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 direction. 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. First go common fractions. 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 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 fractions if they whole part absent, i.e. 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.

Like from decimal fraction to make an ordinary one 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 initial value. That is, endless non-periodic fractions are not converted to ordinary. 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?

by the most simple option it turns out the number in the denominator of which 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 first with fractions 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 fewer more. 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 it turned out improper fraction, then it is necessary 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 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 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 translating, you get finite fractions. 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.

§ 102. Preliminary clarifications.

In the previous part, we considered fractions with all possible denominators and called them ordinary fractions. We were interested in any fraction that arose in the process of measurement or division, regardless of what denominator we got.

Now, from the whole set of fractions, we will select fractions with denominators: 10, 100, 1,000, 10,000, etc., i.e. such fractions, the denominators of which are only numbers represented by unity (1) followed by zeros (one or several). Such fractions are called decimal.

Here are examples of decimals:

We have met with decimal fractions before, but did not indicate any special properties inherent in them. Now we will show that they have some remarkable properties, which simplifies all calculations with fractions.

§ 103. Image of a decimal fraction without a denominator.

Decimal fractions are usually written not in the same way as ordinary fractions, but according to the rules by which whole numbers are written.

To understand how to write a decimal without a denominator, you need to remember how to write in decimal system any integer. If, for example, we write three-digit number using only the number 2, i.e. the number 222, then each of these twos will have special meaning depending on the place it occupies in the number. The first two from the right stands for units, the second for tens, and the third for hundreds. Thus, any digit to the left of any other digit denotes units ten times larger than those indicated by the previous digit. If any digit is missing, then zero is written in its place.

So, in a whole number, units are in the first place on the right, tens are in the second place, etc.

Now let's raise the question of what category of units will be obtained if, for example, we are in the number 222 with right side we will add one more number. To answer this question, you need to take into account that the last two (the first from the right) denotes units.

Therefore, if after the deuce, denoting units, we, a little retreat, write some other number, for example 3, then it will denote units, ten times smaller than the previous ones, in other words, it will denote tenths units; the result is a number containing 222 whole units and 3 tenths of a unit.

It is customary to put a comma between the integer and fractional parts of the number, i.e., write like this:

If after the triple in this number we add another number, for example 4, then it will mean 4 hundredths fractions of a unit; number will look like:

and is pronounced: two hundred and twenty-two point, thirty-four hundredths.

A new digit, for example 5, being assigned to this number, gives us thousandths: 222.345 (two hundred and twenty-two point, three hundred and forty-five thousandths).

For greater clarity, the arrangement in the number of integer and fractional digits can be presented in the form of a table:

Thus, we have explained how decimal fractions are written without a denominator. Let's write some of these fractions.

To write a fraction without a denominator 5/10, you need to take into account that it does not have integers and, therefore, the place of integers must be occupied by zero, i.e. 5/10 = 0.5.

The fraction 2 9/100 without a denominator will be written like this: 2.09, that is, zero must be put in place of the tenths. If we skipped this 0, we would get a completely different fraction, namely 2.9, that is, two whole points and nine tenths.

This means that when writing decimal fractions, it is necessary to denote the missing integer and fractional digits by zero:

0.325 - no integers,
0.012 - no integers and no tenths,
1.208 - no hundredths,
0.20406 - no integers, no hundredths, and no ten-thousandths.

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

In order to avoid mistakes when writing decimal fractions, you need to remember that after the decimal point in the image of a decimal fraction there should be as many digits as there will be zeros in the denominator if we wrote this fraction with a denominator, i.e.

0.1 \u003d 1 / 10 (the denominator has one zero and one digit after the decimal point);

§ 104. Assigning zeros to a decimal fraction.

In the previous paragraph, it was described how decimal fractions without denominators are displayed. Great importance when writing decimal fractions, it has a zero. Every regular decimal fraction has a zero in place of integers to indicate that such a fraction does not have integers. We will now write several different decimals using the numbers: 0, 3 and 5.

0.35 - 0 integers, 35 hundredths,
0.035 - 0 integers, 35 thousandths,
0.305 - 0 integers, 305 thousandths,
0.0035 - 0 integers, 35 ten-thousandths.

Let us now find out what is the meaning of the nulls placed at the end of the decimal fraction, i.e. on the right.

