Subtract the difference of two numbers from a number. Subtracting Natural Numbers
To fully analyze the topic of the article, we will introduce terms and definitions, denote the meaning of the subtraction action, and derive a rule according to which the subtraction action can lead to the addition action. Let's look at practical examples. We will also consider the action of subtraction in a geometric interpretation - on the coordinate line.
In general, the basic terms used to describe the action of subtraction are the same for any type of number.
Yandex.RTB R-A-339285-1 Definition 1
Minuend– an integer from which the subtraction will be performed.
Subtrahend– an integer that we will subtract.
Difference– the result of the subtraction action performed.
To denote the action itself, a minus sign is used, placed between the minuend and the subtrahend. All components of the action indicated above are written in the form of equality. That is, if integers a and b are given, and when subtracting the first from the second, the number c is obtained, the subtraction action will be written as follows: a – b = c.
We will also denote an expression of the form a – b as a difference, as well as the final value of this expression itself.
The meaning of subtracting integers
In the topic of subtracting natural numbers, a relationship was established between the actions of addition and subtraction, which made it possible to define subtraction as the search for one of the terms by a known sum and the second term. Let us assume that subtracting integers has the same meaning: given the sum and one of the terms, the second term is determined.
The indicated meaning of the action of subtracting integers makes it possible to state that c - b = a and c - a = b, if a + b = c, where a, b, c are integers.
Let's look at simple examples to reinforce the theory:
Let us know that - 5 + 11 = 6, then the difference is 6 - 11 = - 5;
Let's say it is known that - 13 + (- 5) = - 18, then - 18 – (- 5) = - 13, and - 18 – (- 13) = - 5.
Rule for subtracting integers
The above meaning of the subtraction action does not indicate for us a specific way to calculate the difference. Those. we can assert that one of the known terms is the result of subtracting another known term from the sum. But, if one of the terms turns out to be unknown, then we cannot know what the difference between the sum and the known term will be. Therefore, to perform the subtraction action we need the rule for subtracting integers:
Definition 1
In order to determine the difference between two numbers, it is necessary to add to the minuend the number opposite to the subtrahend, i.e. a – b = a + (- b), where a and b are integers; b and – b are opposite numbers.
Let us prove the indicated subtraction rule, i.e. Let us prove the validity of the equality specified in the rule. To do this, according to the meaning of subtracting integers, we add the subtrahend b to a + (- b) and make sure that we get the minuend a as a result, i.e. Let's check the validity of the equality (a + (- b)) + b = a. Based on the properties of addition of integers, we can write a chain of equalities: (a + (- b)) + b = a + ((- b) + b) = a + 0 = a, it will be a proof of the rule for subtracting integers.
Let's look at the application of the rule for subtracting integers using specific examples.
Subtracting a positive integer, examples
Example 1It is necessary to subtract the positive integer 45 from the integer 15.
Solution
According to the rule, in order to subtract the positive integer 45 from a given number 15, you need to add the number - 45 to the minuend 15, i.e. opposite to the given 45. Thus, the required difference will be equal to the sum of the integers 15 and - 45. Having calculated the required sum of numbers with opposite signs, we get the number - 30. Those. the result of subtracting the number 45 from the number 15 is the number - 30. Let's write the entire solution in one line: 15 - 45 = 15 + (- 45) = - 30.
Answer: 15 - 45 = - 30.
Example 2
It is necessary to subtract the positive integer 25 from the negative integer - 150.
Solution
According to the rule, we add to the number being reduced - 150 - the number - 25 (i.e., the opposite of the given subtrahend 25). Let's find the sum of negative integers: - 150 + (- 25) = - 175. Thus, the required difference is equal. Let's write the whole solution like this: - 150 - 25 = - 150 + (- 25) = - 175.
Answer: - 150 - 25 = - 175.
Subtracting zero, examples
The rule for subtracting integers makes it possible to derive the principle of subtracting zero from an integer - subtracting zero from any integer does not change this number, i.e. a - 0 = a, where a is an arbitrary integer.
