A lesson in mastering and consolidating new knowledge
Subject : Comparison decimal fractions
Dambaeva Valentina Matveevna
Mathematic teacher
MAOU "Secondary School No. 25", Ulan-Ude
Subject. Comparison of decimal fractions.
Didactic goal: teach students to compare two decimal fractions. Introduce students to the rule of comparison. To form the ability to find a large (smaller) fraction.
educational goal. To develop the creative activity of students in the process of solving examples. Cultivate interest in mathematics, selection various types assignments. Cultivate ingenuity, ingenuity, develop flexible thinking. To continue to develop in students the ability to self-critically relate to the results of the work performed.
Lesson equipment. Handout. Signal cards, task cards, carbon paper.
Visual aids. Task tables, poster rules.
Class type. Assimilation of new knowledge. Consolidation of new knowledge.
Lesson plan
Organizing time. 1 min.
Examination homework. 3 min.
Repetition. 8 min.
Explanation new topic. 18-20 min.
Consolidation. 25-27 min.
Summing up the work. 3 min.
Homework. 1 min.
Express dictation. 10-13 min
During the classes.
1. Organizational moment.
2. Checking homework. Collection of notebooks.
3. Repetition(orally).
a) compare ordinary fractions (work with signal cards).
4/5 and 3/5; 4/4 and 13/40; 1 and 3/2; 4/2 and 12/20; 3 5/6 and 5 5/6;
b) In which category are 4 units, 2 units ... ..?
57532, 4081
c) compare natural numbers
99 and 1111; 5 4 4 and 5 3 4, 556 and 55 9 ; 4 366 and 7 366;
How to compare numbers with the same number of digits?
(Numbers with the same number of digits are compared bit by bit, starting with the most significant digit. Poster-rule).
It can be imagined that the digits of the same name “compete”, whose digit term is greater: one with ones, tens with tens, etc.
4. Explanation of the new topic.
a) What sign (>,< или =) следует заменить вопросительный знак между десятичными дробями на рисунке.
Poster assignment
3425, 672678 ? 3425, 672478
14, 24000 ? 14, 24
To answer this question, you need to learn how to compare decimal fractions.
12, 3 < 15,3
72.1 > 68.4 Why?
Of two decimal fractions, the one with the larger integer part is larger.
13,5 > 13,4
0, 327 > 0,321
Why?
If the integer parts of the compared fractions are equal to each other, then their fractional part is compared by digits.
3. 0,800 ? 0,8
1,32 ? 1,3
But what if there are different numbers of these numbers? If one or more zeros are added to the decimal fraction on the right, then the value of the fraction will not change.
Conversely, if the decimal fraction ends in zeros, then these zeros can be discarded, the value of the fraction will not change from this.
Consider three decimals:
1,25 1,250 1,2500
How do they differ from each other?
Only the number of zeros at the end of the record.
What numbers do they represent?
To find out, you need to write down for each of the fractions the sum of the bit terms.
1,25 = 1+ 2/10 + 5/100
1,250 = 1+ 2/10 + 5/100 1 25/100 = 1,25
1,2500 = 1+ 2/10 + 5/100
In all equalities, the same amount is written on the right. So all three fractions represent the same number. Otherwise, these three fractions are equal: 1.25 = 1.250 = 1.2500.
Decimals can be shown in coordinate beam just like regular fractions. For example, to depict the decimal fraction 0.5 on the coordinate beam. we first represent it in the form common fraction: 0.5 = 5/10. Then we set aside five tenths of a single segment from the beginning of the beam. Get point A(0.5)
Equal decimal fractions are depicted on the coordinate ray by the same point.
The smaller decimal fraction lies on the coordinate ray to the left of the larger one, and the larger one lies to the right of the smaller one.
b) Work with the textbook, with the rule.
Now try to answer the question that was posed at the beginning of the explanation: what sign (>,< или =) следует заменить вопросительный знак.
5. Fixing.
№1
Compare: Working with signal cards
85.09 and 67.99
55.7 and 55.700
0.0025 and 0.00247
98.52 m and 65.39 m
149.63 kg and 150.08 kg
3.55 0 С and 3.61 0 С
6.784 h and 6.718 h
№ 2
Write a decimal
a) with four decimal places, equal to 0.87
b) with five decimal places, equal to 0.541
c) with three decimal places, equal to 35
d) with two decimal places, equal to 8.40000
2 students work on individual boards
№ 3
Smekalkin got ready to do the task of comparing numbers and copied several pairs of numbers into a notebook, between which you need to put a sign > or<. Вдруг он нечаянно уронил тетрадь на мокрый пол. Записи размазались, и некоторые цифры стало невозможно разобрать. Вот что получилось:
a) 4.3** and 4.7**
b) **, 412 and *, 9*
c) 0.742 and 0.741*
d)*, *** and **,**
e) 95.0** and *4.*3*
Smekalkin liked that he was able to complete the task with smeared numbers. After all, instead of a task, riddles turned out. He himself decided to come up with riddles with smeared numbers and offers you. In the following entries, some numbers are smeared. You need to guess what these numbers are.
a) 2.*1 and 2.02
b) 6.431 and 6.4 * 8
c) 1.34 and 1.3*
d) 4.*1 and 4.41
e) 4.5 * 8 and 4, 593
f) 5.657* and 5.68
Task on the poster and on individual cards.
Verification-justification of each set mark.
№ 4
I affirm:
a) 3.7 is less than 3.278
because the first number has fewer digits than the second.
b) 25.63 is equal to 2.563
After all, they have the same numbers in the same order.
Correct my statement
"Counterexample" (oral)
№ 5
What natural numbers are between numbers (in writing).
a) 3, 7 and 6.6
b) 18.2 and 19.8
c) 43 and 45.42
d) 15 and 18
6. The result of the lesson.
How to compare two decimals with different integers?
