home-Ramps-In the form of an ordinary fraction, a decimal fraction. Converting mixed numbers to decimals

In the form of an ordinary fraction, a decimal fraction. Converting mixed numbers to decimals


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 give detailed solutions 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 .

After preparing the correct common fraction You can start converting it to a decimal fraction.

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, perform " preliminary preparation» fractional part of the original mixed number, adding required amount zeros on 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, when solving it we will do everything necessary steps to represent a mixed number as a decimal.

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 integer 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 fraction. With this approach, the solution looks like this: .

Answer:

23,0017 .

Converting fractions to finite and infinite periodic decimals

You can convert not only ordinary fractions with denominators 10, 100, ... into a decimal fraction, but also 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 to a decimal, which we now move on 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 infinite periodic decimal fraction. Let's show this with an example.

Example.

Write the fraction 19/44 as a decimal.

Solution.

To convert a common 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 others in the expansion of the denominator prime numbers, 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. In this expansion there are 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 denominator of the fraction 7/12 is factorized into prime factors as 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 a common 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:

  • firstly, 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

We will devote this material to such an important topic as decimal fractions. First, let's define the basic definitions, give examples and dwell on the rules of decimal notation, as well as what the digits of decimal fractions are. Next, we highlight the main types: finite and infinite, periodic and non-periodic fractions. In the final part we will show how the points corresponding to fractional numbers are located on the coordinate axis.

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What is decimal notation of fractional numbers

The so-called decimal notation of fractional numbers can be used for both natural and fractional numbers. It looks like a set of two or more numbers with a comma between them.

The decimal point is needed to separate the whole part from the fractional part. As a rule, the last digit of a decimal fraction is not a zero, unless the decimal point appears immediately after the first zero.

What are some examples of fractional numbers in decimal notation? This could be 34, 21, 0, 35035044, 0, 0001, 11,231,552, 9, etc.

In some textbooks you can find the use of a period instead of a comma (5. 67, 6789. 1011, etc.). This option is considered equivalent, but it is more typical for English-language sources.

Definition of decimals

Based on the above concept of decimal notation, we can formulate the following definition of decimal fractions:

Definition 1

Decimals represent fractional numbers in decimal notation.

Why do we need to write fractions in this form? It gives us some advantages over ordinary ones, for example, more compact recording, especially in cases where the denominator contains 1000, 100, 10, etc. or a mixed number. For example, instead of 6 10 we can specify 0.6, instead of 25 10000 - 0.0023, instead of 512 3 100 - 512.03.

How to correctly represent ordinary fractions with tens, hundreds, thousands in the denominator in decimal form will be discussed in a separate material.

How to read decimals correctly

There are some rules for reading decimal notations. Thus, those decimal fractions that correspond to their regular ordinary equivalents are read almost the same way, but with the addition of the words “zero tenths” at the beginning. Thus, the entry 0, 14, which corresponds to 14,100, is read as “zero point fourteen hundredths.”

If a decimal fraction can be associated with a mixed number, then it is read in the same way as this number. So, if we have the fraction 56, 002, which corresponds to 56 2 1000, we read this entry as “fifty-six point two thousandths.”

The meaning of a digit in a decimal fraction depends on where it is located (the same as in the case of natural numbers). So, in the decimal fraction 0.7, seven is tenths, in 0.0007 it is ten thousandths, and in the fraction 70,000.345 it means seven tens of thousands of whole units. Thus, in decimal fractions there is also the concept of place value.

The names of the digits located before the decimal point are similar to those that exist in natural numbers. The names of those located after are clearly presented in the table:

Let's look at an example.

Example 1

We have the decimal fraction 43,098. She has a four in the tens place, a three in the units place, a zero in the tenths place, 9 in the hundredths place, and 8 in the thousandths place.

It is customary to distinguish the ranks of decimal fractions by precedence. If we move through the numbers from left to right, then we will go from the most significant to the least significant. It turns out that hundreds are older than tens, and parts per million are younger than hundredths. If we take the final decimal fraction that we cited as an example above, then the highest, or highest, place in it will be the hundreds place, and the lowest, or lowest, place will be the 10-thousandth place.

