Lesson "comparing decimals." Comparison of decimal fractions – Knowledge Hypermarket
A lesson in mastering and consolidating new knowledge
Subject : Comparison decimals
Dambaeva Valentina Matveevna
Math teacher
MAOU "Secondary School No. 25" Ulan-Ude
Subject. Comparing decimals.
Didactic goal: teach students to compare two decimals. Introduce students to the rule of comparison. Develop the ability to find larger (smaller) fractions.
Educational purpose. To develop students' creative activity in the process of solving examples. Cultivate an interest in mathematics by selecting various types tasks. Cultivate intelligence, ingenuity, and develop flexible thinking. Continue to develop in students the ability to be self-critical about the results of their work.
Lesson equipment. Handout material. Signal cards, task cards, carbon paper.
Visual aids. Tables-tasks, poster-rules.
Type of lesson. Assimilation of new knowledge. Consolidation of new knowledge.
Lesson Plan
Organizational moment. 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
Lesson progress.
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 there 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 bitwise, starting with the most significant digit. Poster rule).
One can imagine that the digits of the same name “compete” whose digit term is larger: one with ones, tens with tens, etc.
4. Explanation of a new topic.
A) What sign (>,< или =) следует заменить question mark between decimals in the figure.
Poster task
3425, 672678 ? 3425, 672478
14, 24000 ? 14, 24
To answer this question you need to learn how to compare decimals.
12, 3 < 15,3
72.1 > 68.4 Why?
Of two decimal fractions, the one with the larger whole part is greater.
13,5 > 13,4
0, 327 > 0,321
Why?
If the whole parts of the fractions being compared 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 you add one or more zeros to the right side of a decimal fraction, the value of the fraction will not change.
Conversely, if a decimal fraction ends in zeros, then these zeros can be discarded, the value of the fraction will not change.
Let's look at three decimal fractions:
1,25 1,250 1,2500
How are they different 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 the sum of the digit terms for each fraction.
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 sum is written on the right. This means that all three fractions represent the same number. Otherwise, these three fractions are equal: 1.25 = 1.250 = 1.2500.
Decimal fractions can be represented on a coordinate ray in the same way as ordinary fractions. For example, to depict the decimal fraction 0.5 on a coordinate ray. Let's first represent it in the form common fraction: 0.5 = 5/10. Then we set aside five tenths of a unit segment from the beginning of the ray. We get point A(0.5)
Equal decimal fractions are represented 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) Working with a textbook, with a rule.
Now try to answer the question that was posed at the beginning of the explanation: what sign (>,< или =) следует заменить вопросительный знак.
5. Consolidation.
№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 C and 3.61 0 C
6.784 h and 6.718 h
№ 2
Write the 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 prepared to complete 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, we got riddles. He himself decided to come up with riddles with smeared numbers and offers them to you. In the following entries, some numbers are blurred. You need to guess what numbers these 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
d) 4.5*8 and 4.593
e) 5.657* and 5.68
The task is on the poster and on individual cards.
Checking and justifying each sign placed.
№ 4
I affirm:
a) 3.7 is less than 3.278
After all, the first number has fewer digits than the second.
b) 25.63 equals 2.563
After all, they have the same numbers in the same order.
Correct my statement
"Counterexample" (oral)
№ 5
What natural numbers are between the 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. Lesson summary.
How to compare two decimal fractions with different integers?
How to compare two decimal fractions with the same integers?
How do you 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 natural numbers are between the numbers?
7.5 and 9.1 3.25 and 5.5
84 and 85.001 0.3 and 4
Enter 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 mathematics teacher” Moscow “Enlightenment” 1984
Kabalevsky Yu.D. " Independent work students in the process of learning mathematics" 1988
Bulanova L.M., Dudnitsyn Yu.P. “Test tasks in mathematics”,
Moscow “Dedication” 1992
V.G. Kovalenko " Didactic games in mathematics lessons" Moscow "Enlightenment" 1990
Minaeva S.S. “Calculations in lessons and extracurricular activities in mathematics” Moscow “Enlightenment” 1983
Objective of the lesson:
- create conditions for deriving the rule for comparing decimal fractions and the ability to apply it;
- repeat writing common fractions as decimals, rounding decimals;
- develop logical thinking, ability to generalize, research skills, speech.