How to compare two decimals with the same integers?
How to compare two decimals with the same number of decimal places?
7. Homework.
8. Express dictation.
Write the numbers shorter
0,90 1,40
10,72000 61,610000
Compare fractions
0.3 and 0.31 0.4 and 0.43
0.46 and 0.5 0.38 and 0.4
55.7 and 55.700 88.4 and 88.400
Arrange in order
Descending Ascending
3,456; 3465; 8,149; 8,079; 0,453
What are the natural numbers between the numbers?
7.5 and 9.1 3.25 and 5.5
84 and 85.001 0.3 and 4
Put the numbers to make the inequality true:
15,*2 > 15,62 4,60 < 4,*3
6,99 6,8
Checking express dictation from the board
Additional task.
1. Write 3 examples to your neighbor and check!
Literature:
Stratilatov P.V. "On the system of work of a teacher of mathematics" Moscow "Enlightenment" 1984
Kabalevsky Yu.D. " Independent work students in the process of teaching mathematics "1988
Bulanova L.M., Dudnitsyn Yu.P. "Testing tasks in mathematics",
Moscow "Dedication" 1992
V.G. Kovalenko " Didactic games in mathematics lessons "Moscow" Enlightenment "1990
Minaeva S.S. "Calculations in the classroom and extracurricular activities in mathematics" Moscow "Prosveshchenie" 1983
In this article, we will cover the topic decimal comparison". Let's discuss first general principle comparing decimals. After that, we will figure out which decimal fractions are equal and which are unequal. Next, we will learn how to determine which decimal fraction is greater and which is less. To do this, we will study the rules for comparing finite, infinite periodic and infinite non-periodic fractions. Let's provide the whole theory with examples with detailed decisions. In conclusion, let us dwell on the comparison of decimal fractions with natural numbers, common fractions and mixed numbers.
Let's say right away that here we will only talk about comparing positive decimal fractions (see positive and negative numbers). The remaining cases are analyzed in the articles comparing rational numbers and comparison of real numbers.
Page navigation.
General principle for comparing decimal fractions
Based on this principle of comparison, the rules for comparing decimal fractions are derived, which make it possible to do without converting the compared decimal fractions into ordinary fractions. These rules, as well as examples of their application, we will analyze in the following paragraphs.
By a similar principle, finite decimal fractions or infinite periodic decimal fractions are compared with natural numbers, ordinary fractions and mixed numbers: the compared numbers are replaced by their corresponding ordinary fractions, after which ordinary fractions are compared.
Concerning comparisons of infinite non-recurring decimals, then it usually comes down to comparing final decimal fractions. To do this, consider such a number of signs of compared infinite non-periodic decimal fractions, which allows you to get the result of the comparison.
Equal and unequal decimals
First we introduce definitions of equal and unequal final decimals.
Definition.
The two trailing decimals are called equal if their corresponding common fractions are equal, otherwise these decimal fractions are called unequal.
Based on this definition, it is easy to justify the following statement: if at the end of a given decimal fraction we attribute or discard several digits 0, then we get a decimal fraction equal to it. For example, 0.3=0.30=0.300=… and 140.000=140.00=140.0=140 .
Indeed, adding or discarding zero at the end of the decimal fraction on the right corresponds to multiplying or dividing by 10 the numerator and denominator of the corresponding ordinary fraction. And we know the basic property of a fraction, which says that multiplying or dividing the numerator and denominator of a fraction by the same natural number gives a fraction equal to the original one. This proves that adding or discarding zeros to the right in the fractional part of a decimal fraction gives a fraction equal to the original one.
For example, a decimal fraction 0.5 corresponds to an ordinary fraction 5/10, after adding zero to the right, a decimal fraction 0.50 is obtained, which corresponds to an ordinary fraction 50/100, and. So 0.5=0.50 . Conversely, if in decimal fraction 0.50 discard 0 on the right, then we get a fraction 0.5, so from an ordinary fraction 50/100 we will come to a fraction 5/10, but 
. Therefore, 0.50=0.5 .
Let's move on to definition of equal and unequal infinite periodic decimal fractions.
Definition.
Two infinite periodic fractions equal, if the ordinary fractions corresponding to them are equal; if the ordinary fractions corresponding to them are not equal, then the compared periodic fractions are also not equal.
Three conclusions follow from this definition:
- If the records of periodic decimal fractions are exactly the same, then such infinite periodic decimal fractions are equal. For example, the periodic decimals 0.34(2987) and 0.34(2987) are equal.
- If the periods of the compared decimal periodic fractions start from the same position, the first fraction has a period of 0 , the second has a period of 9 , and the value of the digit preceding period 0 is one more than the value of the digit preceding period 9 , then such infinite periodic decimal fractions are equal. For example, the periodic fractions 8.3(0) and 8.2(9) are equal, and the fractions 141,(0) and 140,(9) are also equal.
- Any two other periodic fractions are not equal. Here are examples of unequal infinite periodic decimal fractions: 9.0(4) and 7,(21) , 0,(12) and 0,(121) , 10,(0) and 9.8(9) .
It remains to deal with equal and unequal infinite non-periodic decimal fractions. As you know, such decimal fractions cannot be converted into ordinary fractions (such decimal fractions represent irrational numbers), so the comparison of infinite non-periodic decimal fractions cannot be reduced to a comparison of ordinary fractions.
Definition.
Two infinite non-recurring decimals equal if their entries match exactly.
But there is one caveat: it is impossible to see the “finished” record of infinite non-periodic decimal fractions, therefore, it is impossible to be sure of the complete coincidence of their records. How to be?