Let me explain. According to the rule of subtraction, subtracting zero is adding the opposite number of zero to the minuend. Zero is a number opposite to itself, i.e. Subtracting zero is the same as adding zero. Based on the corresponding property of addition, adding zero to any integer does not change that number. Thus,
a - 0 = a + (- 0) = a + 0 = a .
Let's look at simple examples of subtracting zero from various integers. For example, the difference 61 - 0 is equal to 61. If you subtract zero from a negative integer - 874, you get - 874. If we subtract zero from zero, we get zero.
Subtracting a negative integer, examples
Example 3It is necessary to subtract the negative integer - 324 from the integer 0.
Solution
According to the subtraction rule, the difference 0 - (- 324) must be determined by adding to the number being reduced 0 the number opposite to the one being subtracted - 324. Then: 0 - (- 324) = 0 + 324 = 324
Answer: 0 - (- 324) = 324
Example 4
Determine the difference - 6 - (- 13) .
Solution
Let's subtract from the negative integer - 6 the negative integer - 13. To do this, we calculate the sum of two numbers: the minuend - 6 and the number 13 (i.e., the opposite of the given subtrahend - 13). We get: - 6 - (- 13) = - 6 + 13 = 7.
Answer: - 6 - (- 13) = 7.
Subtracting equal integers
If the given minuend and subtrahend are equal, then their difference will be equal to zero, i.e. a - a = 0, where a is any integer.
Let me explain. According to the rule for subtracting integers a - a = a + (- a) = 0, which means: in order to subtract an equal number from an integer, you need to add to this number the opposite number, which will result in zero.
For example, the difference between equal integers - 54 and - 54 is zero; performing the action of subtracting the number 513 from the number 513, we get zero; subtracting zero from zero, we also get zero.
Checking the result of subtracting integers
The necessary check is carried out using the addition action. To do this, we add the subtrahend to the resulting difference: the result should be a number equal to the one being reduced.
Example 5
The integer - 112 was subtracted from the integer - 300, and the difference was obtained - 186. Was the subtraction done correctly?
Solution
Let's check according to the above principle. Let's add the subtrahend to the given difference: - 186 + (- 112) = - 298. We received a number different from the specified one to be reduced; therefore, an error was made when calculating the difference.
Answer: no, the subtraction was performed incorrectly.
In conclusion, consider the geometric interpretation of the action of subtracting integers. Let's draw a horizontal coordinate line directed to the right:
Above we derived the rule for performing the subtraction action, according to it: a - b = a + (- b), then the geometric interpretation of subtracting numbers a and b will coincide with the geometric meaning of adding integers a and – b. It follows from this that to subtract an integer b from an integer a, you need:
Move from a point with coordinate a by b unit segments to the left, if b is a positive number;
Move from the point with coordinate a to | b | (modulus of number b) of unit segments to the right, if b is a negative number;
Stay at the point with coordinate a if b = 0.
Let's look at an example using a graphic image:
Let it be necessary to subtract the positive integer 2 from the integer - 2. To do this, according to the above scheme, we move to the left by 2 unit segments, thus ending up at the point with coordinate - 4, i.e. - 2 - 2 = - 4 .
Another example: subtract the negative integer - 3 from the integer 2. Then, according to the diagram, let's move to the right to | - 3 | = 3 unit segments, thus ending up at the point with coordinate 5. We get the equality: 2 - (- 3) = 5 and an illustration for it:
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subtraction), the inverse of addition. Designated using the minus sign “−”. This is an action by which the sum and one of the terms can be used to find the second term.The number from which it is subtracted is called minuend, and the number we are subtracting is subtrahend. The result of subtraction operations is called difference.
Let us know: the sum of 2 numbers c And b equals a, which means the difference a−c will b, and the difference a−b will c.
It is most convenient to subtract using the “column” method.
Subtraction table.
To make it easier and faster to master the subtraction process, review and memorize the subtraction table up to ten for grade 2:
Properties of subtraction of natural numbers.
- Subtraction, as a process, does NOT have the commutative property: a−b≠b−a.
