How to understand decimals. How to solve decimals

The decimal fraction is used when you need to perform operations on non-integer numbers. This may seem irrational. But this type of numbers greatly facilitates the mathematical operations that must be performed with them. This understanding comes with time, when their writing becomes familiar, and reading does not cause difficulties, and the rules of decimal fractions are mastered. Moreover, all actions are repeated already known, which are learned with natural numbers. You just need to remember some features.

Decimal definition

A decimal is a special representation of a non-integer number with a denominator that is divisible by 10 and the answer is one and possibly zeros. In other words, if the denominator is 10, 100, 1000, and so on, it is more convenient to rewrite the number using a comma. Then the integer part will be located before it, and then the fractional part. Moreover, the record of the second half of the number will depend on the denominator. The number of digits that are in the fractional part must be equal to the denominator.

The above can be illustrated with these numbers:

9/10=0,9; 178/10000=0,0178; 3,05; 56 003,7006.

Reasons for using decimals

Mathematicians needed decimals for several reasons:

Simplify recording. Such a fraction is located along one line without a dash between the denominator and numerator, while the visibility does not suffer.

Simplicity in comparison. It is enough just to correlate the numbers that are in the same positions, while with ordinary fractions one would have to bring them to a common denominator.

Simplification of calculations.

Calculators are not designed for the introduction of ordinary fractions, they use decimal notation for all operations.

How to read such numbers correctly?

The answer is simple: just like an ordinary mixed number with a denominator that is a multiple of 10. The only exception is fractions without an integer value, then when reading, you need to say “zero integers”.

For example, 45/1000 should be pronounced as forty five thousandths, while 0.045 will sound like zero point forty-five thousandths.

A mixed number with an integer part equal to 7 and a fraction of 17/100, which will be written as 7.17, in both cases will be read as seven point seventeen hundredths.

The role of digits in the notation of fractions

It is true to note the discharge - this is what mathematics requires. Decimals and their meaning can change significantly if you write the number in the wrong place. However, this has been true before.

To read the digits of the integer part of a decimal fraction, you just need to use the rules known for natural numbers. And on the right side they are mirrored and read differently. If "tens" sounded in the whole part, then after the decimal point it will be already "tenths".

This can be clearly seen in this table.

Decimal Places Table

Class	thousands			units			,	fractional part
discharge	hundred	dec.	units	hundred	dec.	units	,	tenth	hundredth	thousandth	ten thousandth

How to write a mixed number as a decimal?

If the denominator contains a number equal to 10 or 100, and others, then the question of how to convert a fraction to a decimal is simple. To do this, it is enough to rewrite all its constituent parts in a different way. The following points will help with this:

write the numerator of the fraction a little aside, at this moment the decimal point is located on the right, after the last digit;

move the comma to the left, the most important thing here is to correctly count the numbers - you need to move it as many positions as there are zeros in the denominator;

if there are not enough of them, then zeros should appear in empty positions;

zeros that were at the end of the numerator are no longer needed, and they can be crossed out;

add an integer part before the comma, if it was not there, then zero will also appear here.

Attention. You can not cross out zeros that are surrounded by other numbers.

About how to be in a situation where the denominator contains a number not only from one and zeros, how to convert a fraction to a decimal, you can read a little lower. This is important information that you should definitely read.

How to convert a fraction to a decimal if the denominator is an arbitrary number?

There are two options here:

When the denominator can be represented as a number that is ten to any power.

If such an operation cannot be done.

How to check it? You need to factorize the denominator. If only 2 and 5 are present in the product, then everything is fine, and the fraction is easily converted to a final decimal. Otherwise, if 3, 7 and other prime numbers appear, then the result will be infinite. It is customary to round such a decimal fraction for ease of use in mathematical operations. This will be discussed a little lower.

Studying how such decimal fractions are obtained, Grade 5. Examples will be very helpful here.

Let the denominators contain numbers: 40, 24 and 75. Decomposition into prime factors for them it will be:

40=2 2 2 5;
24=2 2 2 3;
75=5 5 3.

