Repeating digit in an infinite decimal fraction. Infinite periodic fractions

The fact that many square roots are irrational numbers, does not at all detract from their significance; in particular, the number $\sqrt2$ is very often used in various engineering and scientific calculations. This number can be calculated with the accuracy required in each specific case. You can get this number to as many decimal places as you have the patience for.

For example, the number $\sqrt2$ can be determined with an accuracy of six decimal places: $\sqrt2=1.414214$. This value is not very different from true meaning, since $1.414214 \times 1.414214=2.000001237796$. This answer differs from 2 by barely more than one millionth. Therefore, the value of $\sqrt2$ equal to $1.414214$ is considered quite acceptable for solving most practical problems. In cases where greater precision is required, it is not difficult to obtain as many significant digits after the decimal point as needed in this case.

However, if you show rare stubbornness and try to extract Square root from the number $\sqrt2$ until you achieve exact result, you will never finish your work. It's a never-ending process. No matter how many decimal places you get, there will always be a few more left.

This fact may surprise you just as much as turning $\frac13$ into an infinite decimal $0.333333333…$ and so on indefinitely, or turning $\frac17$ into $0.142857142857142857…$ and so on indefinitely. At first glance it may seem that these infinite and irrational square roots are phenomena of the same order, but this is not at all the case. After all, these infinite fractions has a fractional equivalent, while $\sqrt2$ does not. Why exactly? The fact is that the decimal equivalent of $\frac13$ and $\frac17$, as well as an infinite number of other fractions, are periodic demons final fractions.

At the same time, the decimal equivalent of $\sqrt2$ is a non-periodic fraction. This statement is also true for any irrational number.

The problem is that any decimal that is an approximation of the square root of 2 is non-periodic fraction. No matter how far we go in our calculations, any fraction we get will be non-periodic.

Imagine a fraction with a huge amount non-periodic decimal places. If suddenly after the millionth digit the entire sequence of decimal places is repeated, it means decimal- periodic and there is an equivalent for it in the form of a ratio of integers. If a fraction with a huge number (billions or millions) of non-recurring decimal places at some point has an endless series of repeating digits, such as $...55555555555...$, this also means that given fraction- periodic and there is an equivalent for it in the form of a ratio of integers.

However, in case, their decimal equivalents are completely non-periodic and cannot become periodic.

Of course you can ask next question: “Who can know and say for sure what happens to a fraction, say, after the trillion sign? Who can guarantee that a fraction will not become periodic?” There are ways to prove irrefutably that irrational numbers are non-periodic, but such proofs require complex mathematical apparatus. But if it suddenly turned out that rational number becomes periodic fraction, that would mean complete collapse basics mathematical sciences. And in fact this is hardly possible. It’s not easy for you to throw it from side to side on your knuckles, there is a complex mathematical theory here.

This article is about decimals. Here we will deal with the decimal notation of fractional numbers and introduce the concept decimal and give examples of decimal fractions. Next we’ll talk about the digits of decimal fractions and give the names of the digits. After this, we will focus on infinite decimal fractions, let's talk about periodic and non-periodic fractions. Next we list the basic operations with decimal fractions. In conclusion, let us establish the position of decimal fractions on the coordinate beam.

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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 proper ordinary fractions, are read in the same way as these ordinary fractions, only “zero integer” is first added. For example, the decimal fraction 0.12 corresponds to the common fraction 12/100 (read “twelve hundredths”), therefore, 0.12 is read as “zero point twelve hundredths”.

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

Places in decimals

In writing decimal fractions, as well as in writing 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 the decimal fraction up to the decimal point completely coincide with the names of the digits in natural numbers. And the names of the decimal places after the decimal point can be seen from the following table.

For example, in the decimal fraction 37.051, the digit 3 is in the tens place, 7 is in the units place, 0 is in the tenths place, 5 is in the hundredths place, and 1 is in the thousandths place.

Places in decimal fractions also differ in precedence. If in writing a decimal fraction we move from digit to digit from left to right, then we will move from senior To junior ranks. For example, the hundreds place is older than the tenths place, and the millions place is lower than the hundredths place. In a given final decimal fraction we can talk about the major and minor digits. For example, in decimal fraction 604.9387 senior (highest) the place is the hundreds place, and junior (lowest)- digit of ten thousandths.

