How to add numbers with different signs rule. Addition of numbers with different signs - Knowledge Hypermarket

This article is devoted to numbers with different signs. We will parse the material and try to subtract between these numbers. In the paragraph, we will get acquainted with the basic concepts and rules that will be useful when solving exercises and problems. The article also provides detailed examples that will help you better understand the material.

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How to do subtraction correctly

In order to better understand the process of subtraction, one should start with basic definitions.

Definition 1

If you subtract the number b from the number a, then this can be converted as the addition of the numbers a and - b, where b and - b are numbers with opposite signs.

If we express this rule in letters, then it looks like this a − b = a + (− b) , where a and b are any real numbers.

This rule for subtracting numbers with different signs works for real, rational, and integer numbers. It can be proved on the basis of properties of actions with real numbers. Thanks to them, we can represent numbers as several equalities (a + (− b)) + b = a + ((− b) + b) = a + 0 = a . Since addition and subtraction are closely related, the expression a − b = a + (− b) will also be equal. This means that the subtraction rule in question is also true.

This rule, which is used to subtract numbers with different signs, allows you to work with both positive and negative numbers. It is also possible to perform the process of subtracting from a negative number from a positive one, which goes into addition.

In order to consolidate the information received, we will consider typical examples and, in practice, consider the subtraction rule for numbers with different signs.

Examples of subtraction exercises

Let's consolidate the material by considering typical examples.

Example 1

You need to subtract 4 from − 16 .

In order to perform a subtraction, you should take the number opposite to the subtrahend 4 , there is − 4 . According to the subtraction rule discussed above, (− 16) − 4 = (− 16) + (− 4) . Next, we must add the resulting negative numbers. We get: (- 16) + (- 4) = - (16 + 4) = - 20 . (− 16) − 4 = − 20 .

To subtract fractions, you need to represent numbers as fractions or decimals. It depends on what kind of numbers it will be more convenient to carry out calculations.

Example 2

It is necessary to subtract − 0 , 7 from 3 7 .

We resort to the rule of subtracting numbers. We replace subtraction with addition: 3 7 - (- 0 , 7) = 3 7 + 0 , 7 .

We add fractions and get the answer as a fractional number. 3 7 - (- 0 , 7) = 1 9 70 .

When any number is represented as a square root, logarithm, basic and trigonometric functions, then often the result of subtraction can be written as a numeric expression. To clarify this rule, consider the following example.

Example 3

It is necessary to subtract the number 5 from the number - 2 .

Let's use the subtraction rule described above. Let's take the opposite number to subtracted 5 - this is - 5. According to work with numbers with different signs - 2 - 5 = - 2 + (- 5) .

Now let's do the addition: we get - 2 + (- 5) = 2 + 5.

The resulting expression is the result of subtracting the original numbers with different signs: - 2 + 5 .

The value of the resulting expression can be calculated as accurately as possible only if necessary. For more information, see other topics related to this topic.

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If the air temperature was equal to 9°С, and then it changed by -6°С (i.e., decreased by 6°С), then it became equal to 9 + (-6) degrees (Fig. 83).

Rice. 83

To add the numbers 9 and -6 using the coordinate line, you need to move the point A (9) to the left by 6 unit segments (Fig. 84). We get point B(3).

Rice. 84

Hence, 9 + (-6) = 3. The number 3 has the same sign as the term 9, and its modulus is equal to the difference between the modules of the terms 9 and -6.

Indeed, |3| = 3 and |9| - |-6| = 9 - 6 = 3.

If the same air temperature of 9°С changed by -12°С (i.e., decreased by 12°С), then it became equal to 9 + (-12) degrees (Fig. 85).

Rice. 85

Adding the numbers 9 and -12 using the coordinate line (Fig. 86), we get 9 + (-12) \u003d -3. The number -3 has the same sign as the term -12, and its modulus is equal to the difference between the modules of the terms -12 and 9.

Rice. 86

Indeed, |-3| = 3 and |-12| - |-9| = 12 - 9 = 3.

Usually, the sign of the sum is first determined and written down, and then the difference of the modules is found.

For example:

When adding positive and negative numbers, you can use a calculator. To enter a negative number into the microcalculator, you must enter the modulus of this number, then press the "sign change" key. For example, to enter the number -56.81, you must press the keys in sequence: . Operations on numbers of any sign are performed on a microcalculator in the same way as on positive numbers. For example, the sum -6.1 + 3.8 is calculated by the program

In short, this program is written like this: .

Questions for self-examination

• The numbers a and b have different signs. What sign will the sum of these numbers have if the larger modulus has a negative number? if the smaller modulus has a negative number? if the larger modulus has a positive number? if the smaller modulus has a positive number?
• Formulate a rule for adding numbers with different signs.
• How to enter a negative number into a microcalculator?

Do the exercises

1061. The number 6 was changed to -10. On which side from the origin is the resulting number? How far from the origin is it? What is the sum of 6 and -10?

1062. The number 10 was changed to -6. On which side from the origin is the resulting number? How far from the origin is it? What is the sum of 10 and -6?

1063. The number -10 was changed to 3. On which side of the origin is the resulting number? How far from the origin is it? What is the sum of -10 and 3?

1064. The number -10 was changed to 15. On which side of the origin is the resulting number? How far from the origin is it? What is the sum of -10 and 15?

1065. In the first half of the day, the temperature changed by -4°С, and in the second - by +12°С. By how many degrees did the temperature change during the day?

