As:

± d m … d 1 d 0 , d -1 d -2 …

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

, - decimal point, which serves as a separator between the integer and fractional parts of the number,

dk- decimal digits.

At the same time, the order of the digits before the comma (to the left of it) has an end (like min 1-per digit), and after the comma (to the right) it can be finite (as an option, there may be no digits after the comma 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.

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

± d m … d 1 d 0 ,

And this coincides with the record of 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 notation of integers are built. Due to this, the notation and rules for decimal fractions are almost the same as for integers.

When writing decimal fractions, you do not need to mark the denominator, it is determined by the place occupied by the corresponding figure. First, write the integer part of the number, then 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 after the decimal point are decimal places.

For example:

One of the advantages of decimal fractions is that they can be very easily converted to ordinary fractions: the number after the decimal point (ours is 5047) is numerator; denominator equals n th degree 10, where n- the number of decimal places (we have this n=4):

When there is no integer part in the decimal fraction, then we put zero in front of the decimal point:

Properties of decimal fractions.

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

13.6 =13.6000.

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

0.00123000 = 0.00123.

Attention! Zeros that are NOT at the end of a decimal must not be removed!

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

3.675 → 367.5 (the fraction has increased a hundred times).

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

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

Types of decimals.

Decimals are divided by final, endless And periodic decimals.

End decimal - this is a fraction containing a finite number of digits after the decimal point (or they are not there 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 divisors other than 2 and 5.

Infinite decimal.

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

Periodic decimal- this is such 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 is a decimal that looks like this:

Such a fraction is usually briefly written like this:

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

When in a periodic fraction the period comes immediately after the decimal point, then the fraction is pure periodic. When there are numbers between the comma and the 1st period, then the fraction is mixed periodic, and a group of digits after the decimal point up to the 1st period sign - 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 takes place:

Any infinite recurring decimal represents a rational number. Conversely, when a rational number is decomposed into an infinite decimal fraction, then this fraction will be periodic.

fractional number.

Decimal notation of a fractional number is a set of two or more digits from $0$ to $9$, between which is the so-called \textit (decimal point).

Example 1

For example, $35.02; $100.7; $123 \ $456.5; $54.89.

The leftmost digit in the decimal representation of a number cannot be zero, except when the decimal point is immediately after the first digit $0$.

Example 2

For example, $0.357; $0.064.

Often the decimal point is replaced by a decimal point. For example, $35.02$; $100.7$; $123 \ $456.5; $54.89.

Decimal definition

Definition 1

Decimals are fractional numbers that are represented in decimal notation.

For example, $121.05; $67.9; $345.6700.

Decimals are used for a more compact representation of regular fractions whose denominators are the numbers $10$, $100$, $1\000$, etc. and mixed numbers whose denominators are $10$, $100$, $1\000$, etc.

For example, the common fraction $\frac(8)(10)$ can be written as the decimal $0.8$, and the mixed number $405\frac(8)(100)$ as the decimal $405.08$.

Reading decimals

Decimals that correspond to regular fractions are read the same as ordinary fractions, only the phrase "zero integers" is added in front. For example, the common fraction $\frac(25)(100)$ (read "twenty-five hundredths") corresponds to the decimal fraction $0.25$ (read "zero point twenty-five hundredths").

Decimals that correspond to mixed numbers are read the same way as mixed numbers. For example, the mixed number $43\frac(15)(1000)$ corresponds to the decimal fraction $43.015$ (read "forty-three point fifteen thousandths").

Places in decimals

In decimal notation, the value of each digit depends on its position. Those. in decimal fractions, the concept also takes place discharge.

The digits in decimal fractions up to the decimal point are called the same as the digits in natural numbers. The digits in decimal fractions after the decimal point are listed in the table:

Picture 1.

Example 3

For example, in the decimal fraction $56,328$, $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenth place, $2$ is in the hundredth place, $8$ is in the thousandth place.

The digits in decimal fractions are distinguished by seniority. When reading a decimal fraction, they move from left to right - from senior discharge to junior.

Example 4

For example, in decimal $56.328$, the most significant (highest) digit is the tens digit, and the least significant (lowest) digit is the thousandths digit.

A decimal fraction can be expanded into digits in the same way as expansion into digits of a natural number.