- The difference of identical numbers is zero: a−a=0.
- Subtracting the sum of 2 integers from an integer: a−(b+c)=(a−b)−c.
- Subtracting a number from the sum of 2 numbers: (a+b)−c=(a−c)+b=a+(b−c).
- Distributive property of multiplication relative to subtraction: a·(b−c)=a·b−a·c and (a−b)·c=a·c−b·c.
- And all other properties of subtracting integers (natural numbers).
Let's look at some of them:
The property of subtracting two equal natural numbers.
The difference between 2 identical natural numbers is zero.
a−a=0,
Where a- any natural number.
Subtracting natural numbers does NOT have the commutative property.
From the property described above, it is clear that for 2 identical natural numbers, the commutative property of subtraction works. In all other cases (if the minuend ≠ the subtrahend), subtracting natural numbers does not have a commutative property. Or, to put it another way, the minuend and subtrahend do not change places.
When the minuend is greater than the subtrahend and we decide to swap them, it means that we will subtract from the natural number that is smaller the natural number that is larger. This system does not correspond to the essence of subtracting natural numbers.
If a And b unequal natural numbers, then a−b≠b−a. For example, 45−21≠21−45.
The property of subtracting the sum of two numbers from a natural number.
Subtracting the required sum of 2 natural numbers from the specified natural number is the same thing as subtracting the 1st term of the required sum from the specified natural number, then subtracting the 2nd term from the calculated difference.
Using letters this can be expressed this way:
a−(b+c)=(a−b)−c,
Where a, b And c- natural numbers, conditions must be met a>b+c or a=b+c.
The property of subtracting a natural number from the sum of two numbers.
Subtracting a natural number from the sum of 2 numbers is the same as subtracting a number from one of the terms, and then adding the difference and the other term. The number being subtracted cannot be greater than the term from which the number is being subtracted.
Let a, b And c- integers. So if a more or equal c, equality (a+b)−c=(a−c)+b will correspond to the truth, and if b more or equal c, That: (a+b)−c=a+(b−c). When and a And b more or equal c, which means both last equalities hold, and they can be written like this:
(a+b)−c=(a−c)+b= a+(b−c).
The concept of subtraction is best understood with an example. You decide to drink tea with sweets. There were 10 sweets in the vase. You ate 3 candies. How many candies are left in the vase? If we subtract 3 from 10, there will be 7 sweets left in the vase. Let's write the problem mathematically:
Let's look at the entry in detail:
10 is the number from which we subtract or decrease, which is why it is called reducible.
3 is the number we are subtracting. That's why they call him deductible.
7 is the result of subtraction or is also called difference. The difference shows how much the first number (10) is greater than the second number (3) or how much the second number (3) is less than the first number (10).
If you doubt whether you found the difference correctly, you need to do check. Add the second number to the difference: 7+3=10
When subtracting l, the minuend cannot be less than the subtrahend.
We draw a conclusion from what has been said. Subtraction- this is an action by which the second term is found from the sum and one of the terms.
In literal form, this expression will look like this:
a—b =c
a – minuend,
b – subtrahend,
c – difference.
Properties of subtracting a sum from a number.
13 — (3 + 4)=13 — 7=6
13 — 3 — 4 = 10 — 4=6
The example can be solved in two ways. The first way is to find the sum of the numbers (3+4), and then subtract from the total number (13). The second way is to subtract the first term (3) from the total number (13), and then subtract the second term (4) from the resulting difference.
In literal form, the property of subtracting a sum from a number will look like this:
a - (b + c) = a - b - c
The property of subtracting a number from a sum.
(7 + 3) — 2 = 10 — 2 = 8
7 + (3 — 2) = 7 + 1 = 8
(7 — 2) + 3 = 5 + 3 = 8
To subtract a number from a sum, you can subtract this number from one term, and then add the second term to the resulting difference. The condition is that the summand will be greater than the number being subtracted.