For decimal fractions, expansion into digits takes place. It is similar to expansion into digits of natural numbers. For example, the expansion into decimal places of 45.6072 is as follows: 45.6072=40+5+0.6+0.007+0.0002. And the properties of addition from the decomposition of a decimal fraction into digits allow you to move on 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.

Ending decimals

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

Definition.

Ending 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 fraction can be represented as a final decimal. For example, the fraction 5/13 cannot be replaced by an equal fraction with one of the denominators 10, 100, ..., therefore, cannot be converted into a final decimal fraction. We will talk more about this in the theory section, converting ordinary fractions to decimals.

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 decimals- these are decimal fractions, the recording of which contains infinite set numbers

It is clear that we cannot write down infinite decimal fractions in full form, so in their recording we limit ourselves 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 infinite decimal fractions, then in the fraction 2.111111111... the endlessly repeating number 1 is clearly visible, and in the fraction 69.74152152152..., starting from the third decimal place, a 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 endless decimal fractions, in the recording of which, starting from a certain decimal place, some number or group of numbers is endlessly repeated, which is called period of the fraction.

For example, the period of the periodic fraction 2.111111111... is the digit 1, and the period of the fraction 69.74152152152... is a group of digits of the form 152.

For infinite periodic decimal fractions it is accepted special shape records. For brevity, we agreed to write down 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 considered as a fraction 0.7(3) with a period of 3, and also 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 discrepancies, 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 into decimal fractions ordinary fractions whose denominators contain prime factors, different from 2 and 5.

Here it is worth mentioning periodic fractions with a period of 9. Let us give examples of such fractions: 6.43(9) , 27,(9) . These fractions are another notation for periodic fractions with period 0, and they are usually replaced by 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 7.25. Another example: 4,(9)=5,(0)=5. The equality of a fraction with period 9 and its corresponding fraction with period 0 is easily established after replacing these decimal fractions with equal ordinary fractions.

Finally, let's take a closer look at infinite decimal fractions, which do not contain an endlessly repeating sequence of digits. They are called non-periodic.

Definition.

Non-recurring decimals(or simply non-periodic fractions ) are infinite decimal fractions that have 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 do not convert to ordinary fractions; infinite non-periodic decimal fractions represent irrational numbers.

Operations with decimals

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

Comparison of decimals essentially based on comparison of ordinary fractions corresponding to the decimal fractions being compared. However, converting decimal fractions into ordinary fractions is a rather labor-intensive process, and infinite non-periodic fractions cannot be represented as an ordinary fraction, so it is convenient to use a place-wise comparison of decimal fractions. Place-wise comparison of decimal fractions is similar to comparison of natural numbers. For more detailed information, we recommend studying the article: comparison of decimal fractions, rules, examples, solutions.

Let's move on to the next step - multiplying decimals. Multiplication of finite decimal fractions is carried out similarly to 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 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 for further study the material in the article: multiplication of decimal fractions, rules, examples, solutions.

Decimals on a coordinate ray

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

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

We can replace finite decimal fractions and infinite periodic decimal fractions with equal ordinary fractions, and then construct the corresponding ordinary fractions on the coordinate ray. For example, the decimal fraction 1.4 corresponds to the common fraction 14/10, so the point with 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 a coordinate ray, starting from the decomposition of a given decimal fraction into digits. For example, let us need to build a point with coordinate 16.3007, since 16.3007=16+0.3+0.0007, then in this point you can get there by sequentially laying off from the origin 16 unit segments, 3 segments whose length is equal to a tenth of a unit segment, and 7 segments whose length is equal to a ten-thousandth of a unit segment.

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

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 a point coordinate ray, removed from the origin by the length of the diagonal of a square with a side of 1 unit segment.

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

Let our task be to get from the origin to a given point on the coordinate line (or to infinitely approach it if we can’t get to it). With the decimal measurement of a segment, we can sequentially lay off from the origin any number of unit segments, 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. By recording the number of segments of each length laid aside, we obtain 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 a tenth of a unit. Thus, point M corresponds to the decimal fraction 1.4.

