In this article we will look at the topic “ comparing decimals" First, let's discuss the general principle of comparing decimal fractions. After this, we will figure out which decimal fractions are equal and which are unequal. Next, we will learn 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. We will provide the entire theory with examples with detailed solutions. In conclusion, let's look at the comparison of decimal fractions with natural numbers, ordinary 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 discussed in the articles comparison of rational numbers and comparison of real numbers.

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General principle for comparing decimal fractions

Based on this principle of comparison, rules for comparing decimal fractions are derived that make it possible to do without converting the compared decimal fractions into common fractions. We will discuss these rules, as well as examples of their application, in the following paragraphs.

A similar principle is used to compare finite decimal fractions or infinite periodic decimal fractions with natural numbers, ordinary fractions and mixed numbers: the compared numbers are replaced by their corresponding ordinary fractions, after which the ordinary fractions are compared.

Regarding comparisons of infinite non-periodic decimals, then it usually comes down to comparing finite decimal fractions. To do this, consider the number of signs of the compared infinite non-periodic decimal fractions that allows you to obtain the result of the comparison.

Equal and unequal decimals

First we introduce definitions of equal and unequal decimal fractions.

Definition.

The two final decimal fractions are called equal, if their corresponding ordinary fractions are equal, otherwise these decimal fractions are called unequal.

Based on this definition, it is easy to justify the following statement: if you add or discard several digits 0 at the end of a given decimal fraction, you will 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 a zero at the end of a 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 states 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 gives a fraction equal to the original one.

For example, the decimal fraction 0.5 corresponds to the common fraction 5/10, after adding a zero to the right, the decimal fraction 0.50 corresponds, which corresponds to the common fraction 50/100, and. Thus, 0.5=0.50. Conversely, if in the decimal fraction 0.50 we discard 0 on the right, then we get the fraction 0.5, so from the ordinary fraction 50/100 we come to the fraction 5/10, but . Therefore, 0.50=0.5.

Let's move on to determination of equal and unequal infinite periodic decimal fractions.

Definition.

Two infinite periodic fractions equal, if the corresponding ordinary fractions are equal; if the ordinary fractions corresponding to them are not equal, then the compared periodic fractions are also not equal.

From this definition Three conclusions follow:

• If the notations of periodic decimal fractions completely coincide, 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 begin with 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 greater 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 is known, such decimal fractions cannot be converted into ordinary fractions (such decimal fractions represent irrational numbers), therefore the comparison of infinite non-periodic decimal fractions cannot be reduced to the comparison of ordinary fractions.

Definition.

Two infinite non-periodic decimals equal, if their records completely match.

But there is one caveat: it is impossible to see the “finished” record of endless non-periodic decimal fractions, therefore, it is impossible to be sure of the complete coincidence of their records. How can this be?

When comparing infinite non-periodic decimal fractions, only a finite number of signs of the fractions being compared is considered, which allows one 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 digit in question. Let's give examples. The infinite non-periodic decimals 5.45839... and 5.45839... are equal to the nearest hundred thousandths, since the finite decimals 5.45839 and 5.45839 are equal; non-periodic decimal fractions 19.54... and 19.54810375... are equal to the nearest hundredth, since they are equal to the fractions 19.54 and 19.54.

With this approach, the inequality of infinite non-periodic decimal fractions is established quite definitely. For example, the infinite non-periodic decimals 5.6789... and 5.67732... are not equal, since differences in their notations are obvious (the finite decimals 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 unequal, you often need to find out which of these fractions is greater and which is less than the other. Now we will look at the rules for comparing decimal fractions, allowing us to answer the question posed.

In many cases, it is sufficient to compare whole parts of the decimal fractions being compared. The following is true rule for comparing decimals: the greater is the decimal fraction whose whole part is greater, and the less is the decimal fraction whose whole part is less.

This rule applies to both finite and infinite decimal fractions. Let's look at the solutions to the examples.

Example.

Compare the decimals 9.43 and 7.983023….

Solution.

Obviously, these decimals are not equal. The integer part of the finite 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 decimal fraction 49.43(14) and 1045.45029... is smaller?

Solution.

The integer part of the periodic fraction 49.43(14) is less than the integer part of the infinite non-periodic decimal fraction 1045.45029..., therefore, 49.43(14)<1 045,45029… .

Answer:

49,43(14) .

If the whole parts of the decimal fractions being compared are equal, then to find out which of them is greater and which is less, you have 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 lower ones.

First, let's look at an example of comparing two decimal fractions.

Example.

Compare the ending decimals 0.87 and 0.8521.

Solution.

The integer parts of these decimal fractions are equal (0=0), so we 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 a fraction is 0.87 greater than the value of the hundredths place of a fraction 0.8521 (7>5). Therefore, 0.87>0.8521.

Answer:

0,87>0,8521 .

Sometimes, to perform a comparison of ending decimal fractions with different quantities decimal places, fractions with fewer decimal places must be appended with a number of zeros to 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 ending decimals 18.00405 and 18.0040532.

Solution.

Obviously, these fractions are unequal, since their notations are different, but at the same time they have equal integer parts (18 = 18).

Before the bitwise comparison of the fractional parts of these fractions, we equalize the number of decimal places. To do this, we add two digits 0 at the end of the fraction 18.00405, and we get an equal value decimal 18,0040500 .

The values ​​of the decimal places of the fractions 18.0040500 and 18.0040532 are equal up to the hundred thousandths, and the value of the millionths place of the fraction is 18.0040500 less than value 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 finite fraction is replaced by an equal infinite periodic fraction with a period of 0, after which a comparison is made by digit.

