How to calculate in a rational way 0.35. Rational ways of computing
In this article, we will begin to study rational numbers. Here we give definitions of rational numbers, give the necessary explanations and give examples of rational numbers. After that, we will focus on how to determine whether a given number is rational or not.
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Definition and examples of rational numbers
In this subsection we give several definitions of rational numbers. Despite differences in wording, all these definitions have the same meaning: rational numbers unite integers and fractional numbers, just as integers unite natural numbers, their opposite numbers, and the number zero. In other words, rational numbers generalize whole and fractional numbers.
Let's start with definitions of rational numbers which is perceived as the most natural.
From the sounded definition it follows that a rational number is:
- Any natural number n . Indeed, any natural number can be represented as an ordinary fraction, for example, 3=3/1.
- Any integer, in particular the number zero. Indeed, any integer can be written as either a positive common fraction, either as a negative common fraction or as zero. For example, 26=26/1 , .
- Any ordinary fraction (positive or negative). This is directly stated by the given definition of rational numbers.
- Any mixed number. Indeed, one can always imagine mixed number in the form of an improper fraction. For example, and .
- Any finite decimal or infinite periodic fraction. This is so because the specified decimal fractions are converted to ordinary fractions. For example, , and 0,(3)=1/3 .
It is also clear that any infinite non-repeating decimal is NOT a rational number, since it cannot be represented as a common fraction.
Now we can easily bring examples of rational numbers. The numbers 4, 903, 100,321 are rational numbers, since they are natural numbers. The integers 58 , −72 , 0 , −833 333 333 are also examples of rational numbers. Ordinary fractions 4/9, 99/3, are also examples of rational numbers. Rational numbers are also numbers.
It can be seen from the above examples that there are both positive and negative rational numbers, and rational number zero is neither positive nor negative.
The above definition of rational numbers can be formulated in a shorter form.
Definition.
Rational numbers call numbers that can be written as a fraction z/n, where z is an integer and n is a natural number.
Let us prove that this definition of rational numbers is equivalent to the previous definition. We know that we can consider the bar of a fraction as a sign of division, then from the properties of dividing integers and the rules for dividing integers, the following equalities follow and . Thus, which is the proof.
Let us give examples of rational numbers, based on this definition. The numbers −5 , 0 , 3 , and are rational numbers, since they can be written as fractions with an integer numerator and a natural denominator of the form and respectively.
The definition of rational numbers can also be given in the following formulation.
Definition.
Rational numbers are numbers that can be written as a finite or infinite periodic decimal fraction.
This definition is also equivalent to the first definition, since any ordinary fraction corresponds to a finite or periodic decimal fraction and vice versa, and any integer can be associated decimal with zeros after the decimal point.
For example, the numbers 5 , 0 , −13 , are examples of rational numbers because they can be written as the following decimals 5.0 , 0.0 , −13.0 , 0.8 and −7,(18) .
We finish the theory of this section with the following statements:
- integer and fractional numbers (positive and negative) make up the set of rational numbers;
- each rational number can be represented as a fraction with an integer numerator and a natural denominator, and each such fraction is a rational number;
- every rational number can be represented as a finite or infinite periodic decimal fraction, and each such fraction represents some rational number.
Is this number rational?
In the previous paragraph, we found out that any natural number, any integer, any ordinary fraction, any mixed number, any final decimal fraction, and also any periodic decimal fraction is a rational number. This knowledge allows us to "recognize" rational numbers from the set of written numbers.
But what if the number is given as some , or as , etc., how to answer the question, is the given number rational? In many cases, it is very difficult to answer it. Let us point out some directions for the course of thought.
If a number is given as a numeric expression that contains only rational numbers and signs arithmetic operations(+, −, · and:), then the value of this expression is a rational number. This follows from how operations on rational numbers are defined. For example, after performing all the operations in the expression, we get a rational number 18 .
Sometimes, after simplifying expressions and more complex type, it becomes possible to determine whether a given number is rational.
Let's go further. The number 2 is a rational number, since any natural number is rational. What about number? Is it rational? It turns out that no - it is not a rational number, it is an irrational number (the proof of this fact by contradiction is given in the textbook on algebra for grade 8, indicated below in the list of references). It has also been proven that Square root from a natural number is a rational number only in those cases when the root is a number that is the perfect square of some natural number. For example, and are rational numbers, since 81=9 2 and 1024=32 2 , and the numbers and are not rational, since the numbers 7 and 199 are not perfect squares of natural numbers.
Is the number rational or not? In this case, it is easy to see that, therefore, this number is rational. Is the number rational? It is proved that the kth root of an integer is a rational number only if the number under the root sign is the kth power of some integer. Therefore, it is not a rational number, since there is no integer whose fifth power is 121.
The method of contradiction allows us to prove that the logarithms of some numbers, for some reason, are not rational numbers. For example, let's prove that - is not a rational number.
Assume the opposite, that is, suppose that is a rational number and can be written as an ordinary fraction m/n. Then and give the following equalities: . The last equality is impossible, since on its left side there is odd number 5 n , and on the right side there is an even number 2 m . Therefore, our assumption is wrong, thus is not a rational number.
In conclusion, it is worth emphasizing that when clarifying the rationality or irrationality of numbers, one should refrain from sudden conclusions.
For example, one should not immediately assert that the product of irrational numbers π and e is an irrational number, this is “as if obvious”, but not proven. This raises the question: “Why would the product be a rational number”? And why not, because you can give an example of irrational numbers, the product of which gives a rational number:.
It is also unknown whether the numbers and many other numbers are rational or not. For example, there are irrational numbers, whose irrational degree is a rational number. To illustrate, let's give a degree of the form , the base of this degree and the exponent are not rational numbers, but , and 3 is a rational number.
Bibliography.
- Mathematics. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
- Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Kozhinova Anastasia
MUNICIPAL NON-TYPICAL BUDGET
GENERAL EDUCATIONAL INSTITUTION
"LYCEUM №76"
WHAT IS THE SECRET OF RATIONAL COUNTING?
Performed:
Student 5 "B" class
Kozhinova Anastasia
Supervisor:
Mathematic teacher
Shiklina Tatiana
Nikolaevna
Novokuznetsk 2013
Introduction………………………………………………………… 3
The main part....……………………………………….......... 5-13
Conclusion and Conclusions………………………………...................... 13-14
References………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….