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

Bibliography.

• Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
• Mathematics. 6th grade: educational. 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 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by 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 those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Already in primary school students encounter fractions. And then they appear in every topic. You cannot forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are not complicated, the main thing is to understand everything in order.

Why are fractions needed?

The world around us consists of entire 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 consists of several pieces. Consider a situation where his tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It can easily be divided into three. But it will not be possible to give five people a whole number of chocolate slices.

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

What is a "fraction"?

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

Essentially, the slash turns out to be a division sign. That is, the numerator can be called the dividend, and the denominator can be called the divisor.

What fractions are there?

In mathematics there are only two types: ordinary and decimal fractions. Schoolchildren first meet in primary school, calling them simply "fractions". The latter will be learned in 5th grade. That's when these names appear.

Common fractions are all those that are written as two numbers separated by a line. For example, 4/7. A decimal is a number in which the fractional part has a positional notation and is separated from the whole number by a comma. For example, 4.7. Students need to clearly understand that the two examples given are completely different numbers.

Every simple fraction can be written in decimal form. This statement is almost always true in reverse direction. There are rules that allow you to write a decimal fraction as a common fraction.

What subtypes do these types of fractions have?

It's better to start in chronological order, since they are being studied. Common fractions come first. Among them, 5 subspecies can be distinguished.

Correct. Its numerator is always less than its denominator.

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

Reducible/irreducible. It may turn out to be either right or wrong. Another important thing is whether the numerator and denominator have common factors. If there are, then it is necessary to divide both parts of the fraction by them, that is, reduce it.

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

Composite. It is formed from two fractions divided by each other. That is, it contains three fractional lines at once.

Decimal fractions have only two subtypes:

finite, that is, one whose 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 fraction to a common fraction?

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

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

How to convert decimal fractions to ordinary fractions if whole part 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. All that remains is to write down the fractional parts. The first number will have a denominator of 10, the second will have a denominator of 100. That is, the given examples will have the following numbers as answers: 9/10, 5/100. Moreover, it turns out that the latter can be reduced by 5. Therefore, the result for it needs to be written as 1/20.

How can you convert a decimal fraction into an ordinary fraction if its integer part is different from zero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its value is written. In the first case it is 5, in the second it is 13. Then you need to move on to the fractional part. The same operation is supposed to be carried out with them. The first number appears 23/100, the second - 108/100000. The second value needs to be reduced again. The answer looks like this mixed fractions: 5 23/100 and 13 27/25000.

How to convert an infinite decimal fraction to an ordinary fraction?

If it is non-periodic, then such an operation will not be possible. This fact is due to the fact that each decimal fraction is always converted to either a finite or a periodic fraction.

The only thing you can do with such a fraction is 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 initial value. That is, infinite non-periodic fractions are not converted into ordinary fractions. This needs to be remembered.

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

In these numbers, there are always one or more digits after the decimal point that are repeated. They are called a period. For example, 0.3(3). Here "3" is in the period. They are classified as rational because they can be converted into ordinary fractions.

Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts 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 as a common fraction will be different for the two types of numbers indicated. It is quite easy to write pure periodic fractions as ordinary fractions. As with finite ones, they need to be converted: write down the period in the numerator, and the denominator will be the number 9, repeated as many times as the number of digits 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. Write 5 as the numerator and 9 as the denominator. That is, the answer will be the fraction 5/9.

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

Look at the length of the period. That's how many 9s the denominator will have.

Write down the denominator: first nines, then zeros.

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

For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period contains one digit. So there will be one zero. 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, you need to subtract 5 from 58. It turns out 53. For example, you would have to write the answer as 53/90.

How are fractions converted to decimals?

The most simple option turns out to be a number whose denominator contains the number 10, 100, etc. 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. You just need to multiply not only the denominator, but also the numerator by the same number.

For all other cases, a simple rule is useful: divide the numerator by the denominator. In this case, you may get two possible answers: a finite or a periodic decimal fraction.

Operations with ordinary fractions

Addition and subtraction

Students become acquainted with them earlier than others. And first for fractions same denominators, and then different. General rules can be reduced to such a plan.

Find the least common multiple of the denominators.

Write additional factors for all ordinary fractions.

