In this article we will talk about addition of negative numbers. First, we give a rule for adding negative numbers and prove it. After that, we will analyze typical examples of adding negative numbers.
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Negative addition rule
Before giving the formulation of the rule for adding negative numbers, let's turn to the material of the article positive and negative numbers. There we mentioned that negative numbers can be perceived as debt, and in this case determines the amount of this debt. Therefore, the addition of two negative numbers is the addition of two debts.
This conclusion makes it possible to understand negative addition rule. To add two negative numbers, you need:
- stack their modules;
- put a minus sign in front of the received amount.
Let's write down the rule for adding negative numbers −a and −b in literal form: (−a)+(−b)=−(a+b).
It is clear that the voiced rule reduces the addition of negative numbers to the addition of positive numbers (the modulus of a negative number is a positive number). It is also clear that the result of adding two negative numbers is a negative number, as evidenced by the minus sign that is placed in front of the sum of the modules.
The rule for adding negative numbers can be proved based on properties of actions with real numbers(or the same properties of operations with rational or integer numbers). To do this, it suffices to show that the difference between the left and right parts of the equality (−a)+(−b)=−(a+b) is equal to zero.
Since subtracting a number is the same as adding the opposite number (see the rule for subtracting integers), then (−a)+(−b)−(−(a+b))=(−a)+(−b)+(a+b). By virtue of the commutative and associative properties of addition, we have (−a)+(−b)+(a+b)=(−a+a)+(−b+b). Since the sum of opposite numbers is equal to zero, then (−a+a)+(−b+b)=0+0 , and 0+0=0 due to the property of adding a number to zero. This proves the equality (−a)+(−b)=−(a+b) , and hence the rule for adding negative numbers.
It remains only to learn how to apply the rule of adding negative numbers in practice, which we will do in the next paragraph.
Examples of Adding Negative Numbers
Let's analyze examples of adding negative numbers. Let's start with the simplest case - the addition of negative integers, the addition will be carried out according to the rule discussed in the previous paragraph.
Example.
Add negative numbers -304 and -18007 .
Solution.
Let's follow all the steps of the rule of adding negative numbers.
First, we find the modules of the added numbers: and 
. Now you need to add the resulting numbers, here it is convenient to perform column addition: 
Now we put a minus sign in front of the resulting number, as a result we have −18 311 .
Let's write the whole solution in short form: (−304)+(−18 007)= −(304+18 007)=−18 311 .
Answer:
−18 311 .
The addition of negative rational numbers, depending on the numbers themselves, can be reduced either to the addition of natural numbers, or to the addition of ordinary fractions, or to the addition of decimal fractions.
Example.
Add a negative number and a negative number −4,(12) .
Solution.
According to the rule of adding negative numbers, you first need to calculate the sum of modules. The modules of the added negative numbers are 2/5 and 4,(12) respectively. The addition of the resulting numbers can be reduced to the addition of ordinary fractions. To do this, we translate the periodic decimal fraction into an ordinary fraction:. So 2/5+4,(12)=2/5+136/33 . Now let's execute
Mastery of negative numbers is an optional skill if you are going to enter the 5th grade of a physics and mathematics school. However, this will greatly simplify, which will further affect the overall result. entrance olympiad.
So let's get started.
First you need to understand that there are numbers less than zero, which are called negative: for example, one less than this 
, one more less than 1, then , and then, etc. Any natural number has its own "negative brother", a number that, together with the original number, gives . 
All natural, "minus natural" numbers and "0" together make up the set of integers.
Addition and subtraction
If you imagine a number line, you can easily master the rules addition and subtraction of negative numbers:
First, find on the line the number to which or from which you will subtract / add. Next, if you need:
- Add a negative number, then you need to move to the left
- Add a positive number - shift to the right
- Subtract negative - shift right
- Subtract positive - shift left
Of course, tasks for for admission to 5th grade it will be possible to solve without using negative numbers, but this will improve your math level in general. Over time, you will not draw or represent a number line, but will do it "on the machine", but for this it is worth practicing: come up with any numbers (negative or positive) and try to add them first, then subtract them. By repeating this exercise once a day, in a day you will feel that you have fully learned add and subtract any whole numbers.
Multiplication and division
Here the situation is even simpler: you just need to remember how the signs change when multiplying or dividing:
Instead of the word "on" can be both multiplication and division.With a sign, we will decide, and the number itself is the result of respectively multiplying or dividing the original numbers without signs.
Let's start with a simple example. Let's determine what the expression 2-5 is equal to. From the point +2, let's put down five divisions, two to zero and three below zero. Let's stop at point -3. That is 2-5=-3. Now notice that 2-5 does not equal 5-2 at all. If in the case of addition of numbers their order does not matter, then in the case of subtraction, everything is different. Number order matters.