When comparing infinite non-periodic decimal fractions, only a finite number of signs of the compared fractions are considered, which allows us to draw the necessary conclusions. Thus, the comparison of infinite non-periodic decimal fractions is reduced to the comparison of finite decimal fractions.
With this approach, we can talk about the equality of infinite non-periodic decimal fractions only up to the considered digit. Let's give examples. Infinite non-periodic decimal fractions 5.45839 ... and 5.45839 ... are equal to within hundred thousandths, since the final decimal fractions 5.45839 and 5.45839 are equal; non-recurring decimal fractions 19.54 ... and 19.54810375 ... are equal to the nearest hundredth, since the fractions 19.54 and 19.54 are equal.
The inequality of infinite non-periodic decimal fractions with this approach is established quite definitely. For example, the infinite non-periodic decimal fractions 5.6789… and 5.67732… are not equal, since the differences in their records are obvious (the final decimal fractions 5.6789 and 5.6773 are not equal). The infinite decimals 6.49354... and 7.53789... are also not equal.
Rules for comparing decimal fractions, examples, solutions
After establishing the fact that two decimal fractions are not equal, it is often necessary to find out which of these fractions is greater and which is less than the other. Now we will analyze the rules for comparing decimal fractions, allowing us to answer the question posed.
In many cases, it is sufficient to compare the integer parts of the compared decimals. The following is true decimal comparison rule: greater than the decimal fraction, the integer part of which is greater, and less than the decimal fraction, the integer part of which is less.
This rule applies to both finite decimals and infinite decimals. Let's consider examples.
Example.
Compare decimals 9.43 and 7.983023….
Decision.
Obviously, these decimal fractions are not equal. The integer part of the final decimal fraction 9.43 is equal to 9, and the integer part of the infinite non-periodic fraction 7.983023 ... is equal to 7. Since 9>7 (see comparison of natural numbers), then 9.43>7.983023.
Answer:
9,43>7,983023 .
Example.
Which of the decimals 49.43(14) and 1,045.45029... is less?
Decision.
The integer part of the periodic fraction 49.43(14) is less than the integer part of the infinite non-periodic decimal fraction 1 045.45029…, therefore, 49.43(14)<1 045,45029… .
Answer:
49,43(14) .
If the integer parts of the compared decimal fractions are equal, then to find out which of them is greater and which is less, one has to compare the fractional parts. Comparison of fractional parts of decimal fractions is carried out bit by bit- from the category of tenths to the younger ones.
First, let's look at an example of comparing two final decimal fractions.
Example.
Compare the end decimals 0.87 and 0.8521 .
Decision.
The integer parts of these decimal fractions are equal (0=0 ), so let's move on to comparing the fractional parts. The values of the tenths place are equal (8=8 ), and the value of the hundredths place of the fraction 0.87 is greater than the value of the hundredths place of the fraction 0.8521 (7>5 ). Therefore, 0.87>0.8521 .
Answer:
0,87>0,8521 .
Sometimes, in order to compare trailing decimals with different amount decimal places, a fraction with fewer decimal places has to be appended with a certain number of zeros on the right. It is quite convenient to equalize the number of decimal places before starting to compare the final decimal fractions by adding a certain number of zeros to the right of one of them.
Example.
Compare the trailing decimals 18.00405 and 18.0040532.
Decision.
Obviously, these fractions are unequal, since their records are different, but at the same time they have equal integer parts (18=18).
Before bitwise comparison of the fractional parts of these fractions, we equalize the number of decimal places. To do this, we assign two digits 0 at the end of the fraction 18.00405, while we get the decimal fraction equal to it 18.0040500.
Values decimal places fractions 18.0040500 and 18.0040532 are equal up to hundred-thousandths, and the value of the millionth place is 18.0040500 less value the corresponding digit of the fraction 18.0040532 (0<3 ), поэтому, 18,0040500<18,0040532 , следовательно, 18,00405<18,0040532 .
Answer:
18,00405<18,0040532 .
When comparing a finite decimal fraction with an infinite one, the final fraction is replaced by an infinite periodic fraction equal to it with a period of 0, after which a comparison is made by digits.
Example.
Compare the ending decimal 5.27 with the infinite non-recurring decimal 5.270013….
Decision.
The integer parts of these decimals are equal. The values of the digits of the tenths and hundredths of these fractions are equal, and in order to perform further comparison, we replace the final decimal fraction with an infinite periodic fraction equal to it with a period of 0 of the form 5.270000 ... . Before the fifth decimal place, the values of the decimal places 5.270000... and 5.270013... are equal, and on the fifth decimal place we have 0<1 . Таким образом, 5,270000…<5,270013… , откуда следует, что 5,27<5,270013… .
Answer:
5,27<5,270013… .
Comparison of infinite decimal fractions is also carried out bit by bit, and ends as soon as the values of some bit are different.
Example.
Compare the infinite decimals 6.23(18) and 6.25181815….
Decision.
The integer parts of these fractions are equal, the values of the tenth place are also equal. And the value of the hundredths place of the periodic fraction 6.23(18) is less than the hundredths place of the infinite non-periodic decimal fraction 6.25181815…, therefore, 6.23(18)<6,25181815… .
Answer:
6,23(18)<6,25181815… .
Example.
Which of the infinite periodic decimals 3,(73) and 3,(737) is greater?
Decision.
It is clear that 3,(73)=3.73737373… and 3,(737)=3.737737737… . At the fourth decimal place, the bitwise comparison ends, since there we have 3<7 . Таким образом, 3,73737373…<3,737737737… , то есть, десятичная дробь 3,(737) больше, чем дробь 3,(73) .
Answer:
3,(737) .
Compare decimals with natural numbers, common fractions and mixed numbers.