Example 5

For example, let's expand the decimal fraction $37,851$ into digits:

$37,851=30+7+0,8+0,05+0,001$

End decimals

Definition 2

End decimals are called decimal fractions, the records of which contain a finite number of characters (digits).

For example, $0.138; $5.34; $56.123456; $350,972.54.

Any final decimal fraction can be converted to a common fraction or a mixed number.

Example 6

For example, the final decimal fraction $7.39$ corresponds to the fractional number $7\frac(39)(100)$, and the final decimal fraction $0.5$ corresponds to the proper fraction $\frac(5)(10)$ (or any fraction, which is equal to it, for example, $\frac(1)(2)$ or $\frac(10)(20)$.

Converting an ordinary fraction to a decimal fraction

Convert common fractions with denominators $10, 100, \dots$ to decimals

Before converting some proper ordinary fractions to decimals, they must first be “prepared”. The result of such preparation should be the same number of digits in the numerator and the number of zeros in the denominator.

The essence of the “preliminary preparation” of correct ordinary fractions for conversion to decimal fractions is to add on the left in the numerator such a number of zeros that the total number of digits becomes equal to the number of zeros in the denominator.

Example 7

For example, let's prepare the common fraction $\frac(43)(1000)$ for conversion to decimal and get $\frac(043)(1000)$. And the ordinary fraction $\frac(83)(100)$ does not need to be prepared.

Let's formulate rule for converting a proper common fraction with denominator $10$, or $100$, or $1\000$, $\dots$ to a decimal fraction:

    write $0$;

    put a decimal point after it;

    write down the number from the numerator (together with added zeros after preparation, if necessary).

Example 8

Convert proper fraction $\frac(23)(100)$ to decimal.

Solution.

The denominator is the number $100$, which contains $2$ two zeros. The numerator contains the number $23$, which contains $2$.digits. this means that preparation for this fraction for conversion to decimal is not necessary.

Let's write $0$, put a decimal point and write the number $23$ from the numerator. We get the decimal fraction $0.23$.

Answer: $0,23$.

Example 9

Write the proper fraction $\frac(351)(100000)$ as a decimal.

Solution.

The numerator of this fraction has $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction needs to be prepared for conversion to decimal. To do this, add $5-3=2$ zeros to the left in the numerator: $\frac(00351)(100000)$.

Now we can form the desired decimal fraction. To do this, write $0$, then put a comma and write the number from the numerator. We get the decimal fraction $0.00351$.

Answer: $0,00351$.

Let's formulate rule for converting improper common fractions with denominators $10$, $100$, $\dots$ to decimals:

    write a number from the numerator;

    separate with a decimal point as many digits on the right as there are zeros in the denominator of the original fraction.

Example 10

Convert improper common fraction $\frac(12756)(100)$ to decimal.

Solution.

Let's write the number from the numerator $12756$, then separate the digits on the right with a decimal point $2$, because the denominator of the original fraction $2$ is zero. We get the decimal fraction $127.56$.

A common fraction (or mixed number) whose denominator is one followed by one or more zeros (i.e. 10, 100, 1000, etc.):

can be written in a simpler form: without a denominator, separating the integer and fractional parts from each other with a comma (in this case, it is believed that the integer part of a proper fraction is 0). First, the integer part is written, then a comma is placed, and after it the fractional part is written.:

Ordinary fractions (or mixed numbers) written in this form are called decimals.

Reading and writing decimals

Decimal fractions are written according to the same rules by which natural numbers are written in the decimal number system. This means that in decimals, as in natural numbers, each digit expresses units that are ten times larger than the neighboring units on the right.

Consider the following entry:

The number 8 means simple units. The number 3 means units that are 10 times smaller than simple units, i.e. tenths. 4 means hundredths, 2 means thousandths, etc.

The numbers to the right after the decimal point are called decimal places.

Decimal fractions are read as follows: first the whole part is called, then the fractional part. When reading the integer part, it must always answer the question: how many integer units are there in the integer part? . The word whole (or whole) is added to the answer, depending on the number of whole units. For example, one integer, two integers, three integers, etc. When reading the fractional part, the number of shares is called and at the end they add the name of those shares with which the fractional part ends:

3:1 reads: three point one tenth.