Applications……………………………………………………. 16-31
I. Introduction
Problem: finding the values of numeric expressions
Goal of the work: search, study of existing methods and techniques of rational counting, their application in practice.
Tasks:
1. Conduct a mini survey in the form of a questionnaire among parallel classes.
2. Analyze on the research topic: the literature available in school library, information in a scientific manual on mathematics for grade 5, on the Internet.
3.Choose the most effective methods and means of rational accounting.
4. Conduct a classification of existing methods of rapid oral and written counting.
5. Create memos containing rational counting techniques for use in parallel 5 classes.
Object of study: rational account.
Subject of study: ways of rational counting.
For efficiency research work I used the following techniques: analysis of information obtained from various resources, synthesis, generalization; opinion poll in the form of a questionnaire. The questionnaire was developed by me in accordance with the purpose and objectives of the study, the age of the respondents and is presented in the main part of the work.
In the course of the research work, issues related to the methods and techniques of rational counting were considered, and recommendations were given to eliminate problems with computing skills, to form a computing culture.
II. Main part
Formation of the computing culture of students
5-6 grades.
It is obvious that the methods of rational counting are a necessary element of the computational culture in the life of every person, primarily because of their practical significance, and students need it in almost every lesson.
Computational culture is the foundation of the study of mathematics and other academic disciplines, because in addition to the fact that calculations activate memory, attention, help rationally organize activities and significantly affect human development.
IN Everyday life, on training sessions When every minute is valued, it is very important to quickly and rationally carry out oral and written calculations without making mistakes and without using any additional computing tools.
We, schoolchildren, face this problem everywhere: in the classroom, at home, in the store, etc. In addition, after grades 9 and 11, we will have to take exams in the form of the IGA and the Unified State Examination, where the use of a microcalculator is not allowed. Therefore, the problem of the formation of a computational culture in each person, an element of which is mastering the methods of rational counting, becomes extremely important.
It is especially necessary to master the methods of rational counting.
in the study of such subjects as mathematics, history, technology, computer science, etc., that is, rational counting helps to master related subjects, better navigate the material being studied, life situations. So what are we waiting for? Let's go to the world of secrets of Rational methods of counting!!!
What problems do students have when doing calculations?
Often, peers of my age have problems when performing various tasks in which it is necessary to perform calculations in a quick and convenient way. . Why???
Here are some guesses:
1. The student did not master the topic studied well
2. The student does not repeat the material
3. Student has poor numeracy skills
4. The student does not want to study this topic
5. The student believes that it will not be useful to him.
I took all these assumptions from my experience and the experience of my classmates and peers. However, in computational exercises important role skills of rational counting play, therefore I have studied, applied and want to present you some tricks of rational counting.
Rational methods of oral and written calculations.
At work and at home, there is a constant need different kind computing. Using the simplest methods of mental counting reduces fatigue, develops attention and memory. Application rational methods calculations are necessary to increase labor, accuracy and speed of calculations. The speed and accuracy of calculations can be achieved only with the rational use of methods and means of mechanizing calculations, as well as with the correct use of mental counting methods.
I. Simplified Number Addition Techniques
There are four methods of addition that allow you to speed up the calculations.
Sequential bitwise addition method used in mental calculations, as it simplifies and speeds up the summation of terms. When using this method, the addition begins with the highest digits: the corresponding digits of the second term are added to the first term.
Example. Let's find the sum of the numbers 5287 and 3564 using the method of sequential bitwise addition.
Solution. We will calculate in the following order:
5 287 + 3 000 = 8 287;
8 287 + 500 = 8 787;
8 787 + 60 = 8 847;
8 847 + 4 = 8 851.
Answer: 8 851
Another way of sequential bitwise addition consists in the fact that the highest rank of the second term is added to the highest digit of the first term, then the next digit of the second term is added to the next digit of the first term, and so on.
Let's consider this solution in the given example, we get:
5 000 + 3 000 = 8 000;
200 + 500 = 700;
Answer: 8851.
round number method . A number that has one significant digit and ends with one or more zeros is called a round number. This method is used when two or more terms can be chosen that can be completed to a round number. The difference between the round number and the number specified in the calculation condition is called the complement. For example, 1000 - 978 = 22. In this case, the number 22 is the arithmetic addition of the number 978 to 1000.
To add by the round number method, one or more terms close to round numbers must be rounded off, add round numbers, and subtract arithmetic additions from the resulting sum.
Example. Find the sum of the numbers 1238 and 193 using the round number method.
Solution. Round the number 193 to 200 and add as follows: 1 238 + 193 \u003d (1 238 + 200) - 7 \u003d 1 431. (associative law)
Method of grouping terms . This method is used when the terms, when grouped together, give round numbers, which are then added together.
Example. Find the sum of the numbers 74, 32, 67, 48, 33 and 26.
Solution. Let's sum the numbers grouped as follows: (74 + 26) + (32 + 48) + (67 + 33) = 280.
(associative-displacement law)
or, when grouping numbers results in equal sums:
Example: 1+2+3+4+5+…+97+98+99+100= (1+100)+(2+99)+(3+98)+…=101x50=5050
(associative-displacement law)
II. Techniques for simplified subtraction of numbers
The method of sequential bitwise subtraction. This method sequentially subtracts each digit subtracted from the reduced one. It is used when numbers cannot be rounded.
Example. Find the difference between the numbers 721 and 398.
Solution. Let's perform actions to find the difference of given numbers in the following sequence:
represent the number 398 as a sum: 300 + 90 + 8 = 398;
do bitwise subtraction:
721 - 300 = 421; 421 - 90 = 331; 331 - 8 = 323.
round number method . This method is used when the subtrahend is close to a round number. To calculate, it is necessary to subtract the subtrahend, taken as a round number, from the reduced, and add the arithmetic addition to the resulting difference.
Example. Let's calculate the difference between the numbers 235 and 197 using the round number method.
Solution. 235 - 197 = 235 - 200 + 3 = 38.