In these examples, only the first fraction can be represented as a final fraction.

Algorithm for converting an ordinary fraction to a final decimal

Check the factorization of the denominator into prime factors and make sure that it will consist of 2 and 5.

Add to these numbers so many 2 and 5 that they become an equal number. They will give the value of the additional multiplier.

Multiply the denominator and numerator by this number. The result will be common fraction, below the line which is 10 to some extent.

If in the task these actions are performed with a mixed number, then it must first be represented as an improper fraction. And only then act according to the described scenario.

Representation of a common fraction as a rounded decimal

This way of how to convert a fraction to a decimal will seem even easier to someone. Because it doesn't have a lot of action. You just need to divide the numerator by the denominator.

Any number with a decimal part to the right of the decimal point can be assigned an infinite number of zeros. This property should be used.

First, write down the whole part and put a comma after it. If the fraction is correct, write zero.

Then it is necessary to perform the division of the numerator by the denominator. So that they have the same number of digits. That is, assign the required number of zeros to the right of the numerator.

Perform division in a column until the required number of digits is dialed. For example, if you need to round up to hundredths, then there should be 3 of them in the answer. In general, there should be one more digits than you need to get in the end.

Record the intermediate answer after the decimal point and round according to the rules. If the last digit is from 0 to 4, then you just need to discard it. And when it is equal to 5-9, then the one in front of it must be increased by one, discarding the last one.

Return from decimal to ordinary

In mathematics, there are problems when it is more convenient to represent decimal fractions in the form of ordinary ones, in which there is a numerator with a denominator. You can breathe a sigh of relief: this operation is always possible.

For this procedure, you need to do the following:

write down the integer part, if it is equal to zero, then nothing needs to be written;

draw a fractional line;

above it, write the numbers from the right side, if the first are zeros, then they must be crossed out;

under the line, write a unit with as many zeros as there are digits after the decimal point in the original fraction.

That's all you need to do to convert a decimal to a common fraction.

What can you do with decimals?

In mathematics, this will be certain actions with decimal fractions that were previously performed for other numbers.

They are:

comparison;

addition and subtraction;

multiplication and division.

The first action, comparison, is similar to how it was done for natural numbers. To determine which is greater, you need to compare the digits of the integer part. If they turn out to be equal, then they switch to the fractional one and compare them in the same way by digits. The number with the largest digit in the highest order will be the answer.

Adding and subtracting decimals

These are perhaps the simplest steps. Because they are performed according to the rules for natural numbers.

So, in order to add decimal fractions, they need to be written one under the other, placing commas in a column. With such a record, integer parts appear to the left of the commas, and fractional parts to the right. And now you need to add the numbers bit by bit, as is done with natural numbers, moving the comma down. You need to start adding from the smallest digit of the fractional part of the number. If there are not enough numbers in the right half, then add zeros.

Subtraction works in the same way. And here the rule applies, which describes the possibility of taking a unit from the highest digit. If the reduced fraction has fewer digits after the decimal point than the subtrahend, then zeros are simply assigned to it.

The situation is a little more complicated with tasks where you need to perform multiplication and division of decimal fractions.

How to multiply decimal in different examples?

The rule for multiplying decimal fractions by a natural number is as follows:

write them down in a column, ignoring the comma;

multiply as if they were natural;

separate with a comma as many digits as there were in the fractional part of the original number.

A special case is an example in which a natural number is equal to 10 to any power. Then, to get an answer, you just need to move the comma to the right by as many positions as there are zeros in another factor. In other words, when multiplying by 10, the comma shifts by one digit, by 100 - there will be two of them, and so on. If there are not enough digits in the fractional part, then you need to write zeros in empty positions.

The rule that is used when in the task you need to multiply decimal fractions by another of the same number:

write them down one under the other, ignoring the commas;

multiply as if they were natural numbers;

separate with a comma as many digits as there were in the fractional parts of both original fractions together.

As a special case, examples are distinguished in which one of the factors is equal to 0.1 or 0.01 and so on. In them, you need to move the comma to the left by the number of digits in the presented factors. That is, if multiplied by 0.1, then the comma is shifted by one position.