In literal form, the property of subtracting a number from a sum will look like this:
(7 + 3) — 2 = 7 + (3 — 2)
(a+b) —c=a + (b - c), provided b > c
(7 + 3) — 2=(7 — 2) + 3
(a + b) - c=(a - c) + b, provided a > c
Subtraction property with zero.
10 — 0 = 10
a - 0 = a
If you subtract zero from a number then it will be the same number.
10 — 10 = 0
a—a = 0
If you subtract the same number from a number then it will be zero.
Related questions:
In example 35 - 22 = 13, name the minuend, subtrahend and difference.
Answer: 35 – minuend, 22 – subtrahend, 13 – difference.
If the numbers are the same, what is their difference?
Answer: zero.
Do the subtraction test 24 - 16 = 8?
Answer: 16 + 8 = 24
Subtraction table for natural numbers from 1 to 10.
Examples for problems on the topic “Subtraction of natural numbers.”
Example #1:
Insert the missing number: a) 20 - ... = 20 b) 14 - ... + 5 = 14
Answer: a) 0 b) 5
Example #2:
Is it possible to subtract: a) 0 - 3 b) 56 - 12 c) 3 - 0 d) 576 - 576 e) 8732 - 8734
Answer: a) no b) 56 - 12 = 44 c) 3 - 0 = 3 d) 576 - 576 = 0 e) no
Example #3:
Read the expression: 20 - 8
Answer: “Subtract eight from twenty” or “subtract eight from twenty.” Pronounce words correctly
Sections: Primary School
Class: 2
Basic goals:
1) form an idea of the property of subtracting a sum from a number, the ability to use this property to rationalize calculations;
2) train mental calculation skills, the ability to independently analyze and solve compound problems;
3) cultivate accuracy.
Demo material:
1) image of Dunno. <Рисунок1 >
2) cards with the statement: yes - bark - success - hov.
3) hourglass.
4) a standard for subtracting a sum from a number.
a-(b+c) = (a-b)-c = (a-c)-b
5) standard of procedure. a – (b+c)
6) Self-test sample for step 6:
7) sample for self-test for the 7th stage.
1) 45 -15=30 (m) – left with Denis
2) 30 - 13 =17 (m)
Answer: Denis has 17 stamps left.
Handout:
1) a beige card with an individual task for stage 2 for each student:
2) a green card with an individual task for stage 5.
3) independent work for stage 6.
4) traffic lights: red, yellow, green.
During the classes:
I. Self-determination for educational activities.
1) motivate activities in the lesson through the introduction of a fairy-tale character;
2) determine the content of the lesson: subtracting an amount from a number.
Organization of the educational process at stage I.
What did you repeat in the last lesson? (Properties of addition)
What properties of addition were repeated? (Commutative and associative)
Why do we need to know the properties of addition? (It’s more convenient to solve examples)
Today our guest is the fairy-tale hero Dunno .<Рисунок1 >
He has prepared many interesting tasks and will watch how we work in class. Ready?
II. Updating knowledge and fixing difficulties in activities.
1) train a mental operation - generalization;
2) repeat the rules for the order of actions in expressions with brackets;
3) organize a difficulty in individual activity and its recording by students in loud speech.
Organization of the educational process at stage II.
1) Oral counting.
Look at the board and complete the steps orally. <Приложение 1 >
If we fulfill them correctly, we will read the wish that Dunno encrypted for us:
(Add 19 to 27, you get 46;
From 46 subtract 24 you get 22;
Add 38 to 22 to get 60;
Subtract 5 from 60 to get 55)
Increase 55 by 200. (200+55=255)
Give a description of the number 255. (255 is a three-digit number, contains two hundreds, five tens and five units. The previous number is 254, the next is 256, the sum of the digit terms is 200+50+5, the sum of the digits is 12).
Express the number 255 in different units of counting. (255=2s 5d 5ed = 25d 5ed = 2s 55ed)
Express 255 cm in different units of measurement. (255=2m 5dm 5cm=25dm 5cm=2m 55cm)
2) Repetition of the rule for the order of actions in expressions with brackets. <Приложение 2 >
How are the expressions similar? (Components of actions, same order of actions)
How are the expressions different? (Miscellaneous deductible)
How are subtrahends represented? (Subtrahends are represented by the sum of two numbers)
What did we repeat when finding the meanings of expressions? (Procedure).