To get the result of comparing a decimal fraction with a natural number, you can compare the integer part of this fraction with a given natural number. In this case, periodic fractions with periods of 0 or 9 must first be replaced with their equal final decimal fractions.
The following is true rule for comparing decimal fraction and natural number: if the integer part of a decimal fraction is less than a given natural number, then the whole fraction is less than this natural number; if the integer part of a fraction is greater than or equal to a given natural number, then the fraction is greater than the given natural number.
Consider examples of the application of this comparison rule.
Example.
Compare natural number 7 with decimal fraction 8.8329….
Decision.
Since the given natural number is less than the integer part of the given decimal fraction, then this number is less than the given decimal fraction.
Answer:
7<8,8329… .
Example.
Compare the natural number 7 and the decimal 7.1.
This topic will consider both a general scheme for comparing decimal fractions and a detailed analysis of the principle of comparing finite and infinite fractions. Let us fix the theoretical part by solving typical problems. We will also analyze with examples the comparison of decimal fractions with natural or mixed numbers, and ordinary fractions.
Let's make a clarification: in the theory below, only positive decimal fractions will be compared.
Yandex.RTB R-A-339285-1
General principle for comparing decimal fractions
For each finite decimal and infinite recurring decimal fraction, there are certain common fractions corresponding to them. Consequently, the comparison of finite and infinite periodic fractions can be made as a comparison of their corresponding ordinary fractions. Actually, this statement is the general principle for comparing decimal periodic fractions.
Based on the general principle, the rules for comparing decimal fractions are formulated, adhering to which it is possible not to convert the compared decimal fractions into ordinary ones.
The same can be said about the cases when a periodic decimal fraction is compared with natural numbers or mixed numbers, ordinary fractions - the given numbers must be replaced with their corresponding ordinary fractions.
If we are talking about comparing infinite non-periodic fractions, then it is usually reduced to comparing finite decimal fractions. For consideration, such a number of signs of the compared infinite non-periodic decimal fractions is taken, which will make it possible to obtain the result of the comparison.
Equal and unequal decimals
Definition 1Equal Decimals- these are two final decimal fractions, which have the same ordinary fractions corresponding to them. Otherwise, decimals are unequal.
Based on this definition, it is easy to justify such a statement: if at the end of a given decimal fraction we sign or, conversely, discard several digits 0, then we get a decimal fraction equal to it. For example: 0 , 5 = 0 , 50 = 0 , 500 = ... . Or: 130 , 000 = 130 , 00 = 130 , 0 = 130 . In fact, adding or discarding zero at the end of the fraction on the right means multiplying or dividing by 10 the numerator and denominator of the corresponding ordinary fraction. Let us add to what has been said the main property of fractions (by multiplying or dividing the numerator and denominator of a fraction by the same natural number, we obtain a fraction equal to the original one) and we have a proof of the above statement.
For example, the decimal fraction 0, 7 corresponds to an ordinary fraction 7 10. Adding zero to the right, we get the decimal fraction 0, 70, which corresponds to the ordinary fraction 70 100, 7 70 100: 10 . I.e.: 0 , 7 = 0 , 70 . And vice versa: discarding zero in the decimal fraction 0, 70 on the right, we get the fraction 0, 7 - thus, from the decimal fraction 70 100 we go to the fraction 7 10, but 7 10 \u003d 70: 10 100: 10 Then: 0, 70 \u003d 0 , 7 .
Now consider the content of the concept of equal and unequal infinite periodic decimal fractions.
Definition 2
Equal infinite periodic fractions are infinite periodic fractions that have equal ordinary fractions corresponding to them. If the ordinary fractions corresponding to them are not equal, then the periodic fractions given for comparison are also unequal.
This definition allows us to draw the following conclusions:
If the records of the given periodic decimal fractions are the same, then such fractions are equal. For example, the periodic decimals 0, 21 (5423) and 0, 21 (5423) are equal;
If in the given decimal periodic fractions the periods start from the same position, the first fraction has a period of 0, and the second - 9; the value of the digit preceding period 0 is one more than the value of the digit preceding period 9 , then such infinite periodic decimal fractions are equal. For example, periodic fractions 91 , 3 (0) and 91 , 2 (9) are equal, as well as fractions: 135 , (0) and 134 , (9) ;
Any two other periodic fractions are not equal. For example: 8 , 0 (3) and 6 , (32) ; 0 , (42) and 0 , (131) etc.
It remains to consider equal and unequal infinite non-periodic decimal fractions. Such fractions are irrational numbers, and they cannot be converted to ordinary fractions. Therefore, the comparison of infinite non-periodic decimal fractions is not reduced to the comparison of ordinary ones.
Definition 3
Equal infinite non-recurring decimals are non-periodic decimal fractions, the entries of which are exactly the same.
The question would be logical: how to compare records if it is impossible to see the “finished” record of such fractions? When comparing infinite non-periodic decimal fractions, it is necessary to consider only a certain finite number of signs of the fractions specified for comparison so that this allows us to draw a conclusion. Those. in essence, comparing infinite non-recurring decimals is comparing finite decimals.
This approach makes it possible to assert the equality of infinite non-periodic fractions only up to the considered digit. For example, the fractions 6, 73451 ... and 6, 73451 ... are equal to within hundred thousandths, because the end decimals 6, 73451 and 6, 7345 are equal. Fractions 20, 47 ... and 20, 47 ... are equal to within hundredths, because the fractions 20, 47 and 20, 47 are equal, and so on.
The inequality of infinite non-periodic fractions is established quite concretely with obvious differences in the records. For example, fractions 6, 4135 ... and 6, 4176 ... or 4, 9824 ... and 7, 1132 ... and so on are unequal.