How to divide a decimal fraction in different tasks?

The division of decimal fractions by a natural number is performed according to the following rule:

write them down for division in a column, as if they were natural;

divide according to the usual rule until the whole part ends;

put a comma in the answer;

continue dividing the fractional component until the remainder is zero;

if necessary, you can assign the desired number of zeros.

If the integer part is equal to zero, then it will not be in the answer either.

Separately, there is a division into numbers equal to ten, one hundred, and so on. In such problems, you need to move the comma to the left by the number of zeros in the divisor. It happens that there are not enough digits in the integer part, then zeros are used instead. It can be seen that this operation is similar to multiplying by 0.1 and similar numbers.

To perform division of decimals, you need to use this rule:

turn the divisor into a natural number, and to do this, move the comma in it to the right to the end;

move the comma and in the divisible by the same number of digits;

follow the previous scenario.

The division by 0.1 is highlighted; 0.01 and other similar numbers. In such examples, the comma is shifted to the right by the number of digits in the fractional part. If they are over, then you need to assign the missing number of zeros. It is worth noting that this action repeats the division by 10 and similar numbers.

Conclusion: it's all about practice

Nothing in learning is easy or effortless. It takes time and practice to master new material reliably. Mathematics is no exception.

So that the topic of decimal fractions does not cause difficulties, you need to solve as many examples as possible with them. After all, there was a time when the addition of natural numbers was confusing. And now everything is fine.

Therefore, to paraphrase a well-known phrase: decide, decide and decide again. Then tasks with such numbers will be performed easily and naturally, like another puzzle.

By the way, puzzles are difficult to solve at first, and then you need to do the usual movements. The same is true in mathematical examples: after going along the same path several times, then you will no longer think about where to turn.

Already in primary school students are dealing 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 into 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 are first introduced to primary school, 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?

Better start at 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 between the fractional and whole parts a comma is placed.

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 point 1).

In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written in the form 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.

Instruction

If in form fractions must represent the whole number, then use one as the denominator, and put the original value in the numerator. This form of writing is called an improper ordinary fraction, since the modulus of its numerator is greater than the modulus of the denominator. For example, number 74 can be written as 74/1, and number-12 is like -12/1. Optionally you can numerator and denominator the same number of times - value fractions in this case will still match the original number. For example, 74=74/1=222/3 or -12=-12/1=-84/7.

If the original number presented in decimal format fractions, then leave its integer part unchanged, and replace the separating comma with a space. Put the fractional part in the numerator, and use the ten raised to a power with an indicator equal to the number of digits in the fractional of the original number as the denominator. The resulting fractional part can be reduced by dividing the numerator and denominator by the same number. For example, decimal fractions 7.625 will correspond to an ordinary fraction 7 625/1000, which, after reduction, will take on the value 7 5/8. This form of notation is ordinary fractions mixed. If necessary, it can be reduced to an incorrect ordinary form by multiplying the integer part by the denominator and adding the result to the numerator: 7.625 \u003d 7 625/1000 \u003d 7 5/8 \u003d 61/8.

If the original decimal fraction is also periodic, then use, for example, a system of equations to calculate its equivalent in the format fractions ordinary. Say, if the original fraction is 3.5(3), then the identity is possible: 100*x-10*x=100*3.5(3)-10*3.5(3). From it, you can derive the equality 90 * x \u003d 318, and that the desired fraction will be equal to 318/90, which, after reduction, will give an ordinary fraction 3 24/45.

Sources:

Can the number 450,000 be represented as a product of 2 numbers?

In everyday life, non-natural numbers are most often found: 1, 2, 3, 4, etc. (5 kg. potatoes), and fractional, non-integer numbers (5.4 kg of onions). Most of them are presented in form decimal fractions. But represent the decimal in form fractions simple enough.

Instruction

For example, given the number "0.12". If not this fraction and present it as it is, then it will look like this: 12/100 ("twelve"). To get rid of hundreds in , you need to divide both the numerator and the denominator by the number that divides their numbers. This number is 4. Then, dividing the numerator and denominator, the number is obtained: 3/25.