If we take an integer, for example 5, put a comma after it, and then write zero after the comma, then this zero will mean zero tenths. Therefore, this zero assigned to the right will not affect the value of the number, i.e.

Now let's take the number 6.1 and add zero to it on the right, we get 6.10, i.e. we had 1/10 after the decimal point, and it became 10/100, but 10/100 are equal to 1/10. This means that the value of the number has not changed, and from the assignment to the right of zero, only the type of number and pronunciation have changed (6.1 - six point one tenth; 6.10 - six point ten hundredths).

By similar reasoning, we can make sure that assigning zeros to the right to a decimal fraction does not change its value. Therefore, we can write the following equalities:

1 = 1,0,
2,3 = 2,300,
6.7 = 6.70000 etc.

If we assign zeros to the left of the decimal fraction, then they will not have any meaning. Indeed, if we write zero to the left of the number 4.6, then the number will take the form 04.6. Where is zero? It stands in the place of tens, that is, it shows that there are no tens in this number, but this is clear even without zero.

However, it should be remembered that sometimes zeros are assigned to decimal fractions on the right. For example, there are four fractions: 0.32; 2.5; 13.1023; 5,238. We assign zeros on the right to those fractions that have fewer decimal places after the decimal point: 0.3200; 2.5000; 13.1023; 5.2380.

What is it for? Assigning zeros to the right, we got four digits after the decimal point for each number, which means that each fraction will have a denominator of 10,000, and before assigning zeros, the denominator of the first fraction was 100, the second 10, the third 10,000 and the fourth 1,000. So Thus, by assigning zeros, we equalized the number of decimal places of our fractions, i.e., brought them to a common denominator. Therefore, reduction of decimal fractions to a common denominator is carried out by assigning zeros to these fractions.

On the other hand, if some decimal fraction has zeros on the right, then we can discard them without changing its value, for example: 2.60 = 2.6; 3.150 = 3.15; 4.200 = 4.2.

How should one understand such a discarding of zeros to the right of the decimal fraction? It is equivalent to its reduction, and this can be seen if we write these decimal fractions with a denominator:

§ 105. Comparison of decimal fractions in magnitude.

When using decimal fractions, it is very important to be able to compare fractions with each other and answer the question of which of them are equal, which are greater and which are less. Comparing decimals is done differently than comparing integers. For example, an integer two-digit number always greater than a single digit, no matter how many units there are in a single digit; a three-digit number is more than a two-digit number, and even more so a one-digit number. But when comparing decimal fractions, it would be a mistake to count all the signs with which fractions are written.

Let's take two fractions: 3.5 and 2.5, and compare them in size. They have the same decimal places, but the first fraction has 3 integers, and the second has 2. The first fraction is greater than the second, i.e.

Let's take other fractions: 0.4 and 0.38. To compare these fractions, it is useful to assign zero to the right of the first fraction. Then we will compare the fractions 0.40 and 0.38. Each of them has two digits after the decimal point, which means that these fractions have the same denominator 100.

We only need to compare their numerators, but the numerator 40 is greater than 38. So the first fraction is greater than the second, i.e.

The first fraction has more tenths than the second, however, the second fraction has 8 more hundredths, but they are less than one tenth, because 1/10 \u003d 10/100.

Now let's compare such fractions: 1.347 and 1.35. We assign zero to the right of the second fraction and compare the decimal fractions: 1.347 and 1.350. The integer parts are the same, so you only need to compare the fractional parts: 0.347 and 0.350. The denominator of these fractions is common, but the numerator of the second fraction is greater than the numerator of the first, which means that the second fraction is greater than the first, i.e. 1.35\u003e 1.347.

Finally, let's compare two more fractions: 0.625 and 0.62473. We add two zeros to the first fraction so that the digits are equal, and compare the resulting fractions: 0.62500 and 0.62473. Their denominators are the same, but the numerator of the first fraction 62 500 is greater than the numerator of the second fraction 62 473. Therefore, the first fraction is greater than the second, i.e. 0.625\u003e 0.62473.

On the basis of the foregoing, we can draw the following conclusion: of two decimal fractions, the one with the greater number of integers is greater; when the integers are equal, that fraction is greater, in which the number of tenths is greater; when integers and tenths are equal, that fraction is greater, in which the number of hundredths is greater, etc.

§ 106. Increase and decrease of a decimal fraction by 10, 100, 1,000, etc. times.

