How to write it as a decimal. Common and decimal fractions and operations on them

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.

Page navigation.

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 only the numbers 2 and (or) 5 are present, 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 of 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 repeated 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 multipliers contains a multiplier 2 or 5 , turns to mixed periodic fraction.

Write numbers as decimals.

A decimal fraction is a fraction in which the denominator is a natural power of 10. This, for example, is the fraction This fraction can be written in the following form: write down the digits of the numerator on a line and separate as many of them with a comma to the right as there are zeros in the denominator, namely :

In such a notation, the numbers to the left of the decimal form the integer part, and the numbers to the right of the decimal form the fractional part of the given decimal fraction.

Let p/q be some positive rational number. From arithmetic, the division process is well known, allowing you to represent a number as a decimal fraction. The essence of the division process is to first find the largest integer number of times q is contained in p; if p is a multiple of q, then this is where the division process ends. Otherwise, a remainder appears. Next, they find how many tenths of q this residue contains, and at this step the process may end, or a new residue will appear. In the latter case, find how many hundredths of q it contains, etc.

If the denominator q has no other prime factors other than 2 or 5, then after a finite number of steps the remainder will be equal to zero, the division process will end and the given ordinary fraction will turn into a final decimal fraction. In fact, in this case, it is always possible to select an integer such that after multiplying the numerator and denominator of a given fraction by it, an equal fraction will be obtained, in which the denominator will represent a natural power of ten. For example, this is the fraction

which can be represented like this:

However, without making these transformations, dividing the numerator by the denominator, the reader will get the same result:

If the denominator of an irreducible fraction has at least one prime divisor other than 2 or 5, then the process of division by q will never end (none of the subsequent remainders will go to zero).

Having performed the division, we find

To write the result obtained in this example, periodically repeating numbers 0 and 6 are enclosed in parentheses and written:

In this example and other similar cases, the action of division does not result in a final result as a decimal. It is possible, generalizing the concept of a decimal fraction, to still say that the quotient 965/132 is represented by an infinite periodic fraction. The repeating numbers 06 are called the period of this fraction, and their number, equal in our example, is the length of the period.

To understand the reason for the phenomenon of periodicity of a fraction, let us analyze, for example, the process of dividing by 7. If the division is not completely performed, then a remainder appears, which can only have one of the following values: 1, 2, 3, 4, 5, 6. And on each from the following steps the remainder will again have one of these six values. Therefore, no later than the seventh step, we will inevitably encounter one of the remainder values ​​that have already appeared before. Starting from this point, the division process will become periodic. Both the values ​​of the balances and the numbers of the quotient will be repeated periodically. The same reasoning applies to any other divisor.

Thus, every ordinary fraction is represented as a finite or infinite periodic decimal fraction. It is remarkable that, conversely, every periodic decimal fraction can be represented as an ordinary fraction. Let's show how this action is performed. In this case, the formula for the sum of an infinitely decreasing geometric progression is used (clause 92).

can be understood this way:

here the terms on the right side, starting from the second, form an infinite geometric progression with the denominator and the first term

Using formula (92.2):

It is clear that the same process will allow any given infinite periodic fraction to be represented in the form of an ordinary fraction (and, as can be shown, precisely the one from which, in the process of division, the given infinite periodic fraction in turn is obtained). However, there is one exception here. Consider the fraction

and apply the process of converting it to a common fraction:

We have arrived at the number 1/2, which appears to be a finite decimal fraction

A similar result will be obtained whenever the period of a given infinite fraction has the form (9). Therefore, we identify such pairs of numbers as, for example,

Sometimes it is useful to also allow records of the form

formally representing finite decimal fractions as infinite with period (0).

Everything that was said about converting an ordinary fraction into a periodic decimal fraction and vice versa applied to positive rational numbers. In the case of a negative number, you can do it in two ways.

1) Take the positive number opposite the given negative number, convert it to a decimal, and then put a minus sign in front of it. For example, for - 5/3 we get

2) Present a given negative rational number as the sum of its integer part (negative) and its fractional part (non-negative), and then convert only this fractional part of the number into a decimal fraction. For example:

To write numbers presented as the sum of their negative integer part and a finite or infinite decimal fraction, the following notation is accepted (an artificial form of writing a negative number):

Here the minus sign is placed not in front of the entire fraction, but above its whole part, to emphasize that only the whole part is negative, and the fractional part following the decimal point is positive.

This notation creates uniformity in the notation of positive and negative decimal fractions and will be used in the future in the theory of decimal logarithms (section 28). For practice, we invite the reader to check the transition from one record to another in the examples:

Now we can formulate the final conclusion: every rational number can be represented by an infinite decimal periodic fraction, and, conversely, every such fraction specifies a rational number. The finite decimal fraction also allows two forms of writing in the form of an infinite decimal fraction: with a period (0) and with a period (9).

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 a 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.

Already in elementary school, students are exposed to 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 one. 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 become acquainted with the first ones in elementary 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 as a decimal. This statement is almost always true in reverse. There are rules that allow you to write a decimal fraction as an ordinary fraction.

What subtypes do these types of fractions have?

It is better to start in chronological order, as they are 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 is applied based on the rule - 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 into ordinary fractions if their integer part is missing, 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 non-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 gives the following 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 the 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 simplest option is a number whose denominator is 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 periodic decimal fraction.

Operations with ordinary fractions

Addition and subtraction

Students become acquainted with them earlier than others. Moreover, at first the fractions have the same denominators, and then they have different ones. General rules can be reduced to this 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 subtracting a larger number from a smaller number. 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 a 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 a whole number, the latter should be written as an 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 fraction by a 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.