Decimal fraction- variety fractions, which has a “round” number in the denominator: 10, 100, 1000, etc., For example, fraction 5/10 has a decimal notation of 0.5. Based on this principle, fraction can be represented in form decimal fractions.

Instructions

Let's say we need to imagine in form decimal fraction 18/25.
First you need to make sure that one of the “round” numbers appears in the denominator: 100, 1000, etc. To do this, you need to multiply the denominator by 4. But you will need to multiply both the numerator and the denominator by 4.

Multiplying the numerator and denominator fractions 18/25 by 4, it turns out 72/100. This is recorded fraction in decimal form so: 0.72.

In mathematics, a fraction is a rational number equal to one or more parts into which the unit is divided. In this case, the record of the fraction must contain an indication of two numbers: one of them indicates exactly how many shares the unit was divided into when creating this fraction, and the other indicates how many of these shares the fraction includes. If these two numbers are written as a numerator and denominator separated by a line, then this recording format is called a “common” fraction. However, there is another format for writing fractions called "decimal".

The three-story form of writing numbers, in which the denominator is located above the numerator, and there is also a dividing line between them, is not always convenient. This inconvenience especially began to manifest itself with the massive spread of personal computers. The decimal form of representing fractions does not have this drawback - it does not require specifying the numerator, since by definition it is always equal to ten to the negative power. Therefore, a fractional number can be written on one line, although its length in most cases will be much larger than the length of the corresponding ordinary fraction.

Another advantage of writing numbers as decimals is that they are much easier to compare. Since the denominator of each digit of two such numbers is the same, it is enough to compare only two digits of the corresponding digits, while when comparing ordinary fractions it is necessary to take into account both the numerator and the denominator of each of them. This advantage is important not only for people, but also for computers - comparing numbers in decimal format is quite easy to program.

There are centuries-old rules for addition, multiplication and other mathematical operations that allow you to make calculations on paper or in your head with numbers in decimal format. This is another advantage of this format over ordinary fractions. Although with the development of computer technology, when even watches have a calculator, it is becoming less and less noticeable.

The described advantages of the decimal format for recording fractional numbers show that its main purpose is to simplify working with mathematical quantities. This format also has disadvantages - for example, in order to write periodic fractions to a decimal fraction, you also have to add a number in parentheses, and non-rational numbers in decimal format always have an approximate value. However, at the current level of development of people and their technologies, it is much more convenient to use than the usual format for writing fractions.


In this article we will look at how converting fractions to decimals, and also consider the reverse process - converting decimal fractions into ordinary fractions. Here we will outline the rules for converting fractions and provide detailed solutions to typical examples.

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Converting fractions to decimals

Let us denote the sequence in which we will deal with converting fractions to decimals.

First, we'll look at how to represent fractions with denominators 10, 100, 1,000, ... as decimals. This is explained by the fact that decimal fractions are essentially a compact form of writing ordinary fractions with denominators 10, 100, ....

After that, we will go further and show how to write any ordinary fraction (not just those with denominators 10, 100, ...) as a decimal fraction. When ordinary fractions are treated in this way, both finite decimal fractions and infinite periodic decimal fractions are obtained.

Now let's talk about everything in order.

Converting common fractions with denominators 10, 100, ... to decimals

Some proper fractions require "preliminary preparation" before being converted to decimals. This applies to ordinary fractions, the number of digits in the numerator of which is less than the number of zeros in the denominator. For example, the common fraction 2/100 must first be prepared for conversion to a decimal fraction, but the fraction 9/10 does not need any preparation.

“Preliminary preparation” of proper ordinary fractions for conversion to decimal fractions consists of adding so many zeros to the left in the numerator that the total number of digits there becomes equal to the number of zeros in the denominator. For example, a fraction after adding zeros will look like .

Once you have a proper fraction prepared, you can begin converting it to a decimal.