III. Techniques for simplified multiplication of numbers
Multiplication by one followed by zeros. When multiplying a number by a number that includes a unit followed by zeros (10; 100; 1,000, etc.), as many zeros are assigned to it on the right as there are in the multiplier after the unit.
Example. Find the product of the numbers 568 and 100.
Solution. 568 x 100 = 56,800.
bitwise multiplication method . This method is used when multiplying a number by any one-digit number. If you need to multiply a two-digit (three-, four-digit, etc.) number by a single-digit one, then first the single-digit multiplier is multiplied by tens of another factor, then by its units and the resulting products are summed up.
Example. Find the product of the numbers 39 and 7.
Solution. 39 x 7 \u003d (30 + 9) x 7 \u003d (30 x 7) + (9 x 7) \u003d 210 + 63 \u003d 273. (distributive law of multiplication with respect to addition)
round number method . This method is used only when one of the factors is close to a round number. The multiplier is multiplied by a round number, and then by the arithmetic addition, and at the end the second is subtracted from the first product.
Example. Find the product of the numbers 174 and 69.
174 x 69 \u003d 174 x (70-1) \u003d 174 x 70 - 174 x 1 \u003d 12 180 - 174 \u003d 12 006. (distributive law of multiplication with respect to subtraction)
A way to expand one of the factors. In this method, one of the factors is first decomposed into parts (terms), then the second factor is multiplied in turn by each part of the first factor, and the resulting products are summed up.
Example. Find the product of the numbers 13 and 325.
Let's decompose the number 13 into terms: 13 \u003d 10 + 3. Let's multiply each of the terms obtained by 325: 10 x 325 \u003d 3 250; 3 x 325 = 975. Summing up the resulting products: 3250 + 975 = 4225
Mastering the skills of rational mental counting will make your work more efficient. This is possible only with a good mastery of all the above arithmetic operations. The use of rational methods of counting speeds up calculations and provides the necessary accuracy. But not only you need to be able to calculate, but you also need to know the multiplication table, the laws of arithmetic operations, classes and digits.
There are mental counting systems that allow you to count quickly and rationally orally. We will look at some of the most commonly used techniques.
- Multiplying a two-digit number by 11.
We have studied this method, but we have not studied it to the end. the secret of this method is that it can be considered the laws of arithmetic operations.
Examples:
23x11 \u003d 23x (10 + 1) \u003d 23x10 + 23x1 \u003d 253 (distributive law of multiplication with respect to addition)
23x11=(20+3)x 11= 20x11+3x11=253 (distributive law and round number method)
We studied this method, but we didn't know another one. The secret of multiplying two-digit numbers by 11.
By observing the results obtained when multiplying two-digit numbers by 11, I noticed that you can get the answer in a more convenient way. : when multiplying a two-digit number by 11, the digits of this number are moved apart and the sum of these digits is put in the middle.
a) 23 11=253, since 2+3=5;
b) 45 11=495, because 4+5=9;
c) 57 11=627, because 5+7=12, two was placed in the middle, and one was added to the hundreds place;
d) 78 11=858, since 7+8=15, then the number of tens will be equal to 5, and the number of hundreds will increase by one and will be equal to 8.
I found confirmation of this method on the Internet.
2) The product of two-digit numbers that have the same number of tens, and the sum of units is 10, i.e. 23 27; 34 36; 52 58 etc.
rule: the digit of tens is multiplied by the next digit in the natural series, the result is recorded and the product of units is attributed to it.
a) 23 27 = 621. How did you get 621? We multiply the number 2 by 3 (the “two” is followed by the “three”), it will be 6, and next we will assign the product of units: 3 7 \u003d 21, it turns out 621.
b) 34 36 = 1224, since 3 4 = 12, we attribute 24 to the number 12, this is the product of units of these numbers: 4 6.
c) 52 58 \u003d 3016, since we multiply the tens number 5 by 6, it will be 30, we attribute the product of 2 and 8, i.e. 16.
d) 61 69=4209. It is clear that 6 was multiplied by 7 and got 42. And where does the zero come from? We multiplied the units and got: 1 9 \u003d 9, but the result must be two-digit, so we take 09.
3) Divide three-digit numbers that have the same digits by 37. The result is the sum of these identical digits of the three-digit number (or a number equal to three times the digit of the three-digit number).
Examples: a) 222:37=6. This is the sum of 2+2+2=6; b) 333:37=9, because 3+3+3=9.
c) 777:37=21, i.e. to 7+7+7=21.
d) 888:37=24, because 8+8+8=24.
We also take into account the fact that 888:24=37.
III. Conclusion
To unravel the main secret in the topic of my work, I had to work hard - to search, analyze information, question classmates, repeat the early known methods and find many unfamiliar methods of rational counting, and, finally, understand what is his secret? And I realized that the main thing is to know and be able to apply the known ones, find new rational methods of counting, the multiplication table, the composition of the number (classes and digits), the laws of arithmetic operations. Besides,
look for new ways to do this:
- Simplified Number Addition Techniques: (method of sequential bitwise addition; method of a round number; method of decomposing one of the factors into terms);
-Techniques for simplified subtraction of numbers(method of sequential bitwise subtraction; round number method);
-Techniques for simplified multiplication of numbers(multiplication by one followed by zeros; bitwise multiplication method; round number method; expansion method of one of the factors ;
- Secrets of fast mental counting(multiplying a two-digit number by 11: when multiplying a two-digit number by 11, the digits of this number are moved apart and the sum of these digits is put in the middle; the product of two-digit numbers that have the same number of tens, and the sum of units is 10; Division of three-digit numbers consisting of identical digits, on the number 37. There are probably many more such ways, so I will continue to work on this topic next year.
IV. Bibliography
- Savin A. P. Mathematical miniatures / A. P. Savin. - M .: Children's literature, 1991
2. Zubareva I.I., Mathematics, Grade 5: a textbook for students educational institutions/ I.I. Zubareva, A.G. Mordkovich. – M.: Mnemosyne, 2011
4. http:/ / www. xreferat.ru
5. http:/ / www. biografia.ru
6. http:/ / www. Mathematics-repetition. en
V. Applications
Mini study (survey in the form of a questionnaire)
To identify students' knowledge of rational counting, I conducted a survey in the form of a questionnaire on the following questions:
* Do you know what rational methods of counting are?
