In this article we will talk about addition negative numbers . First we give the rule for adding negative numbers and prove it. After this, we will look at typical examples of adding negative numbers.
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Rule for adding negative numbers
Before formulating the rule for adding negative numbers, let us turn to the material in 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 allows us to realize rule for adding negative numbers. To add two negative numbers, you need:
- fold 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 letter form: (−a)+(−b)=−(a+b).
It is clear that the stated 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 proven based on properties of operations with real numbers(or the same properties of operations with rational or integer numbers). To do this, it is enough to show that the difference between the left and right sides 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). Due to 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 with zero. This proves the equality (−a)+(−b)=−(a+b) , and hence the rule for adding negative numbers.
All that remains is 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 sort it out examples of adding negative numbers. Let's start from the very beginning simple case– addition of negative integers; addition will be carried out according to the rule discussed in the previous paragraph.
Example.
Add the negative numbers −304 and −18,007.
Solution.
Let's follow all the steps of the rule for adding negative numbers.
First we find the modules of the numbers being added: 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 .
Addition of negative rational numbers depending on the numbers themselves, it 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 for adding negative numbers, you first need to calculate the sum of the modules. The modules of the negative numbers being added are equal to 2/5 and 4, (12) respectively. The addition of the resulting numbers can be reduced to addition ordinary fractions. To do this, we convert the periodic decimal fraction into an ordinary fraction: . Thus, 2/5+4,(12)=2/5+136/33. Now let's do it
In developing computing skills - the most important goal, pursued by mathematics programs from grades 1 to 6. How quickly and correctly a child learns to perform arithmetic operations will determine the speed at which he performs logical (semantic) operations in high school and the level of understanding of the subject as a whole. A math tutor quite often encounters students’ computing problems that prevent them from achieving good results.
What kind of students does a tutor have to work with? Parents need preparation for the Unified State Exam in mathematics, but their child cannot understand ordinary fractions or is confused by negative numbers. What actions should a math tutor take in such cases? How to help a student? The tutor does not have time for a leisurely and consistent study of the rules, so traditional methods often have to be replaced with some artificial “semi-finished accelerators,” so to speak. In this article I will describe one of the possible ways to develop the skill of performing actions with negative numbers, namely subtracting them.
Let's assume that a math tutor has the pleasure of working with a very weak student whose knowledge does not extend beyond the simplest calculations with positive numbers. Let's also assume that the tutor managed to explain the laws of addition and come close to the rule a-b=a+(-b). What points should a math tutor take into account?
Reducing subtraction to addition is not a simple and obvious transformation. Textbooks offer strict and precise mathematical formulations: “In order to subtract the number “b” from the number “a”, you need to add the opposite number to “b” to the number “a”. Formally, you can’t find fault with the text, but as soon as a math tutor starts using it as instructions for performing specific calculations, problems arise. The phrase alone is worth it: “To subtract, you must add.” Without a clear comment from the tutor, the student will not understand. In fact, what should you do: subtract or add?
If you work with the rule according to the intention of the authors of the textbook, then in addition to practicing the concept of “opposite number”, you need to teach the student to relate the notations “a” and “b” to the real numbers in the example. And this will take time. Considering also the fact that the student thinks and writes at the same time, the task of a math tutor becomes even more complicated. A weak student does not have good visual, semantic and motor memory, and therefore it is better to offer an alternative text of the rule:
To subtract the second from the first number, you need
A) Rewrite the first number
B) Put a plus
B) Replace the sign of the second number with the opposite one
D) Add the resulting numbers
Here the stages of the algorithm are clearly divided into points and are not tied to letter designations.
In the course of solving a practical task on translations, the mathematics tutor rereads this text to the student several times (for memorization). I advise you to write it down in your theory notebook. Only after working out the rule for transition to addition can we write down general shape a-b=a+(-b)
The movement of the minus and plus signs in the head of a child (both a small one and a weak adult) is somewhat reminiscent of Brownian. The math tutor needs to bring order to this chaos as quickly as possible. In the process of solving examples, supporting clues (verbal and visual) are used, which, combined with neat and detailed formatting, do their job. It must be remembered that every word uttered by a math tutor at the moment of solving any problem carries either a hint or a hindrance. Each phrase is analyzed by the child to establish a connection with one or another mathematical object (phenomenon) and its image on paper.