Lesson progress
Guys, let's remember what we did with you in previous lessons?
Answer: studied decimal fractions, wrote ordinary fractions as decimals and vice versa, rounded decimals.
What would you like to do today?
(Students answer.)
But you will find out in a few minutes what we will be doing in class. Open your notebooks and write down the date. A student will go to the board and work with reverse side boards. I will offer you tasks that you complete orally. Write down your answers in your notebook on a line separated by a semicolon. A student at the blackboard writes in a column.
I read the tasks that are written in advance on the board:
Let's check. Who has other answers? Remember the rules.
Received: 1,075; 2,175; 3,275; 4,375; 5,475; 6,575; 7,675.
Establish a pattern and continue the resulting series for another 2 numbers. Let's check.
Take the transcript and under each number (the person answering at the board puts a letter next to the number) put the corresponding letter. Read the word.
Explanation:
So, what will we do in class?
Answer: comparison.
By comparison! Okay, for example, I’ll now start comparing my hands, 2 textbooks, 3 rulers. What do you want to compare?
Answer: decimal fractions.
What topic of the lesson will we write down?
I write the topic of the lesson on the board, and the students write it in their notebooks: “Comparing decimals.”
Exercise: compare the numbers (written on the board)
| 18.625 and 5.784 | 15,200 and 15,200 | |
| 3.0251 and 21.02 | 7.65 and 7.8 | |
| 23.0521 and 0.0521 | 0.089 and 0.0081 |
First we open the left side. Whole parts are different. We draw a conclusion about comparing decimal fractions with different integer parts. Open the right side. Whole parts are equal numbers. How to compare?
Offer: write decimals as fractions and compare.
Write a comparison of ordinary fractions. If you convert each decimal fraction into a common fraction and compare 2 fractions, it will take a lot of time. Maybe we can come up with a comparison rule? (Students suggest.) I wrote out the rule for comparing decimal fractions, which the author suggests. Let's compare.
There are 2 rules printed on a piece of paper:
- If the whole parts of decimal fractions are different, then the fraction with the larger whole part is larger.
- If the whole parts of decimal fractions are the same, then the larger fraction is the one whose first of the mismatched decimal places is larger.
You and I have made a discovery. And this discovery is the rule for comparing decimal fractions. It coincided with the rule proposed by the author of the textbook.
I noticed that the rules say which of the 2 fractions is greater. Can you tell me which of the 2 decimal fractions is smaller?
Complete in notebook No. 785(1, 2) on page 172. The task is written on the board. Students comment and the teacher makes signs.
Exercise: compare
3.4208 and 3.4028
So what did we learn to do today? Let's check ourselves. Work on pieces of paper with carbon paper.
Students compare decimal fractions using >,<, =. Когда ученики выполнят задание, то листок сверху оставляют себе, а листок снизу сдают учителю.
Independent work.
(Check - answers on the back of the board.)
Compare
148.05 and 14.805
6.44806 and 6.44863
35.601 and 35.6010
The first one to do it receives task (performs from the back of the board) No. 786(1, 2):
Find the pattern and write down the next number in the sequence. In which sequences are the numbers arranged in ascending order, and in which are they in descending order?
Answer:
- 0.1; 0.02; 0.003; 0.0004; 0.00005; (0.000006) – decreasing
- 0.1 ; 0.11; 0.111; 0.1111; 0.11111; (0.111111) – increases.
After the last student submits the work, check it.
Students compare their answers.
Those who did everything correctly will give themselves a mark of “5”, those who made 1-2 mistakes – “4”, 3 mistakes – “3”. Find out in which comparisons errors were made, on which rule.
Write down your homework: No. 813, No. 814 (clause 4, p. 171). Comment. If you have time, complete No. 786(1, 3), No. 793(a).