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

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

If the numerator of the minuend is less than the subtrahend, then we need to find out whether we have a mixed number or a proper fraction.

In the first case, you need to borrow one 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 subtraction from smaller number more. That is, from the module of the subtrahend, subtract the module of the minuend, and in response put a “-” sign.

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

Multiplication and division

To perform them, fractions do not need to be reduced to common denominator. This makes it easier to perform actions. But they still require you to follow the rules.

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

Multiply the numerators.

Multiply the denominators.

If the result is a reducible fraction, then it must be simplified again.

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

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

In tasks where you need to multiply (divide) by an integer, the latter should be written in the form improper fraction. That is, with a denominator of 1. Then act as described above.



Operations with decimals

Addition and subtraction

Of course, you can always convert a decimal into a fraction. And act according to the plan already described. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction 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 the fractions so that the comma is below the comma.

Add (subtract) like natural numbers.

Remove the comma.

Multiplication and division

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

To multiply, you need to write the fractions one below the other, ignoring the commas.

Multiply like natural numbers.

Place 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 must first 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 natural number.

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



What if one example contains both types of fractions?

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

First way: represent ordinary decimals

It is suitable if division or translation results in finite fractions. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you don’t like working with ordinary fractions, you will have to count them.

Second way: write decimal fractions as ordinary

This technique turns out to be convenient if the part after the decimal point contains 1-2 digits. If there are more of them, you may end up with a very large common fraction and decimal notation will make the task faster and easier to calculate. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.

Remember how in the very first lesson about decimals I said that there are numerical fractions that cannot be represented as decimals (see lesson “Decimals”)? We also learned how to factor the denominators of fractions to see if there were any numbers other than 2 and 5.

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

A periodic decimal is any decimal that:

1. The significant part consists of an infinite number of digits;
2. At certain intervals, the numbers in the significant part are repeated.

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

Since there are many definitions, it is worth considering a few of these fractions in detail:

This fraction appears 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 a number. Therefore, if you have forgotten what it is, I recommend repeating it - see the lesson “”.

Transition to periodic decimal fraction

Let's consider common fraction type a/b. Let's factorize its denominator into prime factors. There are two options:

1. The expansion contains only factors 2 and 5. These fractions are easily converted to decimals - see the lesson “Decimals”. We are not interested in such people;
2. There is something else in the expansion other than 2 and 5. In this case, the fraction cannot be represented as a decimal, but it can be converted into a periodic decimal.

To define a periodic decimal fraction, you need to find its periodic and non-periodic parts. How? Convert the fraction to an improper fraction, and then divide the numerator by the denominator using a corner.

The following will happen:

1. Will split first whole part, if it exists;
2. There may be several numbers after the decimal point;
3. After a while the numbers will start repeat.

That's all! Repeating numbers after the decimal point are denoted by the periodic part, and those in front are denoted by the non-periodic part.

Task. Convert ordinary fractions to periodic decimals:

All fractions without an integer part, so we simply divide the numerator by the denominator with a “corner”:

As you can see, the remainders 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 it in normal form: 4.0909 ... = 4,(09).

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

Transition from periodic decimal fraction to ordinary fraction

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

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

Task. Convert the number to an ordinary improper fraction:

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

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

The parentheses contain only one digit, so the period is k = 1. Next, 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 look at the second fraction. So X = 32,(39) = 32.393939...

Period k = 2, so 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.

Let's move on to the third fraction: X = 0.30(5) = 0.30555... The diagram 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.

It is known that if the denominator P irreducible fraction in its canonical expansion has a prime factor not equal to 2 and 5, then this fraction cannot be represented as a finite decimal fraction. If we try in this case to write down the original irreducible fraction as a decimal, dividing the numerator by the denominator, then the division process cannot end, because if it were completed after a finite number of steps, we would get a finite decimal fraction, which contradicts the previously proven theorem. So in this case decimal notation positive rational number A= appears to be an infinite fraction.

For example, fraction = 0.3636... . It is easy to notice that the remainders when dividing 4 by 11 are periodically repeated, therefore, the decimal places will be periodically repeated, i.e. it turns out infinite periodic decimal fraction, which can be written as 0,(36).