A typical problem for weak schoolchildren is separating the sign of an action from the sign of the number involved in it. The same visual image makes it difficult to recognize the minuend "a" and the subtrahend "b" in differences a-b. When a math tutor reads an expression during an explanation, you need to make sure that the word “subtract” is used instead of “-”. It is necessary! For example, the entry should read: “Out of minus five subtract minus three." We must not forget about the rule of translation into addition: “So that from the number “a” subtract the number “b” is necessary...”
If a math tutor constantly says “minus 5 minus minus 3”, then it is clear that it will be more difficult for the student to imagine the structure of the example. A one-to-one correspondence between a word and an arithmetic operation helps a math tutor accurately convey information.
How can a tutor explain the transition to addition?
Of course, you can refer to the definition of “subtract” and look for the number that must be added to “b” to get “a”. However, a weak student thinks far from strict mathematics and the tutor will need some analogies with him when working with him. simple actions. I often tell my sixth-graders: “In mathematics there is no such arithmetic operation as difference.” The notation 5 – 3 is a simple notation for the result of addition 5+(-3). The plus sign is simply omitted and not written.”
Children are surprised by the tutor’s words and involuntarily remember that they cannot subtract numbers directly. The math tutor declares 5 and -3 terms, and to make his words more persuasive, compares the results of actions 5-3 and 5+(-3). After this, the identity a-b=a+(-b) is written
Whatever the type of student, and no matter how much time the math tutor has to work with him, you need to work out the concept of “opposite number” in time. The entry “-x” deserves special attention from a math tutor. A 6th grade student must learn that it does not represent a negative number, but the opposite of X.
It is necessary to dwell separately on calculations with two minus signs located next to each other. The problem arises of understanding the operation of their simultaneous removal. You need to carefully go through all the points of the outlined algorithm for transition to addition. It will be better if, when working with the difference -5- (-3), before making any comments, the math tutor will highlight the numbers -5 and -3 in a frame or underline them. This will help the student identify the components of the action.
Math tutor's focus on memorization
Reliable memorization is the result practical application mathematical rules, so it is important for the tutor to provide a good density of independently solved examples. To improve the stability of memorization, you can call for help with visual cues - chips. For example, interesting way converting the subtraction of a negative number into addition. 
The math tutor connects two minuses with one line (as shown in the figure), and the student’s gaze opens to a plus sign (at the intersection with the bracket).
To prevent distraction, I recommend that math tutors highlight the minuend and subtrahend with boxes. 
If a math tutor uses frames or circles to highlight the components of an arithmetic operation, then the student will be able to more easily and quickly see the structure of the example and relate it to the corresponding rule. When drawing up solutions, you should not place pieces of the whole object on different lines of a notebook sheet, and also start adding until it is written down. All actions and transitions are necessarily shown (at least at the start of studying the topic).
Some math tutors strive for 100% accurate justification of translation rules, considering this strategy the only correct and useful for developing computational skills. However, practice shows that this path does not always bring good dividends. The need to understand what a person is doing most often appears after memorizing the stages of the algorithm used and practical consolidation of computational operations.
It is extremely important to practice the transition to a sum in a long numerical expression with several subtractions, for example. Before counting or converting, I have the student circle the numbers along with their signs on the left. The figure shows an example of how a math tutor identifies terms. For very weak sixth graders, you can additionally color the circles. Use one color for positive terms and another color for negative terms. IN special cases I pick up scissors and cut the expression into pieces. They can be rearranged arbitrarily, thus simulating the rearrangement of terms. The child will see that the signs move along with the terms themselves. That is, if the minus sign was to the left of the number 5, then no matter where we move the corresponding card, it will not come off the five.