Lesson summary.
- What did you guys learn to do in class?
- Did you like it or not?
- What were the difficulties?
Take the sheets and fill them out, indicating the degree of your assimilation of the material:
- fully mastered, I can perform;
- I have completely mastered it, but find it difficult to use;
- partially mastered;
- not learned.
Thanks for the lesson.
A fraction is one or more equal parts of one whole. A fraction is written using two natural numbers separated by a line. For example, 1/2, 14/4, ¾, 5/9, etc.
The number written above the line is called the numerator of the fraction, and the number written below the line is called the denominator of the fraction.
For fractional numbers whose denominator is 10, 100, 1000, etc. We agreed to write down 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.
This notation is usually called a decimal fraction.
How to compare two decimals
Let's figure out how to compare two decimal fractions. To do this, let us 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 we add or discard a zero at the end of a decimal fraction, we get a fraction equal to the given one. In other words, the value of the fraction will not change.
Fractions with like denominators
Let's say we need to compare two decimal fractions 4.345 and 4.36.
First you need to equalize the number of decimal places by adding or discarding zeros on the right. The results will be 4.345 and 4.360.
Now you need to write them down as improper fractions:
- 4,345 = 4345 / 1000 ;
- 4,360 = 4360 / 1000 .
The resulting fractions have the same denominators. According to the rule for comparing fractions, we know that in this case the fraction with the larger numerator is greater. This means that the fraction 4.36 is greater than the fraction 4.345.
Thus, in order to compare two decimal fractions, you must first equalize the number of decimal places in them by adding zeros to one of them on the right, and then, discarding the comma, compare the resulting natural numbers.
Decimal fractions 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 represented by the same point on the number line.
This topic will consider both the general scheme for comparing decimal fractions and a detailed analysis of the principle of comparing finite and infinite fractions. We will strengthen the theoretical part by solving typical problems. We will also look at examples of comparison of decimal fractions with natural or mixed numbers, and ordinary fractions.
Let us make a clarification: in theory, the comparison of only positive decimal fractions will be considered below.
Yandex.RTB R-A-339285-1
General principle for comparing decimal fractions
For every finite decimal and infinite periodic decimal, there are certain ordinary fractions corresponding to them. Consequently, a comparison of finite and infinite periodic fractions can be made as a comparison of the corresponding ordinary fractions. Actually, this statement is the general principle for comparing decimal periodic fractions.
Based on the general principle, 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 cases when a decimal periodic 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 finite decimal fractions whose corresponding ordinary fractions are equal. Otherwise the decimals are unequal.
Based on this definition, it is easy to justify the following statement: if you sign or, conversely, discard several digits 0 at the end of a given decimal fraction, you will get a decimal fraction equal to it. For example: 0, 5 = 0, 50 = 0, 500 = …. Or: 130, 000 = 130, 00 = 130, 0 = 130. Essentially, adding or dropping a zero at the end of a fraction on the right means multiplying or dividing by 10 the numerator and denominator of the corresponding ordinary fraction. Let's add to what has been said the basic 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 the common fraction 7 10. By adding zero to the right, we get the decimal fraction 0, 70, which corresponds to the common fraction 70 100, 7 70 100: 10 . That is: 0.7 = 0.70. And vice versa: discarding the zero on the right in the decimal fraction 0, 70, we get the fraction 0, 7 - thus, from the decimal fraction 70 100 we go to the fraction 7 10, but 7 10 = 70: 10 100: 10 Then: 0, 70 = 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 whose corresponding ordinary fractions are equal. 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 notations of the given periodic decimal fractions coincide, then such fractions are equal. For example, the periodic decimal fractions 0.21 (5423) and 0.21 (5423) are equal;
If in the given decimal periodic fractions the periods begin from the same position, the first fraction has a period of 0, and the second has a period of 9; the value of the digit preceding period 0 is one greater than the value of the digit preceding period 9, then such infinite periodic decimal fractions are equal. For example, the periodic fractions 91, 3 (0) and 91, 2 (9), as well as the fractions: 135, (0) and 134, (9) are equal;
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 cannot be converted into ordinary fractions. Consequently, the comparison of infinite non-periodic decimal fractions is not reduced to the comparison of ordinary ones.