Rules for comparing decimal fractions. Solution of examples
If it is established that two decimal fractions are not equal, it is usually also necessary to determine which of them is greater and which is less. Consider the rules for comparing decimal fractions, which make it possible to solve the above problem.
Very often, it is enough just to compare the integer parts of the decimal fractions given for comparison.
Definition 4
That decimal fraction, which has a larger integer part, is larger. The smaller fraction is the one whose integer part is smaller.
This rule applies to both finite decimal fractions and infinite ones.
Example 1
It is necessary to compare decimal fractions: 7, 54 and 3, 97823 ....
Decision
It is quite obvious that the given decimal fractions are not equal. Their whole parts are equal respectively: 7 and 3 . Because 7 > 3, then 7, 54 > 3, 97823 … .
Answer: 7 , 54 > 3 , 97823 … .
In the case when the integer parts of the fractions given for comparison are equal, the solution of the problem is reduced to comparing the fractional parts. The fractional parts are compared bit by bit - from the tenth place to the lower ones.
Consider first the case when you need to compare trailing decimal fractions.
Example 2
You want to compare the end decimals 0.65 and 0.6411.
Decision
Obviously, the integer parts of the given fractions are (0 = 0) . Let's compare the fractional parts: in the tenth place, the values are (6 \u003d 6) , but in the hundredth place, the value of the fraction 0, 65 is greater than the value of the hundredth place in the fraction 0, 6411 (5 > 4) . So 0.65 > 0.6411 .
Answer: 0 , 65 > 0 , 6411 .
In some tasks for comparing final decimal fractions with a different number of decimal places, it is necessary to attribute the required number of zeros to the right to a fraction with fewer decimal places. It is convenient to equalize in this way the number of decimal places in given fractions even before the start of the comparison.
Example 3
It is necessary to compare the final decimals 67 , 0205 and 67 , 020542 .
Decision
These fractions are obviously not equal, because their records are different. Moreover, their integer parts are equal: 67 \u003d 67. Before proceeding to the bitwise comparison of the fractional parts of the given fractions, we equalize the number of decimal places by adding zeros to the right in fractions with fewer decimal places. Then we get fractions for comparison: 67, 020500 and 67, 020542. We carry out a bitwise comparison and see that in the hundred-thousandth place the value in the fraction 67 , 020542 is greater than the corresponding value in the fraction 67 , 020500 (4 > 0) . So 67.020500< 67 , 020542 , а значит 67 , 0205 < 67 , 020542 .
Answer: 67 , 0205 < 67 , 020542 .
If it is necessary to compare a finite decimal fraction with an infinite one, then the final fraction is replaced by an infinite one equal to it with a period of 0. Then a bitwise comparison is made.
Example 4
It is necessary to compare the final decimal fraction 6, 24 with an infinite non-periodic decimal fraction 6, 240012 ...
Decision
We see that the integer parts of the given fractions are (6 = 6) . In the tenth and hundredth places, the values of both fractions are also equal. To be able to draw a conclusion, we continue the comparison, replacing the final decimal fraction equal to it with an infinite one with a period of 0 and get: 6, 240000 ... . Having reached the fifth decimal place, we find the difference: 0< 1 , а значит: 6 , 240000 … < 6 , 240012 … . Тогда: 6 , 24 < 6 , 240012 … .
Answer: 6, 24< 6 , 240012 … .
When comparing infinite decimal fractions, a bitwise comparison is also used, which will end when the values in some digit of the given fractions turn out to be different.
Example 5
It is necessary to compare the infinite decimal fractions 7, 41 (15) and 7, 42172 ... .
Decision
In the given fractions, there are equal whole parts, the values of the tenths are also equal, but in the hundredth place we see the difference: 1< 2 . Тогда: 7 , 41 (15) < 7 , 42172 … .
Answer: 7 , 41 (15) < 7 , 42172 … .
Example 6
It is necessary to compare the infinite periodic fractions 4 , (13) and 4 , (131) .
Decision:
Equalities are clear and correct: 4 , (13) = 4 , 131313 … and 4 , (133) = 4 , 131131 … . We compare integer parts and bitwise fractional parts, and fix the discrepancy at the fourth decimal place: 3 > 1 . Then: 4 , 131313 … > 4 , 131131 … , and 4 , (13) > 4 , (131) .
Answer: 4 , (13) > 4 , (131) .
To get the result of comparing a decimal fraction with a natural number, you need to compare the integer part of a given fraction with a given natural number. In this case, it should be taken into account that periodic fractions with periods of 0 or 9 must first be represented as final decimal fractions equal to them.
Definition 5
If the integer part of a given decimal fraction is less than a given natural number, then the whole fraction is smaller with respect to a given natural number. If the integer part of a given fraction is greater than or equal to a given natural number, then the fraction is greater than the given natural number.
Example 7
It is necessary to compare the natural number 8 and the decimal fraction 9, 3142 ... .
Decision:
The given natural number is less than the integer part of the given decimal fraction (8< 9) , а значит это число меньше заданной десятичной дроби.
Answer: 8 < 9 , 3142 … .
Example 8
It is necessary to compare the natural number 5 and the decimal fraction 5, 6.
Decision
The integer part of a given fraction is equal to a given natural number, then, according to the above rule, 5< 5 , 6 .
Answer: 5 < 5 , 6 .
Example 9
It is necessary to compare the natural number 4 and the periodic decimal fraction 3 , (9) .
Decision
The period of the given decimal fraction is 9, which means that before comparing, it is necessary to replace the given decimal fraction with a finite or natural number equal to it. In this case: 3 , (9) = 4 . Thus, the original data are equal.
Answer: 4 = 3 , (9) .