Let's give rule for converting a proper common fraction with a denominator of 10, or 100, or 1,000, ... into a decimal fraction. It consists of three steps:

  • write 0;
  • after it we put a decimal point;
  • We write down the number from the numerator (along with added zeros, if we added them).

Let's consider the application of this rule when solving examples.

Example.

Convert the proper fraction 37/100 to a decimal.

Solution.

The denominator contains the number 100, which has two zeros. The numerator contains the number 37, its notation has two digits, therefore, this fraction does not need to be prepared for conversion to a decimal fraction.

Now we write 0, put a decimal point, and write the number 37 from the numerator, and we get the decimal fraction 0.37.

Answer:

0,37 .

To strengthen the skills of converting proper ordinary fractions with numerators 10, 100, ... into decimal fractions, we will analyze the solution to another example.

Example.

Write the proper fraction 107/10,000,000 as a decimal.

Solution.

The number of digits in the numerator is 3, and the number of zeros in the denominator is 7, so this common fraction needs to be prepared for conversion to a decimal. We need to add 7-3=4 zeros to the left in the numerator so that the total number of digits there becomes equal to the number of zeros in the denominator. We get.

All that remains is to create the required decimal fraction. To do this, firstly, we write 0, secondly, we put a comma, thirdly, we write the number from the numerator together with zeros 0000107, as a result we have a decimal fraction 0.0000107.

Answer:

0,0000107 .

Improper fractions do not require any preparation when converting to decimals. The following should be adhered to rules for converting improper fractions with denominators 10, 100, ... into decimals:

  • write down the number from the numerator;
  • We use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.

Let's look at the application of this rule when solving an example.

Example.

Convert the improper fraction 56,888,038,009/100,000 to a decimal.

Solution.

Firstly, we write down the number from the numerator 56888038009, and secondly, we separate the 5 digits on the right with a decimal point, since the denominator of the original fraction has 5 zeros. As a result, we have the decimal fraction 568880.38009.

Answer:

568 880,38009 .

To convert a mixed number into a decimal fraction, the denominator of the fractional part of which is the number 10, or 100, or 1,000, ..., you can convert the mixed number into an improper ordinary fraction, and then convert the resulting fraction into a decimal fraction. But you can also use the following the rule for converting mixed numbers with a fractional denominator of 10, or 100, or 1,000, ... into decimal fractions:

  • if necessary, we perform “preliminary preparation” of the fractional part of the original mixed number by adding the required number of zeros to the left in the numerator;
  • write down the integer part of the original mixed number;
  • put a decimal point;
  • We write down the number from the numerator along with the added zeros.

Let's look at an example in which we complete all the necessary steps to represent a mixed number as a decimal fraction.

Example.

Convert the mixed number to a decimal.

Solution.

The denominator of the fractional part has 4 zeros, but the numerator contains the number 17, consisting of 2 digits, therefore, we need to add two zeros to the left in the numerator so that the number of digits there becomes equal to the number of zeros in the denominator. Having done this, the numerator will be 0017.

Now we write down the whole part of the original number, that is, the number 23, put a decimal point, after which we write the number from the numerator along with the added zeros, that is, 0017, and we get the desired decimal fraction 23.0017.

Let's write down the whole solution briefly: .

Of course, it was possible to first represent the mixed number as an improper fraction and then convert it to a decimal. With this approach, the solution looks like this: .

Answer:

23,0017 .

Converting fractions to finite and infinite periodic decimals

Not only ordinary fractions with denominators 10, 100, ... can be converted into decimal fractions, but ordinary fractions with other denominators. Now we will figure out how this is done.

In some cases, the original ordinary fraction is easily reduced to one of the denominators 10, or 100, or 1,000, ... (see bringing an ordinary fraction to a new denominator), after which it is not difficult to represent the resulting fraction as a decimal fraction. For example, it is obvious that the fraction 2/5 can be reduced to a fraction with a denominator 10, for this you need to multiply the numerator and denominator by 2, which will give the fraction 4/10, which, according to the rules discussed in the previous paragraph, is easily converted to the decimal fraction 0, 4.

