Decimals. Common fraction What is the difference between a common fraction and a decimal

Topic: The concept of a decimal fraction.

Reading and writing decimals.

1. 
   The purpose of the lesson: the formation of skills for writing and reading decimal fractions, the ability to translate ordinary fractions with denominators 10, 100, 1000, etc. into a decimal.

1. 
   Tasks:

- educational to teach read and write decimal fractions;

- developing - to develop the skills of self-assessment and self-analysis of educational activities, to develop students' mathematical speech;

- educational - to cultivate a culture of mathematical thinking, the ability to work independently.
3. Type of lesson - knowledge consolidation lesson
4. Teaching methods: verbal, visual, practical
5. Forms of student work - frontal, individual, group

6. Necessary technical equipment - multimedia projector, computer, screen

7. Educational and methodological support: textbook "Mathematics 5", I. I. Zubareva, A. G. Mordkovich

Lesson structure:

1. 
   Org. moment.
2. 
   Repetition of previous topics, oral work.
3. 
   Mathematical dictation.
4. 
   Fizkultpauza.
5. 
   Main part.
6. 
   Reflection.
7. 
   Homework.

During the classes:

1. 
   Org. moment.

• 
  Mutual greetings between teacher and students.
• 
  Checking jobs.
• 
  Communication to students of the lesson plan.

- Hello guys!

It's good that I came to you. I was told that you would certainly help in my investigation.

My investigative committee received a complaint from two drivers who became participants in a traffic accident.

Let's turn to the case file.

^ INFORMATION OF VICTIMS.

From two points A and B, a car and a truck left towards each other. The speed of a car is 60 km/h and the speed of a truck is 40 km/h. How long will it take for them to meet if the distance between the points is 350 km?

- Consider the solution .

1) 40 + 60 \u003d 100 (km / h) - the total speed of cars (convergence speed)

2) 350: 100 = 35 (h)

Answer: the cars will meet in 35 hours.
- Guys, pay attention to all the data, and answer: “Does this result cause you doubt?”
- Yes, there is a doubt, in this problem the time cannot be 35 hours.
- So, as a result of the decision, a mistake was made. We will find out what the answer should be by conducting an investigation and examining all the facts, documents and evidence.
- For our investigation, I took a magnifying glass, scales and books.

FIRST TASK. (evidence one)
Delete from these numbers:

• 
  Integers
• 
  Proper fractions
• 
  Improper fractions
• 
  mixed numbers

8 45/1000; 1000; 12; 3/2; 0,12; 1/6; 15/15; 30/24; 12/1000; 21,032; 1 2/3.

What numbers are left?

Numbers written in a new way appeared on our mathematical horizon. These are decimals.
- Let's turn to scientific documents.

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

For example:

^ Decimal fractions are separated from ordinary fractions into a separate form.
Any number of zeros can be added to the fractional part of the decimal fraction on the right, this does not change the value of the fraction.

^ The fractional part of the decimal fraction is read by the last significant digit.

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

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

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

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

For example:
10.7 - ten point seven

0.67 - zero point sixty-seven hundredths.

Decimals are the fractional digits. The fractional part is not read by digits (unlike natural numbers), but as a whole, so the fractional part of the decimal fraction is determined last on the right significant rank.

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

• 
  1st place after the decimal point - tenth place
• 
  2nd place after the decimal point - hundredth place
• 
  3rd place after the decimal point - thousandth place
• 
  4th place after the decimal point - ten-thousandth place
• 
  5th place after the decimal point - hundred-thousandth place
• 
  6th place after the decimal point - millionth place
• 
  7th place after the decimal point - ten-millionth place
• 
  8th place after the decimal point - one hundred millionth place

What information did you get about our object of study?

Let's turn to archival materials.
Explore historical evidence. How were these fractions written before?

In the 5th century, the Chinese scientist Tszyu-Chun-Zhi recorded a fraction of the form 2.135436 as follows:

2 chi, 1 cun, 3 shares, 5 ordinals, 4 hairs, 3 finest, 6 gossamer.

Uzbek scientist Jemshid Giyaseddin al-Kashi in the book

"Key to Arithmetic" (1424) showed the recording of a fraction in one line with numbers in the decimal system.