Periodically repeating numbers 3 and 6 form a period. It may turn out that there are several digits between the decimal point and the beginning of the first period. These numbers form the pre-period. For example,

0.1931818... The process of dividing 17 by 88 is endless. The numbers 1, 9, 3 form the pre-period; 1, 8 – period. The examples we have considered reflect a pattern, i.e. any positive rational number can be represented as either a finite or an infinite periodic decimal fraction.

Theorem 1. Let the ordinary fraction be irreducible in the canonical expansion of the denominator n is a prime factor different from 2 and 5. Then the common fraction can be represented as an infinite periodic decimal fraction.

Proof. We already know that the process of dividing a natural number m to a natural number n will be endless. Let us show that it will be periodic. In fact, when dividing m on n the resulting balances will be smaller n, those. numbers of the form 1, 2, ..., ( n– 1), from which it is clear that the number of different remainders is finite and therefore, starting from a certain step, some remainder will be repeated, which will entail the repetition of the decimal places of the quotient, and the infinite decimal fraction becomes periodic.

Two more theorems hold.

Theorem 2. If the expansion of the denominator of an irreducible fraction into prime factors does not include the numbers 2 and 5, then when this fraction is converted into an infinite decimal fraction, a pure periodic fraction will be obtained, i.e. a fraction whose period begins immediately after the decimal point.

Theorem 3. If the expansion of the denominator includes factors 2 (or 5) or both, then the infinite periodic fraction will be mixed, i.e. between the decimal point and the beginning of the period there will be several digits (pre-period), namely as many as the largest of the exponents of the factors 2 and 5.

Theorems 2 and 3 are proposed to the reader to prove independently.

28. Methods of transition from infinite periodic
decimal fractions to common fractions

Let a periodic fraction be given A= 0,(4), i.e. 0.4444... .

Let's multiply A by 10, we get

10A= 4.444…4…Þ 10 A = 4 + 0,444….

Those. 10 A = 4 + A, we obtained an equation for A, having solved it, we get: 9 A= 4 Þ A = .

We note that 4 is both the numerator of the resulting fraction and the period of the fraction 0,(4).

Rule converting a pure periodic fraction into an ordinary fraction is formulated as follows: the numerator of the fraction is equal to the period, and the denominator consists of the same number of nines as there are digits in the period of the fraction.

Let us now prove this rule for a fraction whose period consists of P

A= . Let's multiply A on 10 n, we get:

10n × A = = + 0, ;

10n × A = + a;

(10n – 1) A = Þ a = = .

So, the previously formulated rule has been proven for any pure periodic fraction.

Let us now give a fraction A= 0.605(43) – mixed periodic. Let's multiply A by 10 with the same indicator, how many digits are in the pre-period, i.e. by 10 3, we get

10 3 × A= 605 + 0,(43) Þ 10 3 × A = 605 + = 605 + = = ,

those. 10 3 × A= .

Rule converting a mixed periodic fraction into an ordinary fraction is formulated as follows: the numerator of the fraction is equal to the difference between the number written in digits before the beginning of the second period and the number written in digits before the beginning of the first period, the denominator consists of the number of nines equal to the number of digits in the period and such number of zeros how many digits there are before the start of the first period.

Let us now prove this rule for a fraction whose preperiod consists of P numbers, and the period is from To numbers Let a periodic fraction be given

Let's denote V= ; r= ,

With= ; Then With=in × 10k + r.

Let's multiply A by 10 with such an exponent how many digits are in the preperiod, i.e. on 10 n, we get:

A×10 n = + .

Taking into account the notations introduced above, we write:

a× 10n= V+ .

So, the rule formulated above has been proven for any mixed periodic fraction.

Every infinite periodic decimal fraction is a form of writing some rational number.

For the sake of consistency, sometimes a finite decimal is also considered an infinite periodic decimal with period "zero". For example, 0.27 = 0.27000...; 10.567 = 10.567000...; 3 = 3,000... .

Now the following statement becomes true: every rational number can (and in a unique way) be expressed by an infinite periodic decimal fraction, and every infinite periodic decimal fraction expresses exactly one rational number (periodic decimal fractions with a period of 9 are not considered).