We already know that adding zeros to a decimal does not affect its value. When we studied integers, we saw that every zero assigned to the right increased the number by 10 times. It is not difficult to understand why this happened. If we take an integer, for example 25, and add zero to the right of it, then the number will increase by 10 times, the number 250 is 10 times greater than 25. When zero appeared on the right, the number 5, which used to denote units, now began to denote tens, and the number 2, which used to stand for tens, now stands for hundreds. So, thanks to the appearance of zero, the old digits were replaced by new ones, they became larger, they moved one place to the left. When it is necessary to increase a decimal fraction, for example, by 10 times, then we also have to move the digits one place to the left, but such a movement cannot be achieved with zero. A decimal fraction consists of an integer part and a fractional part, separated by a comma. To the left of the decimal point is the lowest integer digit, to the right is the highest fractional digit. Consider a fraction:

How can we move the digits in it, at least by one place, i.e., in other words, how can we increase it 10 times? If we move the comma one place to the right, then first of all this will affect the fate of the five: it is from the region fractional numbers falls into the realm of integers. The number will then take the form: 12345.678. The change happened with all other numbers, and not just with the five. All numbers included in the number began to play new role, the following happened (see table):

All ranks changed their name, and all rank units, so to speak, rose one place. From this, the whole number increased by 10 times. Thus, moving the comma one character to the right increases the number by 10 times.

Let's look at some more examples:

1) Take the fraction 0.5 and move the comma one place to the right; we get the number 5, which is 10 times more than 0.5, because before the five meant tenths of a unit, and now it means whole units.

2) Move the comma in the number 1.234 two digits to the right; the number becomes 123.4. This number is 100 times larger than the previous one, because in it the number 3 began to denote units, the number 2 - tens, and the number 1 - hundreds.

Thus, to increase the decimal fraction by 10, you need to move the comma in it one place to the right; to increase it by 100 times, you need to move the comma two places to the right; to increase 1,000 times - three digits to the right, etc.

If at the same time there are not enough signs for the number, then zeros are assigned to it on the right. For example, let's increase the fraction 1.5 by 100 times by moving the comma by two digits; we get 150. Let's increase the fraction 0.6 by 1,000 times; we get 600.

back if required decrease decimal fraction by 10, 100, 1,000, etc. times, then you need to move the comma to the left in it by one, two, three, etc. characters. Let the fraction 20.5 be given; let's reduce it by 10 times; to do this, we move the comma one sign to the left, the fraction will take the form 2.05. Let's reduce the fraction 0.015 by 100 times; we get 0.00015. Let's reduce the number 334 by 10 times; we get 33.4.

In this article, we will analyze how converting common fractions to decimals, and also consider the reverse process - the conversion of decimal fractions to ordinary fractions. Here we will voice the rules for inverting fractions and give detailed solutions typical examples.

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Converting common fractions to decimals

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

First, we will look at how to represent ordinary fractions with denominators 10, 100, 1000, ... as decimal fractions. This is because decimal fractions are essentially a compact form of ordinary fractions with denominators 10, 100, ....

After that, we will go further and show how any ordinary fraction (not only with denominators 10, 100, ...) can be written as a decimal fraction. With this conversion of ordinary fractions, both finite decimal fractions and infinite periodic decimal fractions are obtained.

Now about everything in order.

Converting ordinary fractions with denominators 10, 100, ... to decimal fractions

Some regular fractions need "preliminary preparation" before converting 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 to be prepared.

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

After preparing the correct ordinary fraction, you can begin to convert 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 down 0 ;
• put a decimal point after it;
• write down the number from the numerator (together with added zeros, if we added them).

Consider the application of this rule in solving examples.

Example.

Convert the proper fraction 37/100 to decimal.

Decision.

The denominator contains the number 100, which has two zeros in its entry. The numerator contains the number 37, there are two digits in its record, 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, while we get the decimal fraction 0.37.

Answer:

0,37 .

To consolidate the skills of translating regular ordinary fractions with numerators 10, 100, ... into decimal fractions, we will analyze the solution of another example.

Example.

write down proper fraction 107/10,000,000 as a decimal.

Decision.

The number of digits in the numerator is 3, and the number of zeros in the denominator is 7, so this ordinary fraction needs to be prepared for conversion to 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 .

It remains to form the desired decimal fraction. To do this, firstly, we write down 0, secondly, we put a comma, thirdly, we write down the number from the numerator together with zeros 0000107 , as a result we have a decimal fraction 0.0000107 .