2.017 reads like this: two point seventeen thousandths.

To better understand the rules for writing and reading decimal fractions, consider the table of digits and the examples of writing numbers given in it:

Please note that after a decimal point in a decimal fraction, there are as many digits as there are zeros in the denominator of the corresponding ordinary fraction:

A decimal fraction differs from an ordinary fraction in that its denominator is a bit unit.

For example:

Decimal fractions have been separated from ordinary fractions into a separate form, which has led to its own rules for comparing, adding, subtracting, multiplying and dividing these fractions. In principle, you can work with decimal fractions according to the rules of ordinary fractions. Own rules for converting decimal fractions simplify calculations, and rules for converting ordinary fractions to decimals, and vice versa, serve as a link between these types of fractions.

Writing and reading decimal fractions allows you to write, compare and operate on them according to rules very similar to the rules for operations with natural numbers.

For the first time, the system of decimal fractions and operations on them was described in the 15th century. Samarkand mathematician and astronomer Jamshid ibn-Masudal-Kashi in the book "The Key to the Art of Accounting".

The integer part of the decimal fraction is separated from the fractional part by a comma, in some countries (USA) they put a period. If there is no integer part in the decimal fraction, then put the number 0 before the decimal point.

Any number of zeros can be added to the fractional part of the decimal fraction on the right, this does not change the value of the fraction. The fractional part of the decimal fraction is read by the last significant digit.

For example:
0.3 - three tenths
0.75 - seventy-five hundredths
0.000005 - five millionths.

Reading the integer part of a decimal is the same as reading natural numbers.

For example:
27.5 - twenty-seven ...;
1.57 - one...

After the integer part of the decimal fraction, the word "whole" is pronounced.

For example:
10.7 - ten point seven

0.67 - zero point sixty-seven hundredths.

Decimals are fractional digits. The fractional part is read not by digits (unlike natural numbers), but as a whole, therefore the fractional part of a decimal fraction is determined by the last significant digit to the right. The bit system of the fractional part of a decimal fraction is somewhat different than that of natural numbers.

  • 1st digit after busy - tenths digit
  • 2nd place after the decimal point - hundredth place
  • 3rd place after the decimal point - thousandth place
  • 4th place after the decimal point - ten-thousandth place
  • 5th place after the decimal point - hundred-thousandth place
  • 6th place after the decimal point - millionth place
  • 7th place after the decimal point - ten-millionth place
  • The 8th place after the decimal point is the hundred-millionth place

In calculations, the first three digits are most often used. The large bit depth of the fractional part of decimal fractions is used only in specific branches of knowledge, where infinitesimal values ​​are calculated.

Decimal to mixed fraction conversion consists of the following: write the number before the decimal point as the integer part of the mixed fraction; the number after the decimal point is the numerator of its fractional part, and in the denominator of the fractional part, write one with as many zeros as there are digits after the decimal point.

Already in elementary school, students are faced with fractions. And then they appear in every topic. It is impossible to forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are simple, the main thing is to understand everything in order.

Why are fractions needed?

The world around us consists of whole 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 slices. Consider the situation where its tile is formed by twelve rectangles. If you divide it by two, you get 6 parts. It will be well divided into three. But the five will not be able to give a whole number of slices of chocolate.

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

In fact, the fractional bar turns out to be a division sign. That is, the numerator can be called a dividend, and the denominator can be called a divisor.

What are the fractions?

In mathematics, there are only two types of them: ordinary and decimal fractions. Schoolchildren get acquainted with the first ones in the elementary grades, calling them simply “fractions”. The second learn in the 5th grade. That's when these names appear.

Common fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the integer with a comma. For example, 4.7. Students need to be clear 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 as well. There are rules that allow you to write a decimal fraction as an ordinary fraction.

What subspecies do these types of fractions have?

It is 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 the denominator.

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

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

    Mixed. An integer is assigned to its usual correct (incorrect) fractional part. And it always stands on the left.

    Composite. It is formed from two fractions divided into each other. That is, it has three fractional features at once.

Decimals have only two subspecies:

    final, that is, one in which the 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 decimal to ordinary?