Why did you repeat the procedure?
Where can we repeat the rule of procedure? (In the textbook or standards <Приложение 3 > )
3) Individual task.
Take a pen and a piece of beige paper. <Приложение 4 >
Now let's take a moment to solve examples. At my command, you stop your decision.
Attention! Let's start! ...
Raise your hand, who solved all the examples?
Raise your hand, who solved one example?
Suggest a standard by which you solved the examples. (We don’t know the standard).
Who hasn't solved the examples?
III.Identification of the causes of difficulties and setting goals for the activity.
1) identify and record the location and cause of the difficulty;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage III.
Repeat, what was the task?
Why did the problem arise? (Little time, no suitable property)
What to do? (Children's guess). Set the sheets aside.
Try to formulate the purpose of the lesson.
Formulate the topic of the lesson.
Lesson topic: Subtracting a sum from a number. Say the topic of the lesson to yourself, in a low voice. (The topic of the lesson is written on the board)
IV. Building a project for getting out of a problem.
1) organize children’s construction of a new way of action using introductory dialogue;
2) fix the new method of action symbolically and in speech.
Organization of the educational process at stage IV.
Look and read the expression: 87 – (7+15).
Which term is more convenient to subtract first? (It’s more convenient to subtract the first term – 7)
We subtracted the first term, but we need to subtract two terms. What need to do? (Subtract the second term)
The teacher writes on the board. <Приложение5 >
Look, I replace the number 87 with the letter a, the number 7 with the letter b, the number 15 with the letter c, and we get an equality. <Приложение 6 >
Let's get a look. Read the expression: 87 – (15+7)
Which is more convenient to subtract the term from the number 87? (It’s more convenient to subtract the second term 7)
The teacher writes on the board.
We subtracted the second term, but we need to subtract two terms. What need to do? (Subtract the first term)
The teacher writes on the board. <Приложение 7 >
Let's get a look. I will replace the number 87 with the letter a, the number 7 with the letter b, the number 15 with the letter c, and we get an equality. <Приложение 8 >
Draw a conclusion about how you can subtract the amount from a number. (Children's answers are listened to)
Where can we check whether we have made the right conclusions? (In the textbook)
Open your textbook to page 44. Read the rule. <Приложение 9 >
V. Primary consolidation in external speech.
Goal: to create conditions for fixing the learned method of action in external speech.
Organization of the educational process at stage V.
Who will repeat the rule?
Why did the problem arise? (We couldn't decide quickly)
Can we do it now?
What helped us? (Rule of subtracting an amount from a number)
Take a green sheet and solve the examples at my command. <Приложение10 >
Attention! Let's start! Stop!
Frontal survey.
How much did you get in the first example?
Raise your hand like that.
Who has the error?
How much did you get in the second example?
Raise your hand like that.
Who has the error?
How did you decide? Where is the mistake? What is the reason?
Can you say that you have learned to solve? (Yes)
What helped? (We know the rule, the speed of solution has increased)
Where can we apply this new technique? (When solving problems, examples).
At home, solve on page 44, task No. 4, on the new rule. Come up with and write down your own example. (The task is written on the board). <Приложение11 >
Who will remind you of the rule?
VI. Independent work with self-test.
1) organize independent completion by students of standard tasks for a new method of action with self-test according to the model;
2) organize children’s self-assessment of the correctness of the task.
Organization of the educational process at stage VI.
And now Dunno will see how we have learned to apply the new rule.
Independent work. <Приложение12 >
Why do we do independent work? (Find out difficulties and overcome them, test your strength)
What methods of subtracting a sum from a number have you studied? (It is convenient to subtract one term and then the other)
Take a white sheet. At my command, we begin to decide.
Start...Stop.
Take a simple pencil and compare it with the sample. <Приложение13 >
For those who have this, put “+”.
If anyone has an error, put “-”.
