As part of this material, we will touch on such an important topic as the addition of negative numbers. In the first paragraph, we will describe the basic rule for this action, and in the second, we will analyze specific examples of solving such problems.
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Basic rule for adding natural numbers
Before deriving the rule, let's recall what we generally know about positive and negative numbers. Earlier we agreed that negative numbers should be perceived as a debt, a loss. The modulus of a negative number expresses the exact size of this loss. Then the addition of negative numbers can be thought of as the addition of two losses.
Using this reasoning, we formulate the basic rule for adding negative numbers.
Definition 1
In order to fulfill addition of negative numbers, you need to add 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 the addition of negative numbers is similar to the addition of positive ones, only in the end we must definitely get a negative number, because we must put a minus sign in front of the sum of modules.
What evidence can be given for this rule? To do this, we need to recall the basic properties of operations with real numbers (either with integers or with rational ones - they are the same for all these types of numbers). To prove it, we only need to demonstrate that the difference between the left and right sides of the equation (− 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) \u003d 0 + 0, and 0 + 0 \u003d 0. Our equality can be considered proven, which means that the rule for adding negative numbers we also proved.
In the second paragraph, we will take specific problems where you need to add negative numbers, and try to apply the learned rule in them.
Example 1
Find the sum of two negative numbers - 304 and - 18007.
Decision
Let's do the steps step by step. First we need to find the modules of the numbers to be added: - 304 = 304 , - 180007 = 180007 . Next, we need to perform the addition action, for which we use the column count method:
All we have left is to put a minus in front of the result and get - 18 311 .
Answer: - - 18 311 .
It depends on what numbers we have, to what we can reduce the action of addition: to finding the sum of natural numbers, to adding ordinary or decimal fractions. Let's analyze the problem with such numbers.
Example N
Find the sum of two negative numbers - 2 5 and − 4 , (12) .
Decision
We find the modules of the desired numbers and get 2 5 and 4 , (12) . We have two different fractions. We reduce the problem to the addition of two ordinary fractions, for which we represent the periodic fraction in the form of an ordinary:
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 got a fraction that will be easy to add to the first original term (if you forgot how to add fractions with different denominators correctly, 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 need to put a minus. This completes the calculations.
Answer: - 4 86 105 .
Real negative numbers are added in the same way. The result of such an action is usually written as a numerical expression. Its value can 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 devoted a separate material to the addition of real numbers, in which you can find other examples.
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Negative addition rule
If you recall the math lesson and the topic “Addition and subtraction of numbers with different signs”, then to add two negative numbers you need:
- perform the addition of their modules;
- add the sign "-" to the received amount.
According to the addition rule, we can write:
$(−a)+(−b)=−(a+b)$.
The negative addition rule applies to negative integers, rational numbers, and real numbers.
Example 1
Add negative numbers $−185$ and $−23 \ 789.$
Decision.
Let's use the rule of adding negative numbers.
Let's find the modules of these numbers:
$|-23 \ 789|=23 \ 789$.
Let's add the resulting numbers:
$185+23 \ 789=23 \ 974$.
We put the sign $"–"$ in front of the found number and get $−23 \ 974$.
Brief solution: $(−185)+(−23 \ 789)=−(185+23 \ 789)=−23 \ 974$.
Answer: $−23 \ 974$.
When adding negative rational numbers, they must be converted to the form of natural numbers, ordinary or decimal fractions.
Example 2
Add the negative numbers $-\frac(1)(4)$ and $−7.15$.
Decision.
According to the rule of adding negative numbers, you first need to find the sum of the modules:
$|-\frac(1)(4)|=\frac(1)(4)$;
It is convenient to reduce the obtained values to decimal fractions and perform their addition:
$\frac(1)(4)=0.25$;
$0,25+7,15=7,40$.
Let's put the sign $"-"$ in front of the received value and get $-7.4$.
Solution summary:
$(-\frac(1)(4))+(−7.15)=−(\frac(1)(4)+7.15)=–(0.25+7.15)=−7, 4$.
To add positive and negative numbers:
- calculate modules of numbers;
- if they are equal, then the original numbers are opposite and their sum is equal to zero;
- if they are not equal, then you need to remember the sign of the number whose modulus is greater;
subtract the smaller one from the larger one;
- before the received value, put the sign of the number whose modulus is greater.
compare the received numbers:
Adding numbers with opposite signs is reduced to subtracting a smaller negative number from a larger positive number.