For recording, he used the vertical line,

the ink is black and red.

In the book "Mathematical Canon" by the French mathematician F. Vieta (1540-1603), the decimal fraction is written as follows 2 135436 - the fractional part was underlined and written above the line of the integer part of the number

1571 G. - Johannes Kepler proposed a modern notation for decimal fractions, i.e. separating the whole part of the comma.

Before him, there were other options:

3,7 were written as 3(0)7 or 3\ 7 or in different inks the integer and fractional parts.
- So, describe what the decimal fraction looks like now.
^ WE WILL CONTINUE THE INVESTIGATIVE ACTIONS.
Second TASK. (two evidence)
Specify the least significant digit of the number and read it:

1,25 12, 54 3,06 1410,05

THIRD TASK. (Evidence Three)
How are decimals written?

46,5 80,35 4,65 8,035 40,065 83,05 0,465 0,0835

^ LET'S CARRY OUT AN INVESTIGATIVE EXPERIMENT.
MATHEMATICAL DICTATION.
- For the next task, we need a magnifying glass, because we need to find the comma in the numbers.
4735,62 123,456 54,5454 230,032 74635,2

Exchange work with your colleagues and check

PHYSICAL MINUTE.

^ MAIN PART.

Let's hear the testimony of witnesses:

Mom bought 2¼ kg of apples and 3.5 kg of pears. How many kilograms of fruit did mom buy?
What fractions are found in the document? ( ordinary And decimal)

Do you think it's possible to add these fractions? ( No)

What needs to be done to answer the question of the problem? ( calculate either in ordinary or decimal fractions).

To do this, you need to convert one fraction to another. This is where I need the scales.

What are scales for? ( weigh, compare, equalize)

On our mathematical scales, we will compare the number of decimal places (in the fractional part) and zeros in a bit unit.
^ A). Express the number as a fraction:

0,13 6,013 0, 05 14,007 51, 3 830,0026

(Each group receives one number. After completing the task, they defend their answer, supplementing with their own example).

B). Express the number as a decimal:

1 1 / 10 , 25 / 100 , 98 3 / 10 , 2 56 / 1000 , 75 108 / 10000

R B O A B
Arrange the common fractions in ascending order.

BRAVO
4. REFLECTION.
Our investigation is coming to an end. All materials of the case were considered, facts were compared, documents were studied.
- Back to our violation.
- What should be the number in the problem in order to get the correct answer? "What have you lost in this number?" (COMMA)
- What is the correct answer?
How do you write down the answer as a fraction?
- Convert to hours and minutes?
- Thank you, well done. I take my hat off to you. We coped with the task.

5. Homework.

Prepare messages on topics:

"History of decimal fractions"

Where are decimals used?
THANK YOU FOR THE LESSON.

Studying the queen of all sciences - mathematics, at some point everyone is faced with fractions. Although this concept (like the types of fractions themselves or mathematical operations with them) is not difficult at all, it must be treated carefully, because in real life outside of school it will be very useful. So, let's refresh our knowledge of fractions: what they are, what they are for, what types they are and how to perform various arithmetic operations with them.

Her Majesty the fraction: what is it

Fractions in mathematics are numbers, each of which consists of one or more parts of the unit. Such fractions are also called ordinary, or simple. As a rule, they are written as two numbers, which are separated by a horizontal or slash bar, it is called a "fractional". For example: ½, ¾.
The top, or first of these numbers is the numerator (shows how many fractions of the number are taken), and the bottom, or second, is the denominator (demonstrates how many parts the unit is divided into).
The fractional bar actually functions as a division sign. For example, 7:9=7/9
Traditionally, common fractions are less than one. While decimals can be larger than it.

What are fractions for? Yes, for everything, because in the real world, not all numbers are integers. For example, two schoolgirls in the cafeteria bought together one delicious chocolate bar. When they were about to share dessert, they met a friend and decided to treat her as well. However, now it is necessary to correctly divide the chocolate bar, given that it consists of 12 squares.
At first, the girls wanted to share everything equally, and then each would get four pieces. But, after thinking it over, they decided to treat their girlfriend, not 1/3, but 1/4 chocolates. And since schoolgirls did not study fractions well, they did not take into account that in such a situation, as a result, they would have 9 pieces that are very poorly divided into two. This rather simple example shows how important it is to be able to correctly find the part of a number. But in life there are many more such cases.