Raise your hand, who succeeded?
Raise your hand, who has a mistake? Where did the problem arise? (Computational technique)
You did a great job.
What did you learn in the lesson? (learned a convenient way to subtract an amount from a number)
Draw a conclusion. (Children's answers)
Physical exercise.
VII. Inclusion in the knowledge system and repetition.
Goal: repeat the solution to the problem, find a convenient way to solve it.
Organization of the educational process at stage VII.
Where can you apply the rules you have learned? (When solving problems, examples)
Look and read task No. 3 to yourself.
Analyze the task. (In the problem it is known that Denis had 45 marks. He gave Petya 15 marks, and Kolya 13 marks. We need to find out how many marks he has left.
To answer the question in the problem, you need to subtract the number of stamps that Denis gave to Petya and Kolya from the total number of stamps. We cannot immediately answer the question of the problem, since we do not know how many stamps Denis gave to Petya and Kolya. And we can find out by adding the number of stamps that he gave to Petya to the number of stamps that he gave to Kolya).
If there is difficulty in analyzing a problem, the teacher helps with questions presented below:
What is known about the problem?
What do you need to know?
How to answer the task question?
Can we immediately answer the question of the problem? Why?
Can we find out? How?
Tell us your plan for solving the problem. (The first action is to find out how many stamps Denis gave in total, then we will answer the question in the problem). <Приложение 14 >
Who solved the problem differently? (To answer the question of the problem, you need to subtract from the total number of stamps the number of stamps that Denis gave to Petya, and then the number of stamps that he gave to Kolya)
Tell us your plan for solving the problem using the second method. (The first action is to find out how many stamps Denis has left after he gave Petya, and then we find out how many stamps he has left after he gave Kolya 13 stamps and answer the question of the problem). <Приложение15 >
What is the most convenient way to solve the problem? Why? (Secondly, it is more convenient to subtract one part from the whole, and then the other part)
Write down the solution to the problem in a convenient way. Self-test according to the example. <Приложение16 >
VIII. Reflection of activity.
1) record in speech a new method of action learned in the lesson: subtracting an amount from a number;
2) record the difficulties that remain and ways to overcome them;
3) evaluate your own activities in class and agree on homework.
Organization of the educational process at stage VIII.
So, today in the lesson one more rule was added to our knowledge, remember it. (Today in the lesson we learned how to subtract a sum from a number. To subtract a sum from a number, you can first subtract one term and then the other)
Who's having trouble?
Did you manage to overcome them? How?
What else needs to be worked on?
Grading by the teacher for work in the lesson.
Homework: p.44, no.4. Come up with and solve your own example on a new topic.
Literature
1) Textbook “Mathematics 2nd grade, part 2”; L.G. Peterson. Publishing house “Yuventa”, 2008.
3) L.G. Peterson, I.G. Lipatnikova “Oral exercises in mathematics lessons, grade 2.” M.: “School 2000...”
The difference between non-negative integers a andb is the number of elements in the complement of set B to set A, provided thatn(A)= a, n(B)= b, B.A., i.e. A -b = n(A B). This is due to the fact that A = B (AB), i.e.n(A)= n(B) + n(A B).
Let's prove it. Since by condition IN- proper subset of the set A, then they can be represented as in Fig. 3.
Subtraction of natural (non-negative integer) numbers is defined as the inverse operation of addition: A -b = c () b + c = a.
Difference AB shaded in this figure. We see that there are many IN And AB are not suppressed and their union is equal A. Therefore, the number of elements in the set A can be found using the formula n(A)=n(B) + n(AB), from which, by the definition of subtraction as the inverse operation of addition, we obtain n(AB) = A -b.
The subtraction of zero receives a similar interpretation, as well as the subtraction A from A. Because A=A, AA=, That A - 0= a And a - a = 0.
Difference A -b non-negative integers exist if and only if .
The action by which the difference is found A -b, called by subtraction, number A- reducible, b- deductible.