In other cases, you have to use another method of converting an ordinary fraction into a decimal, which we now proceed to consider.

To convert an ordinary fraction to a decimal fraction, the numerator of the fraction is divided by the denominator, the numerator is first replaced by an equal decimal fraction with any number of zeros after the decimal point (we talked about this in the section equal and unequal decimal fractions). In this case, division is performed in the same way as division by a column of natural numbers, and in the quotient a decimal point is placed when the division of the whole part of the dividend ends. All this will become clear from the solutions to the examples given below.

Example.

Convert the fraction 621/4 to a decimal.

Solution.

Let's represent the number in the numerator 621 as a decimal fraction, adding a decimal point and several zeros after it. First, let's add 2 digits 0, later, if necessary, we can always add more zeros. So, we have 621.00.

Now let's divide the number 621,000 by 4 with a column. The first three steps are no different from dividing natural numbers by a column, after which we arrive at the following picture:

This is how we get to the decimal point in the dividend, and the remainder is different from zero. In this case, we put a decimal point in the quotient and continue dividing in a column, not paying attention to the commas:

This completes the division, and as a result we get the decimal fraction 155.25, which corresponds to the original ordinary fraction.

Answer:

155,25 .

To consolidate the material, consider the solution to another example.

Example.

Convert the fraction 21/800 to a decimal.

Solution.

To convert this common fraction to a decimal, we divide with a column of the decimal fraction 21,000... by 800. After the first step, we will have to put a decimal point in the quotient, and then continue the division:

Finally, we got the remainder 0, this completes the conversion of the common fraction 21/400 to a decimal fraction, and we arrived at the decimal fraction 0.02625.

Answer:

0,02625 .

It may happen that when dividing the numerator by the denominator of an ordinary fraction, we still do not get a remainder of 0. In these cases, division can be continued indefinitely. However, starting from a certain step, the remainders begin to repeat periodically, and the numbers in the quotient also repeat. This means that the original fraction is converted to an infinitely periodic decimal fraction. Let's show this with an example.

Example.

Write the fraction 19/44 as a decimal.

Solution.

To convert an ordinary fraction to a decimal, perform division by column:

It is already clear that during division the residues 8 and 36 began to be repeated, while in the quotient the numbers 1 and 8 are repeated. Thus, the original common fraction 19/44 is converted into a periodic decimal fraction 0.43181818...=0.43(18).

Answer:

0,43(18) .

To conclude this point, we will figure out which ordinary fractions can be converted into finite decimal fractions, and which ones can only be converted into periodic ones.

Let us have an irreducible ordinary fraction in front of us (if the fraction is reducible, then we first reduce the fraction), and we need to find out which decimal fraction it can be converted into - finite or periodic.

It is clear that if an ordinary fraction can be reduced to one of the denominators 10, 100, 1,000, ..., then the resulting fraction can be easily converted into a final decimal fraction according to the rules discussed in the previous paragraph. But to the denominators 10, 100, 1,000, etc. Not all ordinary fractions are given. Only fractions whose denominators are at least one of the numbers 10, 100, ... can be reduced to such denominators. And what numbers can be divisors of 10, 100, ...? The numbers 10, 100, ... will allow us to answer this question, and they are as follows: 10 = 2 5, 100 = 2 2 5 5, 1,000 = 2 2 2 5 5 5, .... It follows that the divisors are 10, 100, 1,000, etc. There can only be numbers whose decompositions into prime factors contain only the numbers 2 and (or) 5.

Now we can make a general conclusion about converting ordinary fractions to decimals:

  • if in the decomposition of the denominator into prime factors there are only numbers 2 and (or) 5, then this fraction can be converted into a final decimal fraction;
  • if, in addition to twos and fives, there are other prime numbers in the expansion of the denominator, then this fraction is converted to an infinite decimal periodic fraction.