To compare a decimal fraction with an ordinary fraction or a mixed number, you must:
Write a common fraction or mixed number as a decimal and then compare the decimals or
- write the decimal fraction as a common fraction (except for infinite non-periodic), and then perform a comparison with a given common fraction or mixed number.
Example 10
It is necessary to compare the decimal fraction 0, 34 and the common fraction 1 3 .
Decision
Let's solve the problem in two ways.
- We write the given ordinary fraction 1 3 as a periodic decimal fraction equal to it: 0 , 33333 ... . Then it becomes necessary to compare the decimal fractions 0, 34 and 0, 33333…. We get: 0 , 34 > 0 , 33333 ... , which means 0 , 34 > 1 3 .
- Let's write the given decimal fraction 0, 34 in the form of an ordinary equal to it. I.e.: 0 , 34 = 34 100 = 17 50 . Compare ordinary fractions with different denominators and get: 17 50 > 1 3 . Thus, 0 , 34 > 1 3 .
Answer: 0 , 34 > 1 3 .
Example 11
You need to compare an infinite non-repeating decimal 4 , 5693 ... and a mixed number 4 3 8 .
Decision
An infinite non-periodic decimal fraction cannot be represented as mixed number, but it is possible to convert the mixed number to improper fraction, and, in turn, write it as a decimal fraction equal to it. Then: 4 3 8 = 35 8 and
Those.: 4 3 8 = 35 8 = 4, 375 . Let's compare decimal fractions: 4, 5693 ... and 4, 375 (4, 5693 ... > 4, 375) and get: 4, 5693 ... > 4 3 8 .
Answer: 4 , 5693 … > 4 3 8 .
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
We will call a fraction one or more equal parts of one whole. A fraction is written using two natural numbers, which are separated by a line. For example, 1/2, 14/4, ¾, 5/9, etc.
The number above the bar is called the numerator of the fraction, and the number below the bar is called the denominator of the fraction.
For fractional numbers whose denominator is 10, 100, 1000, etc. agreed to write the number without a denominator. To do this, first write the integer part of the number, put a comma and write the fractional part of this number, that is, the numerator of the fractional part.
For example, instead of 6 * (7/10) they write 6.7.
Such a record is called decimal fraction.
How to compare two decimals
Let's figure out how to compare two decimal fractions. To do this, we first verify one auxiliary fact.
For example, the length of a certain segment is 7 centimeters or 70 mm. Also 7 cm = 7 / 10 dm or in decimal notation 0.7 dm.
On the other hand, 1 mm = 1/100 dm, then 70 mm = 70/100 dm, or in decimal notation 0.70 dm.
Thus, we get that 0.7 = 0.70.
From this we conclude that if zero is added or discarded at the end of the decimal fraction, then a fraction equal to the given one will be obtained. In other words, the value of the fraction will not change.
Fractions with the same denominators
Let's say we need to compare two decimals 4.345 and 4.36.
First, you need to equalize the number of decimal places by adding or discarding zeros to the right. You get 4.345 and 4.360.
Now you need to write them as improper fractions:
- 4,345 = 4345 / 1000 ;
- 4,360 = 4360 / 1000 .
The resulting fractions have the same denominators. By the rule of comparing fractions, we know that in this case, the larger fraction is the one with the larger numerator. So the fraction 4.36 is greater than the fraction 4.345.
Thus, in order to compare two decimal fractions, you must first equalize their number of decimal places, assigning zeros to one of them on the right, and then discarding the comma to compare the resulting natural numbers.
Decimals can be represented as dots on a number line. And therefore, sometimes in the case when one number is greater than another, they say that this number is located to the right of the other, or if it is less, then to the left.
If two decimal fractions are equal, then they are depicted on the number line by the same point.
SECTION 7 DECIMAL FRACTIONS AND ACTIONS WITH THEM
In the section you will learn:
what is a decimal fraction and what is its structure;
how to compare decimals;
what are the rules for adding and subtracting decimal fractions;
how to find the product and the quotient of two decimal fractions;
what is rounding a number and how to round numbers;
how to apply the learned material in practice
§ 29. WHAT IS A DECIMAL FRACTION. COMPARISON OF DECIMAL FRACTIONS
Look at Figure 220. You can see that the length of the segment AB is 7 mm, and the length of the segment DC is 18 mm. To give the lengths of these segments in centimeters, you need to use fractions: 
You know many other examples where fractions with denominators 10,100, 1000, and the like are used. So, 
Such fractions are called decimals. To record them, they use a more convenient form, which is suggested by the ruler from your accessories. Let's look at the example in question.
You know that the length of the segment DC (Fig. 220) can be expressed as a mixed number
If we put a comma after the integer part of this number, and after it the numerator of the fractional part, then we get a more compact notation: 1.8 cm. For the segment AB, then we get: 0.7 cm. Indeed, the fraction is correct, it is less than one, therefore its integer part is 0. The numbers 1.8 and 0.7 are examples of decimals.
The decimal fraction 1.8 is read like this: "one point eight", and the fraction 0.7 - "zero point seven".
How to write fractions 
in decimal form? To do this, you need to know the structure of the decimal notation.
In decimal notation, there is always an integer and a fractional part. they are separated by a comma. In the integer part, classes and digits are the same as for natural numbers. You know that these are classes of units, thousands, millions, etc., and each of them has 3 digits - units, tens and hundreds. In the fractional part of a decimal fraction, classes are not distinguished, and there can be as many digits as you like, their names correspond to the names of the denominators of fractions - tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths, etc. The tenth place is the oldest in the fractional part of a decimal.