Any decimal fraction can be expanded into individual digits, that is, presented as a sum. This action is performed in the same way as for natural numbers.

Example 2

Let's try to expand the fraction 56, 0455 into digits.

We will get:

56 , 0455 = 50 + 6 + 0 , 4 + 0 , 005 + 0 , 0005

If we remember the properties of addition, we can represent this fraction in other forms, for example, as the sum 56 + 0, 0455, or 56, 0055 + 0, 4, etc.

What are trailing decimals?

All the fractions we talked about above are finite decimals. This means that the number of digits after the decimal point is finite. Let's derive the definition:

Definition 1

Trailing decimals are a type of decimal fraction that has a finite number of decimal places after the decimal sign.

Examples of such fractions can be 0, 367, 3, 7, 55, 102567958, 231 032, 49, etc.

Any of these fractions can be converted either to a mixed number (if the value of their fractional part is different from zero) or to an ordinary fraction (if the integer part is zero). We dedicated to how this is done separate material. Here we’ll just point out a couple of examples: for example, we can reduce the final decimal fraction 5, 63 to the form 5 63 100, and 0, 2 corresponds to 2 10 (or any other fraction equal to it, for example, 4 20 or 1 5.)

But the reverse process, i.e. writing a common fraction in decimal form may not always be possible. So, 5 13 cannot be replaced by an equal fraction with the denominator 100, 10, etc., which means that a final decimal fraction cannot be obtained from it.

Main types of infinite decimal fractions: periodic and non-periodic fractions

We indicated above that finite fractions are so called because they have a finite number of digits after the decimal point. However, it may well be infinite, in which case the fractions themselves will also be called infinite.

Definition 2

Infinite decimal fractions are those that have an infinite number of digits after the decimal point.

Obviously, such numbers simply cannot be written down in full, so we indicate only part of them and then add an ellipsis. This sign indicates an infinite continuation of the sequence of decimal places. Examples of infinite decimal fractions include 0, 143346732…, ​​3, 1415989032…, 153, 0245005…, 2, 66666666666…, 69, 748768152…. etc.

The “tail” of such a fraction may contain not only seemingly random sequences of numbers, but also a constant repetition of the same character or group of characters. Fractions with alternating numbers after the decimal point are called periodic.

Definition 3

Periodic decimal fractions are those infinite decimal fractions in which one digit or a group of several digits is repeated after the decimal point. The repeating part is called the period of the fraction.

For example, for the fraction 3, 444444…. the period will be the number 4, and for 76, 134134134134... - the group 134.

What is the minimum number of characters that can be left in the notation of a periodic fraction? For periodic fractions, it will be enough to write the entire period once in parentheses. So, fraction 3, 444444…. It would be correct to write it as 3, (4), and 76, 134134134134... – as 76, (134).

In general, entries with several periods in brackets will have exactly the same meaning: for example, the periodic fraction 0.677777 is the same as 0.6 (7) and 0.6 (77), etc. Records of the form 0, 67777 (7), 0, 67 (7777), etc. are also acceptable.

To avoid mistakes, we introduce uniformity of notation. Let's agree to write down only one period (the shortest possible sequence of numbers), which is closest to the decimal point, and enclose it in parentheses.

That is, for the above fraction, we will consider the main entry to be 0, 6 (7), and, for example, in the case of the fraction 8, 9134343434, we will write 8, 91 (34).

If the denominator of an ordinary fraction contains prime factors that are not equal to 5 and 2, then when converted to decimal notation, they will result in infinite fractions.

In principle, we can write any finite fraction as a periodic one. To do this, we just need to add an infinite number of zeros to the right. What does it look like in recording? Let's say we have the final fraction 45, 32. In periodic form it will look like 45, 32 (0). This action is possible because adding zeros to the right of any decimal fraction gives us the result of a fraction equal to it.