The rule of adding numbers with opposite signs is carried out for integer, rational and real numbers.
Example 3
Add the numbers $4$ and $−8$.
Decision.
You need to add numbers with opposite signs. Let's use the appropriate addition rule.
Let's find the modules of these numbers:
The modulus of the number $−8$ is greater than the modulus of the number $4$, i.e. remember the sign $"-"$.
We put the sign $"–"$, which we memorized, in front of the resulting number, and we get $−4.$
Solution summary:
$4+(–8) = –(8–4) = –4$.
Answer: $4+(−8)=−4$.
To add rational numbers with opposite signs, it is convenient to represent them as ordinary or decimal fractions.
Subtraction of numbers with different and negative signs
Rule for subtracting negative numbers:
To subtract a negative number $b$ from the number $a$, it is necessary to add to the minuend $a$ the number $−b$, which is the opposite of the subtracted $b$.
According to the subtraction rule, we can write:
$a−b=a+(−b)$.
This rule is valid for integer, rational and real numbers. The rule can be used when subtracting a negative number from a positive number, from a negative number, and from zero.
Example 4
Subtract from the negative number $−28$ the negative number $−5$.
Decision.
The opposite number for the number $–5$ is the number $5$.
According to the rule for subtracting negative numbers, we get:
$(−28)−(−5)=(−28)+5$.
Let's add numbers with opposite signs:
$(−28)+5=−(28−5)=−23$.
Answer: $(−28)−(−5)=−23$.
When subtracting negative fractional numbers, you must convert the numbers to the form of ordinary fractions, mixed numbers, or decimal fractions.
Addition and subtraction of numbers with different signs
The rule for subtracting numbers with opposite signs is the same as the rule for subtracting negative numbers.
Example 5
Subtract the positive number $7$ from the negative number $−11$.
Decision.
The opposite number for the number $7$ is the number $–7$.
According to the rule for subtracting numbers with opposite signs, we get:
$(−11)−7=(–11)+(−7)$.
Let's add negative numbers:
$(−11)+(–7)=−(11+7)=−18$.
Brief solution: $(−28)−(−5)=(−28)+5=−(28−5)=−23$.
Answer: $(−11)−7=−18$.
When subtracting fractional numbers with different signs, it is necessary to convert the numbers to the form of ordinary or decimal fractions.
The development of numeracy skills is a major goal pursued by mathematics programs from grades 1 to 6. How quickly and correctly the child learns to perform arithmetic operations will depend on the speed of his logical (semantic) operations in the senior classes and the level of understanding of the subject as a whole. It is not uncommon for a math tutor to encounter student computational problems that prevent them from achieving high scores.
What kind of students does not have to work with a tutor. Parents need preparation for the exam in mathematics, and their child cannot understand ordinary fractions or gets confused in negative numbers. What actions should the 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 by some artificial "semi-finished products-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.
Suppose 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 conversion. Textbooks offer strict and precise mathematical formulations: “In order to subtract the number “b” from the number “a”, you need to add the number opposite to “b” to the number “a”. Formally, you can’t find fault with the text, but as soon as it begins to be used by a math tutor as an instruction for performing specific calculations, problems arise. One phrase alone is worth something: “To subtract, you must add.” Without a clear commentary from the tutor, the student will not understand. In fact, what to do: subtract or add?
If you work with the rule according to the intention of the authors of the textbook, then in addition to working out the concept of “opposite number”, you need to teach the student to correlate the designations “a” and “b” with 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 a 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,
A) Rewrite the first number
B) put a plus
B) Change the sign of the second number to the opposite
D) Add the resulting numbers
Here, the stages of the algorithm are clearly separated by points and are not tied to letter designations.
In the course of solving a practical assignment for translations, the math tutor rereads this text to the student several times (for memorization). I advise you to write it down in a theoretical notebook. Only after working out the rule of transition to addition, you can write the general form a-b=a+(-b)
The movement of the minus and plus signs in the head of a child (both a small and a weak adult) is somewhat reminiscent of Brownian. A math tutor needs to put things in order in this chaos as quickly as possible. In the process of solving examples, reference prompts (verbal and visual) are used, which, in combination with accurate and detailed layout, do their job. It must be remembered that every word uttered by a math tutor at the time of solving any problem carries either a hint or a hindrance. Each phrase is analyzed by the child in order to establish a connection with one or another mathematical object (phenomenon) and its image on paper.
A typical problem of weak schoolchildren is the separation of the sign of an action from the sign of the number involved in it. The same visual image makes it difficult to recognize the reduced "a" and the subtracted "b" in the difference a-b. When, in the process of explaining, a math tutor reads an expression, you need to make sure that the word “subtract” is used instead of “-”. It is necessary! For example, the entry should be read like this: “From 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 tutor in mathematics constantly flies off the tongue “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 to accurately convey information.
How can a tutor explain the transition to addition?
Of course, one 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 simple actions when working with him. I often say to my sixth graders, “There is no such arithmetic operation in mathematics as “difference”. Writing 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 at the words of the tutor and involuntarily remember that you cannot subtract numbers directly. The math tutor declares 5 and -3 terms, and for greater motivation of his words, compares the results of actions 5-3 and 5+(-3). After that, the identity a-b=a+(-b) is written
Whatever the student, and no matter how much time is given to the math tutor for classes with him, you need to work out the concept of “opposite number” in time. The record “-x” deserves special attention of a math tutor. A 6th grade student must learn that it does not display a negative number, but the opposite of x.
It is necessary to dwell separately on calculations with two minus signs located side by side. There is a problem of understanding the operation of their simultaneous removal. It is necessary to carefully go through all the points of the stated algorithm for the transition to addition. It will be better if, when working with the difference -5- (-3), before any comments, the math tutor will highlight the numbers -5 and -3 in a box or underline them. This will help the student identify the components of the action.
The focus of the math tutor on memorization
Reliable memorization is the result of the practical application of mathematical rules, so it is important for the tutor to ensure a good density of independently solved examples. To improve the stability of memorization, you can call for help visual cues - chips. For example, an interesting way to translate 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 up the plus sign (at the intersection with the bracket).
To prevent distraction, I recommend that math tutors highlight the minuend and the subtrahend with boxes. 
If a math tutor uses boxes or circles to highlight the components of an arithmetic operation, then the student will more easily and quickly learn to see the structure of the example and correlate it with the corresponding rule. You should not place pieces of the whole object when making decisions on different lines of a notebook sheet, and also start adding until it is written down. All actions and transitions are shown without fail (at least at the start of studying the topic).
Some math tutors strive for a 100% accurate justification of the translation rules, considering this strategy the only correct and useful one for the formation of computational skills. However, practice shows that this path does not always bring good dividends. The need for awareness of what a person is doing most often appears after memorizing the steps of the applied algorithm and practical fixing of computational operations.
It is extremely important to work out the transition to the sum in a long numerical expression with several subtractions, for example. Before proceeding with the counting or conversion, I have the student circle the numbers along with their signs to the left. The figure shows an example of how a math tutor selects terms. For very weak sixth graders, you can additionally tint the circles. Use one color for positive terms and another color for negative terms. In special cases, I take scissors in my hands and cut the expression into pieces. They can be arbitrarily rearranged, thus imitating a permutation 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 wherever we shift the corresponding card, it will not come off the five.
Kolpakov A.N. Mathematics tutor grade 5-6. Moscow. Strogino.
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 perform 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.
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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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 the modulus of the number 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 moduli.
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.
Thus, this addition rule applies to both negative integers and rational numbers, as well as real 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.
Add negative numbers -304 and -18007 .
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 addition in a column: 
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 .
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.
Add a negative number and a negative number −4,(12) .
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 add fractions with different denominators: .
It remains to put a minus sign in front of the resulting number: . This completes the addition of the original negative numbers.
Negative real numbers are added according to the same rule for adding negative numbers. It is worth noting here that the result of adding real numbers is very often written as a numerical expression, and the value of this expression is calculated approximately, and then if necessary.
For example, let's find the sum of negative numbers and -5. The modules of these numbers are equal to the square root of three and five, respectively, and the sum of the original numbers is . This is how the answer is written. Other examples can be found in the article. addition of real numbers.
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How to add two negative numbers
Operations with negative and positive numbers
Absolute value (modulus). Addition.
Subtraction. Multiplication. Division.
Absolute value (modulus). For negative number is a positive number obtained by changing its sign from “-” to “+”; for positive number and zero is the number itself. To denote the absolute value (modulus) of a number, two straight lines are used, inside which this number is written.
EXAMPLES: | – 5 | = 5, | 7 | = 7, | 0 | = 0.
1) when adding two numbers with the same sign, add
their absolute values and the sum is preceded by a common sign.
2) when adding two numbers with different signs, their absolute
values are subtracted (from the larger one the smaller one) and the sign is put
numbers with a larger absolute value.
Subtraction. You can replace the subtraction of two numbers with addition, while the minuend retains its sign, and the subtrahend is taken with the opposite sign.
(+ 8) – (+ 5) = (+ 8) + (– 5) = 3;
(+ 8) – (– 5) = (+ 8) + (+ 5) = 13;
(– 8) – (– 5) = (– 8) + (+ 5) = – 3;
(– 8) – (+ 5) = (– 8) + (– 5) = – 13;
Multiplication. When two numbers are multiplied, their absolute values are multiplied, and the product takes on the “+” sign if the signs of the factors are the same, and the sign “-” if the signs of the factors are different.
The following scheme is useful ( multiplication sign rules):
When multiplying several numbers (two or more), the product has a “+” sign if the number of negative factors is even, and a “-” sign if their number is odd.
Division. When dividing two numbers, the absolute value of the dividend is divided by the absolute value of the divisor, and the quotient takes on the sign "+" if the signs of the dividend and the divisor are the same, and the sign "-" if the signs of the dividend and divisor are different.
There are The same sign rules, as in multiplication:
Adding negative numbers
Addition of positive and negative numbers can be parsed using the number axis.
Adding numbers using the coordinate line
Addition of numbers small in absolute value is conveniently performed on the coordinate line, mentally imagining as a point denoting the number moves along the number axis.
Let's take some number, for example, 3 . Let's designate it on a numerical axis with a point " A ".
Let's add the positive number 2 to the number. This will mean that the point "A" must be moved two unit segments in the positive direction, that is, to the right. As a result, we will get point "B" with coordinate 5.
In order to add a negative number “−5” to a positive number, for example, 3, the point “A” must be moved 5 units of length in the negative direction, that is, to the left.
In this case, the coordinate of the point "B" is equal to - "2".
So, the order of adding rational numbers using the number axis will be as follows:
Moving from the point - 2 to the left (since there is a minus sign in front of 6), we get - 8.
Addition of numbers with the same signs
Adding rational numbers is easier if you use the concept of a modulus.
Suppose we need to add numbers that have the same sign.
To do this, we discard the signs of numbers and take the modules of these numbers. We add the modules and put the sign in front of the sum, which was common to these numbers.
An example of adding negative numbers.
To add numbers of the same sign, you need to add their modules and put the sign in front of the sum that was in front of the terms.
Addition of numbers with different signs
If the numbers have different signs, then we act somewhat differently than when adding numbers with the same signs.
An example of adding a negative and a positive number.
An example of adding mixed numbers.
To add numbers of opposite sign necessary:
- subtract the smaller module from the larger module;
- before the resulting difference, put the sign of the number that has a larger modulus.
- perform the addition of their modules;
- add the sign "-" to the received amount.
- calculate modules of numbers;
- compare the received numbers:
- subtract the smaller one from the larger one;
- before the received value, put the sign of the number whose modulus is greater.
Addition and subtraction of positive and negative numbers
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Negative addition rule
To add two negative numbers:
According to the addition rule, we can write:
The negative addition rule applies to negative integers, rational numbers, and real numbers.
Add negative numbers $−185$ and $−23 \ 789.$
Let's use the rule of adding negative numbers.
Let's add the resulting numbers:
$185+23 \ 789=23 \ 974$.
We put the sign $"–"$ in front of the found number and get $−23 974$.
Brief solution: $(−185)+(−23 \ 789)=−(185+23 \ 789)=−23 \ 974$.
When adding negative rational numbers, they must be converted to the form of natural numbers, ordinary or decimal fractions.
Add the negative numbers $-\frac $ and $−7.15$.
According to the rule of adding negative numbers, you first need to find the sum of modules:
It is convenient to reduce the obtained values to decimal fractions and perform their addition:
Let's put the sign $"-"$ in front of the received value and get $-7.4$.
Solution summary:
Addition of numbers with opposite signs
Rule for adding numbers with opposite signs:
if they are equal, then the original numbers are opposite and their sum is equal to zero;
if they are not equal, then you need to remember the sign of the number whose modulus is greater;
Adding numbers with opposite signs is reduced to subtracting a smaller negative number from a larger positive number.
The rule of adding numbers with opposite signs is carried out for integer, rational and real numbers.
Add the numbers $4$ and $−8$.
You need to add numbers with opposite signs. Let's use the appropriate addition rule.
Let's find the modules of these numbers:
The modulus of the number $−8$ is greater than the modulus of the number $4$, i.e. remember the sign $"-"$.
We put the sign $"–"$, which we memorized, in front of the resulting number, and we get $−4.$
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To add rational numbers with opposite signs, it is convenient to represent them as ordinary or decimal fractions.
Subtraction of negative numbers
Rule for subtracting negative numbers:
To subtract a negative number $b$ from the number $a$, it is necessary to add to the minuend $a$ the number $−b$, which is the opposite of the subtracted $b$.
According to the subtraction rule, we can write:
This rule is valid for integer, rational and real numbers. The rule can be used when subtracting a negative number from a positive number, from a negative number, and from zero.
Subtract from the negative number $−28$ the negative number $−5$.
The opposite number for the number $–5$ is the number $5$.
According to the rule for subtracting negative numbers, we get:
Let's add numbers with opposite signs:
Brief solution: $(−28)−(−5)=(−28)+5=−(28−5)=−23$.
When subtracting negative fractional numbers, you must convert the numbers to the form of ordinary fractions, mixed numbers, or decimal fractions.
Subtraction of numbers with opposite signs
The rule for subtracting numbers with opposite signs is the same as the rule for subtracting negative numbers.
Subtract the positive number $7$ from the negative number $−11$.
The opposite number for the number $7$ is the number $–7$.
According to the rule for subtracting numbers with opposite signs, we get:
Let's add negative numbers:
When subtracting fractional numbers with opposite signs, it is necessary to convert the numbers to the form of ordinary or decimal fractions.
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Addition of negative numbers: rule, examples
As part of this material, we will touch on such an important topic as the addition of negative numbers. In the first paragraph, we will describe the basic rule for this action, and in the second, we will analyze specific examples of solving such problems.
Basic rule for adding natural numbers
Before deriving the rule, let's recall what we generally know about positive and negative numbers. Earlier we agreed that negative numbers should be perceived as a debt, a loss. The modulus of a negative number expresses the exact size of this loss. Then the addition of negative numbers can be thought of as the addition of two losses.
Using this reasoning, we formulate the basic rule for adding negative numbers.
In order to fulfill addition of negative numbers, you need to add 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 the addition of negative numbers is similar to the addition of positive ones, only in the end we must definitely get a negative number, because we must put a minus sign in front of the sum of modules.
What evidence can be given for this rule? To do this, we need to recall the basic properties of operations with real numbers (either with integers or with rational ones - they are the same for all these types of numbers). To prove it, we only need to demonstrate that the difference between the left and right sides of the equation (− 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) \u003d 0 + 0, and 0 + 0 \u003d 0. Our equality can be considered proven, which means that the rule for adding negative numbers we also proved.
Problems for addition of negative numbers
In the second paragraph, we will take specific problems where you need to add negative numbers, and try to apply the learned rule in them.
Find the sum of two negative numbers - 304 and - 18007.
Decision
Let's do the steps step by step. First we need to find the modules of the numbers to be added: - 304 \u003d 304, - 180007 \u003d 180007. Next, we need to perform the addition action, for which we use the column count method:
All we have left is to put a minus in front of the result and get - 18 311 .
Answer: — — 18 311 .
It depends on what numbers we have, to what we can reduce the action of addition: to finding the sum of natural numbers, to adding ordinary or decimal fractions. Let's analyze the problem with such numbers.
Find the sum of two negative numbers - 2 5 and - 4 , (12) .
We find the modules of the desired numbers and get 2 5 and 4 , (12) . We have two different fractions. We reduce the problem to the addition of two ordinary fractions, for which we represent the periodic fraction in the form of an ordinary:
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 got a fraction that will be easy to add to the first original term (if you forgot how to add fractions with different denominators correctly, 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 need to put a minus. This completes the calculations.
Answer: — 4 86 105 .
Real negative numbers are added in the same way. The result of such an action is usually written as a numerical expression. Its value can 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 devoted a separate material to the addition of real numbers, in which you can find other examples.
