Converting a common fraction to a decimal means finding a decimal fraction that would be equal to the given common fraction. When converting ordinary fractions to decimals, we will encounter two cases:
1) when ordinary fractions can be converted to decimals exactly;
2) when ordinary fractions can only be converted to decimals approximately. Let's consider these cases sequentially.
1. How to convert an ordinary irreducible fraction into a decimal, or, in other words, how to replace an ordinary fraction with a decimal equal to it?
In the case where ordinary fractions can be exactly converted to decimal, there is two ways such treatment.
Let us remember how to replace one fraction with another that is equal to the first, or how to move from one fraction to another without changing the value of the first. We did this when we reduced fractions to a common denominator (§86). When we reduce fractions to a common denominator, we proceed as follows: we find the common denominator for these fractions, calculate an additional factor for each fraction, and then multiply the numerator and denominator of each fraction by this factor.
Having noticed this, let's take the irreducible fraction 3/20 and try to convert it to a decimal. The denominator of this fraction is 20, but you need to bring it to another denominator, which would be represented by one with zeros. We will be looking for the smallest denominator of one followed by zeros.
First way converting a fraction to a decimal is based on decomposing the denominator into prime factors.
You need to find out what number you need to multiply 20 by so that the product is expressed as one followed by zeros. To find out, you first need to remember into what prime factors the numbers represented by one and zeros are decomposed. These are the decompositions:
10 = 2 5,
100 = 2 2 5 . 5,
1 000 = 2 2 2 5 5 5,
10 000 = 2 2 2 2 5 5 5 5.
We see that the number represented by one with zeros is decomposed only into twos and fives, and there are no other factors in the expansion. In addition, twos and fives are included in the expansion in the same number. And, finally, the number of those and other factors separately is equal to the number of zeros after the one in the image of a given number.
Now let's see how 20 is decomposed into prime factors: 20 = 2 2 5. From this it is clear that in the decomposition of the number 20 there are two twos, and one fives. This means that if we add one five to these factors, we will get a number represented by one with zeros. In other words, in order for the denominator to have a number represented by one with zeros instead of 20, you need to multiply 20 by 5, and so that the value of the fraction does not change, you need to multiply its numerator by 5, i.e.
Thus, in order to convert an ordinary fraction into a decimal, you need to decompose the denominator of this ordinary fraction into prime factors and then equalize the number of twos and fives in it, introducing into it (and, of course, into the numerator) the missing factors in the required number.
Let's apply this conclusion to some fractions.
Convert 3/50 to a decimal. The denominator of this fraction is expanded as follows:
This means that it is missing one deuce. Let's add it:
Convert 7/40 to a decimal.
The denominator of this fraction is decomposed as follows: 40 = 2 2 2 5, i.e. it is missing two fives. Let's introduce them into the numerator and denominator as factors:
From what has been stated, it is not difficult to conclude which ordinary fractions convert exactly to decimals. It is quite obvious that an irreducible ordinary fraction, the denominator of which does not contain any other prime factors other than 2 and 5, converts exactly to a decimal. A decimal fraction, which is obtained by reversing some ordinary fraction, will have as many decimal places as the number of times the denominator of the ordinary fraction after its reduction includes the numerically predominant factor 2 or 5.
If we take the fraction 9/40, then, firstly, it will turn into a decimal, because its denominator includes the factors 2 2 2 5, and secondly, the resulting decimal fraction will have 3 decimal places, because the numerically dominant factor 2 enters into expansion three times. Indeed:
Second way(by dividing the numerator by the denominator).
Suppose you want to convert 3/4 to a decimal fraction. We know that 3/4 is the quotient of 3 divided by 4. We can find this quotient by dividing 3 by 4. Let's do this:
Thus, 3 / 4 = 0.75.
Another example: convert 5/8 to a decimal fraction.
So 5 / 8 = 0.625.
So, to convert a fraction to a decimal, you just need to divide the numerator of the fraction by its denominator.
2. Let us now consider the second of the cases indicated at the beginning of the paragraph, i.e., the case when an ordinary fraction cannot be converted into an exact decimal.
An ordinary irreducible fraction whose denominator contains any prime factors other than 2 and 5 cannot be converted exactly to a decimal. In fact, for example, the fraction 8/15 cannot be converted into a decimal, since its denominator 15 is decomposed into two factors: 3 and 5.