Answer:

0,0000107 .

Improper common fractions do not need preparation when converting to decimal fractions. The following should be adhered to rules for converting improper common fractions with denominators 10, 100, ... to decimal fractions:

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

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

Example.

Convert improper common fraction 56 888 038 009/100 000 to decimal.

Decision.

Firstly, we write down the number from the numerator 56888038009, and secondly, we separate 5 digits on the right with a decimal point, since there are 5 zeros in the denominator of the original fraction. As a result, we have a decimal fraction 568 880.38009.

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, after which the resulting fraction can be converted into a decimal fraction. But you can also use the following the rule for converting mixed numbers with a denominator of the fractional part 10, or 100, or 1,000, ... into decimal fractions:

• if necessary, we perform “preliminary preparation” of the fractional part of the original mixed number by adding 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 the number from the numerator together with the added zeros.

Let's consider an example, in solving which we will perform all the necessary steps to represent a mixed number as a decimal fraction.

Example.

Convert mixed number to decimal.

Decision.

There are 4 zeros in the denominator of the fractional part, and the number 17 in the numerator, consisting of 2 digits, therefore, we need to add two zeros to the left in the numerator so that the number of characters there becomes equal to the number of zeros in the denominator. By doing 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 together with the added zeros, that is, 0017, while we get the desired decimal fraction 23.0017.

Let's write down the whole solution briefly: .

Undoubtedly, 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 ordinary fractions to finite and infinite periodic decimal fractions

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

In some cases, the original ordinary fraction is easily reduced to one of the denominators 10, or 100, or 1,000, ... (see the reduction of an ordinary fraction to a new denominator), after which it is not difficult to present 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 a fraction 4/10, which, according to the rules discussed in the previous paragraph, can be easily converted into a decimal fraction 0, 4 .

In other cases, you have to use a different way of converting an ordinary fraction into a decimal, which we will now consider.

To convert an ordinary fraction to a decimal fraction, the numerator of the fraction is divided by the denominator, the numerator is previously replaced by a decimal fraction equal to it 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 a decimal point is placed in the quotient when the division of the integer part of the dividend ends. All this will become clear from the solutions of the examples given below.

Example.

Convert the common fraction 621/4 to decimal.

Decision.

We represent the number in the numerator 621 as a decimal fraction by adding a decimal point and a few zeros after it. To begin with, we will 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 by a column. The first three steps are no different from division by a column natural numbers, after which we arrive at the following picture:

So we got 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 the division by a column, ignoring the commas:

This division is completed, and as a result we got the decimal fraction 155.25, which corresponds to the original ordinary fraction.

Answer:

155,25 .

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

Example.

Convert the common fraction 21/800 to decimal.

Decision.

To convert this common fraction to a decimal, let's divide the decimal fraction 21,000 ... by 800 by a column. 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, on this the conversion of the ordinary fraction 21/400 to the decimal fraction is completed, and we have come to the decimal fraction 0.02625.

Answer:

0,02625 .

It may happen that when dividing the numerator by the denominator of an ordinary fraction, we never get a remainder of 0. In these cases, the division can be continued as long as desired. However, starting from a certain step, the remainders begin to repeat periodically, while the digits in the quotient also repeat. This means that the original common fraction translates to an infinite periodic decimal. Let's show this with an example.

Example.

Write the common fraction 19/44 as a decimal.

Decision.

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

It is already clear that when dividing, the remainders 8 and 36 began to repeat, while in the quotient the numbers 1 and 8 are repeated. Thus, the original ordinary fraction 19/44 is translated into a periodic decimal fraction 0.43181818…=0.43(18) .

Answer:

0,43(18) .

In conclusion of this paragraph, we will figure out which ordinary fractions can be converted to final decimal fractions, and which ones can only be converted to periodic ones.

Let us have an irreducible ordinary fraction in front of us (if the fraction is reducible, then we first perform the reduction of the fraction), and we need to find out what 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, 1000, ..., 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 can be reduced to such denominators, the denominators of which are at least one of the numbers 10, 100, ... 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 of 10, 100, 1,000, etc. there can only be numbers whose expansions into prime factors contain only the numbers 2 and (or) 5 .