Special attention should be paid to periodic fractions with a period of 9, for example, 4, 89 (9), 31, 6 (9). They are an alternative notation for similar fractions with a period of 0, so they are often replaced when writing with fractions with a zero period. In this case, one is added to the value of the next digit, and (0) is indicated in parentheses. The equality of the resulting numbers can be easily verified by representing them as ordinary fractions.

For example, the fraction 8, 31 (9) can be replaced with the corresponding fraction 8, 32 (0). Or 4, (9) = 5, (0) = 5.

Infinite decimal periodic fractions are classified as rational numbers. In other words, any periodic fraction can be represented as an ordinary fraction, and vice versa.

There are also fractions that do not have an endlessly repeating sequence after the decimal point. In this case, they are called non-periodic fractions.

Definition 4

Non-periodic decimal fractions include those infinite decimal fractions that do not contain a period after the decimal point, i.e. repeating group of numbers.

Sometimes non-periodic fractions look very similar to periodic ones. For example, 9, 03003000300003 ... at first glance it seems to have a period, however detailed analysis decimal places confirms that this is still a non-periodic fraction. You need to be very careful with such numbers.

Non-periodic fractions refer to irrational numbers. They are not converted to ordinary fractions.

Basic operations with decimals

The following operations can be performed with decimal fractions: comparison, subtraction, addition, division and multiplication. Let's look at each of them separately.

Comparing decimals can be reduced to comparing fractions that correspond to the original decimals. But infinite non-periodic fractions cannot be reduced to this form, and converting decimal fractions into ordinary fractions is often a labor-intensive task. How can we quickly perform a comparison action if we need to do this while solving a problem? It is convenient to compare decimal fractions by digit in the same way as we compare natural numbers. We will devote a separate article to this method.

To add some decimal fractions with others, it is convenient to use the column addition method, as for natural numbers. To add periodic decimal fractions, you must first replace them with ordinary ones and count according to standard scheme. If, according to the conditions of the problem, we need to add infinite non-periodic fractions, then we need to first round them to a certain digit, and then add them. The smaller the digit to which we round, the higher the accuracy of the calculation will be. For subtraction, multiplication and division of infinite fractions, pre-rounding is also necessary.

Finding the difference between decimal fractions is the inverse of addition. Essentially, using subtraction we can find a number whose sum with the fraction we are subtracting will give us the fraction we are minimizing. We will talk about this in more detail in a separate article.

Multiplying decimal fractions is done in the same way as for natural numbers. The column calculation method is also suitable for this. We again reduce this action with periodic fractions to the multiplication of ordinary fractions according to the rules already studied. Infinite fractions, as we remember, must be rounded before calculations.

The process of dividing decimals is the inverse of multiplying. When solving problems, we also use columnar calculations.

You can establish an exact correspondence between the final decimal fraction and a point on the coordinate axis. Let's figure out how to mark a point on the axis that will exactly correspond to the required decimal fraction.

We have already studied how to construct points corresponding to ordinary fractions, but decimal fractions can be reduced to this form. For example, the common fraction 14 10 is the same as 1, 4, so the corresponding point will be removed from the origin in the positive direction by exactly the same distance:

You can do without replacing the decimal fraction with an ordinary one, but use the method of expansion by digits as a basis. So, if we need to mark a point whose coordinate will be equal to 15, 4008, then we will first present this number as the sum 15 + 0, 4 +, 0008. To begin with, let’s set aside 15 whole unit segments in the positive direction from the beginning of the countdown, then 4 tenths of one segment, and then 8 ten-thousandths of one segment. As a result, we get a coordinate point that corresponds to the fraction 15, 4008.

For an infinite decimal fraction, it is better to use this method, since it allows you to get as close as you like to the desired point. In some cases, it is possible to construct an exact correspondence to an infinite fraction on the coordinate axis: for example, 2 = 1, 41421. . . , and this fraction can be associated with a point on coordinate ray, removed from 0 by the length of the diagonal of the square, the side of which will be equal to one unit segment.

If we find not a point on the axis, but a decimal fraction corresponding to it, then this action is called the decimal measurement of a segment. Let's see how to do this correctly.