Kolpakov A.N. Mathematics tutor for grades 5-6. Moscow. Strogino.
This article is devoted to the analysis of such a topic as subtracting negative numbers. The material is useful information about the rule for subtracting negative numbers and other definitions. To reinforce the essence of the paragraph, we will analyze in detail examples of typical exercises and tasks.
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Rule for subtracting negative numbers
In order to understand this topic, you should learn the basic definitions and concepts.
Definition 1
The rule for subtracting negative numbers is formulated as follows: so that from the number a subtract a number b with minus sign, necessary to reduce a add the number − b, which is the opposite of the subtrahend b.
If you imagine this rule subtracting a negative number b from an arbitrary number a in letter form, then it will look like this: a − b = a + (− b) .
In order to use this rule, it is necessary to prove its validity.
Let's take the numbers a And b. To subtract from a number a number b, you need to find such a number With, which adds up to the number b will equal the number a. In other words, if such a number is found c, What c + b = a, then the difference a−b equal to c.
In order to prove the subtraction rule, it is necessary to show that adding a sum a + (− b) with number b- this is a number a. It is necessary to remember the properties operations with real numbers. Since the combinatory property of addition works in this case, the equality (a + (− b)) + b = a + ((− b) + b) will be true.
Since the sum of numbers with opposite signs equals zero, then a + ((− b) + b) = a + 0, and the sum a + 0 = a ( If you add zero to a number, it will not change). Equality a − b = a + (− b) is considered proven, which means that the validity of the given rule for subtracting numbers with a minus sign is also proven.
We looked at how this rule works for real numbers a And b. But it is also considered valid for any rational and integer numbers a And b. Operations with rational and integer numbers also have the properties used in the proof. It should be added that using the parsed rule, you can perform the actions of a number with a minus sign as from positive number, and from negative or zero.
Let's look at the analyzed rule using typical examples.
Examples of using the subtraction rule
Let's look at examples involving subtracting numbers. First, let's look at a simple example that will help you easily understand all the intricacies of the process.
Example 1
Must be subtracted from the number − 13 number − 7 .
Let's take the opposite number to be subtracted − 7 . This number 7 . Then, by the rule for subtracting negative numbers, we have (− 13) − (− 7) = (− 13) + 7 . Let's do the addition. Now we get: (− 13) + 7 = − (13 − 7) = − 6 .
Here is the entire solution: (− 13) − (− 7) = (− 13) + 7 = − (13 − 7) = − 6 . (− 13) − (− 7) = − 6 . Subtraction of fractional negative numbers can also be performed. You need to move on to fractions, mixed numbers or decimals. The choice of number depends on which option is more convenient for you to work with.
Example 2
You need to subtract from a number 3 , 4 numbers - 23 2 3.
We apply the subtraction rule described above, we get 3, 4 - - 23 2 3 = 3, 4 + 23 2 3. Replace the fraction with decimal number: 3, 4 = 34 10 = 17 5 = 3 2 5 (you can see how to translate fractions in the material on the topic), we get 3, 4 + 23 2 3 = 3 2 5 + 23 2 3. Let's do the addition. This completes the subtraction of a negative number - 23 2 3 from the number 3 , 4 completed.
Here is a short summary of the solution: 3, 4 - - 23 2 3 = 27 1 15.
Example 3
You need to subtract a number − 0 , (326) from zero.
According to the subtraction rule we learned above, 0 − (− 0 , (326)) = 0 + 0 , (326) = 0 , (326) .
The last transition is correct, since the property of adding a number with zero works here: 0 − (− 0 , (326)) = 0 , (326) .
From the examples discussed, it is clear that when subtracting a negative number, you can get both a positive and a negative number. Subtracting a negative number can result in the number 0 , this happens when the minuend is equal to the subtrahend.
Example 4
It is necessary to calculate the difference of negative numbers - 5 - - 5.
By the subtraction rule we get - 5 - - 5 = - 5 + 5.