Now let's move on to negative area scales. Suppose you need to add +5 to -2. (From now on, we'll put "+" signs in front of positive numbers and parenthesize both positive and negative numbers so we don't confuse the signs in front of numbers with addition and subtraction signs.) Now our problem can be written as (-2)+ (+5). To solve it, from the point -2 we will go up five divisions and find ourselves at the point +3.
Does this task make any practical sense? Of course have. Let's say you have $2 in debt and you made $5. Thus, after you repay the debt, you will have 3 dollars left.
You can also move down the negative area of the scale. Suppose you need to subtract 5 from -2, or (-2)-(+5). From point -2 on the scale, let's lay down five divisions and find ourselves at point -7. What is the practical meaning of this task? Suppose you had $2 in debt and had to borrow another $5. Now your debt is $7.
We see that with negative numbers one can carry out the same addition and subtraction operations, as well as with positive ones.
True, we have not yet mastered all the operations. We only added to negative numbers and subtracted only positive ones from negative numbers. But what to do if you need to add negative numbers or subtract negative ones from negative numbers?
In practice, this is similar to dealing with debts. Let's say you were charged $5 in debt, which means the same as if you received $5. On the other hand, if I somehow make you accept responsibility for someone's $5 debt, that's the same as taking that $5 away from you. That is, subtracting -5 is the same as adding +5. And adding -5 is the same as subtracting +5.
This allows us to get rid of the subtraction operation. Indeed, "5-2" is the same as (+5)-(+2) or according to our rule (+5)+(-2). In both cases, we get the same result. From the point +5 on the scale, we need to go down two divisions, and we get +3. In the case of 5-2, this is obvious, because the subtraction is a downward movement.
In the case of (+5)+(-2) this is less obvious. We add a number, which means moving up the scale, but we add a negative number, that is, we do the opposite action, and these two factors taken together mean that we need to move not up the scale, but in the opposite direction, that is down.
Thus, we again get the answer +3.
Why is it really necessary replace subtraction with addition? Why move up "in reverse"? Isn't it easier to just move down? The reason is that in the case of addition, the order of the terms does not matter, while in the case of subtraction, it is very important.
We have already found out before that (+5)-(+2) is not at all the same as (+2)-(+5). In the first case, the answer is +3, and in the second -3. On the other hand, (-2)+(+5) and (+5)+(-2) result in +3. Thus, by switching to addition and abandoning subtraction operations, we can avoid random errors associated with the rearrangement of terms.
Similarly, you can act when subtracting a negative. (+5)-(-2) is the same as (+5)+(+2). In both cases, we get the answer +7. We start at the +5 point and move "down in the opposite direction", that is, up. In the same way, we would act when solving the expression (+5) + (+2).
The replacement of subtraction by addition is actively used by students when they begin to study algebra, and therefore this operation is called "algebraic addition". In fact, this is not entirely fair, since such an operation is obviously arithmetic, and not algebraic at all.
This knowledge is unchanged for everyone, so even if you get an education in Austria through www.salls.ru, although studying abroad is valued more, you can still apply these rules there.
Goals and objectives of the lesson:
- General lesson in mathematics in grade 6 "Addition and subtraction positive and negative numbers
- Summarize and systematize students' knowledge on this topic.
- Develop subject and general educational skills and abilities, the ability to use the acquired knowledge to achieve the goal; establish patterns of diversity of connections to achieve a level of systematic knowledge.
- Education of skills of self-control and mutual control; to develop desires and needs to generalize the facts obtained; develop independence, interest in the subject.
During the classes
I. Organizational moment
Guys, we are traveling around the country of "Rational numbers", where positive, negative numbers and zero live. Traveling, we learn a lot of interesting things about them, get acquainted with the rules and laws by which they live. This means that we must abide by these rules and obey their laws.
And what rules and laws did we get acquainted with? (rules of addition and subtraction of rational numbers, laws of addition)
And so the topic of our lesson is "Addition and subtraction of positive and negative numbers."(Students write the number and topic of the lesson in their notebooks)
II. Checking homework
III. Knowledge update.
Let's start the lesson with oral work. You have a series of numbers in front of you.
8,6; 21,8; -0,5; 6,6; 4,7; 7; -18; 0.
Answer the questions:
What is the largest number in the series?
What number has the largest modulus?
What is the smallest number in the series?
What number has the smallest modulus?
How to compare two positive numbers?
How to compare two negative numbers?
How to compare numbers with different signs?
What are the opposite numbers in the series?
List the numbers in ascending order.
IV. find the mistake
a) -47 + 25+ (-18) = 30
c) - 7.2+(- 3.5) + 10.6= - 0.1
d) - 7.2+ (- 2.9) + 7.2= 2.4
V.Task "Guess the word"
In each group, I gave out tasks in which the words were encrypted.