Now we can make a general conclusion about the conversion of ordinary fractions to decimal fractions:

• if only the numbers 2 and (or) 5 are present in the decomposition of the denominator into prime factors, then this fraction can be converted into a final decimal fraction;
• if, in addition to two and fives, there are others in the expansion of the denominator prime numbers, then this fraction is translated 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 to a final decimal fraction, and which can only be converted to a periodic one.

Decision.

The prime factorization of the denominator of the fraction 47/20 has the form 20=2 2 5 . There are only twos and fives in this expansion, so this fraction can be reduced to one of the denominators 10, 100, 1000, ... (in this example, to the denominator 100), therefore, it can be converted to a final decimal fraction.

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

Fraction 21/56 - contractible, after reduction it takes the form 3/8. The decomposition of the denominator into prime factors contains three factors equal to 2, therefore, the ordinary fraction 3/8, and hence the fraction equal to it 21/56, can be translated into a final decimal fraction.

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

Answer:

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

Common fractions do not convert to infinite non-repeating decimals

The information of the previous paragraph raises the question: “Can an infinite non-periodic fraction be obtained when dividing the numerator of a fraction by the denominator”?

Answer: no. When translating an ordinary fraction, either a finite decimal fraction or an infinite periodic decimal fraction can be obtained. Let's explain why this is so.

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

• either we get the remainder 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 on q, equal remainders are obtained, which follows from the already mentioned divisibility theorem), so an infinite periodic decimal fraction will be obtained.

There can be no other options, therefore, when converting an ordinary fraction to a decimal fraction, an infinite non-periodic decimal fraction cannot be obtained.

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

Convert decimals to common fractions

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

Converting end decimals to common fractions

Getting an ordinary fraction, which is written as a final decimal fraction, is quite simple. The rule for converting a final decimal fraction to an ordinary 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 in 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 consider examples.

Example.

Convert the decimal 3.025 to a common fraction.

Decision.

If we remove the decimal point in the original decimal fraction, then we get the number 3025. It has no zeros on the left that we would discard. So, in the numerator of the required fraction we write 3025.

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

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

Answer:

.

Example.

Convert decimal 0.0017 to common fraction.

Decision.

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

In the denominator we write a unit with four zeros, since in the original decimal fraction there are 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 one is completed.

Answer:

.

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

• the number before the decimal point must be written as the 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 zeros on the left in it;
• in the denominator of the fractional part, you need to write the number 1, to which, on the right, add as many zeros as there are digits in the entry of the original decimal fraction after the decimal point;
• if necessary, reduce the fractional part of the resulting mixed number.

Consider an example of converting a decimal fraction to a mixed number.

Example.

Express decimal 152.06005 as a mixed number

Fractions

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

Fractions in high school are not very annoying. For the time being. Until you run into degrees with rational indicators yes logarithms. And there…. You press, you press the calculator, and it shows all the full scoreboard of some numbers. You have to think with your head, like in the third grade.

Let's deal with fractions, finally! Well, how much can you get confused in them!? Moreover, it is all simple and logical. So, what are fractions?

Types of fractions. Transformations.

Fractions happen three types.

1. Common fractions , For example:

Sometimes, instead of a horizontal line, they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens ...), tell yourself the phrase with the expression: " Zzzzz remember! Zzzzz denominator - out zzzz u!" Look, everything will be remembered.)

A dash, which is horizontal, which is oblique, means division top number (numerator) to bottom number (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.

When the division is possible entirely, it must be done. So, instead of the fraction "32/8" it is much more pleasant to write the number "4". Those. 32 is simply divided by 8.

32/8 = 32: 8 = 4

I'm not talking about the fraction "4/1". Which is also just "4". And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.

2. Decimals , For example:

It is in this form that it will be necessary to write down the answers to tasks "B".

3. mixed numbers , For example:

Mixed numbers are practically not used in high school. In order to work with them, they must be converted to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the puzzle and hang ... From scratch. But we remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if there are all sorts of logarithms, sines and other letters in the fraction, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

Basic property of a fraction.

So let's go! First of all, I will surprise you. The whole variety of fraction transformations is provided by a single property! That's what it's called basic property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:

It is clear that you can write further, until you are blue in the face. Do not let sines and logarithms confuse you, we will deal with them further. The main thing to understand is that all these various expressions are the same fraction . 2/3.

And we need it, all these transformations? And how! Now you will see for yourself. First, let's use the basic property of a fraction for fraction abbreviations. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction like 5/10, but fractional expression with all sorts of letters.

