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 ordinary 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.
Concerning 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 ending 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.
Three conclusions follow from this definition:
- 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 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 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 to 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 finite 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, in order to compare trailing decimal fractions with different numbers of 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 decimal fraction 18.0040500.
The values of the decimal places of the fractions 18.0040500 and 18.0040532 are equal up to hundred thousandths, and the value of the millionth place of the fraction 18.0040500 is less than the value of the corresponding place 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 digit of a periodic fraction 6.23(18) is less than the hundredths digit 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.
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.
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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 - 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 numbers of decimal places, it is necessary to add the required number of zeros to the right to the 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 places of tenths and hundredths, 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 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 .
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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 integers.
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?
Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for the year; methodological recommendations; discussion programs Integrated LessonsThe purpose 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.
During the classes
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 from the back of the board. 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.
Got: 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.
Thank you for the lesson.