Types of fractions: ordinary and decimal

All mathematical fractions are divided into two large digits: ordinary and decimal. The features of the first of them were described in the previous paragraph, so now it is worth paying attention to the second.
A decimal is a positional notation of a fraction of a number, which is fixed in a letter separated by a comma, without a dash or slash. For example: 0.75, 0.5.
In fact, a decimal fraction is identical to an ordinary one, however, its denominator is always one followed by zeros - hence its name.
The number preceding the decimal point is the integer part, and everything after the decimal point is the fractional part. Any simple fraction can be converted to a decimal. So, the decimal fractions indicated in the previous example can be written as ordinary ones: ¾ and ½.
It is worth noting that both decimal and ordinary fractions can be both positive and negative. If they are preceded by a "-" sign, this fraction is negative, if "+" - then positive.

Subtypes of ordinary fractions

There are such types of simple fractions.

Correct. Their numerator is always less than the denominator. For example: 7/8. This is a proper fraction because the numerator 7 is less than the denominator 8. Incorrect. In such fractions, either the numerator and denominator are equal to each other (8/8), or the value of the lower number is less than the upper one (9/8). Mixed. This is the name of a proper fraction written together with an integer: 8 ½. It is understood as the sum of this number and a fraction. By the way, it’s quite easy to make an improper fraction appear in its place. To do this, 8 must be written as 16/2+1/2=17/2. Compound. As the name implies, they consist of several fractional features: ½ / ¾. Reducible / irreducible. These can include both proper and improper fractions. It all depends on whether the numerator and denominator can be divided by the same number. For example, 6/9 is a reduced fraction, because both of its components can be divided by 3 and you get 2/3. But 7/9 is irreducible, since 7 and 9 are prime numbers that do not have a common divisor and cannot be reduced.

Subspecies of the decimal fraction

Unlike a simple, decimal fraction is divided into only 2 types.

Final - got its name due to the fact that after the decimal point it has a limited (finite) number of digits: 19.25. An infinite fraction is a number with an endless number of digits after the decimal point. For example, when dividing 10 by 3, the result will be an infinite fraction 3.333 ...

Addition of fractions

Performing various arithmetic manipulations with fractions is a little more difficult than with ordinary numbers. However, if you learn the basic rules, solving any example with them will not be difficult.
So, in order to add fractions together, first of all, you need to make sure that both terms have the same denominators. To do this, you have to find the smallest number that can be divided without a remainder by the denominators of the summands.
For example: 2/3+3/4. The least common multiple for them will be 12, therefore, it is necessary that this number be in each denominator. To do this, we multiply the numerator and denominator of the first fraction by 4, it turns out 8/12, we do the same with the second term, but only multiply by 3 - 9/12. Now you can easily solve the example: 8/12+9/12= 17/12. The resulting fraction is an incorrect value because the numerator is greater than the denominator. It can and should be converted into the correct mixed one by dividing 17:12 = 1 and 5/12.
If mixed fractions are added, first the actions are performed with integers, and then with fractional ones.
If the example contains a decimal fraction and an ordinary one, it is necessary that both become simple, then bring them to the same denominator and add them. For example 3.1+1/2. The number 3.1 can be written as a mixed fraction of 3 and 1/10, or as an improper - 31/10. The common denominator for the terms will be 10, so you need to multiply the numerator and denominator 1/2 by 5 in turn, it turns out 5/10. Then you can easily calculate everything: 31/10+5/10=35/10. The result obtained is an improper contractible fraction, we bring it into normal form, reducing it by 5: 7/2=3 and 1/2, or decimal - 3.5.
When adding 2 decimals, it is important that there are the same number of digits after the decimal point. If this is not the case, you just need to add the required number of zeros, because in a decimal fraction this can be done painlessly. For example, 3.5+3.005. To solve this task, you need to add 2 zeros to the first number and then add in turn: 3.500 + 3.005 = 3.505.