How to reduce fractions correctly and quickly without doing unnecessary work can be found in special Section 555.

A normal student does not bother dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where it hides typical mistake, blooper if you want.

For example, you need to simplify the expression:

There is nothing to think about, we cross out the letter "a" from above and the deuce from below! We get:

Everything is correct. But really you shared the whole numerator and the whole denominator "a". If you are used to just cross out, then, in a hurry, you can cross out the "a" in the expression

and get again

Which would be categorically wrong. Because here the whole numerator on "a" already not shared! This fraction cannot be reduced. By the way, such an abbreviation is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When reducing, it is necessary to divide the whole numerator and the whole denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?

The basic property of a fraction allows you to convert ordinary fractions to decimals and vice versa without calculator! This is important for the exam, right?

How to convert fractions from one form to another.

It's easy with decimals. As it is heard, so it is written! Let's say 0.25. It's zero point, twenty-five hundredths. So we write: 25/100. We reduce (divide the numerator and denominator by 25), we get the usual fraction: 1/4. Everything. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

What if integers are non-zero? It's OK. Write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three whole, seventeen hundredths. We write 317 in the numerator, and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all of the above, a useful conclusion: any decimal fraction can be converted to a common fraction .

And here inverse transformation, ordinary to decimal, some without a calculator can not do. But you must! How will you write down the answer on the exam!? We carefully read and master this process.

What is a decimal fraction? She has in the denominator always is worth 10 or 100 or 1000 or 10000 and so on. If your usual fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if in the answer to the task of section "B" it turned out 1/2? What will we write in response? Decimals are required...

We remember basic property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But, then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.

However, all sorts of denominators come across. For example, the fraction 3/16 will fall. Try it, figure out what to multiply 16 by to get 100, or 1000... Doesn't work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as in lower grades taught. We get 0.1875.

And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 into an exact decimal fraction does not translate. Just like 1/7, 5/6 and so on. Many of them are untranslatable. Hence another useful conclusion. Not every common fraction converts to a decimal. !

By the way, this helpful information for self-test. In section "B" in response, you need to write down a decimal fraction. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere along the way you made a mistake! Come back, check the solution.

So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they all need to be converted to ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always a sixth grader will be at hand ... We will have to do it ourselves. It's not hard. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator ordinary fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.

Let in the problem you saw with horror the number:

Calmly, without panic, we understand. The whole part is 1. One. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of an ordinary fraction. That's all. It looks even simpler in mathematical notation:

Clearly? Then secure your success! Convert to common fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction into a mixed number - is rarely required in high school. Well, if... And if you - not in high school - you can look into the special Section 555. In the same place, by the way, you will learn about improper fractions.

Well, almost everything. You remembered the types of fractions and understood as convert them from one type to another. The question remains: why do it? Where and when to apply this deep knowledge?

I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers, we convert everything into ordinary fractions. It can always be done. Well, if something like 0.8 + 0.3 is written, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is full of decimal fractions, but um ... some kind of evil ones, go to ordinary ones, try it! Look, everything will be fine. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction?

0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Oh, it's shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. Everything!

Let's summarize this lesson.

1. There are three types of fractions. Ordinary, decimal and mixed numbers.

2. Decimals and mixed numbers always can be converted to common fractions. Reverse Translation not always available.

3. The choice of the type of fractions for working with the task depends on this very task. In the presence of different types fractions in one task, the most reliable thing is to switch to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary ones:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

You should get answers like this (in a mess!):

On this we will finish. In this lesson, we brushed up on the key points on fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not mastered it yet ... Those can go to a special Section 555. All the basics are detailed there. Many suddenly understand everything are starting. And they solve fractions on the fly).

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Decimal fractions are the same ordinary fractions, but in the so-called decimal notation. Decimal notation used for fractions with denominators 10, 100, 1000, etc. In this case, instead of fractions 1/10; 1/100; 1/1000; ... write 0.1; 0.01; 0.001;... .

For example, 0.7 ( zero point seven) is a fraction 7/10; 5.43 ( five point forty-three hundredths) is a mixed fraction 5 43/100 (or, equivalently, an improper fraction 543/100).

It may happen that there is one or more zeros immediately after the decimal point: 1.03 is the fraction 1 3/100; 17.0087 is the fraction 1787/10000. General rule is this: there must be as many zeros in the denominator of an ordinary fraction as there are digits after the decimal point in the decimal notation.