Definition 3
Equal infinite non-periodic decimals- these are non-periodic decimal fractions, the entries of which completely coincide.
The logical question would be: how to compare records if it is impossible to see the “finished” record of such fractions? When comparing infinite non-periodic decimal fractions, you need to consider only a certain finite number of signs of the fractions specified for comparison so that this allows you to draw a conclusion. Those. Essentially, comparing infinite non-periodic decimals is comparing finite decimals.
This approach makes it possible to assert the equality of infinite non-periodic fractions only up to the digit in question. For example, the fractions 6, 73451... and 6, 73451... are equal to the nearest hundred thousandths, because the final decimal fractions 6, 73451 and 6, 7345 are equal. The fractions 20, 47... and 20, 47... are equal to the nearest hundredths, because the fractions 20, 47 and 20, 47 and so on are equal.
The inequality of infinite non-periodic fractions is established quite specifically with obvious differences in the notations. For example, the fractions 6, 4135... and 6, 4176... or 4, 9824... and 7, 1132... and so on are unequal.
Rules for comparing decimal fractions. Solving Examples
If it is established that two decimal fractions are unequal, it is usually also necessary to determine which is greater and which is less. Let's consider the rules for comparing decimal fractions, which make it possible to solve the above problem.
Very often it is enough just to compare whole parts of the decimal fractions given for comparison.
Definition 4
The decimal fraction whose whole part is larger is the larger one. The smaller fraction is the one whose whole part is smaller.
This rule applies to both finite and infinite decimal fractions.
Example 1
It is necessary to compare the decimal fractions: 7, 54 and 3, 97823....
Solution
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 whole parts of the fractions given for comparison are equal, the solution of the problem is reduced to comparing the fractional parts. Comparison of fractional parts is carried out bit by bit - from the place of tenths to the lower ones.
Let's first consider the case when we need to compare finite decimal fractions.
Example 2
It is necessary to compare the final decimal fractions 0.65 and 0.6411.
Solution
Obviously, the integer parts of the given fractions are equal (0 = 0). Let's compare fractional parts: in the tenths place the values are equal (6 = 6), but in the hundredths place the value of the fraction 0.65 is greater than the value of the hundredths place in the fraction 0.6411 (5 > 4). Thus, 0.65 > 0.6411.
Answer: 0 , 65 > 0 , 6411 .
In some problems comparing finite decimal fractions with different quantities decimal places, it is necessary to add the required number of zeros on 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 starting the comparison.
Example 3
It is necessary to compare the final decimal fractions 67, 0205 and 67, 020542.
Solution
These fractions are obviously not equal, because their records are different. Moreover, their integer parts are equal: 67 = 67. Before we begin the bitwise comparison of the fractional parts of given fractions, let’s equalize the number of decimal places by adding zeros to the right in fractions with fewer decimal places. Then we get the fractions for comparison: 67, 020500 and 67, 020542. We carry out a bitwise comparison and see that in the place of hundred thousandths the value in the fraction 67.020542 is greater than the corresponding value in the fraction 67.020500 (4 > 0). Thus, 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 finite fraction is replaced by an infinite one, equal to it with a period of 0. Then a bitwise comparison is performed.
Example 4
It is necessary to compare the finite decimal fraction 6, 24 with the infinite non-periodic decimal fraction 6, 240012 ...
Solution
We see that the integer parts of the given fractions are equal (6 = 6). In the tenths and hundredths places, the values of both fractions are also equal. To be able to draw a conclusion, we continue the comparison, replacing the finite decimal fraction with an equal infinite fraction with a period of 0 and we 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 place-by-place comparison is also used, which ends when the values in some place 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....
Solution
In the given fractions there are equal integer parts, the values of the tenths are also equal, but in the place of hundredths we see a 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).