Subtraction of fractions

When subtracting fractions, it is worth doing the same as when adding: reduce to a common denominator, subtract one numerator from another, if necessary, convert the result into a mixed fraction.
For example: 16/20-5/10. The common denominator will be 20. You need to bring the second fraction to this denominator, multiplying both of its parts by 2, you get 10/20. Now you can solve the example: 16/20-10/20= 6/20. However, this result applies to reducible fractions, so it is worth dividing both parts by 2 and the result is 3/10.

Multiplication of fractions

Division and multiplication of fractions are much simpler operations than addition and subtraction. The fact is that when performing these tasks, there is no need to look for a common denominator.
To multiply fractions, you just need to alternately multiply both numerators together, and then both denominators. Reduce the resulting result if the fraction is a reduced value.

For example: 4/9x5/8. After alternate multiplication, the result is 4x5/9x8=20/72. Such a fraction can be reduced by 4, so the final answer in the example is 5/18.

How to divide fractions

Dividing fractions is also a simple action, in fact it still comes down to multiplying them. To divide one fraction by another, you need to flip the second and multiply by the first.

For example, division of fractions 5/19 and 5/7. To solve the example, you need to swap the denominator and numerator of the second fraction and multiply: 5/19x7/5=35/95. The result can be reduced by 5 - it turns out 7/19.
If you need to divide a fraction by a prime number, the technique is slightly different. Initially, it is worth writing this number as an improper fraction, and then dividing according to the same scheme. For example, 2/13:5 should be written as 2/13:5/1. Now you need to flip 5/1 and multiply the resulting fractions: 2/13x1/5= 2/65.
Sometimes you have to divide mixed fractions. You need to deal with them, as with integers: turn them into improper fractions, flip the divisor and multiply everything. For example, 8 ½: 3. Turning everything into improper fractions: 17/2: 3/1. This is followed by a 3/1 flip and multiplication: 17/2x1/3= 17/6. Now you should convert the improper fraction to the correct one - 2 integers and 5/6.
So, having figured out what fractions are and how you can perform various arithmetic operations with them, you need to try not to forget about it. After all, people are always more inclined to divide something into parts than to add, so you need to be able to do it right.

Common fraction

quarters

1. Orderliness. a And b there is a rule that allows you to uniquely identify between them one and only one of the three relations: “< », « >' or ' = '. This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and ; two non-positive numbers a And b are related by the same relation as two non-negative numbers and ; if suddenly a non-negative, and b- negative, then a > b. style="max-width: 98%; height: auto; width: auto;" src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">

   

   summation of fractions

2. addition operation. For any rational numbers a And b there is a so-called summation rule c. However, the number itself c called sum numbers a And b and is denoted , and the process of finding such a number is called summation. The summation rule has the following form: .
3. multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which puts them in correspondence with some rational number c. However, the number itself c called work numbers a And b and is denoted , and the process of finding such a number is also called multiplication. The multiplication rule is as follows: .
4. Transitivity of the order relation. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
5. Associativity of addition. The order in which three rational numbers are added does not affect the result.
6. The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
8. Commutativity of multiplication. By changing the places of rational factors, the product does not change.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
12. Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
13. Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality. max-width: 98% height: auto; width: auto;" src="/pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
14. Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum will exceed a. style="max-width: 98%; height: auto; width: auto;" src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

All other properties inherent in rational numbers are not singled out as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proved on the basis of the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense here to cite just a few of them.

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Set countability

Numbering of rational numbers

To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it suffices to give an algorithm that enumerates rational numbers, that is, establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms is as follows. An infinite table of ordinary fractions is compiled, on each i-th line in each j th column of which is a fraction. For definiteness, it is assumed that the rows and columns of this table are numbered from one. Table cells are denoted , where i- the row number of the table in which the cell is located, and j- column number.

The resulting table is managed by a "snake" according to the following formal algorithm.

These rules are searched from top to bottom and the next position is selected by the first match.

In the process of such a bypass, each new rational number is assigned to the next natural number. That is, fractions 1 / 1 are assigned the number 1, fractions 2 / 1 - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to unity of the greatest common divisor of the numerator and denominator of the fraction.