If we consider a more household one, then often on the price tag you can see that its weight is, for example, 0.478 kg or so. Such a number is also easy to imagine in form fractions:
478/1000 = 239/500. This fraction is rather ugly, and if there was an opportunity, then this decimal fraction could be reduced further. And all by the same method: selecting a number that divides both the numerator and the denominator. This number is the greatest common factor. The "largest" multiplier is because it is much more convenient to divide both the numerator and the denominator by 4 at once (as in the first example) than to divide twice by 2.

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.

This article is about decimals. Here we will deal with decimal notation fractional numbers, we introduce the concept of a decimal fraction and give examples of decimal fractions. Next, let's talk about the digits of decimal fractions, give the names of the digits. After that, we will focus on infinite decimal fractions, say about periodic and non-periodic fractions. Next, we list the main actions with decimal fractions. In conclusion, we establish the position of decimal fractions on the coordinate ray.

Page navigation.

Decimal notation of a fractional number

Reading decimals

Let's say a few words about the rules for reading decimal fractions.

Decimal fractions, which correspond to the correct ordinary fractions, are read in the same way as these ordinary fractions, only “zero whole” is added beforehand. For example, the decimal fraction 0.12 corresponds to the ordinary fraction 12/100 (it reads “twelve hundredths”), therefore, 0.12 is read as “zero point twelve hundredths”.

Decimal fractions, which correspond to mixed numbers, are read in exactly the same way as these mixed numbers. For example, the decimal fraction 56.002 corresponds to a mixed number, therefore, the decimal fraction 56.002 is read as "fifty-six point two thousandths."

Places in decimals

In the notation of decimal fractions, as well as in the notation of natural numbers, the value of each digit depends on its position. Indeed, the number 3 in decimal 0.3 means three tenths, in decimal 0.0003 - three ten thousandths, and in decimal 30,000.152 - three tens of thousands. Thus, we can talk about digits in decimals, as well as about digits in natural numbers.

Names of digits in decimal fraction up to decimal point completely coincide with the names of digits in natural numbers. And the names of the digits in the decimal fraction after the decimal point are visible from the following table.

For example, in the decimal fraction 37.051, the number 3 is in the tens place, 7 is in the units place, 0 is in the tenth place, 5 is in the hundredth place, 1 is in the thousandth place.

The digits in the decimal fraction also differ in seniority. If we move from digit to digit from left to right in the decimal notation, then we will move from senior to junior ranks. For example, the hundreds digit is older than the tenths digit, and the millionths digit is younger than the hundredths digit. In this final decimal fraction, we can talk about the most significant and least significant digits. For example, in decimal 604.9387 senior (highest) the digit is the hundreds digit, and junior (lowest)- ten-thousandth place.

For decimal fractions, expansion into digits takes place. It is analogous to the expansion in digits of natural numbers. For example, the decimal expansion of 45.6072 is: 45.6072=40+5+0.6+0.007+0.0002 . And the properties of addition from the expansion of a decimal fraction into digits allow you to go to other representations of this decimal fraction, for example, 45.6072=45+0.6072 , or 45.6072=40.6+5.007+0.0002 , or 45.6072= 45.0072+0.6 .

End decimals

Up to this point, we have only talked about decimal fractions, in the record of which there is a finite number of digits after the decimal point. Such fractions are called final decimal fractions.

Definition.

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

Here are some examples of final decimals: 0.317 , 3.5 , 51.1020304958 , 230 032.45 .

However, not every common fraction can be represented as a finite decimal fraction. For example, the fraction 5/13 cannot be replaced by an equal fraction with one of the denominators 10, 100, ..., therefore, it cannot be converted to a final decimal fraction. We'll talk more about this in the theory section of converting ordinary fractions to decimal fractions.

Infinite decimals: periodic fractions and non-periodic fractions

In writing a decimal fraction after a decimal point, you can allow the possibility of an infinite number of digits. In this case, we will come to the consideration of the so-called infinite decimal fractions.