Example.

Without converting ordinary fractions to decimals, tell me which of the fractions 47/20, 7/12, 21/56, 31/17 can be converted into a final decimal fraction, and which ones can only be converted into a periodic fraction.

Solution.

The denominator of the fraction 47/20 is factorized into prime factors as 20=2·2·5. This expansion contains only twos and fives, so this fraction can be reduced to one of the denominators 10, 100, 1,000, ... (in this example, to the denominator 100), therefore, can be converted to a final decimal fraction.

The decomposition of the denominator of the fraction 7/12 into prime factors has the form 12=2·2·3. Since it contains a prime factor of 3, different from 2 and 5, this fraction cannot be represented as a finite decimal, but can be converted into a periodic decimal.

Fraction 21/56 – contractile, after contraction it takes the form 3/8. Factoring the denominator into prime factors contains three factors equal to 2, therefore, the common fraction 3/8, and therefore the equal fraction 21/56, can be converted into a final decimal fraction.

Finally, the expansion of the denominator of the fraction 31/17 is 17 itself, therefore this fraction cannot be converted into a finite decimal fraction, but can be converted into an infinite periodic fraction.

Answer:

47/20 and 21/56 can be converted to a finite decimal fraction, but 7/12 and 31/17 can only be converted to a periodic fraction.

Ordinary fractions do not convert to infinite non-periodic decimals

The information in the previous paragraph gives rise to the question: “Can dividing the numerator of a fraction by the denominator result in an infinite non-periodic fraction?”

Answer: no. When converting a common fraction, the result can be either a finite decimal fraction or an infinite periodic decimal fraction. Let us explain why this is so.

From the theorem on divisibility with a remainder, it is clear that the remainder is always less than the divisor, that is, if we divide some integer by an integer q, then the remainder can only be one of the numbers 0, 1, 2, ..., q−1. It follows that after the column has completed dividing the integer part of the numerator of an ordinary fraction by the denominator q, in no more than q steps one of the following two situations will arise:

  • or we will get a remainder of 0, this will end the division, and we will get the final decimal fraction;
  • or we will get a remainder that has already appeared before, after which the remainders will begin to repeat as in the previous example (since when dividing equal numbers by q, equal remainders are obtained, which follows from the already mentioned divisibility theorem), this will result in an infinite periodic decimal fraction.

There cannot be any other options, therefore, when converting an ordinary fraction to a decimal fraction, an infinite non-periodic decimal fraction cannot be obtained.

From the reasoning given in this paragraph it also follows that the length of the period of a decimal fraction is always less than the value of the denominator of the corresponding ordinary fraction.

Converting decimals to fractions

Now let's figure out how to convert a decimal fraction into an ordinary fraction. Let's start by converting final decimal fractions to ordinary fractions. After this, we will consider a method for inverting infinite periodic decimal fractions. In conclusion, let's say about the impossibility of converting infinite non-periodic decimal fractions into ordinary fractions.

Converting trailing decimals to fractions

Obtaining a fraction that is written as a final decimal is quite simple. The rule for converting a final decimal fraction to a common fraction consists of three steps:

  • first, write the given decimal fraction into the numerator, having previously discarded the decimal point and all zeros on the left, if any;
  • secondly, write one into the denominator and add as many zeros to it as there are digits after the decimal point in the original decimal fraction;
  • thirdly, if necessary, reduce the resulting fraction.

Let's look at the solutions to the examples.

Example.

Convert the decimal 3.025 to a fraction.

Solution.

If we remove the decimal point from the original decimal fraction, we get the number 3,025. There are no zeros on the left that we would discard. So, we write 3,025 in the numerator of the desired fraction.

We write the number 1 into the denominator and add 3 zeros to the right of it, since in the original decimal fraction there are 3 digits after the decimal point.