We cannot eliminate the triple from the denominator and cannot select an integer such that, after multiplying the given denominator by it, the product is expressed as one followed by zeros.
In such cases, we can only talk about approximation ordinary fractions to decimals.
How it's done? This is done by dividing the numerator of a common fraction by the denominator, i.e. in this case, the second method of converting a common fraction to a decimal is used. This means that this method is used for both precise and approximate handling.
If a fraction is converted exactly to a decimal, then division produces a final decimal fraction.
If an ordinary fraction does not convert to an exact decimal, then division produces an infinite decimal fraction.
Since we cannot carry out an endless division process, we must stop division at some decimal place, that is, do an approximate division. We can, for example, stop dividing at the first decimal place, that is, limit ourselves to tenths; if necessary, we can stop at the second decimal place, obtaining hundredths, etc. In these cases, we say that we are rounding an infinite decimal fraction. Rounding is done with the accuracy required for solving this problem.
§ 115. The concept of a periodic fraction.
A perpetual decimal fraction in which one or more digits invariably repeat in the same sequence is called a periodic decimal fraction. For example:
0,33333333...; 1,12121212...; 3,234234234...
A set of repeating numbers is called period this fraction. The period of the first of the fractions written above is 3, the period of the second fraction is 12, the period of the third fraction is 234. This means that the period can consist of several digits - one, two, three, etc. The first set of repeating digits is called the first period, the second the totality - the second period, etc., i.e.
Periodic fractions can be pure or mixed. A periodic fraction is called pure if its period begins immediately after the decimal point. This means that the periodic fractions written above will be pure. Against, periodic fraction is called mixed if it has one or more non-repeating digits between the decimal point and the first period, for example:
2,5333333...; 4,1232323232...; 0,2345345345345... 160
To shorten the letter, you can write the period numbers once in brackets and do not put ellipses after the brackets, i.e. instead of 0.33... you can write 0,(3); instead of 2.515151... you can write 2,(51); instead of 0.2333... you can write 0.2(3); instead of 0.8333... you can write 0.8(3).
Periodic fractions are read like this:
0,(3) - 0 integers, 3 in period.
7,2(3) - 7 integers, 2 before the period, 3 in the period.
5.00(17) - 5 integers, two zeros before the period, 17 in the period.
How do periodic fractions arise? We have already seen that when converting fractions to decimals, there can be two cases.
Firstly, the denominator of an ordinary irreducible fraction does not contain any factors other than 2 and 5; in this case, the ordinary fraction becomes a final decimal.
Secondly, the denominator of an ordinary irreducible fraction contains any prime factors other than 2 and 5; in this case, the ordinary fraction does not turn into a final decimal. In this latter case, attempting to convert a fraction to a decimal by dividing the numerator by the denominator results in an infinite fraction that will always be periodic.
To see this, let's look at an example. Let's try to convert the fraction 18/7 to a decimal.
We, of course, know in advance that a fraction with such a denominator cannot be converted into a final decimal, and we are only talking about an approximate conversion. Divide the numerator 18 by the denominator 7.
We got eight decimal places in the quotient. There is no need to continue the division further, because it will not end anyway. But from this it is clear that the division can be continued indefinitely and thus obtain new numbers in the quotient. These new numbers will arise because we will always have leftovers; but no remainder can be greater than the divisor, which for us is 7.
Let's see what balances we had: 4; 5; 1; 3; 2; b, i.e. these were numbers less than 7. Obviously, there cannot be more than six of them, and with further continuation of the division they will have to be repeated, and after them the digits of the quotient will be repeated. The above example confirms this idea: the decimal places in the quotient are in this order: 571428, and after that the numbers 57 appeared again. This means that the first period has ended and the second begins.
Thus, an infinite decimal fraction obtained by inverting a common fraction will always be periodic.
If a periodic fraction is encountered when solving a problem, then it is taken with the accuracy required by the conditions of the problem (to the tenth, to the hundredth, to the thousandth, etc.).
§ 116. Joint actions with ordinary and decimal fractions.
When solving various problems, we will encounter cases where the problem includes both ordinary and decimals.
In these cases, you can go in different ways.
1. Convert all fractions to decimals. This is convenient because calculations with decimal fractions are easier than with ordinary fractions. For example,
Let's convert the fractions 3/4 and 1 1/5 to decimals:
2. Convert all fractions to ordinary fractions. This is most often done in cases where there are ordinary fractions that do not turn into final decimals.