In table 40 you see the names of the decimal places and the number "one hundred and twenty-three integers and four thousand five hundred and six hundred thousandths" or 
The name of the fractional part of "hundred-thousandths" in an ordinary fraction determines its denominator, and in decimal - the last digit of its fractional part. You see that in the numerator of the fractional part of the number 
one fewer digits than zeros in the denominator. If this is not taken into account, then we will get an error in writing the fractional part - instead of 4506 hundred-thousandths we will write 4506 ten-thousandths, but 
Therefore, in writing this number as a decimal fraction, you must put 0 after the decimal point (in the tenth place): 123.04506.
Note:
in a decimal fraction, there should be as many digits after the decimal point as there are zeros in the denominator of the corresponding ordinary fraction.
We can now write fractions
in the form of decimals.
Decimals can be compared in the same way as natural numbers. If there are many digits in decimal fractions, then special rules are used. Consider examples.
Task. Compare fractions: 1) 96.234 and 830.123; 2) 3.574 and 3.547.
Solutions. 1, The integer part of the first fraction is the two-digit number 96, and the integer part of the fraction of the second is the three-digit number 830, so:
96,234 < 830,123.
2. In the entries of fractions 3.574 and 3.547 and the whole parts are equal. Therefore, we compare their fractional parts bit by bit. To do this, we write these fractions one below the other:
Each fraction has 5 tenths. But in the first fraction there are 7 hundredths, and in the second - only 4 hundredths. Therefore, the first fraction is greater than the second: 3.574 > 3.547.
Rules for comparing decimal fractions.
1. Of two decimal fractions, the one with the larger integer part is greater.
2. If the integer parts of decimal fractions are equal, then their fractional parts are compared bit by bit, starting from the most significant digit.
Like common fractions, decimal fractions can be placed on the coordinate line. In Figure 221, you see that points A, B and C have coordinates: A (0.2), B (0.9), C (1.6).
Find out more
Decimals are related to the decimal positional number system. However, their appearance has a longer history and is associated with the name of the outstanding mathematician and astronomer al-Kashi ( full name- Jamshid ibn-Masudal-Kashi). In his work "The Key to Arithmetic" (XV centuries), he first formulated the rules for actions with decimal fractions, gave examples of performing actions with them. Knowing nothing about the discovery of al-Kashi, the Flemish mathematician and engineer Simon Stevin “discovered” decimal fractions for the second time approximately 150 years later. In the work "Decimal" (1585 p.), S. Stevin outlined the theory of decimal fractions. He promoted them in every possible way, emphasizing the convenience of decimal fractions for practical calculations.
Separating the integer part from the fractional decimal fraction was proposed in different ways. So, al-Kashi wrote the integer and fractional parts in different ink or put a vertical line between them. S. Stevin put a zero in a circle to separate the integer part from the fractional one. The comma accepted in our time was proposed by the famous German astronomer Johannes Kepler (1571 - 1630).
SOLVE THE CHALLENGES
1173. Write down in centimeters the length of the segment AB if:
1)AB = 5mm; 2)AB = 8mm; 3)AB = 9mm; 4)AB = 2mm.
1174. Read fractions:
1)12,5; 3)3,54; 5)19,345; 7)1,1254;
2)5,6; 4)12,03; 6)15,103; 8)12,1065.
Name: a) the whole part of the fraction; b) the fractional part of the fraction; c) digits of a fraction.
1175. Give an example of a decimal fraction in which the decimal point is:
1) one digit; 2) two digits; 3) three digits.
1176. How many decimal places does a decimal fraction have if the denominator of the corresponding ordinary fraction is equal to:
1)10; 2)100; 3)1000; 4) 10000?
1177. Which of the fractions has the greater integer part:
1) 12.5 or 115.2; 4) 789.154 or 78.4569;
2) 5.25 or 35.26; 5) 1258.00265 or 125.0333;
3) 185.25 or 56.325; 6) 1269.569 or 16.12?
1178. In the number 1256897, separate the last digit with a comma and read the number you got. Then sequentially rearrange the comma one digit to the left and name the fractions that you received.
1179. Read the fractions and write them as a decimal fraction:
1180 Read the fractions and write them as a decimal:
1181. Write in ordinary fraction:
1) 2,5; 4)0,5; 7)315,89; 10)45,089;
2)125,5; 5)12,12; 8)0,15; 11)258,063;
3)0,9; 6)25,36; 9) 458;,025; 12)0,026.
1182. Write in ordinary fraction:
1)4,6; 2)34,45; 3)0,05; 4)185,342.
1183. Write down in decimal fraction:
1) 8 whole 3 tenths; 5) 145 point 14;
2) 12 whole 5 tenths; 6) 125 point 19;
3) 0 whole 5 tenths; 7) 0 whole 12 hundredths;
4) 12 whole 34 hundredths; 8) 0 whole 3 hundredths.
1184. Write in decimal fraction:
1) zero as many as eight thousandths;
2) twenty point four hundredths;
3) thirteen point five hundredths;
4) one hundred and forty-five point two hundredths.
1185. Write the share as a fraction, and then as a decimal:
1)33:100; 3)567:1000; 5)8:1000;
2)5:10; 4)56:1000; 6)5:100.
1186. Write as a mixed number and then as a decimal:
1)188:100; 3)1567:1000; 5)12548:1000;
2)25:10; 4)1326:1000; 6)15485:100.
1187. Write as a mixed number and then as a decimal:
1)1165:100; 3)2546:1000; 5)26548:1000;
2) 69: 10; 4) 1269: 1000; 6) 3569: 100.
1188. Express in hryvnias:
1) 35 k.; 2) 6 k.; 3) 12 UAH 35 kopecks; 4) 123k.
1189. Express in hryvnias:
1) 58 k.; 2) 2 to.; 3) 56 UAH 55 kopecks; 4) 175k.