Let's say we need to get from zero to a given point on the coordinate axis (or get as close as possible in the case of an infinite fraction). To do this, we gradually postpone unit segments from the origin of coordinates until we get to desired point. After whole segments, if necessary, we measure tenths, hundredths and smaller fractions so that the match is as accurate as possible. As a result, we received a decimal fraction that corresponds to a given point on the coordinate axis.

Above we showed a drawing with point M. Look at it again: to get to this point, you need to measure one unit segment and four tenths of it from zero, since this point corresponds to the decimal fraction 1, 4.

If we cannot get to a point in the process of decimal measurement, then it means that it corresponds to an infinite decimal fraction.

If you notice an error in the text, please highlight it and press Ctrl+Enter

As:

± d m … d 1 d 0 , d -1 d -2 …

where ± is the fraction sign: either +, or -,

, is a decimal point that serves as a separator between the integer and fractional parts of a number,

dk- decimal numbers.

In this case, the order of numbers before the decimal point (to the left of it) has an end (as min 1 per digit), and after the decimal point (to the right) it can be both finite (as an option, there may be no digits after the decimal point at all) and infinite.

Decimal value ± d m … d 1 d 0 , d -1 d -2 … is a real number:

which is equal to the sum of a finite or infinite number of terms.

Performance real numbers using decimal fractions is a generalization of writing integers in the decimal number system. The decimal representation of an integer has no digits after the decimal point, so the representation looks like this:

± d m … d 1 d 0 ,

And this coincides with writing our number in the decimal number system.

Decimal - this is the result of dividing 1 into 10, 100, 1000 and so on parts. These fractions are quite convenient for calculations, because they are based on the same positional system on which counting and recording of integers are based. Thanks to this, the notation and rules for working with decimal fractions are almost the same as for whole numbers.

When writing decimal fractions, you do not need to mark the denominator; it is determined by the place occupied by the corresponding digit. First we write the whole part of the number, then we put a decimal point on the right. The first digit after the decimal point indicates the number of tenths, the second - the number of hundredths, the third - the number of thousandths, and so on. The numbers that are located after the decimal point are decimals.

For example:

One of the advantages of decimal fractions is that they can very easily be reduced to ordinary fractions: the number after the decimal point (for us it is 5047) is numerator; denominator equals n-th power of 10, where n- the number of decimal places (for us this is n=4):

When there is no integer part in a decimal fraction, we put a zero before the decimal point:

Properties of decimal fractions.

1. The decimal does not change when zeros are added to the right:

13.6 =13.6000.

2. The decimal does not change when the zeros at the end of the decimal are removed:

0.00123000 = 0.00123.

Attention! You cannot remove zeros that are NOT located at the end of the decimal fraction!

3. The decimal fraction increases by 10, 100, 1000 and so on times when we move the decimal point to 1, 2, 2 and so on positions to the right, respectively:

3.675 → 367.5 (fraction increased a hundred times).

4. The decimal fraction becomes ten, one hundred, thousand, and so on times smaller when we move the decimal point to 1, 2, 3, and so on positions to the left, respectively:

1536.78 → 1.53678 (the fraction became a thousand times smaller).

Types of decimal fractions.

Decimal fractions are divided into final, endless And periodic decimals.

The final decimal fraction is this is a fraction containing a finite number of digits after the decimal point (or there are none at all), i.e. looks like that:

A real number can be represented as a finite decimal fraction only if this number is rational and when written as an irreducible fraction p/q denominator q has no prime factors other than 2 and 5.

Infinite decimal.

Contains an infinitely repeating group of numbers called period. The period is written in parentheses. For example, 0.12345123451234512345… = 0.(12345).

Periodic decimal- this is an infinite decimal fraction in which the sequence of digits after the decimal point, starting from a certain place, is a periodically repeating group of digits. In other words, periodic fraction- a decimal fraction that looks like this:

Such a fraction is usually written briefly as follows:

Group of numbers b 1 … b l, which is repeated, is period of the fraction, the number of digits in this group is period length.

When in a periodic fraction the period comes immediately after the decimal point, it means the fraction is pure periodic. When there are numbers between the decimal point and the 1st period, then the fraction is mixed periodic, and the group of digits after the decimal point up to the 1st digit of the period is fraction preperiod.