A decimal can end in one or more zeros. It turns out that these zeros are “extra” - they can simply be removed: 1.30 = 1.3; 5.4600 = 5.46; 3,000 = 3. Can you figure out why this is so?

Decimals naturally arise when dividing by "round" numbers - 10, 100, 1000, ... Be sure to understand the following examples:

27:10 = 27/10 = 2 7/10 = 2,7;

579:100 = 579/100 = 5 79/100 = 5,79;

33791:1000 = 33791/1000 = 33 791/1000 = 33,791;

34,9:10 = 349/10:10 = 349/100 = 3,49;

6,35:100 = 635/100:100 = 635/10000 = 0,0635.

Do you notice a pattern here? Try to formulate it. What happens if you multiply a decimal by 10, 100, 1000?

To convert an ordinary fraction to a decimal, you need to bring it to some kind of "round" denominator:

2/5 = 4/10 = 0.4; 11/20 = 55/100 = 0.55; 9/2 = 45/10 = 4.5 etc.

Adding decimal fractions is much more convenient than ordinary fractions. Addition is performed in the same way as with ordinary numbers - according to the corresponding digits. When adding in a column, the terms must be written so that their commas are on the same vertical. The sum comma will also appear on the same vertical. The subtraction of decimal fractions is performed in exactly the same way.

If, when adding or subtracting in one of the fractions, the number of digits after the decimal point is less than in the other, then at the end of this fraction, the required number of zeros should be added. You can not add these zeros, but simply imagine them in your mind.

When multiplying decimal fractions, they should again be multiplied as ordinary numbers (in this case, it is no longer necessary to write a comma under a comma). In the result obtained, you need to separate with a comma the number of characters equal to the total number of decimal places in both factors.

When dividing decimal fractions, you can simultaneously move the comma to the right by the same number of digits in the dividend and divisor: the quotient will not change from this:

2,8:1,4 = 2,8/1,4 = 28/14 = 2;

4,2:0,7 = 4,2/0,7 = 42/7 = 6;

6:1,2 = 6,0/1,2 = 60/12 = 5.

Explain why this is so?

1. Draw a 10x10 square. Paint over some part of it equal to: a) 0.02; b) 0.7; c) 0.57; d) 0.91; e) 0.135 of the area of ​​the whole square.
2. What is 2.43 squares? Draw in the picture.
3. Divide 37 by 10; 795; 4; 2.3; 65.27; 0.48 and write the result as a decimal fraction. Divide these numbers by 100 and 1000.
4. Multiply by 10 the numbers 4.6; 6.52; 23.095; 0.01999. Multiply these numbers by 100 and 1000.
5. Express the decimal as a fraction and reduce it:
   a) 0.5; 0.2; 0.4; 0.6; 0.8;
   b) 0.25; 0.75; 0.05; 0.35; 0.025;
   c) 0.125; 0.375; 0.625; 0.875;
   d) 0.44; 0.26; 0.92; 0.78; 0.666; 0.848.
6. Imagine in the form mixed fraction: 1,5; 3,2; 6,6; 2,25; 10,75; 4,125; 23,005; 7,0125.
7. Write a common fraction as a decimal:
   a) 1/2; 3/2; 7/2; 15/2; 1/5; 3/5; 4/5; 18/5;
   b) 1/4; 3/4; 5/4; 19/4; 1/20; 7/20; 49/20; 1/25; 13/25; 77/25; 1/50; 17/50; 137/50;
   c) 1/8; 3/8; 5/8; 7/8; 11/8; 125/8; 1/16; 5/16; 9/16; 23/16;
   d) 1/500; 3/250; 71/200; 9/125; 27/2500; 1999/2000.
8. Find the sum: a) 7.3 + 12.8; b) 65.14+49.76; c) 3.762+12.85; d) 85.4+129.756; e) 1.44+2.56.
9. Think of a unit as the sum of two decimals. Find twenty more ways to do this.
10. Find the difference: a) 13.4–8.7; b) 74.52–27.04; c) 49.736–43.45; d) 127.24–93.883; e) 67–52.07; f) 35.24–34.9975.
11. Find the product: a) 7.6 3.8; b) 4.8 12.5; c) 2.39 7.4; d) 3.74 9.65.