* If yes, where, and if not, why not?
* How many ways of rational counting do you know?
* Do you have difficulty in mental counting?
* How do you study math? a) on "5"; b) on "4"; c) on "3"
* What do you like most about math?
a) examples; b) tasks; c) fractions
* What do you think, where can mental counting be useful, except for mathematics? * Do you remember the laws of arithmetic operations, if so, which ones?
After conducting a survey, I realized that my classmates do not know enough the laws of arithmetic operations, most of them have problems with rational counting, many students count slowly and with errors, and everyone wants to learn how to count quickly, correctly and in a convenient way. Therefore, the topic of my research work is extremely important for all students and not only.
1. Interesting oral and written methods of calculations that we studied in mathematics lessons, using the examples of the textbook "mathematics, grade 5":
Here are some of them:
to quickly multiply a number by 5, it suffices to note that 5=10:2.
For example, 43x5=(43x10):2=430:2=215;
48x5=(48:2)x10=24x10=240.
To multiply a number by 50 , you can multiply it by 100 and divide by 2.
For example: 122x50=(122x100):2=12200:2=6100
To multiply a number by 25 , you can multiply it by 100 and divide by 4,
For example, 32x25=(32x100):4=3200:4=800
To multiply a number by 125 , you can multiply it by 1000 and divide by 8 ,
For example: 192x125=(192x1000):8=192000:8=24000
To make a round number ending with two 0's divided by 25 , you can divide it by 100 and multiply by 4.
For example: 2400:25=(2400:100) x 4=24 x 4=96
To divide a round number by 50 , can be divided by 100 and multiplied by 2
For example: 4500:50=(4500:100) x 2 =45 x 2 =90
But not only you need to be able to calculate, but you also need to know the multiplication table, the laws of arithmetic operations, the composition of the number (classes and digits) and have the skills to use them
Laws of arithmetic operations.
a + b = b + a
Commutative law of addition
(a + b) + c = a + (b + c)
Associative law of addition
a · b = b · a
Commutative law of multiplication
(a · b) · c = a · (b · c)
Associative law of multiplication
(a = b) · c = a · c = b · c
Distributive law of multiplication (with respect to addition)
Multiplication table.
What is multiplication?
This is smart addition.
After all, it’s smarter to multiply times,
Than to add up everything for an hour.
Multiplication table
We all need it in life.
And not without reason named
MULTIPLY it!
Ranks and classes
In order to make it convenient to read and also remember numbers with large values, they should be divided into so-called “classes”: starting from the right, the number is divided by a space into three digits “first class”, then three more digits are selected, “second class” and etc. Depending on the meaning of the number, the last class can end with three, two or one digit.
For example, the number 35461298 is written as follows:
This number is divided into classes:
482 - first class (class of units)
630 - second class (class of thousands)
35 - third class (class of millions)
Discharge
Each of the digits that make up the class is called its category, the countdown of which also goes to the right.
For example, the number 35 630 482 can be decomposed into classes and digits:
482 - first class
2 - first digit (unit digit)
8 - second digit (tens digit)
4 - third digit (hundreds digit)
630 - second class
0 - first digit (thousands digit)
3 - second digit (digit of tens of thousands)
6 - third digit (hundred thousand digit)
35 - third grade
5 - first digit (digit of units of millions)
3 - second digit (digit of tens of millions)
The number 35 630 482 reads:
Thirty-five million six hundred thirty thousand four hundred and eighty-two.
Problems with rational counting and how to fix them
Rational methods of memorization.
As a result of the survey and observations from the lessons, I noticed that some students solve various problems and exercises poorly because they are not familiar with rational methods of calculation.
1. One of the methods is to bring the studied material into a system that is convenient for memorization and storage in memory.
2. In order for the memorized material to be stored by memory in a certain system, some work must be done on its content.
3. Then you can start mastering each individual part of the text, rereading it and trying to immediately reproduce (repeat to yourself or aloud) what you read.
4. Great value for memorization has a repetition of the material. This is also said folk proverb: "Repetition is the mother of learning." But it must also be repeated reasonably and correctly.
The work of repetition must be revived by drawing on illustrations or examples that did not exist before or have already been forgotten.
Based on the foregoing, we can briefly formulate the following recommendations for the successful assimilation of educational material:
1. Set a task, quickly and firmly remember educational material for a long time.
2. Focus on what needs to be learned.
3. Understand the study material well.
4. Make a plan of the memorized text, highlighting the main thoughts in it, break the text into parts.
5. If the material is large, sequentially assimilate one part after another, and then state everything as a whole.
6. After reading the material, it is necessary to reproduce it (tell what was read).
7. Repeat material until it is forgotten.
8. Distribute the repetition over a longer time.
9. When memorizing, use different types of memory (primarily semantic) and some individual characteristics memory (visual, auditory or motor).
10. Difficult material should be repeated before going to bed, and then in the morning, "for fresh memory."
11. Try to apply the acquired knowledge in practice. This The best way their preservation in memory (not without reason they say: "The real mother of the doctrine is not repetition, but application").
12. It is necessary to acquire more knowledge, to learn something new.
Now you have learned how to quickly and correctly memorize the studied material.
An interesting technique of multiplying some numbers by 9 in combination with the addition of consecutive natural numbers from 2 to 10
12345x9+6=111111
123456x9+7=1111111
1234567x9+8=11111111
12345678x9+9=111111111
123456789x9+10=1111111111
Interesting game "Guess the number"
Have you played the Guess the Number game? This is a very simple game. Let's say I think of a natural number less than 100, write it down on paper (so that there is no way to cheat), and you try to guess it by asking questions that can only be answered with "yes" or "no". Then you guess the number, and I try to guess it. Whoever guesses in the least number of questions wins.
How many questions do you need to guess my number? Do not know? I undertake to guess your number by asking only seven questions. How? But, for example, how. Let you guess the number. I ask, "Is it less than 64?" - "Yes". – “Less than 32?” - "Yes". - "Less than 16?" - "Yes". – “Less than 8?” - "No". - "Less than 12?" - "No". - "Less than 14?" - "Yes". - "Less than 13?" - "No". - "The number 13 is conceived."