1190. Write down in hryvnias and kopecks:
1) 10.34 UAH; 2) UAH 12.03; 3) 0.52 UAH; 4) UAH 126.05
1191. Express in meters and write down the answer as a decimal fraction: 1) 5 m 7 dm; 2) 15m 58cm; 3) 5 m 2 mm; 4) 12 m 4 dm 3 cm 2 mm.
1192. Express in kilometers and write down the answer in decimal fraction: 1) 3 km 175 m; 2) 45 km 47 m; 3) 15 km 2 m.
1193. Write down in meters and centimeters:
1) 12.55 m; 2) 2.06 m; 3) 0.25 m; 4) 0.08 m.
1194. The greatest depth of the Black Sea is 2.211 km. Express the depth of the sea in meters.
1195. Compare fractions:
1) 15.5 and 16.5; 5) 4.2 and 4.3; 9) 1.4 and 1.52;
2) 12.4 and 12.5; 6) 14.5 and 15.5; 10) 4.568 and 4.569;
3) 45.8 and 45.59; 7) 43.04 and 43.1; 11)78.45178.458;
4) 0.4 and 0.6; 8) 1.23 and 1.364; 12) 2.25 and 2.243.
1196. Compare fractions:
1) 78.5 and 79.5; 3) 78.3 and 78.89; 5) 25.03 and 25.3;
2) 22.3 and 22.7; 4) 0.3 and 0.8; 6) 23.569 and 23.568.
1197. Write down the decimal fractions in ascending order:
1) 15,3; 6,9; 18,1; 9,3; 12,45; 36,85; 56,45; 36,2;
2) 21,35; 21,46; 21,22; 21,56; 21,59; 21,78; 21,23; 21,55.
1198. Write down the decimal fractions in descending order:
15,6; 15,9; 15,5; 15,4; 15,45; 15,95; 15,2; 15,35.
1199. Express in square meters and write as a decimal:
1) 5 dm2; 2) 15 cm2; 3)5dm212cm2.
1200 . The room is in the shape of a rectangle. Its length is 90 dm, and its width is 40 dm. Find the area of the room. Write your answer in square meters.
1201 . Compare fractions:
1) 0.04 and 0.06; 5) 1.003 and 1.03; 9) 120.058 and 120.051;
2) 402.0022 and 40.003; 6) 1.05 and 1.005; 10) 78.05 and 78.58;
3) 104.05 and 105.05; 7) 4.0502 and 4.0503; 11) 2.205 and 2.253;
4) 40.04 and 40.01; 8) 60.4007і60.04007; 12) 20.12 and 25.012.
1202. Compare fractions:
1) 0.03 and 0.3; 4) 6.4012 and 6.404;
2) 5.03 and 5.003; 5) 450.025 and 450.2054;
1203. Write down five decimal fractions that are between the fractions on the coordinate beam:
1) 6.2 and 6.3; 2) 9.2 and 9.3; 3) 5.8 and 5.9; 4) 0.4 and 0.5.
1204. Write down five decimal fractions that are between the fractions on the coordinate beam: 1) 3.1 and 3.2; 2) 7.4 and 7.5.
1205. Between which two adjacent natural numbers is a decimal fraction placed:
1)3,5; 2)12,45; 3)125,254; 4)125,012?
1206. Write down five decimal fractions for which the inequality is true:
1)3,41 <х< 5,25; 3) 1,59 < х < 9,43;
2) 15,25 < х < 20,35; 4) 2,18 < х < 2,19.
1207. Write down five decimal fractions for which the inequality is true:
1) 3 < х < 4; 2) 3,2 < х < 3,3; 3)5,22 <х< 5,23.
1208. Write down the largest decimal fraction:
1) with two digits after the decimal point, less than 2;
2) with one digit after the decimal point less than 3;
3) with three digits after the decimal point, less than 4;
4) with four digits after the decimal point, less than 1.
1209. Write down the smallest decimal fraction:
1) with two digits after the decimal point, which is greater than 2;
2) with three digits after the decimal point, which is greater than 4.
1210. Write down all the numbers that can be put instead of an asterisk to get the correct inequality:
1) 0, *3 >0,13; 3) 3,75 > 3, *7; 5) 2,15 < 2,1 *;
2) 8,5* < 8,57; 4) 9,3* < 9,34; 6)9,*4>9,24.
1211. What number can be put instead of an asterisk to get the correct inequality:
1)0,*3 >0,1*; 2) 8,5* <8,*7; 3)3,7*>3,*7?
1212. Write down all decimal fractions, the whole part of which is 6, and the fractional part contains three decimal places, written as 7 and 8. Write these fractions in descending order.
1213. Write down six decimal fractions, the whole part of which is 45, and the fractional part consists of four various numbers: 1, 2, 3, 4. Write these fractions in ascending order.
1214. How many decimal fractions can be formed, the whole part of which is equal to 86, and the fractional part consists of three different digits: 1,2,3?
1215. How many decimal fractions can be formed, the whole part of which is equal to 5, and the fractional part is three-digit, written as 6 and 7? Write these fractions in descending order.
1216. Cross out three zeros in the number 50.004007 so that it forms:
1) largest number; 2) the smallest number.
APPLY IN PRACTICE
1217. Measure the length and width of your notebook in millimeters and write down your answer in decimeters.
1218. Write down your height in meters using a decimal fraction.
1219. Measure the dimensions of your room and calculate its perimeter and area. Write your answer in meters and square meters.
REPETITION TASKS
1220. For what values of x is a fraction improper?
1221. Solve the equation:
1222. The store had to sell 714 kg of apples. For the first day, all apples were sold, and for the second - from what was sold on the first day. How many apples were sold in 2 days?
1223. The edge of a cube was reduced by 10 cm and a cube was obtained, the volume of which is 8 dm3. Find the volume of the first cube.