So we got the common fraction 3,025/1,000. This fraction can be reduced by 25, we get .

Answer:

.

Example.

Convert the decimal fraction 0.0017 to a fraction.

Solution.

Without a decimal point, the original decimal fraction looks like 00017, discarding the zeros on the left we get the number 17, which is the numerator of the desired ordinary fraction.

We write one with four zeros in the denominator, since the original decimal fraction has 4 digits after the decimal point.

As a result, we have an ordinary fraction 17/10,000. This fraction is irreducible, and the conversion of a decimal fraction to an ordinary fraction is complete.

Answer:

.

When the integer part of the original final decimal fraction is non-zero, it can be immediately converted to a mixed number, bypassing the common fraction. Let's give rule for converting a final decimal fraction to a mixed number:

  • the number before the decimal point must be written as an integer part of the desired mixed number;
  • in the numerator of the fractional part you need to write the number obtained from the fractional part of the original decimal fraction after discarding all the zeros on the left;
  • in the denominator of the fractional part you need to write down the number 1, to which add as many zeros to the right as there are digits after the decimal point in the original decimal fraction;
  • if necessary, reduce the fractional part of the resulting mixed number.

Let's look at an example of converting a decimal fraction to a mixed number.

Example.

Express the decimal fraction 152.06005 as a mixed number

To write a rational number m/n as a decimal fraction, you need to divide the numerator by the denominator. In this case, the quotient is written as a finite or infinite decimal fraction.

Write this number as a decimal fraction.

Solution. Divide the numerator of each fraction into a column by its denominator: A) divide 6 by 25; b) divide 2 by 3; V) divide 1 by 2, and then add the resulting fraction to one - the integer part of this mixed number.

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

IN example 1 in case A) denominator 25=5·5; in case V) the denominator is 2, so we get the final decimals 0.24 and 1.5. In case b) the denominator is 3, so the result cannot be written as a finite decimal.

Is it possible, without long division, to convert into a decimal fraction such an ordinary fraction, the denominator of which does not contain other divisors other than 2 and 5? Let's figure it out! What fraction is called a decimal and is written without a fraction bar? Answer: fraction with denominator 10; 100; 1000, etc. And each of these numbers is a product equal number of twos and fives. In fact: 10=2 ·5 ; 100=2 ·5 ·2 ·5 ; 1000=2 ·5 ·2 ·5 ·2 ·5 etc.

Consequently, the denominator of an irreducible ordinary fraction will need to be represented as the product of “twos” and “fives”, and then multiplied by 2 and (or) 5 so that the “twos” and “fives” become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. To ensure that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which we multiplied the denominator.

Express the following common fractions as decimals:

Solution. Each of these fractions is irreducible. Let's factor the denominator of each fraction into prime factors.

20=2·2·5. Conclusion: one “A” is missing.

8=2·2·2. Conclusion: three “A”s are missing.

25=5·5. Conclusion: two “twos” are missing.

Comment. In practice, they often do not use factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is one with zeros (10 or 100 or 1000, etc.). And then the numerator is multiplied by the same number.

So, in case A)(example 2) from the number 20 you can get 100 by multiplying by 5, therefore, you need to multiply the numerator and denominator by 5.

In case b)(example 2) from the number 8 the number 100 will not be obtained, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.

In case V)(example 2) from 25 you get 100 if you multiply by 4. This means that the numerator 8 must be multiplied by 4.

An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodic as a decimal. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosed in parentheses.

In case b)(example 1) there is only one repeating digit and is equal to 6. Therefore, our result 0.66... ​​will be written like this: 0,(6) . They read: zero point, six in period.

If there are one or more non-repeating digits between the decimal point and the first period, then such a periodic fraction is called a mixed periodic fraction.

An irreducible common fraction whose denominator is together with others multiplier contains multiplier 2 or 5 , turns to mixed periodic fraction.

Write numbers as decimals.