Definition.

Endless decimals- These are decimal fractions, in the record of which there is an infinite number of digits.

It is clear that we cannot write the infinite decimal fractions in full, therefore, in their recording they are limited to only a certain finite number of digits after the decimal point and put an ellipsis indicating an infinitely continuing sequence of digits. Here are some examples of infinite decimal fractions: 0.143940932…, 3.1415935432…, 153.02003004005…, 2.111111111…, 69.74152152152….

If you look closely at the last two endless decimal fractions, then in the fraction 2.111111111 ... the infinitely repeating number 1 is clearly visible, and in the fraction 69.74152152152 ..., starting from the third decimal place, the repeating group of numbers 1, 5 and 2 is clearly visible. Such infinite decimal fractions are called periodic.

Definition.

Periodic decimals(or simply periodic fractions) are infinite decimal fractions, in the record of which, starting from a certain decimal place, some digit or group of digits, which is called fraction period.

For example, the period of the periodic fraction 2.111111111… is the number 1, and the period of the fraction 69.74152152152… is a group of numbers like 152.

For infinite periodic decimal fractions, it is accepted special form records. For brevity, we agreed to write the period once, enclosing it in parentheses. For example, the periodic fraction 2.111111111… is written as 2,(1) , and the periodic fraction 69.74152152152… is written as 69.74(152) .

It is worth noting that for the same periodic decimal fraction, you can specify different periods. For example, the periodic decimal 0.73333… can be considered as a fraction 0.7(3) with a period of 3, as well as a fraction 0.7(33) with a period of 33, and so on 0.7(333), 0.7 (3333), ... You can also look at the periodic fraction 0.73333 ... like this: 0.733(3), or like this 0.73(333), etc. Here, in order to avoid ambiguity and inconsistency, we agree to consider as the period of a decimal fraction the shortest of all possible sequences of repeating digits, and starting from the closest position to the decimal point. That is, the period of the decimal fraction 0.73333… will be considered a sequence of one digit 3, and the frequency starts from the second position after the decimal point, that is, 0.73333…=0.7(3) . Another example: the periodic fraction 4.7412121212… has a period of 12, the periodicity starts from the third digit after the decimal point, that is, 4.7412121212…=4.74(12) .

Infinite decimal periodic fractions are obtained by converting to decimal fractions of ordinary fractions whose denominators contain prime factors other than 2 and 5.

Here it is worth mentioning periodic fractions with a period of 9. Here are examples of such fractions: 6.43(9) , 27,(9) . These fractions are another notation for periodic fractions with period 0, and it is customary to replace them with periodic fractions with period 0. To do this, period 9 is replaced by period 0, and the value of the next highest digit is increased by one. For example, a fraction with period 9 of the form 7.24(9) is replaced by a periodic fraction with period 0 of the form 7.25(0) or an equal final decimal fraction of 7.25. Another example: 4,(9)=5,(0)=5 . The equality of a fraction with a period of 9 and its corresponding fraction with a period of 0 is easily established after replacing these decimal fractions with their equal ordinary fractions.

Finally, let's take a closer look at infinite decimals, which do not have an infinitely repeating sequence of digits. They are called non-periodic.

Definition.

Non-recurring decimals(or simply non-periodic fractions) are infinite decimals with no period.

Sometimes non-periodic fractions have a form similar to that of periodic fractions, for example, 8.02002000200002 ... is a non-periodic fraction. In these cases, you should be especially careful to notice the difference.

Note that non-periodic fractions are not converted to ordinary fractions, infinite non-periodic decimal fractions represent irrational numbers.

Operations with decimals

One of the actions with decimals is comparison, and four basic arithmetic are also defined operations with decimals: addition, subtraction, multiplication and division. Consider separately each of the actions with decimal fractions.

Decimal Comparison essentially based on a comparison of ordinary fractions corresponding to the compared decimal fractions. However, converting decimal fractions to ordinary ones is a rather laborious operation, and infinite non-repeating fractions cannot be represented as an ordinary fraction, so it is convenient to use a bitwise comparison of decimal fractions. Bitwise comparison of decimals is similar to comparison of natural numbers. For more detailed information, we recommend that you study the article material comparison of decimal fractions, rules, examples, solutions.