For example, 
Let's convert decimal fractions to ordinary fractions:
3. Calculations are carried out without converting some fractions to others.
This is especially useful when the example involves only multiplication and division. For example,
Let's rewrite the example like this:
4. In some cases convert all fractions to decimals(even those that turn into periodic ones) and find an approximate result. For example,
Let's convert 2/3 to a decimal fraction, limiting ourselves to thousandths.
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 convert absolutely any numerical fraction into a 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:
- 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 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
Consider an ordinary fraction of the form a /b. Let's factorize its denominator into prime factors. There are two options:
- 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;
- 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:
- Will split first whole part , if it exists;
- There may be several numbers after the decimal point;
- 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).
Write to 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:
- Find the period of the fraction, i.e. count how many digits are in the periodic part. Let this be the number k;
- Find the value of the expression X · 10 k. This is equivalent to shifting the decimal point by full period to the right - see the lesson “Multiplying and dividing decimals”;
- The original expression must be subtracted from the resulting number. In this case, the periodic part is “burned” and remains common fraction;
- Find X in the resulting equation. We convert all decimal fractions to ordinary fractions.
Task. Reduce to ordinary improper fraction numbers:
- 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.
The division operation involves the participation of several main components. The first of them is the so-called dividend, that is, a number that is subject to the division procedure. The second is the divisor, that is, the number by which the division is performed. The third is the quotient, that is, the result of the operation of dividing the dividend by the divisor.
Result of division
The most simple option The result that can be obtained when two positive integers are used as the dividend and divisor is another positive integer. For example, when dividing 6 by 2, the quotient will be equal to 3. This situation is possible if the dividend is the divisor, that is, it is divided by it without a remainder.However, there are other options when it is impossible to carry out a division operation without a remainder. In this case, a non-integer number becomes quotient, which can be written as a combination of an integer and a fractional part. For example, when dividing 5 by 2, the quotient is 2.5.
Number in period
One of the options that can result if the dividend is not a multiple of the divisor is the so-called number in period. It can arise as a result of division if the quotient turns out to be an endlessly repeating set of numbers. For example, a number in a period may appear when dividing the number 2 by 3. In this situation, the result, as a decimal fraction, will be expressed as a combination of an infinite number of 6 digits after the decimal point.In order to indicate the result of such a division, a special way of writing numbers in a period was invented: such a number is indicated by placing a repeating digit in brackets. For example, the result of dividing 2 by 3 would be written using this method as 0,(6). This notation is also applicable if only part of the number resulting from division is repeating.
For example, when dividing 5 by 6, the result will be a periodic number of the form 0.8(3). Using this method, firstly, is more effective compared to trying to write down all or part of the digits of a number in a period, and secondly, it has greater accuracy compared to another method of transmitting such numbers - rounding, and in addition, it allows you to distinguish numbers in period from an exact decimal fraction with the corresponding value when comparing the magnitude of these numbers. So, for example, it is obvious that 0.(6) is significantly greater than 0.6.
As is known, 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 whole numbers, rational numbers include fractions.
Why then is the entire set of rational numbers sometimes considered as infinite periodic decimal fractions? Indeed, in addition to fractions, they also include integers, 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 recording method.
How is an infinite periodic decimal represented? In it, a repeating group of numbers after the decimal point is placed in brackets. For example, 1.56(12) is a fraction in which the group of digits 12 is repeated, i.e. the fraction has the value 1.561212121212... and so on endlessly. A repeating group of numbers is called a period.
However, we can represent any number in this form if we consider its period to be the number 0, which also repeats endlessly. For example, the number 2 is the same as 2.00000.... Therefore, it can be written as an infinite periodic fraction, i.e. 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 one. Therefore, they separate finite fractions and infinite periodic ones. Thus, it is more correct to say that the rational numbers include
- all integers
- final fractions,
- infinite periodic fractions.
At the same time, simply remember that integers 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. When it comes to fractions, both finite and infinite decimals can be uniquely represented as a fraction. This means that from the point of view of ordinary fractions, periodic and finite fractions are the same thing. Additionally, whole numbers can also be represented as a fraction by imagining that we are dividing the number by 1.
How to represent a decimal infinite periodic fraction as an ordinary fraction? The most commonly used algorithm is something like this:
- Reduce the fraction so that after the decimal point there is only a period.