Solution:
The following equalities are clear and true: 4, (13) = 4, 131313... and 4, (133) = 4, 131131.... We compare the integer parts and the bitwise fractional parts, and at the fourth decimal place we record the discrepancy: 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 whole part of a given fraction with a given natural number. It should be taken into account that periodic fractions with periods of 0 or 9 must first be represented in the form of finite 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 entire fraction is smaller with respect to the 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....
Solution:
The given natural number is less than the whole 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.
Solution
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).
Solution
The period of a given decimal fraction is 9, which means that before comparison it is necessary to replace the given decimal fraction with a finite or natural number equal to it. IN in this case: 3, (9) = 4. Thus, the original data is equal.
Answer: 4 = 3, (9).
To compare a decimal fraction with a fraction or mixed number, you must:
Write a fraction or mixed number as a decimal, and then compare decimals or
- write a decimal fraction as a common fraction (with the exception of an infinite non-periodic fraction), 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.
Solution
Let's solve the problem in two ways.
- Let's write the given ordinary fraction 1 3 in the form of an equal periodic decimal fraction: 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 as an ordinary fraction equal to it. That is: 0, 34 = 34,100 = 17,50. Let's compare ordinary fractions with different denominators and get: 17 50 > 1 3. Thus, 0, 34 > 1 3.
Answer: 0 , 34 > 1 3 .
Example 11
It is necessary to compare the infinite non-periodic decimal fraction 4, 5693 ... and a mixed number 4 3 8 .
Solution
An infinite non-periodic decimal fraction cannot be represented as a mixed number, but it is possible to convert a mixed number into an improper fraction, and in turn write it as an equal decimal fraction. Then: 4 3 8 = 35 8 and
Those.: 4 3 8 = 35 8 = 4.375. Let's compare the 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 an error in the text, please highlight it and press Ctrl+Enter
SECTION 7 DECIMAL FRACTIONS AND OPERATIONS WITH THEM
In this 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 decimals;
how to find the product and quotient of two decimal fractions;
what is rounding a number and how to round numbers;
how to apply the studied material in practice
§ 29. WHAT IS A DECIMAL? COMPARING DECIMALS
Look at Figure 220. You see that the length of segment AB is 7 mm, and the length of 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 of 10, 100, 1000 and the like are used. So, 
Such fractions are called decimals. To record them, use a more convenient form, which is suggested by the ruler from your accessory. 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, we get a more compact entry: 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 decimal fractions.
The decimal fraction 1.8 is read as follows: “one point eight”, and the fraction 0.7 is “zero point seven”.
How to write fractions 
as decimals? To do this, you need to know the structure of decimal notation.
In notation of a decimal fraction there is always an integer and a fractional part. they are separated by a comma. In the whole part, the classes and ranks are the same as those of 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 the decimal fraction, classes are not distinguished, but there can be as many digits as desired; their names correspond to the names of the denominators of the fractions - tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths, etc. The tenths place is the oldest place in the fractional part of a decimal.
In table 40 you see the names of the decimal places and the number “one hundred twenty-three whole and four thousand five hundred six hundred thousandths” or 
The name of the fractional part “hundred thousandths” in an ordinary fraction determines its denominator, and in the decimal part - the last digit of its fractional part. You see that in the numerator of the fractional part of the number 
There are one fewer digits than zeros in the denominator. If we do not take this into account, then we will get an error in recording the fractional part - instead of 4506 hundred thousandths, we will write 4506 ten thousandths, but 
Therefore, in the recording given number the decimal fraction must be put 0 after the decimal point (in the tenth place): 123.04506.
Please note:
a decimal fraction must have as many digits after the decimal point as there are zeros in the denominator of the corresponding ordinary fraction.
We can now write down the fractions
as decimals.
Decimals can be compared in the same way as natural numbers. If there are many digits in the recording of decimal fractions, then special rules are used. Let's look at examples.
Task. Compare the 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 second fraction is the three-digit number 830, therefore:
96,234 < 830,123.