We have arrived at the sum of opposite numbers, which is always equal to zero: - 5 - - 5 = - 5 + 5 = 0
So, - 5 - - 5 = 0.
In some cases, the result of a subtraction must be written as a numerical expression. This is true in cases where the minuend or subtrahend is an irrational number. For example, subtracting from a negative number − 2 negative number – π carried out like this: (− 2) − (− π) = (− 2) + π = π − 2. The value of the resulting expression can be calculated as accurately as possible only if necessary. For detailed information You can explore other sections related to this topic.
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In this material, we will touch upon such an important topic as adding negative numbers. In the first paragraph we will tell you the basic rule for this action, and in the second we will analyze specific examples solving similar problems.
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Basic rule for adding natural numbers
Before we derive the rule, let us remember what we generally know about positive and negative numbers. Previously, we agreed that negative numbers should be perceived as debt, loss. The modulus of a negative number expresses the exact size of this loss. Then the addition of negative numbers can be represented as the addition of two losses.
Using this reasoning, we formulate the basic rule for adding negative numbers.
Definition 1
In order to complete adding negative numbers, you need to add up the values of their modules and put a minus in front of the result. In literal form, the formula looks like (− a) + (− b) = − (a + b) .
Based on this rule, we can conclude that adding negative numbers is similar to adding positive ones, only in the end we must get a negative number, because we must put a minus sign in front of the sum of the modules.
What evidence can be given for this rule? To do this, we need to remember the basic properties of operations with real numbers (or with integers, or with rational numbers - they are the same for all these types of numbers). To prove it, we just need to demonstrate that the difference between the left and right sides of the equality (− a) + (− b) = − (a + b) will be equal to 0.
Subtracting one number from another is the same as adding the same opposite number to it. Therefore, (− a) + (− b) − (− (a + b)) = (− a) + (− b) + (a + b) . Recall that numerical expressions with addition have two main properties - associative and commutative. Then we can conclude that (− a) + (− b) + (a + b) = (− a + a) + (− b + b) . Since, by adding opposite numbers, we always get 0, then (− a + a) + (− b + b) = 0 + 0, and 0 + 0 = 0. Our equality can be considered proven, which means the rule for adding negative numbers We also proved it.
In the second paragraph, we will take specific problems where we need to add negative numbers, and we will try to apply the learned rule to them.
Example 1
Find the sum of two negative numbers - 304 and - 18,007.
Solution
Let's perform the steps step by step. First we need to find the modules of the numbers being added: - 304 = 304, - 180007 = 180007. Next we need to perform the addition action, for which we use the column counting method:
All we have left is to put a minus in front of the result and get - 18,311.
Answer: - - 18 311 .
What numbers we have depends on what we can reduce the action of addition to: finding the sum natural numbers, to the addition of ordinary or decimals. Let's analyze the problem with these numbers.
Example N
Find the sum of two negative numbers - 2 5 and − 4, (12).
Solution
We find the modules of the required numbers and get 2 5 and 4, (12). We got two different fractions. Let us reduce the problem to the addition of two ordinary fractions, for which we represent the periodic fraction in the form of an ordinary one:
4 , (12) = 4 + (0 , 12 + 0 , 0012 + . . .) = 4 + 0 , 12 1 - 0 , 01 = 4 + 0 , 12 0 , 99 = 4 + 12 99 = 4 + 4 33 = 136 33
As a result, we received a fraction that will be easy to add with the first original term (if you have forgotten how to correctly add fractions with different denominators, repeat the corresponding material).
2 5 + 136 33 = 2 33 5 33 + 136 5 33 5 = 66 165 + 680 165 = 764 165 = 4 86 105
As a result, we got a mixed number, in front of which we only have to put a minus. This completes the calculations.
Answer: - 4 86 105 .
Real negative numbers add up in a similar way. The result of such an action is usually written down as a numerical expression. Its value may not be calculated or limited to approximate calculations. So, for example, if we need to find the sum - 3 + (− 5), then we write the answer as - 3 − 5. We have devoted a separate material to the addition of real numbers, in which you can find other examples.
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