Following this algorithm, one can enumerate all positive rational numbers. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers, simply by assigning to each rational number its opposite. That. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.

The statement about the countability of the set of rational numbers may cause some bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.

Insufficiency of rational numbers

The hypotenuse of such a triangle is not expressed by any rational number

Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates a deceptive impression that rational numbers can measure any geometric distances in general. It is easy to show that this is not true.

It is known from the Pythagorean theorem that the hypotenuse of a right triangle is expressed as the square root of the sum of the squares of its legs. That. the length of the hypotenuse of an isosceles right triangle with a unit leg is equal to, i.e., a number whose square is 2.

If we assume that the number is represented by some rational number, then there is such an integer m and such a natural number n, which, moreover, the fraction is irreducible, i.e., the numbers m And n are coprime.

If , then , i.e. m 2 = 2n 2. Therefore, the number m 2 is even, but the product of two odd numbers is odd, which means that the number itself m also clear. So there is a natural number k, such that the number m can be represented as m = 2k. Number square m In this sense m 2 = 4k 2 but on the other hand m 2 = 2n 2 means 4 k 2 = 2n 2 , or n 2 = 2k 2. As shown earlier for the number m, which means that the number n- exactly like m. But then they are not coprime, since both are divisible in half. The resulting contradiction proves that is not a rational number.

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

Why are fractions needed?

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

For example, chocolate consists of several slices. Consider the situation where its tile is formed by twelve rectangles. If you divide it 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 get acquainted with the first ones in the elementary grades, calling them simply “fractions”. The second learn in the 5th grade. That's when these names appear.

Common fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the integer with a comma. For example, 4.7. Students need to be clear that the two examples given are completely different numbers.

Every simple fraction can be written as a decimal. This statement is almost always true in reverse as well. There are rules that allow you to write a decimal fraction as an ordinary fraction.

What subspecies do these types of fractions have?

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

Correct. Its numerator is always less than the denominator.

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

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

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

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

Decimals have only two subspecies:

final, that is, one in which the fractional part is limited (has an end);

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

How to convert decimal to ordinary?

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

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

How to convert decimal fractions to ordinary ones if their whole part is missing, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. It remains to write down only the fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter turns out to be possible to reduce by 5. Therefore, the result for it must be written 1/20.

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

How to convert an infinite decimal to a common fraction?

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

The only thing that is allowed to be done with such a fraction is to round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal - will never give the initial value. That is, infinite non-periodic fractions are not translated into ordinary fractions. This must be remembered.

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

In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called periods. For example, 0.3(3). Here "3" in the period. They are classified as rational, as they can be converted into ordinary fractions.

Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second, the fractional part begins with any numbers, and then the repetition begins.

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

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

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

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

Write down the denominator: first nines, then zeros.

To determine the numerator, you need to write the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtractable - it is without a period.

For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period is one digit. So zero will be one. There is also only one digit in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.

To determine the numerator from 58, you need to subtract 5. It turns out 53. For example, you will have to write 53/90 as an answer.

How are common fractions converted to decimals?

The simplest option is a number whose denominator is the number 10, 100, and so on. Then the denominator is simply discarded, and a comma is placed between the fractional and integer parts.

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

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

Operations with common fractions

Addition and subtraction

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

Find the least common multiple of the denominators.

Write additional factors to all ordinary fractions.

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

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

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

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

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

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

Multiplication and division

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

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

Multiply numerators.

Multiply the denominators.

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

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

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

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



Operations with decimals

Addition and subtraction

Of course, you can always turn a decimal into a common fraction. And act according to the already described plan. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.

Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Assign the missing number of zeros in it.

Write fractions so that the comma is under the comma.

Add (subtract) like natural numbers.

Remove the comma.

Multiplication and division

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

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

Multiply like natural numbers.

Put a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.

To divide, you must first convert the divisor: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.

Multiply the dividend by the same number.

Divide a decimal by a natural number.

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



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

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

First way: represent ordinary decimals

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

The second way: write decimal fractions as ordinary

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

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

For example:

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

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

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

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

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

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

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

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

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

For example:
10.7 - ten point seven

0.67 - zero point sixty-seven hundredths.

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

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

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

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