After completing all the tasks, you will guess the keywords (flowers, gift, girls)
| 1 row | Answer | Letter |
|
| Answer | Letter |
||
| 54-(-74) | |||
| 2,5-3,6 | |||
| 23,7+23,7 | |||
| 11,2+10,3 | |||
| 3 row | Answer | Letter |
|
| 2,03-7,99 | |||
| 67,34-45,08 | |||
| 10,02 | 112,42 | 50,94 | 50,4 |
VI. Fizminutka
Well done, you did a good job, I think it's time to relax, concentrate, relieve fatigue, and simple exercises will help restore peace of mind
PHYSMINUTE (If the statement is correct, clap your hands, if not, shake your head from side to side):
When adding two negative numbers, the modules of the terms must be subtracted -
The sums of two negative numbers are always negative +
Adding two opposite numbers always results in 0 +
When adding numbers with different signs, you need to add their modules -
The sum of two negative numbers is always less than each of the terms +
When adding numbers with different signs, you need to subtract a smaller module from a larger module +
VII.Solving textbook assignments.
No. 1096 (a, e, i)
VIII. Homework
1 level "3" - №1132
Level 2 - "4" - No. 1139, 1146
IX. Independent work on options.
Level 1, "3"
| 1 option | Option 2 |
2nd level, "4"
| 1 option | Option 2 |
| 1 - (- 3 )+(- 2 ) |
3rd level, "5"
| 1 option | 2 option |
| 4,2-3,25-(-0,6) | 2,4-1,75-(-2,6) |
Mutual check on the board, changing neighbors on the desk
X. Summing up the lesson. Reflection
Let's remember the beginning of our lesson, guys.
What are the objectives of the lesson?
Do you think we have achieved our goals?
Guys, now evaluate your work in the lesson. In front of you is a card with a picture of a mountain. If you think you did a good job at the lesson, everything is fine for you.Okay, then draw yourself on top of a mountain. If something is unclear, draw yourself below, and decide for yourself on the left or right.
Send me your drawings along with the grade card, you will find out the final grade for the work in the next lesson.
In this article, we will analyze how subtraction of negative numbers from arbitrary numbers. Here we will give a rule for subtracting negative numbers, and consider examples of the application of this rule.
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Rule for subtracting negative numbers
The following takes place rule for subtracting negative numbers: in order to subtract a negative number b from the number a, you need to add the number −b to the reduced a, opposite to the subtracted b.
In literal form, the rule for subtracting a negative number b from an arbitrary number a looks like this: a−b=a+(−b) .
Let us prove the validity of this rule for subtracting numbers.
First, let's recall the meaning of subtracting the numbers a and b. To find the difference between the numbers a and b means finding a number c whose sum with the number b is equal to a (see the connection between subtraction and addition). That is, if a number c is found such that c+b=a , then the difference a−b is equal to c .
Thus, in order to prove the announced subtraction rule, it is enough to show that adding the number b to the sum a+(−b) will give the number a . To show this, let's look at properties of operations with real numbers. By virtue of the associative property of addition, the equality (a+(−b))+b=a+((−b)+b) is true. Since the sum of opposite numbers is equal to zero, then a+((−b)+b)=a+0 , and the sum of a+0 is equal to a, since adding zero does not change the number. Thus, the equality a−b=a+(−b) has been proved, which means that the validity of the above rule for subtracting negative numbers has been proved.
We have proved this rule for real numbers a and b . However, this rule is also true for any rational numbers a and b , as well as for any integers a and b , since operations with rational and integer numbers also have the properties that we used in the proof. Note that with the help of the parsed rule, it is possible to subtract a negative number both from a positive number and from a negative number, as well as from zero.
It remains to consider how the subtraction of negative numbers is performed using the parsed rule.
Examples of subtracting negative numbers
Consider examples of subtracting negative numbers. Let's start by solving a simple example to understand all the intricacies of the process without bothering with calculations.
Example.
Subtract negative -13 from negative -7 .
Solution.
The number opposite to the subtracted −7 is the number 7 . Then, by the rule of subtracting negative numbers, we have (−13)−(−7)=(−13)+7 . It remains to perform the addition of numbers with different signs, we get (−13)+7=−(13−7)=−6 .
Here is the whole solution: (−13)−(−7)=(−13)+7=−(13−7)=−6 .
Answer:
(−13)−(−7)=−6 .
Subtraction of fractional negative numbers can be done by jumping to the corresponding common fractions, mixed numbers, or decimals. Here it is worth starting from what numbers it is more convenient to work with.
Example.
Subtract from the number 3.4 a negative number.
Solution.
Applying the rule for subtracting negative numbers, we have 
. Now replace the decimal 3.4 with a mixed number: 
(see the translation of decimal fractions into ordinary fractions), we get 
. It remains to perform the addition of mixed numbers: .
This completes the subtraction of a negative number from the number 3.4. Let's give a brief record of the solution: .
Answer:
.
Example.
Subtract the negative number −0,(326) from zero.
Solution.
By the rule of subtracting negative numbers, we have 0−(−0,(326))=0+0,(326)=0,(326) . The last transition is valid due to the property of adding a number to zero.