It's clear? I divide the set of possible numbers in half, then the remaining half in half again, and so on, until the remainder is one number.
If you liked the game or, on the contrary, you want more, then go to the library and take the book “A. P. Savin (Mathematical miniatures). In this book you will find a lot of interesting and exciting things. Book picture:
Thank you all for your attention
And I wish you success!!!
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Slides captions:
What is the secret of rational counting?
Purpose of the work: search for information, study of existing methods and techniques of rational counting, their application in practice.
Tasks: 1. Conduct a mini survey in the form of a questionnaire among parallel classes. 2. Analyze on the topic of research: the literature available in the school library, information in the textbook on mathematics for grade 5, as well as on the Internet. 3. Choose the most effective methods and means of rational counting. 4. Carry out a classification of existing methods of rapid oral and written counting. 5. Create Memos containing rational counting techniques for use in parallel 5 classes.
As I have already said, the topic of rational counting is relevant not only for students, but for every person, to make sure of this, I conducted a survey among 5th grade students. Questions and answers of the survey are presented to you in the application.
What is a rational account? A rational account is a convenient account (the word rational means convenient, correct)
Why do students have difficulty?
Here are some assumptions: The student: 1. did not master the studied topic well; 2. does not repeat the material; 3. has poor counting skills; 4 . thinks he won't need it.
Rational methods of oral and written calculations. In work and life, the need for various kinds of calculations constantly arises. Using the simplest methods of mental counting reduces fatigue, develops attention and memory.
There are four methods of addition that allow you to speed up the calculations. I. Techniques for simplified addition of numbers
The method of sequential bitwise addition is used in mental calculations, as it simplifies and speeds up the summation of terms. When using this method, the addition begins with the highest digits: the corresponding digits of the second term are added to the first term. Example. Find the sum of the numbers 5287 and 3564 using this method. Solution. We will calculate in the following sequence: 5,287 + 3,000 = 8,287; 8287 + 500 = 8787; 8787 + 60 = 8847; 8847 + 4 = 8851 . Answer: 8 851.
Another way of successive bitwise addition is that the highest digit of the second term is added to the highest digit of the first term, then the next digit of the second term is added to the next digit of the first term, and so on. Let's consider this solution in the given example, we get: 5,000 + 3,000 = 8,000; 200 + 500 = 700; 80 + 60 = 140; 7 + 4 = 11 Answer: 8851.
round number method. A number that ends in one or more zeros is called a round number. This method is used when two or more terms can be chosen that can be completed to a round number. The difference between the round number and the number specified in the calculation condition is called the complement. For example, 1000 - 978 = 22. In this case, the number 22 is the arithmetic complement of the number 978 to 1000. To add by the round number method, one or more terms close to round numbers must be rounded off, add round numbers, and subtract arithmetic additions from the resulting sum. Example. Find the sum of the numbers 1238 and 193 using the round number method. Solution. Round the number 193 to 200 and add as follows: 1238 + 193 = (1238 + 200) - 7 = 1431.
Method for grouping terms. This method is used when the terms, when grouped together, give round numbers, which are then added together. Example. Find the sum of the numbers 74, 32, 67, 48, 33 and 26. Solution. Let's sum the numbers grouped as follows: (74 + 26) + (32 + 48) + (67 + 33) = 280.
Addition method based on the grouping of terms. Example: 1+2+3+4+5+6+7+8+9+…….+97+98+99+100=(1+100)+(2+99)+(3+98)= 101x50=5050.
II. Techniques for simplified subtraction of numbers
The method of sequential bitwise subtraction. This method sequentially subtracts each digit subtracted from the reduced one. It is used when numbers cannot be rounded. Example. Find the difference between the numbers 721 and 398 . Let's perform actions to find the difference of given numbers in the following sequence: represent the number 398 as a sum: 300 + 90 + 8 = 398; perform a bitwise subtraction: 721 - 300 = 421; 421 - 90 = 331; 331 - 8 = 323.
round number method. This method is used when the subtrahend is close to a round number. To calculate, it is necessary to subtract the subtrahend, taken as a round number, from the reduced, and add the arithmetic addition to the resulting difference. Example. Let's calculate the difference between the numbers 235 and 197 using the round number method. Solution. 235 - 197 = 235 - 200 + 3 = 38.
III. Techniques for simplified multiplication of numbers
Multiplication by one followed by zeros. When multiplying a number by a number that includes a unit followed by zeros (10; 100; 1,000, etc.), as many zeros are assigned to it on the right as there are in the multiplier after the unit. Example. Find the product of the numbers 568 and 100. Solution. 568 x 100 = 56,800.
The method of sequential bitwise multiplication. This method is used when multiplying a number by any one-digit number. If you need to multiply a two-digit (three-, four-digit, etc.) number by a single one, then first one of the factors is multiplied by tens of the other factor, then by its units and the resulting products are summed up. Example. Let's find the product of the numbers 39 and 7. Solution. 39 x 7 = (30 x 7) + (9 x 7) = 210 + 63 = 273.
round number method. This method is used only when one of the factors is close to a round number. The multiplier is multiplied by a round number, and then by the arithmetic addition, and at the end the second is subtracted from the first product. Example. Let's find the product of the numbers 174 and 69. Solution. 174 x 69 = (174 x 70) - (174 x 1) = 12,180 - 174 = 12,006.
A way to expand one of the factors. In this method, one of the factors is first decomposed into parts (terms), then the second factor is multiplied in turn by each part of the first factor, and the resulting products are summed up. Example. Let's find the product of the numbers 13 and 325. Solution. Let's decompose the number into terms: 13 \u003d 10 + 3. Let's multiply each of the terms obtained by 325: 10 x 325 \u003d 3 250; 3 x 325 = 975 We sum up the products obtained: 3,250 + 975 = 4,225.
Secrets of fast mental counting. There are mental counting systems that allow you to count quickly and rationally orally. We will look at some of the most commonly used techniques.
Multiplying a two-digit number by 11.
Examples: 23x11= 23x(10+1) = 23x10+23x1=253(distributive law of multiplication with respect to addition) 23x11=(20+3)x 11= 20x11+3x11=253 (distributive law and round number method) We studied this method , but we did not know one more secret of multiplying two-digit numbers by 11.