2. In writing the fractions 3.574 and 3.547 and the integer 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 there are 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 whose whole part is larger is greater.
2. If the integer parts of decimal fractions are equal, then compare their fractional parts bit by bit, starting with the most significant digit.
Like fractions, decimals can be placed on a coordinate ray. 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- Jemshid ibn Masudal-Kashi). In his work “The Key to Arithmetic” (15th century), he first formulated the rules for working with decimal fractions and 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 about 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 whole part from the fractional decimal has been proposed in different ways. Thus, al-Kashi wrote the whole and fractional parts in different inks or put a vertical line between them. S. Stevin put a zero in a circle to separate the whole part from the fractional part. The comma adopted in our time was proposed by the famous German astronomer Johannes Kepler (1571 - 1630).
SOLVE PROBLEMS
1173. Write down the length of segment AB in centimeters if:
1)AB = 5mm; 2)AB = 8mm; 3)AB = 9mm; 4)AB = 2mm.
1174. Read the 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 a fraction; c) fraction digits.
1175. Give an example of a decimal fraction in which after the decimal point there is:
1) one digit; 2) two numbers; 3) three numbers.
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 larger 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. Separate the number 1256897 with a comma last digit and read the number you received. Then successively move the comma one digit to the left and name the fractions you received.
1179. Read the fractions and write them as decimals:
1180 Read the fractions and write them as decimals:
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 in decimal fraction:
1) 8 point 3; 5) 145 point 14;
2) 12 point 5; 6) 125 point 19;
3) 0 point 5; 7) 0 point 12 hundredths;
4) 12 point 34 hundredths; 8) 0 point 3 hundredths.
1184. Write in decimal fraction:
1) zero point eight thousandths;
2) twenty point four;
3) thirteen point five;
4) one hundred forty-five point two hundredths.
1185. Write the fraction 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 k.; 3) 56 UAH 55 kopecks; 4)175k.
1190. Write in hryvnias and kopecks:
1)10.34 UAH; 2) 12.03 UAH; 3) 0.52 UAH; 4) 126.05 UAH.
1191. Express in meters and write the answer as a decimal fraction: 1) 5 m 7 dm; 2) 15 m 58 cm; 3) 5 m 2 mm; 4) 12 m 4 dm 3 cm 2 mm.
1192. Express in kilometers and write the answer as a decimal fraction: 1) 3 km 175 m; 2) 45 km 47 m; 3) 15 km 2 m.
1193. Write 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 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 it in decimal fraction:
1) 5 dm2; 2) 15 cm2; 3)5dm212cm2.
1200. The room is shaped like 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)60.4007i60.04007; 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 the five decimal fractions that are located between the fractions on the coordinate ray:
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 located between the fractions on the coordinate beam: 1) 3.1 and 3.2; 2) 7.4 and 7.5.
1205. Between what 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 holds:
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 holds:
1) 3 < х < 4; 2) 3,2 < х < 3,3; 3)5,22 <х< 5,23.
1208. Write 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 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 in place of the 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 the decimals whose integer part is equal to 6, and whose fractional part contains three decimal places, written as 7 and 8. Write these fractions in descending order.
1213. Write down six decimal fractions, the integer part of which is equal to 45, and the fractional part consists of four different digits: 1, 2, 3, 4. Write these fractions in ascending order.
1214. How many decimal fractions can you make, the integer 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 made, the integer 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 as to form:
1) greatest number; 2) the smallest number.
PUT IT IN PRACTICE
1217. Measure the length and width of your notebook in millimeters and write the answer in decimeters.
1218. Write your height in meters using decimals.
1219. Measure the dimensions of your room and calculate its perimeter and area. Write your answer in meters and square meters.
REVIEW PROBLEMS
1220. At what values of x is the fraction improper?
1221. Solve the equation:
1222. The store had to sell 714 kg of apples. On the first day, all the apples were sold, and on the second day - 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 we obtained a cube whose volume is 8 dm3. Find the volume of the first cube.