Example.

Compare the finite decimal 5.27 with the infinite non-periodic decimal 5.270013... .

Solution.

The whole parts of these decimal fractions are equal. The values ​​of the tenths and hundredths digits of these fractions are equal, and in order to perform further comparison, we replace the finite decimal fraction with an equal infinite periodic fraction with period 0 of the form 5.270000.... Up to the fifth decimal place, the values ​​of the decimal places 5.270000... and 5.270013... are equal, and at 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 placewise, and ends as soon as the values ​​of some digits turn out to be different.

Example.

Compare the infinite decimals 6.23(18) and 6.25181815….

Solution.

The whole parts of these fractions are equal, and the tenths place values ​​are also equal. And the value of the hundredths place of a periodic fraction 6.23(18) is less than the hundredths place of an 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?

Solution.

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, fractions, and mixed numbers.

The result of comparing a decimal fraction with a natural number can be obtained by comparing the integer part of a given fraction with a given natural number. In this case, periodic fractions with periods of 0 or 9 must first be replaced with finite decimal fractions equal to them.

The following is true rule for comparing decimal fractions and natural numbers: if the whole 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.

Let's look at examples of the application of this comparison rule.

Example.

Compare the natural number 7 with the decimal fraction 8.8329….

Solution.

Since a given natural number is less than the integer part of a given decimal fraction, then this number is less than a given decimal fraction.

Answer:

7<8,8329… .

Example.

Compare the natural number 7 and the decimal fraction 7.1.

The segment AB is equal to 6 cm, that is, 60 mm. Since 1 cm = dm, then 6 cm = dm. This means AB is 0.6 dm. Since 1 mm = dm, then 60 mm = dm. This means AB = 0.60 dm.
Thus, AB = 0.6 dm = 0.60 dm. This means that the decimal fractions 0.6 and 0.60 express the length of the same segment in decimeters. These fractions are equal to each other: 0.6 = 0.60.

If you add a zero or discard the zero at the end of the decimal fraction, you get fraction, equal to this.
For example,

0,87 = 0,870 = 0,8700; 141 = 141,0 = 141,00 = 141,000;
26,000 = 26,00 = 26,0 = 26; 60,00 = 60,0 = 60;
0,900 = 0,90 = 0,9.

Let's compare two decimal fractions 5.345 and 5.36. Let's equalize the number of decimal places by adding a zero to the right of the number 5.36. We get the fractions 5.345 and 5.360.

Let's write them in the form of improper fractions:

These fractions have the same denominators. This means that the one with the larger numerator is larger.
Since 5345< 5360, то which means 5.345< 5,360, то есть 5,345 < 5,36.
To compare two decimal fractions, you must first equalize the number of decimal places 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 on a coordinate ray in the same way as ordinary fractions.
For example, to depict the decimal fraction 0.4 on a coordinate ray, we first represent it as an ordinary fraction: 0.4 = Then we set aside four tenths of a unit segment from the beginning of the ray. We obtain point A(0,4) (Fig. 141).

Equal decimal fractions are represented on the coordinate ray by the same point.

For example, the fractions 0.6 and 0.60 are represented by one point B (see Fig. 141).

The smaller decimal fraction lies on coordinate ray to the left of the larger one, and the larger one to the right of the smaller one.

For example, 0.4< 0,6 < 0,8, поэтому точка A(0,4) лежит левее точки B(0,6), а точка С(0,8) лежит правее точки B(0,6) (см. рис. 141).

Will a decimal change if a zero is added to the end?
A6 zeros?
Formulate a comparison rule decimal fractions

1172. Write the decimal fraction:

a) with four decimal places, equal to 0.87;
b) with five decimal places, equal to 0.541;
c) with three digits after occupied, equal to 35;
d) with two decimal places, equal to 8.40000.

1173. By adding zeros to the right, equalize the number of decimal places in decimal fractions: 1.8; 13.54 and 0.789.

1174. Write shorter fractions: 2.5000; 3.02000; 20,010.

85.09 and 67.99; 55.7 and 55.7000; 0.5 and 0.724; 0.908 and 0.918; 7.6431 and 7.6429; 0.0025 and 0.00247.

1176. Arrange the numbers in ascending order:

3,456; 3,465; 8,149; 8,079; 0,453.

0,0082; 0,037; 0,0044; 0,08; 0,0091

arrange in descending order.

a) 1.41< х < 4,75; г) 2,99 < х < 3;
b) 0.1< х < 0,2; д) 7 < х < 7,01;
c) 2.7< х < 2,8; е) 0,12 < х < 0,13.

1184. Compare the values:

a) 98.52 m and 65.39 m; e) 0.605 t and 691.3 kg;
b) 149.63 kg and 150.08 kg; f) 4.572 km and 4671.3 m;
c) 3.55°C and 3.61°C; g) 3.835 hectares and 383.7 a;
d) 6.781 hours and 6.718 hours; h) 7.521 l and 7538 cm3.

Is it possible to compare 3.5 kg and 8.12 m? Give some examples of quantities that cannot be compared.

1185. Calculate orally:

1186. Restore the chain of calculations

1187. Is it possible to say how many digits after the decimal point there are in a decimal fraction if its name ends with the word:

a) hundredths; b) ten thousandths; c) tenths; d) millionths?

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A lesson in mastering and consolidating new knowledge

Subject : Comparison of 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 interest in mathematics by selecting different types of 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.

Checking homework. 3 min.

Repetition. 8 min.

Explanation of a 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 вопросительный знак между десятичными дробями на рисунке.

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. First, let's present it in the form of an ordinary 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 of 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