Observing the results obtained when multiplying two-digit numbers by 11, I noticed that you can get the answer in a more convenient way: when multiplying a two-digit number by 11, the digits of this number are moved apart and the sum of these digits is put in the middle. Examples. a) 23 11=253, since 2+3=5; b) 45 11=495, because 4+5=9; c) 57 11=627, because 5+7=12, two was placed in the middle, and one was added to the hundreds place; I found confirmation of this method on the Internet.
2) The product of two-digit numbers that have the same number of tens, and the sum of units is 10, i.e. 23 27; 34 36; 52 58, etc. Rule: the digit of tens is multiplied by the next digit in the natural series, the result is written down and the product of units is attributed to it. Examples. a) 23 27 = 621. How did you get 621? We multiply the number 2 by 3 (the “two” is followed by the “three”), it will be 6, and next we will assign the product of units: 3 7 \u003d 21, it turns out 621. b) 34 36 = 1224, since 3 4 = 12, we attribute 24 to the number 12, this is the product of units of these numbers: 4 6.
3) Dividing three-digit numbers consisting of the same digits by the number 37. The result is equal to the sum of these identical digits of the three-digit number (or a number equal to three times the digit of the three-digit number). Examples. a) 222:37=6. This is the sum of 2+2+2=6 . b) 333:37=9, because 3+3+3=9. c) 777:37=21, because 7+7+7=21. d) 888:37=24, since 8+8+8=24. We also take into account the fact that 888:24=37.
Mastering the skills of rational mental counting will make your work more efficient. This is possible only with a good mastery of all the above arithmetic operations. The use of rational methods of counting speeds up calculations and provides the necessary accuracy.
Conclusion To unravel the main secret in the topic of my work, I had to work hard - to search, analyze information, question classmates, repeat the early known methods and find many unfamiliar methods of rational counting, and, finally, understand what is its secret? And I realized that the main thing is to know and be able to apply the known ones, to find new rational methods of counting, to know the multiplication table, the composition of the number (classes and digits), the laws of arithmetic operations. Other than that, look for new ways to do this:
Techniques for simplified addition of numbers: (method of sequential bitwise addition; method of a round number; method of decomposing one of the factors into terms); - Techniques for simplified subtraction of numbers (method of sequential bitwise subtraction; method of a round number); - Techniques for simplified multiplication of numbers (multiplication by one followed by zeros; method of sequential bitwise multiplication; method of a round number; method of expanding one of the factors; - Secrets of quick mental counting (multiplication of a two-digit number by 11: when multiplying a two-digit number by 11, the digits of this number are moved apart and in the middle they put the sum of these digits; the product of two-digit numbers that have the same number of tens, and the sum of units is 10; The division of three-digit numbers consisting of the same digits by the number 37. Probably, there are still a lot of such ways, so I will continue to work on this topic next year.
In conclusion, I would like to end my speech with the following words:
Thank you all for your attention, I wish you success!!!
In the distant past, when the calculus system had not yet been invented, people counted everything on their fingers. With the advent of arithmetic and the basics of mathematics, it has become much easier and more practical to keep records of goods, products, and household items. However, what does it look like modern system calculus: what types are divided existing numbers and what does " rational view numbers"? Let's see.
How many types of numbers are there in mathematics?
The very concept of "number" denotes a certain unit of any object, which characterizes its quantitative, comparative or ordinal indicators. In order to correctly calculate the number of certain things or perform certain mathematical operations with numbers (add, multiply, etc.), you should first of all familiarize yourself with the varieties of these same numbers.
So, the existing numbers can be divided into the following categories:
- Natural numbers are those numbers with which we count the number of objects (the smallest natural number is 1, it is logical that the series of natural numbers is infinite, that is, there is no largest natural number). The set of natural numbers is usually denoted by the letter N.
- Whole numbers. This set includes everything, while negative values are added to it, including the number "zero". The designation of the set of integers is written in the form of the Latin letter Z.
- Rational numbers are those that we can mentally convert into a fraction, the numerator of which will belong to the set of integers, and the denominator will belong to natural numbers. Below we will analyze in more detail what "rational number" means, and give a few examples.
- - a set that includes all rational and This set is denoted by the letter R.
- Complex numbers contain part of the real and part of the variable. They are used in solving various cubic equations, which, in turn, can have a negative expression in the formulas (i 2 = -1).
What does "rational" mean: we analyze it with examples
If those numbers that we can represent as a common fraction are considered rational, then it turns out that all positive and negative integers are also included in the set of rational ones. After all, any integer, for example 3 or 15, can be represented as a fraction, where the denominator will be one.
Fractions: -9/3; 7/5, 6/55 are examples of rational numbers.
What does "rational expression" mean?
Go ahead. We have already discussed what the rational form of numbers means. Let's now imagine a mathematical expression that consists of the sum, difference, product, or quotient of various numbers and variables. Here is an example: a fraction, in the numerator of which is the sum of two or more integers, and the denominator contains both an integer and some variable. It is this expression that is called rational. Based on the rule "you cannot divide by zero", you can guess that the value of this variable cannot be such that the value of the denominator becomes zero. Therefore, when solving a rational expression, you must first determine the range of the variable. For example, if the denominator contains the following expression: x+5-2, then it turns out that "x" cannot be equal to -3. Indeed, in this case, the entire expression turns into zero, therefore, when solving, it is necessary to exclude the integer -3 for this variable.
How to solve rational equations correctly?
Rational expressions can contain quite a large number of numbers and even 2 variables, so sometimes their solution becomes difficult. To facilitate the solution of such an expression, it is recommended to perform certain operations in a rational way. So, what does "in a rational way" mean, and what rules should be applied when deciding?
- The first type, when it is enough just to simplify the expression. To do this, you can resort to the operation of reducing the numerator and denominator to an irreducible value. For example, if the numerator contains the expression 18x, and the denominator 9x, then, reducing both indicators by 9x, we get just an integer equal to 2.