Using the definitions, we will show that 8 - 5 = 3 . Let two sets be given such that n(A) = 8, n(B) = 5. And let the multitude IN is a subset of the set A. For example, A ={a, s, d, f, g, h, j, k} , B={a, s, d, f, g} .
Let's find the complement of the set IN to many A: AB ={h, j, k). We get that n(AB) = 3.
Hence , 8 - 5 = 3.
The relationship between subtracting numbers and subtracting sets allows us to justify the choice of action when solving word problems. Let’s find out why the following problem is solved using subtraction and solve it: “There were 7 trees near the school, 3 of them were birches, the rest were lindens. How many linden trees grew near the school?
Let us present the problem conditions visually by depicting each tree planted near the school in a circle (Fig. 4). Among them there are 3 birch trees - in the picture we will highlight them with shading. Then the remaining trees - the unshaded circles - are lindens. That is, there are as many of them as it would be to subtract 3 from 7 , i.e. . 4.
The problem considers three sets: the set A all the trees, lots of them IN- birches, which is a subset A, and many WITH lip - it represents the complement of a set IN before A. The problem requires finding the number of elements in this addition.
By condition n(A) = 7, n(B)= 3 and BA. Let A ={a, b, c, d, e, f, g} , B={a, b, c} . Let's find the complement of the set A before IN: AB ={d, e, f, g) And n(AB) = 4.
Means, n(C) = n(AB) = n(A)- n(B)= 7 - 3 = 4.
Consequently, the school had 4 linden trees.
The considered approach to the addition and subtraction of non-negative integers allows us to interpret various rules from a set-theoretic point of view.
The rule for subtracting a number from a sum: to subtract a number from a sum, it is enough to subtract this number from one of the terms and add another term to the resulting result, i.e. at ac we have that (a+b)-c=(a-c)+b; at bc we have that (a+b)-c=a+(b-c); at ac And bc You can use any of these formulas.
Let's find out the meaning of this rule: Let A, B, C- such sets that n(A)=a, n(B)=b And AB= , SA(Fig. 5).
It is not difficult to prove with the help of Euler circles that the equality holds for these sets.
The right side of the equality looks like:
The left side of the equality has the form: Therefore (a + b) - c = (a- c) + b,at provided that a>c.
Rule for subtracting an amount from a number : to subtract the sum of numbers from a number, it is enough to subtract from this number each term one by one, i.e. provided that a b +c, we have A - (b + c) = (a - b) - c.
Let's find out the meaning of this rule. For these sets the equality holds.
Then we get that the right side of the equality has the form:. The left side of the equality looks like: .
Hence (a + b) - c = (a- c) + b, at provided that a>c.
The rule for subtracting the difference from a number:
to subtract from a number A difference b - c, it is enough to add the subtrahend to this number With and subtract the minuend from the result obtained b; at a>b you can subtract the minuend b from the number a and add the subtracted c to the result obtained, i.e. A - (b - c) = (a + c) - b = (a - b) +c.
Means, A(BC) = .
Hence, n(A(BC)) = n( ) And A - (b - c) = (a + c) - b.
The rule for subtracting a number from a difference: to subtract a third number from the difference of two numbers, It is enough to subtract the sum of two other numbers from the minuend, i.e. (A -b) - c = a - (b + c). The proof is similar to the rule for subtracting a sum from a number.
Example. In what ways can you find the difference: a) 15 - (5 + 6); b) (12 + 6) - 2?
Solution. a) We use the rule for subtracting a sum from a number: 15 - (5 + 6) = (15 - 5) - 6 = 10 - 6 = 4.
Or 15 - (5 + 6) = (15 - 6) - 5 = 9 - 4 = 4.
Or 15 - (5 + 6) = 15 - 11= 4 .
b) We use the rule for subtracting a number from a sum: (12 + 6) - 2 = (12 - 2) + 6 = 10 + 6 = 16.
Or (12 + 6) - 2 = 12 + (6 - 2) = 12 + 4 = 16 .
Or (12 + 6) - 2 = 18 - 2 = 16.
These rules make it possible to simplify calculations and are widely used in elementary mathematics courses.