- Multiply an infinite periodic fraction by 10 or 100 or ... so that the decimal point moves to the right by one period (i.e., one period ends up in the whole 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. From b I subtract a. That is, they make up the equation Nx – x = b – a.
- When solving an equation, the result is an ordinary fraction.
An example of converting an infinite periodic decimal fraction into 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 =
Periodic fraction an infinite decimal fraction in which, starting from a certain point, there is only a periodically repeated certain group of digits. For example, 1.3181818...; In short, this fraction is written like this: 1.3(18), that is, they place the period in brackets (and say: “18 in the period”). P. is called pure if the period begins immediately after the decimal point, for example 2(71) = 2.7171..., and mixed if after the decimal point there are numbers preceding the period, for example 1.3(18). The role of decimal fractions in arithmetic is due to the fact that when rational numbers, that is, ordinary (simple) fractions, are represented by decimal fractions, either finite or periodic fractions are always obtained. More precisely: a final decimal fraction is obtained when the denominator of an irreducible simple fraction does not contain other prime factors other than 2 and 5; in all other cases, the result is a P. fraction, and, moreover, it is pure if the denominator of a given irreducible fraction does not contain the factors 2 and 5 at all, and mixed if at least one of these factors is contained in the denominator. Any fractional fraction can be converted into a simple fraction (that is, it is equal to some rational number). A pure fraction is equal to a simple fraction, the numerator of which is the period, and the denominator is represented by the number 9, written as many times as there are digits in the period; When converting a mixed fraction into a simple fraction, the numerator is the difference between the number represented by the numbers preceding the second period and the number represented by the numbers preceding the first period; To compose the denominator, you need to write the number 9 as many times as there are numbers in the period, and add as many zeros to the right as there are numbers before the period. These rules assume that the given P. is correct, that is, it does not contain whole units; otherwise the whole part is given special consideration. The rules for determining the length of the period of a fraction corresponding to a given ordinary fraction are also known. For example, for a fraction a/p, Where R - prime number and 1 ≤ a ≤ p- 1, period length is a divisor R - 1. So, for known approximations to a number (see Pi)
22/7 and 355/113 periods are equal to 6 and 112 respectively.
Big Soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
Synonyms:See what “Periodic fraction” is in other dictionaries:
An infinite decimal fraction in which, starting from a certain place, a certain group of digits (period) is periodically repeated, for example. 0.373737... pure periodic fraction or 0.253737... mixed periodic fraction... Big encyclopedic Dictionary
Fraction, infinite fraction Dictionary of Russian synonyms. periodic fraction noun, number of synonyms: 2 infinite fraction (2) ... Synonym dictionary
A decimal fraction in which a series of digits are repeated in the same order. For example, 0.135135135... is a p.d. whose period is 135 and which is equal to the simple fraction 135/999 = 5/37. Dictionary of foreign words included in the Russian language. Pavlenkov F... Dictionary of foreign words of the Russian language
A decimal is a fraction with a denominator of 10n, where n natural number. It has special form entries: an integer part in the decimal number system, then a comma and then a fractional part in the decimal number system, and the number of digits of the fractional part ... Wikipedia
An infinite decimal fraction in which, starting from a certain point, a certain group of digits (period) is periodically repeated; for example, 0.373737... pure periodic fraction or 0.253737... mixed periodic fraction. * * * PERIODIC… … encyclopedic Dictionary
An endless decimal fraction in which, starting from a certain place, the definition is periodically repeated. group of digits (period); for example, 0.373737... pure P. d. or 0.253737... mixed P. d. ... Natural science. encyclopedic Dictionary
See part... Dictionary of Russian synonyms and similar expressions. under. ed. N. Abramova, M.: Russian Dictionaries, 1999. fraction trifle, part; dunst, ball, meal, buckshot; a fractional number Dictionary of Russian synonyms... Synonym dictionary
periodic decimal- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information Technology in general EN circulating decimalrecurring decimalperioding decimalperiodic decimalperiodical decimal ... Technical Translator's Guide
If some integer a is divided by another integer b, i.e., a number x is sought that satisfies the condition bx = a, then two cases can arise: either in the series of integers there is a number x that satisfies this condition, or it turns out ,… … Encyclopedic Dictionary F.A. Brockhaus and I.A. Ephron
A fraction whose denominator is whole degree numbers 10. D. are written without a denominator, separating as many digits in the numerator on the right with a comma as there are zeros in the denominator. For example, In such a record, the part on the left... ... Great Soviet Encyclopedia
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