- The second method is practical when we have a monomial in the numerator and a polynomial in the denominator. Let's look at an example: in the numerator we have 5x, and in the denominator - 5x + 20x 2 . In this case, it is best to take the variable in the denominator out of brackets, we get the following form of the denominator: 5x(1+4x). And now you can use the first rule and simplify the expression by reducing 5x in the numerator and denominator. As a result, we get a fraction of the form 1/1+4x.
What operations can be performed with rational numbers?
The set of rational numbers has a number of its own peculiarities. Many of them are very similar to the characteristic that is present in integers and natural numbers, in view of the fact that the latter are always included in the rational set. Here are a few properties of rational numbers, knowing which, you can easily solve any rational expression.
- The commutativity property allows you to sum two or more numbers, regardless of their order. Simply put, the sum does not change from a change in the places of the terms.
- The distributivity property allows solving problems using the distributive law.
- And finally, the operations of addition and subtraction.
Even schoolchildren know what the "rational type of numbers" means and how to solve problems based on such expressions, so an educated adult simply needs to remember at least the basics of the set of rational numbers.
IN this lesson addition and subtraction of rational numbers are considered. The topic is classified as complex. Here it is necessary to use the entire arsenal of previously acquired knowledge.
The rules for adding and subtracting integers are also valid for rational numbers. Recall that rational numbers are numbers that can be represented as a fraction, where a - is the numerator of a fraction b is the denominator of the fraction. Wherein, b should not be null.
In this lesson, we will increasingly refer to fractions and mixed numbers as one common phrase - rational numbers.
Lesson navigation:Example 1 Find the value of an expression:
We enclose each rational number in brackets along with its signs. We take into account that the plus which is given in the expression is the sign of the operation and does not apply to fractions. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract a smaller module from a larger module, and put the sign of the rational number whose module is larger in front of the answer. And in order to understand which module is greater and which is less, you need to be able to compare the modules of these fractions before calculating them:
The modulus of a rational number is greater than the modulus of a rational number. Therefore, we subtracted from . Got an answer. Then, reducing this fraction by 2, we got the final answer.
Some primitive actions, such as putting numbers in brackets and putting down modules, can be skipped. This example can be written in a shorter way:
Example 2 Find the value of an expression:
We enclose each rational number in brackets along with its signs. We take into account that the minus between rational numbers and is the sign of the operation and does not apply to fractions. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
Let's replace subtraction with addition. Recall that for this you need to add to the minuend the number opposite to the subtrahend:
We got the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus before the answer:
Note. It is not necessary to enclose every rational number in parentheses. This is done for convenience, in order to clearly see what signs rational numbers have.
Example 3 Find the value of an expression:
In this expression, the fractions have different denominators. To make things easier for ourselves, we reduce these fractions to common denominator. We will not go into detail on how to do this. If you experience difficulties, be sure to repeat the lesson.
After reducing the fractions to a common denominator, the expression will take the following form:
This is the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the received answer we put the sign of the rational number, the module of which is greater:
Let's write down the solution of this example in a shorter way:
Example 4 Find the value of an expression
We calculate this expression in the following way: we add the rational numbers and , then subtract the rational number from the result obtained.
First action:
Second action:
Example 5. Find the value of an expression:
Let's represent the integer −1 as a fraction, and translate the mixed number into an improper fraction:
We enclose each rational number in brackets along with its signs:
We got the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the received answer we put the sign of the rational number, the module of which is greater:
Got an answer.
There is also a second solution. It consists in putting together whole parts separately.
So, back to the original expression:
Enclose each number in parentheses. For this mixed number temporarily:
Let's calculate the integer parts:
(−1) + (+2) = 1
In the main expression, instead of (−1) + (+2), we write the resulting unit:
The resulting expression. To do this, write the unit and the fraction together:
Let's write the solution in this way in a shorter way:
Example 6 Find the value of an expression
Convert the mixed number to an improper fraction. We rewrite the rest without change:
We enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
Let's write down the solution of this example in a shorter way:
Example 7 Find value expression
Let's represent the integer −5 as a fraction, and translate the mixed number into an improper fraction:
Let's bring these fractions to a common denominator. After bringing them to a common denominator, they will take the following form:
We enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:
Thus, the value of the expression is .
We will decide given example the second way. Let's go back to the original expression:
Let's write the mixed number in expanded form. We rewrite the rest without changes:
We enclose each rational number in brackets together with its signs:
Let's calculate the integer parts:
In the main expression, instead of writing the resulting number −7
The expression is an expanded form of writing a mixed number. Let's write the number −7 and the fraction together, forming the final answer:
Let's write this solution shortly:
Example 8 Find the value of an expression
We enclose each rational number in brackets together with its signs:
Let's replace subtraction with addition:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:
Thus, the value of the expression is
This example can be solved in the second way. It consists in adding the whole and fractional parts separately. Let's go back to the original expression:
We enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer. But this time we add separately the integer parts (−1 and −2), and the fractional and
Let's write this solution shortly:
Example 9 Find expression expressions
Convert mixed numbers to improper fractions:
We enclose the rational number in brackets together with its sign. A rational number does not need to be enclosed in brackets, since it is already in brackets:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:
Thus, the value of the expression is
Now let's try to solve the same example in the second way, namely by adding the integer and fractional parts separately.
This time, in order to get a short solution, let's try to skip some actions, such as writing a mixed number in expanded form and replacing subtraction with addition:
Note that the fractional parts have been reduced to a common denominator.
Example 10 Find the value of an expression
Let's replace subtraction with addition:
The resulting expression does not negative numbers which are the main cause of errors. And since there are no negative numbers, we can remove the plus in front of the subtrahend, and also remove the parentheses:
The result is a simple expression that is easy to calculate. Let's calculate it in any way convenient for us:
Example 11. Find the value of an expression
This is the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and put the sign of the rational number, the module of which is greater, in front of the received answers:
Example 12. Find the value of an expression
The expression consists of several rational numbers. According to, first of all, you need to perform the actions in brackets.
First, we calculate the expression , then the expression We add the results obtained.
First action:
Second action:
Third action:
Answer: expression value equals
Example 13 Find the value of an expression
Convert mixed numbers to improper fractions:
We enclose the rational number in brackets along with its sign. A rational number does not need to be enclosed in parentheses, since it is already in parentheses:
Let's give these fractions in a common denominator. After bringing them to a common denominator, they will take the following form:
Let's replace subtraction with addition:
We got the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and put the sign of the rational number, the module of which is greater, in front of the received answers:
Thus, the value of the expression equals
Consider the addition and subtraction of decimal fractions, which are also rational numbers and which can be both positive and negative.