For example, the fraction 1,(23) = 1.2323... is pure periodic, and the fraction 0.1(23) = 0.12323... is mixed periodic.

The main property of periodic fractions, due to which they are distinguished from the entire set of decimal fractions, lies in the fact that periodic fractions and only they represent rational numbers. More precisely, the following occurs:

Any infinitely periodic decimal fraction represents rational number. Conversely, when a rational number is expanded into an infinite decimal fraction, it means that this fraction will be periodic.

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

Why are fractions needed?

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

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

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

What is a "fraction"?

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

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

What fractions are there?

In mathematics there are only two types: ordinary and decimal fractions. Schoolchildren 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 a common fraction.

What subtypes do these types of fractions have?

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

    Correct. Its numerator is always less than its denominator.

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

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

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

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

Decimal fractions have only two subtypes:

    finite, that is, one whose fractional part is limited (has an end);

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

How to convert a decimal fraction to a common fraction?

If this is a finite number, then an association 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 different from zero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its value is written. In the first case it is 5, in the second it is 13. Then you need to move on to the fractional part. The same operation is supposed to be carried out with them. The first number appears 23/100, the second - 108/100000. The second value needs to be reduced again. The answer 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 most simple option turns out to be a number whose denominator contains the number 10, 100, etc. Then the denominator is simply discarded, and a comma is placed between the fractional and integer parts.

There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. You just need to multiply not only the denominator, but also the numerator by the same number.

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

Operations with ordinary fractions

Addition and subtraction

Students become acquainted with them earlier than others. Moreover, at first the fractions have the same denominators, and then they have different ones. General rules can be reduced to such a plan.

    Find the least common multiple of the denominators.

    Write additional factors for all ordinary fractions.

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

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

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

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

    In the second, it is necessary to apply the rule of 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 by natural number.

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

    What if one example contains both types of fractions?

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

    First way: represent ordinary decimals

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

    Second way: write decimal fractions as ordinary

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

Fractions written in the form 0.8; 0.13; 2.856; 5.2; 0.04 is called decimal. In fact, decimal fractions are a simplified notation ordinary fractions. This notation is convenient to use for all fractions whose denominators are 10, 100, 1000, and so on.

Let's look at examples (0.5 is read as zero point five);

(0.15 read as, zero point fifteen);

(5.3 read as, five point three).

Please note that in the notation of a decimal fraction, a comma separates the integer part of a number from the fractional part, the integer part of a proper fraction is 0. The notation of the fractional part of a decimal fraction contains as many digits as there are zeros in the notation of the denominator of the corresponding ordinary fraction.

Let's look at an example, , , .

In some cases, it may be necessary to treat a natural number as a decimal whose fractional part is zero. It is customary to write that 5 = 5.0; 245 = 245.0 and so on. Note that in the decimal notation of a natural number, the unit of the least significant digit is 10 times less than the unit of the adjacent most significant digit. Writing decimal fractions has the same property. Therefore, immediately after the decimal point there is a place of tenths, then a place of hundredths, then a place of thousandths, and so on. Below are the names of the digits of the number 31.85431, the first two columns are the integer part, the remaining columns are the fractional part.

This fraction is read as thirty-one point eighty-five thousand four hundred and thirty-one hundred thousandths.

Adding and subtracting decimals

The first way is to convert decimal fractions into ordinary fractions and perform addition.

As can be seen from the example, this method is very inconvenient and it is better to use the second method, which is more correct, without converting decimal fractions into ordinary ones. In order to add two decimal fractions, you need to:

  • equalize the number of digits after the decimal point in the terms;
  • write the terms one below the other so that each digit of the second term is under the corresponding digit of the first term;
  • add the resulting numbers the same way you add natural numbers;
  • Place a comma in the resulting sum under the commas in the terms.

Let's look at examples:

  • equalize the number of digits after the decimal point in the minuend and subtrahend;
  • write the subtrahend under the minuend so that each digit of the subtrahend is under the corresponding digit of the minuend;
  • perform subtraction in the same way as natural numbers are subtracted;
  • put a comma in the resulting difference under the commas in the minuend and subtrahend.

Let's look at examples:

In the examples discussed above, it can be seen that the addition and subtraction of decimal fractions was performed bit by bit, that is, in the same way as we performed similar operations with natural numbers. This is the main advantage of the decimal form of writing fractions.

Multiplying Decimals

In order to multiply a decimal fraction by 10, 100, 1000, and so on, you need to move the decimal point in this fraction to the right by 1, 2, 3, and so on, respectively. Therefore, if the comma is moved to the right by 1, 2, 3 and so on digits, then the fraction will increase accordingly by 10, 100, 1000 and so on times. In order to multiply two decimal fractions, you need to:

  • multiply them as natural numbers, ignoring commas;
  • in the resulting product, separate as many digits on the right with a comma as there are after the commas in both factors together.

There are cases when a product contains fewer digits than is required to be separated by a comma; the required number of zeros are added to the left before this product, and then the comma is moved to the left by the required number of digits.

Let's look at examples: 2 * 4 = 8, then 0.2 * 0.4 = 0.08; 23 * 35 = 805, then 0.023 * 0.35 = 0.00805.

There are cases when one of the multipliers is equal to 0.1; 0.01; 0.001 and so on, it is more convenient to use the following rule.

  • To multiply a decimal by 0.1; 0.01; 0.001 and so on, in this decimal fraction you need to move the decimal point to the left by 1, 2, 3, and so on, respectively.

Let's look at examples: 2.65 * 0.1 = 0.265; 457.6 * 0.01 = 4.576.

The properties of multiplication of natural numbers also apply to decimal fractions.

  • ab = ba- commutative property of multiplication;
  • (ab) c = a (bc)- the associative property of multiplication;
  • a (b + c) = ab + ac is a distributive property of multiplication relative to addition.

Decimal division

It is known that if you divide a natural number a to a natural number b means to find such a natural number c, which when multiplied by b gives a number a. This rule remains true if at least one of the numbers a, b, c is a decimal fraction.

Let's look at an example: you need to divide 43.52 by 17 with a corner, ignoring the comma. In this case, the comma in the quotient should be placed immediately before the first digit after the decimal point in the dividend is used.

There are cases when the dividend is less than the divisor, then the integer part of the quotient is equal to zero. Let's look at an example:

Let's look at another interesting example.

The division process has stopped because the digits of the dividend have run out and the remainder does not have a zero. It is known that a decimal fraction will not change if any number of zeros are added to it on the right. Then it becomes clear that the numbers of the dividend cannot end.

In order to divide a decimal fraction by 10, 100, 1000, and so on, you need to move the decimal point in this fraction to the left by 1, 2, 3, and so on digits. Let's look at an example: 5.14: 10 = 0.514; 2: 100 = 0.02; 37.51: 1000 = 0.03751.

If the dividend and divisor are increased simultaneously by 10, 100, 1000, and so on times, then the quotient will not change.

Consider an example: 39.44: 1.6 = 24.65, increase the dividend and divisor by 10 times 394.4: 16 = 24.65 It is fair to note that dividing a decimal fraction by a natural number in the second example is easier.

In order to divide a decimal fraction by a decimal, you need to:

  • move the commas in the dividend and divisor to the right by as many digits as there are after the decimal point in the divisor;
  • divide by a natural number.

Let's consider an example: 23.6: 0.02, note that the divisor has two decimal places, therefore we multiply both numbers by 100 and get 2360: 2 = 1180, divide the result by 100 and get the answer 11.80 or 23.6: 0, 02 = 11.8.

Comparison of decimals

There are two ways to compare decimals. Method one, you need to compare two decimal fractions 4.321 and 4.32, equalize the number of decimal places and start comparing place by place, tenths with tenths, hundredths with hundredths, and so on, in the end we get 4.321 > 4.320.

The second way to compare decimal fractions is done using multiplication; multiply the above example by 1000 and compare 4321 > 4320. Which method is more convenient, everyone chooses for themselves.

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