Let's move on to the next step - multiplying decimals. Multiplication of final decimal fractions is carried out similarly to the subtraction of decimal fractions, rules, examples, solutions to multiplication by a column of natural numbers. In the case of periodic fractions, multiplication can be reduced to the multiplication of ordinary fractions. In turn, the multiplication of infinite non-periodic decimal fractions after their rounding is reduced to the multiplication of finite decimal fractions. We recommend further study of the material of the article multiplication of decimal fractions, rules, examples, solutions.

Decimals on the coordinate beam

There is a one-to-one correspondence between dots and decimals.

Let's figure out how points are constructed on the coordinate ray corresponding to a given decimal fraction.

We can replace finite decimal fractions and infinite periodic decimal fractions with ordinary fractions equal to them, and then construct the corresponding ordinary fractions on the coordinate ray. For example, a decimal fraction 1.4 corresponds to an ordinary fraction 14/10, therefore, the point with coordinate 1.4 is removed from the origin in the positive direction by 14 segments equal to a tenth of a single segment.

Decimal fractions can be marked on the coordinate beam, starting from the expansion of this decimal fraction into digits. For example, let's say we need to build a point with coordinate 16.3007 , since 16.3007=16+0.3+0.0007 , then in given point can be reached by sequentially laying 16 unit segments from the origin, 3 segments, the length of which is equal to a tenth of a unit segment, and 7 segments, the length of which is equal to a ten-thousandth fraction of a unit segment.

This way of building decimal numbers on the coordinate ray allows you to get as close as you like to the point corresponding to an infinite decimal fraction.

It is sometimes possible to accurately plot a point corresponding to an infinite decimal. For example, , then this infinite decimal fraction 1.41421... corresponds to the point of the coordinate ray, remote from the origin by the length of the diagonal of a square with a side of 1 unit segment.

The reverse process of obtaining a decimal fraction corresponding to a given point on the coordinate beam is the so-called decimal measurement of a segment. Let's see how it is done.

Let our task be to get from the origin to a given point on the coordinate line (or infinitely approach it if it is impossible to get to it). With a decimal measurement of a segment, we can sequentially postpone any number of unit segments from the origin, then segments whose length is equal to a tenth of a single segment, then segments whose length is equal to a hundredth of a single segment, etc. By writing down the number of plotted segments of each length, we get the decimal fraction corresponding to a given point on the coordinate ray.

For example, to get to point M in the above figure, you need to set aside 1 unit segment and 4 segments, the length of which is equal to the tenth of the unit. Thus, the point M corresponds to the decimal fraction 1.4.

It is clear that the points of the coordinate beam, which cannot be reached during the decimal measurement, correspond to infinite decimal fractions.

Bibliography.

Maths: studies. for 5 cells. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 p.: ill. ISBN 5-346-00699-0.
Maths. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Decimal definition

Reasons for using decimals

How to read such numbers correctly?

The role of digits in the notation of fractions

How to write a mixed number as a decimal?

How to convert a fraction to a decimal if the denominator is an arbitrary number?

Algorithm for converting an ordinary fraction to a final decimal

Representation of a common fraction as a rounded decimal

Return from decimal to ordinary

What can you do with decimals?

Adding and subtracting decimals

How to multiply decimal in different examples?

How to divide a decimal fraction in different tasks?

Conclusion: it's all about practice

Why are fractions needed?

What is a "fraction"?

What are the fractions?

What subspecies do these types of fractions have?

How to convert decimal to ordinary?

How to convert an infinite decimal to a common fraction?

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

How are common fractions converted to decimals?

Operations with common fractions

Operations with decimals

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

Properties of decimal fractions.

Types of decimals.

Decimal notation of a fractional number

Reading decimals

Places in decimals

End decimals

Infinite decimals: periodic fractions and non-periodic fractions

Operations with decimals

Decimals on the coordinate beam

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