Example 14 Find the value of the expression −3.2 + 4.3
We enclose each rational number in brackets along with its signs. We take into account that the plus that is given in the expression is the sign of the operation and does not apply to the decimal fraction 4.3. This decimal has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
(−3,2) + (+4,3)
This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract a smaller module from a larger module, and put the sign of the rational number whose module is larger in front of the answer. And in order to understand which modulus is larger and which is smaller, you need to be able to compare the moduli of these decimal fractions before calculating them:
(−3,2) + (+4,3) = |+4,3| − |−3,2| = 1,1
The modulus of 4.3 is greater than the modulus of −3.2, so we subtracted 3.2 from 4.3. Got the answer 1.1. The answer is yes, because the answer must be preceded by the sign of the rational number whose modulus is greater. And the modulus of 4.3 is greater than the modulus of −3.2
Thus, the value of the expression −3.2 + (+4.3) is 1.1
−3,2 + (+4,3) = 1,1
Example 15 Find the value of the expression 3.5 + (−8.3)
This is the addition of rational numbers with different signs. As in the previous example, we subtract the smaller one from the larger module and put the sign of the rational number, the module of which is greater, before the answer:
3,5 + (−8,3) = −(|−8,3| − |3,5|) = −(8,3 − 3,5) = −(4,8) = −4,8
Thus, the value of the expression 3.5 + (−8.3) is equal to −4.8
This example can be written shorter:
3,5 + (−8,3) = −4,8
Example 16 Find the value of the expression −7.2 + (−3.11)
This is the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus before the answer.
You can skip the entry with modules to avoid cluttering up the expression:
−7,2 + (−3,11) = −7,20 + (−3,11) = −(7,20 + 3,11) = −(10,31) = −10,31
Thus, the value of the expression −7.2 + (−3.11) is equal to −10.31
This example can be written shorter:
−7,2 + (−3,11) = −10,31
Example 17. Find the value of the expression −0.48 + (−2.7)
This is the addition of negative rational numbers. We add their modules and put a minus before the received answer. You can skip the entry with modules to avoid cluttering up the expression:
−0,48 + (−2,7) = (−0,48) + (−2,70) = −(0,48 + 2,70) = −(3,18) = −3,18
Example 18. Find the value of the expression −4.9 − 5.9
We enclose each rational number in brackets along with its signs. We take into account that the minus which is located between the rational numbers −4.9 and 5.9 is the sign of the operation and does not apply to the number 5.9. This rational number has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
(−4,9) − (+5,9)
Let's replace subtraction with addition:
(−4,9) + (−5,9)
We got the addition of negative rational numbers. We add their modules and put a minus before the received answer:
(−4,9) + (−5,9) = −(4,9 + 5,9) = −(10,8) = −10,8
Thus, the value of the expression −4.9 − 5.9 is equal to −10.8
−4,9 − 5,9 = −10,8
Example 19. Find the value of the expression 7 − 9.3
Enclose in brackets each number along with its signs
(+7) − (+9,3)
Let's replace subtraction with addition
(+7) + (−9,3)
(+7) + (−9,3) = −(9,3 − 7) = −(2,3) = −2,3
Thus, the value of the expression 7 − 9.3 is −2.3
Let's write down the solution of this example in a shorter way:
7 − 9,3 = −2,3
Example 20. Find the value of the expression −0.25 − (−1.2)
Let's replace subtraction with addition:
−0,25 + (+1,2)
We got the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the answer we put the sign of the number whose module is greater:
−0,25 + (+1,2) = 1,2 − 0,25 = 0,95
Let's write down the solution of this example in a shorter way:
−0,25 − (−1,2) = 0,95
Example 21. Find the value of the expression -3.5 + (4.1 - 7.1)
Perform the actions in brackets, then add the received answer with the number −3.5
First action:
4,1 − 7,1 = (+4,1) − (+7,1) = (+4,1) + (−7,1) = −(7,1 − 4,1) = −(3,0) = −3,0
Second action:
−3,5 + (−3,0) = −(3,5 + 3,0) = −(6,5) = −6,5
Answer: the value of the expression −3.5 + (4.1 − 7.1) is −6.5.
Example 22. Find the value of the expression (3.5 - 2.9) - (3.7 - 9.1)
Let's do the parentheses. Then, from the number that resulted from the execution of the first brackets, subtract the number that resulted from the execution of the second brackets:
First action:
3,5 − 2,9 = (+3,5) − (+2,9) = (+3,5) + (−2,9) = 3,5 − 2,9 = 0,6
Second action:
3,7 − 9,1 = (+3,7) − (+9,1) = (+3,7) + (−9,1) = −(9,1 − 3,7) = −(5,4) = −5,4
Third act
0,6 − (−5,4) = (+0,6) + (+5,4) = 0,6 + 5,4 = 6,0 = 6
Answer: the value of the expression (3.5 - 2.9) - (3.7 - 9.1) is 6.
Example 23. Find the value of an expression −3,8 + 17,15 − 6,2 − 6,15
Enclose in brackets every rational number along with its signs
(−3,8) + (+17,15) − (+6,2) − (+6,15)
Let's replace subtraction with addition where possible:
(−3,8) + (+17,15) + (−6,2) + (−6,15)
The expression consists of several terms. According to the associative law of addition, if the expression consists of several terms, then the sum will not depend on the order of actions. This means that the terms can be added in any order.
We will not reinvent the wheel, but add all the terms from left to right in the order in which they appear:
First action:
(−3,8) + (+17,15) = 17,15 − 3,80 = 13,35
Second action:
13,35 + (−6,2) = 13,35 − −6,20 = 7,15
Third action:
7,15 + (−6,15) = 7,15 − 6,15 = 1,00 = 1
Answer: the value of the expression −3.8 + 17.15 − 6.2 − 6.15 is equal to 1.
Example 24. Find the value of an expression
Let's convert the decimal fraction -1.8 to a mixed number. We will rewrite the rest without change: