In this article, we will deal with adding numbers with different signs. Here we give a rule for adding a positive and a negative number, and consider examples of the application of this rule when adding numbers with different signs.
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Rule for adding numbers with different signs
Examples of adding numbers with different signs
Consider examples of adding numbers with different signs according to the rule discussed in the previous paragraph. Let's start with a simple example.
Example.
Add the numbers −5 and 2 .
Solution.
We need to add numbers with different signs. Let's follow all the steps prescribed by the rule of adding positive and negative numbers.
First, we find the modules of the terms, they are equal to 5 and 2, respectively.
The modulus of the number −5 is greater than the modulus of the number 2, so remember the minus sign.
It remains to put the memorized minus sign in front of the resulting number, we get −3. This completes the addition of numbers with different signs.
Answer:
(−5)+2=−3 .
To add rational numbers with different signs that are not integers, they should be represented as ordinary fractions (you can work with decimal fractions, if it is convenient). Let's take a look at this point in the next example.
Example.
Add a positive number and a negative number −1.25.
Solution.
Let's represent the numbers in the form of ordinary fractions, for this we will perform the transition from a mixed number to an improper fraction: , and translate the decimal fraction into an ordinary one: 
.
Now you can use the rule for adding numbers with different signs.
The modules of the added numbers are 17/8 and 5/4. For the convenience of performing further actions, we reduce the fractions to a common denominator, as a result we have 17/8 and 10/8.
Now we need to compare the common fractions 17/8 and 10/8. Since 17>10 , then . Thus, the term with a plus sign has a larger modulus, therefore, remember the plus sign.
Now we subtract the smaller one from the larger module, that is, we subtract fractions with the same denominators: 
.
It remains to put a memorized plus sign in front of the resulting number, we get, but - this is the number 7/8.
In this lesson, we will learn what a negative number is and what numbers are called opposites. We will also learn how to add negative and positive numbers (numbers with different signs) and analyze several examples of adding numbers with different signs.
Look at this gear (see Fig. 1).
Rice. 1. Clock gear
This is not an arrow that directly shows the time and not a dial (see Fig. 2). But without this detail, the clock does not work.
Rice. 2. Gear inside the watch
What does the letter Y stand for? Nothing but the sound Y. But without it, many words will not “work”. For example, the word "mouse". So are negative numbers: they do not show any amount, but without them the calculation mechanism would be much more difficult.
We know that addition and subtraction are equal operations, and they can be performed in any order. In direct order, we can calculate: , but there is no way to start with subtraction, since we have not agreed yet, but what is .
It is clear that increasing the number by and then decreasing by means, as a result, a decrease by three. Why not designate this object and count it this way: to add is to subtract. Then .
The number can mean, for example, apples. The new number does not represent any real quantity. By itself, it does not mean anything, like the letter Y. It's just a new tool to simplify calculations.
Let's name new numbers negative. Now we can subtract a larger number from a smaller number. Technically, you still need to subtract the smaller number from the larger number, but put a minus sign in the answer: .
Let's look at another example: 
. You can do all the actions in a row:.
However, it is easier to subtract the third number from the first number, and then add the second number:
Negative numbers can be defined in another way.
For each natural number, for example , let's introduce a new number, which we denote , and determine that it has the following property: the sum of the number and is equal to : .
The number will be called negative, and the numbers and - opposite. Thus, we got an infinite number of new numbers, for example:
The opposite of number ;
The opposite of ;
The opposite of ; 
The opposite of ;
Subtract the larger number from the smaller number: Let's add to this expression: . We got zero. However, according to the property: a number that adds up to five gives zero is denoted minus five:. Therefore, the expression can be denoted as .
Every positive number has a twin number, which differs only in that it is preceded by a minus sign. Such numbers are called opposite(See Fig. 3).
Rice. 3. Examples of opposite numbers
Properties of opposite numbers
1. The sum of opposite numbers is equal to zero:.
2. If you subtract a positive number from zero, then the result will be the opposite negative number: .
1. Both numbers can be positive, and we already know how to add them: .
2. Both numbers can be negative.
We have already covered the addition of such numbers in the previous lesson, but we will make sure that we understand what to do with them. For example: .
To find this sum, add opposite positive numbers and put a minus sign.
3. One number can be positive and another negative.
We can replace the addition of a negative number, if it is convenient for us, with the subtraction of a positive one:.
One more example: . Again, write the sum as a difference. You can subtract a larger number from a smaller number by subtracting a smaller number from a larger one, but putting a minus sign.
The terms can be interchanged: .
Another similar example: .
In all cases, the result is a subtraction.
To briefly formulate these rules, let's recall another term. Opposite numbers, of course, are not equal to each other. But it would be strange not to notice they have something in common. This common we called modulus of number. The modulus of opposite numbers is the same: for a positive number it is equal to the number itself, and for a negative one it is the opposite, positive. For example: , .
To add two negative numbers, add their modulus and put a minus sign:
To add a negative and a positive number, you need to subtract the smaller module from the larger module and put the sign of the number with the larger module:
Both numbers are negative, therefore, add their modules and put a minus sign:
Two numbers with different signs, therefore, from the modulus of the number (larger modulus) we subtract the modulus of the number and put a minus sign (the sign of the number with a larger modulus):
Two numbers with different signs, therefore, from the modulus of the number (larger modulus) we subtract the modulus of the number and put a minus sign (the sign of the number with a large modulus): .
Two numbers with different signs, therefore, subtract the module of the number from the module of the number (larger module) and put a plus sign (sign of the number with a large module): .
Positive and negative numbers have historically different roles.
First, we introduced natural numbers for counting objects:
Then we introduced other positive numbers - fractions, for counting non-integer quantities, parts: .
Negative numbers appeared as a tool to simplify calculations. There was no such thing that in life there were some quantities that we could not count, and we invented negative numbers.
That is, negative numbers did not originate from the real world. They just turned out to be so convenient that in some places they were used in life. For example, we often hear about negative temperatures. In this case, we never encounter a negative number of apples. What is the difference?
The difference is that in real life negative values are only used for comparison, not for quantities. If a basement was equipped in the hotel and an elevator was launched there, then in order to leave the usual numbering of ordinary floors, a minus the first floor may appear. This minus one means only one floor below ground level (see Fig. 1).
Rice. 4. Minus the first and minus the second floors
A negative temperature is negative only compared to zero, which was chosen by the author of the scale, Anders Celsius. There are other scales, and the same temperature may no longer be negative there.
At the same time, we understand that it is impossible to change the starting point so that there are not five, but six apples. Thus, in life, positive numbers are used to determine quantities (apples, cake).
We also use them instead of names. Each phone could be given its own name, but the number of names is limited, and there are no numbers. That's why we use phone numbers. Also for ordering (century follows century).
Negative numbers in life are used in the last sense (minus the first floor below the zero and first floors)
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. "Gymnasium", 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. Moscow: Education, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A guide for students in grade 6 of the MEPhI correspondence school. M.: ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. M .: Education, Mathematics Teacher Library, 1989.
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Homework
"Addition of numbers with different signs" - Mathematics textbook Grade 6 (Vilenkin)
Short description:
In this section, you will learn the rules for adding numbers with different signs: that is, learn how to add negative and positive numbers.
You already know how to add them on a coordinate line, but in each example you won’t draw a line and count along it? Therefore, you need to learn how to add without it.
Let's try with you to add a negative number to a positive number, for example add eight minus six: 8+(-6). You already know that adding a negative number causes the original number to decrease by the value of the negative number. This means that eight must be reduced by six, that is, six should be subtracted from eight: 8-6=2, it turns out two. In this example, everything seems to be clear, we subtract six from eight.
And if we take this example: add a positive number to a negative number. For example, minus eight add six: -8+6. The essence remains the same: we reduce the positive number by the value of the negative, we get six subtracting eight will be minus two: -8+6=-2.
As you noticed, both in the first and in the second example, subtraction is performed with numbers. Why? Because they have different signs (plus and minus). In order not to make mistakes when adding numbers with different signs, you should perform the following algorithm of actions:
1. find modules of numbers;
2. subtract the smaller module from the larger module;
3. before the result, put a number sign with a large modulus (usually only a minus sign is put, and a plus sign is not put).
If you add numbers with different signs, following this algorithm, then you will have much less chance of making a mistake.
This lesson covers addition and subtraction of rational numbers. The topic is classified as complex. Here it is necessary to use the entire arsenal of previously acquired knowledge.
The rules for adding and subtracting integers are also valid for rational numbers. Recall that rational numbers are numbers that can be represented as a fraction, where a - is the numerator of a fraction b is the denominator of the fraction. Wherein, b should not be null.
In this lesson, we will increasingly refer to fractions and mixed numbers as one common phrase - rational numbers.
Lesson navigation:Example 1 Find the value of an expression:
We enclose each rational number in brackets along with its signs. We take into account that the plus which is given in the expression is the sign of the operation and does not apply to fractions. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract a smaller module from a larger module, and put the sign of the rational number whose module is larger in front of the answer. And in order to understand which module is greater and which is less, you need to be able to compare the modules of these fractions before calculating them:
The modulus of a rational number is greater than the modulus of a rational number. Therefore, we subtracted from . Got an answer. Then, reducing this fraction by 2, we got the final answer.
Some primitive actions, such as putting numbers in brackets and putting down modules, can be skipped. This example can be written in a shorter way:
Example 2 Find the value of an expression:
We enclose each rational number in brackets along with its signs. We take into account that the minus between rational numbers and is the sign of the operation and does not apply to fractions. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
Let's replace subtraction with addition. Recall that for this you need to add to the minuend the number opposite to the subtrahend:
We got the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus before the answer:
Note. It is not necessary to enclose every rational number in parentheses. This is done for convenience, in order to clearly see what signs rational numbers have.
Example 3 Find the value of an expression:
In this expression, the fractions have different denominators. To make it easier for ourselves, let's bring these fractions to a common denominator. We will not go into detail on how to do this. If you experience difficulties, be sure to repeat the lesson.
After reducing the fractions to a common denominator, the expression will take the following form:
This is the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the received answer we put the sign of the rational number, the module of which is greater:
Let's write down the solution of this example in a shorter way:
Example 4 Find the value of an expression
We calculate this expression in the following way: we add the rational numbers and , then subtract the rational number from the result obtained. 
First action:
Second action:
Example 5. Find the value of an expression:
Let's represent the integer −1 as a fraction, and translate the mixed number into an improper fraction:
We enclose each rational number in brackets along with its signs:
We got the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the received answer we put the sign of the rational number, the module of which is greater:
Got an answer.
There is also a second solution. It consists in putting together whole parts separately.
So, back to the original expression:
Enclose each number in parentheses. For this mixed number temporarily:
Let's calculate the integer parts:
(−1) + (+2) = 1
In the main expression, instead of (−1) + (+2), we write the resulting unit:
The resulting expression. To do this, write the unit and the fraction together:
Let's write the solution in this way in a shorter way:
Example 6 Find the value of an expression
Convert the mixed number to an improper fraction. We rewrite the rest without change:
We enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
Let's write down the solution of this example in a shorter way:
Example 7 Find value expression
Let's represent the integer −5 as a fraction, and translate the mixed number into an improper fraction:
Let's bring these fractions to a common denominator. After bringing them to a common denominator, they will take the following form:
We enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:
Thus, the value of the expression is .
Let's solve this example in the second way. Let's go back to the original expression:
Let's write the mixed number in expanded form. We rewrite the rest without changes:
We enclose each rational number in brackets together with its signs:
Let's calculate the integer parts:
In the main expression, instead of writing the resulting number −7
The expression is an expanded form of writing a mixed number. Let's write the number −7 and the fraction together, forming the final answer:
Let's write this solution shortly:
Example 8 Find the value of an expression
We enclose each rational number in brackets together with its signs:
Let's replace subtraction with addition:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:
Thus, the value of the expression is
This example can be solved in the second way. It consists in adding the whole and fractional parts separately. Let's go back to the original expression:
We enclose each rational number in brackets along with its signs:
Let's replace subtraction with addition:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer. But this time we add separately the integer parts (−1 and −2), and the fractional and
Let's write this solution shortly:
Example 9 Find expression expressions 
Convert mixed numbers to improper fractions:
We enclose the rational number in brackets together with its sign. A rational number does not need to be enclosed in brackets, since it is already in brackets:
We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:
Thus, the value of the expression is
Now let's try to solve the same example in the second way, namely by adding the integer and fractional parts separately.
This time, in order to get a short solution, let's try to skip some actions, such as writing a mixed number in expanded form and replacing subtraction with addition:
Note that the fractional parts have been reduced to a common denominator.
Example 10 Find the value of an expression 
Let's replace subtraction with addition:
The resulting expression does not contain negative numbers, which are the main cause of errors. And since there are no negative numbers, we can remove the plus in front of the subtrahend, and also remove the parentheses:
The result is a simple expression that is easy to calculate. Let's calculate it in any way convenient for us:
Example 11. Find the value of an expression
This is the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and put the sign of the rational number, the module of which is greater, in front of the received answers:
Example 12. Find the value of an expression 
The expression consists of several rational numbers. According to, first of all, you need to perform the actions in brackets.
First, we calculate the expression , then the expression We add the results obtained.
First action:
Second action:
Third action:
Answer: expression value equals
Example 13 Find the value of an expression 
Convert mixed numbers to improper fractions:
We enclose the rational number in brackets along with its sign. A rational number does not need to be enclosed in parentheses, since it is already in parentheses:
Let's give these fractions in a common denominator. After bringing them to a common denominator, they will take the following form:
Let's replace subtraction with addition:
We got the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and put the sign of the rational number, the module of which is greater, in front of the received answers:
Thus, the value of the expression equals
Consider the addition and subtraction of decimal fractions, which are also rational numbers and which can be both positive and negative.
Example 14 Find the value of the expression −3.2 + 4.3
We enclose each rational number in brackets along with its signs. We take into account that the plus that is given in the expression is the sign of the operation and does not apply to the decimal fraction 4.3. This decimal has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
(−3,2) + (+4,3)
This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract a smaller module from a larger module, and put the rational number whose module is larger in front of the answer. And in order to understand which modulus is larger and which is smaller, you need to be able to compare the moduli of these decimal fractions before calculating them:
(−3,2) + (+4,3) = |+4,3| − |−3,2| = 1,1
The modulus of 4.3 is greater than the modulus of −3.2, so we subtracted 3.2 from 4.3. Got the answer 1.1. The answer is yes, because the answer must be preceded by the sign of the rational number whose modulus is greater. And the modulus of 4.3 is greater than the modulus of −3.2
Thus, the value of the expression −3.2 + (+4.3) is 1.1
−3,2 + (+4,3) = 1,1
Example 15 Find the value of the expression 3.5 + (−8.3)
This is the addition of rational numbers with different signs. As in the previous example, we subtract the smaller one from the larger module and put the sign of the rational number, the module of which is greater, before the answer:
3,5 + (−8,3) = −(|−8,3| − |3,5|) = −(8,3 − 3,5) = −(4,8) = −4,8
Thus, the value of the expression 3.5 + (−8.3) is equal to −4.8
This example can be written shorter:
3,5 + (−8,3) = −4,8
Example 16 Find the value of the expression −7.2 + (−3.11)
This is the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus before the answer.
You can skip the entry with modules to avoid cluttering up the expression:
−7,2 + (−3,11) = −7,20 + (−3,11) = −(7,20 + 3,11) = −(10,31) = −10,31
Thus, the value of the expression −7.2 + (−3.11) is equal to −10.31
This example can be written shorter:
−7,2 + (−3,11) = −10,31
Example 17. Find the value of the expression −0.48 + (−2.7)
This is the addition of negative rational numbers. We add their modules and put a minus before the received answer. You can skip the entry with modules to avoid cluttering up the expression:
−0,48 + (−2,7) = (−0,48) + (−2,70) = −(0,48 + 2,70) = −(3,18) = −3,18
Example 18. Find the value of the expression −4.9 − 5.9
We enclose each rational number in brackets along with its signs. We take into account that the minus which is located between the rational numbers −4.9 and 5.9 is the sign of the operation and does not apply to the number 5.9. This rational number has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:
(−4,9) − (+5,9)
Let's replace subtraction with addition:
(−4,9) + (−5,9)
We got the addition of negative rational numbers. We add their modules and put a minus before the received answer:
(−4,9) + (−5,9) = −(4,9 + 5,9) = −(10,8) = −10,8
Thus, the value of the expression −4.9 − 5.9 is equal to −10.8
−4,9 − 5,9 = −10,8
Example 19. Find the value of the expression 7 − 9.3
Enclose in brackets each number along with its signs
(+7) − (+9,3)
Let's replace subtraction with addition
(+7) + (−9,3)
(+7) + (−9,3) = −(9,3 − 7) = −(2,3) = −2,3
Thus, the value of the expression 7 − 9.3 is −2.3
Let's write down the solution of this example in a shorter way:
7 − 9,3 = −2,3
Example 20. Find the value of the expression −0.25 − (−1.2)
Let's replace subtraction with addition:
−0,25 + (+1,2)
We got the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the answer we put the sign of the number whose module is greater:
−0,25 + (+1,2) = 1,2 − 0,25 = 0,95
Let's write down the solution of this example in a shorter way:
−0,25 − (−1,2) = 0,95
Example 21. Find the value of the expression -3.5 + (4.1 - 7.1)
Perform the actions in brackets, then add the received answer with the number −3.5
First action:
4,1 − 7,1 = (+4,1) − (+7,1) = (+4,1) + (−7,1) = −(7,1 − 4,1) = −(3,0) = −3,0
Second action:
−3,5 + (−3,0) = −(3,5 + 3,0) = −(6,5) = −6,5
Answer: the value of the expression −3.5 + (4.1 − 7.1) is −6.5.
Example 22. Find the value of the expression (3.5 - 2.9) - (3.7 - 9.1)
Let's do the parentheses. Then, from the number that resulted from the execution of the first brackets, subtract the number that resulted from the execution of the second brackets:
First action:
3,5 − 2,9 = (+3,5) − (+2,9) = (+3,5) + (−2,9) = 3,5 − 2,9 = 0,6
Second action:
3,7 − 9,1 = (+3,7) − (+9,1) = (+3,7) + (−9,1) = −(9,1 − 3,7) = −(5,4) = −5,4
Third act
0,6 − (−5,4) = (+0,6) + (+5,4) = 0,6 + 5,4 = 6,0 = 6
Answer: the value of the expression (3.5 - 2.9) - (3.7 - 9.1) is 6.
Example 23. Find the value of an expression −3,8 + 17,15 − 6,2 − 6,15
Enclose in brackets every rational number along with its signs
(−3,8) + (+17,15) − (+6,2) − (+6,15)
Let's replace subtraction with addition where possible:
(−3,8) + (+17,15) + (−6,2) + (−6,15)
The expression consists of several terms. According to the associative law of addition, if the expression consists of several terms, then the sum will not depend on the order of actions. This means that the terms can be added in any order.
We will not reinvent the wheel, but add all the terms from left to right in the order in which they appear:
First action:
(−3,8) + (+17,15) = 17,15 − 3,80 = 13,35
Second action:
13,35 + (−6,2) = 13,35 − −6,20 = 7,15
Third action:
7,15 + (−6,15) = 7,15 − 6,15 = 1,00 = 1
Answer: the value of the expression −3.8 + 17.15 − 6.2 − 6.15 is equal to 1.
Example 24. Find the value of an expression
Let's convert the decimal fraction -1.8 to a mixed number. We will rewrite the rest without change:
Practically the entire course of mathematics is based on operations with positive and negative numbers. After all, as soon as we begin to study the coordinate line, numbers with plus and minus signs begin to meet us everywhere, in every new topic. There is nothing easier than adding ordinary positive numbers together, it is not difficult to subtract one from the other. Even arithmetic with two negative numbers is rarely a problem.
However, many people get confused in adding and subtracting numbers with different signs. Recall the rules by which these actions occur.
Addition of numbers with different signs
If to solve the problem we need to add a negative number "-b" to a certain number "a", then we need to act as follows.
- Let's take modules of both numbers - |a| and |b| - and compare these absolute values with each other.
- Note which of the modules is larger and which is smaller, and subtract the smaller value from the larger value.
- We put before the resulting number the sign of the number whose modulus is greater.
This will be the answer. It can be put more simply: if in the expression a + (-b) the modulus of the number "b" is greater than the modulus of "a", then we subtract "a" from "b" and put a "minus" in front of the result. If the modulus "a" is greater, then "b" is subtracted from "a" - and the solution is obtained with a "plus" sign.
It also happens that the modules are equal. If so, then you can stop at this point - we are talking about opposite numbers, and their sum will always be zero.
Subtraction of numbers with different signs
We figured out the addition, now consider the rule for subtraction. It is also quite simple - and besides, it completely repeats a similar rule for subtracting two negative numbers.
In order to subtract from a certain number "a" - arbitrary, that is, with any sign - a negative number "c", you need to add to our arbitrary number "a" the number opposite to "c". For example:
- If “a” is a positive number, and “c” is negative, and “c” must be subtracted from “a”, then we write it like this: a - (-c) \u003d a + c.
- If “a” is a negative number, and “c” is positive, and “c” must be subtracted from “a”, then we write as follows: (- a) - c \u003d - a + (-c).
Thus, when subtracting numbers with different signs, we eventually return to the rules of addition, and when adding numbers with different signs, we return to the rules of subtraction. Remembering these rules allows you to solve problems quickly and easily.
