decide what such a record means. So if we accept the notation 0, (9) at all, then this notation, of course, means the number one.

It remains only to add that if we used, say, the ternary number system, then when dividing by a column of one (1 3) by three (10 3), we would get 0.1 3 (read “zero point one third”), and when dividing one to two would be 0, (1) 3.

So the frequency of the fraction-record is not some objective characteristic of the fraction-number, but just a side effect of using one or another number system.