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

As a hint about the required denominator, remember that it is always a one and a few zeros. The latter need to be written as many as the digits in the fractional part of the number in question.

How to convert decimal fractions to ordinary ones if their whole 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. It remains to write down only the fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter turns out to be possible to reduce by 5. Therefore, the result for it must be written 1/20.

How to make an ordinary fraction from a decimal if its integer part is different from zero? For example, 5.23 or 13.00108. Both examples read the integer part and write its value. In the first case, this is 5, in the second, 13. Then you need to move on to the fractional part. With them it is necessary to carry out the same operation. The first number has 23/100, the second has 108/100000. The second value needs to be reduced again. The answer is mixed fractions: 5 23/100 and 13 27/25000.

How to convert an infinite decimal to a common fraction?

If it is non-periodic, then such an operation cannot be carried out. This fact is due to the fact that each decimal fraction is always converted to either final or periodic.

The only thing that is allowed to be done with such a fraction is to 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 translated into ordinary fractions. This must be remembered.

How to write an infinite periodic fraction in the form of an ordinary?

In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called periods. For example, 0.3(3). Here "3" in the period. They are classified as rational, as 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 any numbers, and then the repetition begins.

The rule by which you need to write an infinite decimal in the form of an ordinary fraction will be different for these two types of numbers. It is quite easy to write pure periodic fractions as ordinary fractions. As with the final ones, they need to be converted: write the period into the numerator, and the number 9 will be the denominator, repeating as many times as there are digits in the period.

For example, 0,(5). The number does not have an integer part, so you need to immediately proceed to the fractional part. Write 5 in the numerator, and write 9 in the denominator. That is, the answer will be the fraction 5/9.

A rule on how to write a common decimal fraction that is a mixed fraction.

    Look at the length of the period. So much 9 will have a denominator.

    Write down the denominator: first nines, then zeros.

    To determine the numerator, you need to write the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtractable - 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 is one digit. So zero will be one. There is also only one digit 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 from 58, you need to subtract 5. It turns out 53. For example, you will have to write 53/90 as an answer.

How are common fractions converted to decimals?

The simplest option is a number whose denominator is the number 10, 100, and so on. 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. Only it is necessary to multiply not only the denominator, but also the numerator by the same number.

For all other cases, a simple rule will come in handy: divide the numerator by the denominator. In this case, you may get two answers: a final or a periodic decimal fraction.

Operations with common fractions

Addition and subtraction

Students get to know them earlier than others. And at first the fractions have the same denominators, and then different. General rules can be reduced to such a plan.

    Find the least common multiple of the denominators.

    Write additional factors to all ordinary fractions.

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

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

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

    In the first case, the integer part needs to take one. Add a denominator to the numerator of a fraction. And then do the subtraction.

    In the second - it is necessary to apply the rule of subtraction from a smaller number to a larger one. That is, subtract the modulus of the minuend from the modulus of the subtrahend, and put the “-” sign in response.

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

    Multiplication and division

    For their implementation, fractions do not need to be reduced to a common denominator. This makes it easier to take action. But they still have to follow the rules.

      When multiplying ordinary fractions, it is necessary to consider the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.

      Multiply numerators.

      Multiply the denominators.

      If you get a reducible fraction, then it is supposed to be simplified again.

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

      Then proceed as in multiplication (starting from step 1).

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

    Operations with decimals

    Addition and subtraction

    Of course, you can always turn a decimal into a common fraction. And act according to the already described plan. 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. Assign the missing number of zeros in it.

      Write fractions so that the comma is under the comma.

      Add (subtract) like natural numbers.

      Remove the comma.

    Multiplication and division

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

      For multiplication, you need to write fractions one under the other, not paying attention to commas.

      Multiply like natural numbers.

      Put 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 convert 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 a natural number.

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

    What if there are both types of fractions in one example?

    Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. There are two possible solutions to these problems. You need to objectively weigh the numbers and choose the best one.

    First way: represent ordinary decimals

    It is suitable if, when dividing or converting, final fractions are obtained. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to count them.

    The second way: write decimal fractions as ordinary

    This technique is convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, a very large ordinary fraction can turn out and decimal entries will allow you to calculate the task faster and easier. Therefore, it is always necessary to soberly evaluate the task and choose the simplest solution method.