1066. Perform addition:

• a) 26 + (-6);
• b) -70 + 50;
• c) -17 + 30;
• d) 80 + (-120);
• e) -6.3 + 7.8;
• f) -9 + 10.2;
• g) 1 + (-0.39);
• h) 0.3 + (-1.2);

1067. Add:

• a) to the sum of -6 and -12 the number 20;
• b) to the number 2.6 the sum is -1.8 and 5.2;
• c) to the sum of -10 and -1.3 the sum of 5 and 8.7;
• d) to the sum of 11 and -6.5 the sum of -3.2 and -6.

1068. Which of the numbers 8; 7.1; -7.1; -7; -0.5 is the root of the equation -6 + x = -13.1?

1069. Guess the root of the equation and check:

• a) x + (-3) = -11;
• b) -5 + y = 15;
• c) t + (-12) = 2;
• d) 3 + n = -10.

1070. Find the value of the expression:

1071. Follow the steps using the calculator:

• a) -3.2579 + (-12.308);
• b) 7.8547 + (-9.239);
• c) -0.00154 + 0.0837;
• d) -3.8564 + (-0.8397) + 7.84;
• e) -0.083 + (-6.378) + 3.9834;
• f) -0.0085 + 0.00354 + (-0.00921).

1072. Find the value of the sum:

1073. Find the value of the expression:

1074. How many integers are located between the numbers:

• a) 0 and 24;
• b) -12 and -3;
• c) -20 and 7?

1075. Express the number -10 as the sum of two negative terms so that:

• a) both terms were integers;
• b) both terms were decimal fractions;
• c) one of the terms was a proper ordinary fraction.

1076. What is the distance (in unit segments) between the points of the coordinate line with coordinates:

• a) 0 and a;
• b) -a and a;
• c) -a and 0;
• d) a and -za?

1077. The radii of the geographic parallels of the earth's surface, on which the cities of Athens and Moscow are located, are respectively 5040 km and 3580 km (Fig. 87). How much shorter is the Moscow parallel than the Athens parallel?

Rice. 87

1078. Make an equation for solving the problem: “A field with an area of ​​2.4 hectares was divided into two sections. Find the area of ​​each parcel if one of the parcels is known to be:

1079. Solve the problem:

1. On the first day the travelers traveled 240 km, on the second day 140 km, on the third day they traveled 3 times more than on the second, and on the fourth day they rested. How many kilometers did they drive on the fifth day if they averaged 230 kilometers a day in 5 days?
2. A farmer with two sons placed the collected apples in 4 containers, on average 135 kg each. The farmer collected 280 kg of apples, and the youngest son - 4 times less. How many kilograms of apples did the eldest son collect?

1080. Follow these steps:

1. (2,35 + 4,65) 5,3: (40 - 2,9);
2. (7,63 - 5,13) 0,4: (3,17 + 6,83).

1081. Perform addition:

1082. Present as a sum of two equal terms each of the numbers: 10; -eight; -6.8; .

1083. Find the value a + b if:

1084. There were 8 apartments on one floor of the residential building. Living area of ​​22.8 m 2 had 2 apartments, 16.2 m 2 - 3 apartments, 34 m 2 - 2 apartments. What living area did the eighth apartment have if on this floor, on average, each apartment had 24.7 m 2 of living space?

1085. The freight train consisted of 42 wagons. There were 1.2 times more covered wagons than platforms, and the number of tanks was equal to the number of platforms. How many wagons of each type were in the train?

1086. Find the value of an expression

In this lesson we will learn addition and subtraction of whole numbers, as well as rules for their addition and subtraction.

Recall that integers are all positive and negative numbers, as well as the number 0. For example, the following numbers are integers:

−3, −2, −1, 0, 1, 2, 3

Positive numbers are easy , and . Unfortunately, this cannot be said about negative numbers, which confuse many beginners with their minuses before each digit. As practice shows, mistakes made due to negative numbers upset students the most.

Integer addition and subtraction examples

The first thing to learn is to add and subtract whole numbers using the coordinate line. It is not necessary to draw a coordinate line. It is enough to imagine it in your thoughts and see where the negative numbers are and where the positive ones are.

Consider the simplest expression: 1 + 3. The value of this expression is 4:

This example can be understood using the coordinate line. To do this, from the point where the number 1 is located, you need to move three steps to the right. As a result, we will find ourselves at the point where the number 4 is located. In the figure you can see how this happens:

The plus sign in the expression 1 + 3 tells us that we should move to the right in the direction of increasing numbers.

Example 2 Let's find the value of the expression 1 − 3.

The value of this expression is −2

This example can again be understood using the coordinate line. To do this, from the point where the number 1 is located, you need to move three steps to the left. As a result, we will find ourselves at the point where the negative number −2 is located. The figure shows how this happens:

The minus sign in the expression 1 − 3 tells us that we should move to the left in the direction of decreasing numbers.

In general, we must remember that if addition is carried out, then we need to move to the right in the direction of increase. If subtraction is carried out, then you need to move to the left in the direction of decrease.

Example 3 Find the value of the expression −2 + 4

The value of this expression is 2

This example can again be understood using the coordinate line. To do this, from the point where the negative number -2 is located, you need to move four steps to the right. As a result, we will find ourselves at the point where the positive number 2 is located.

It can be seen that we have moved from the point where the negative number −2 is located to the right by four steps, and ended up at the point where the positive number 2 is located.

The plus sign in the expression -2 + 4 tells us that we should move to the right in the direction of increasing numbers.

Example 4 Find the value of the expression −1 − 3

The value of this expression is −4

This example can again be solved using a coordinate line. To do this, from the point where the negative number −1 is located, you need to move three steps to the left. As a result, we will find ourselves at the point where the negative number -4 is located

It can be seen that we have moved from the point where the negative number −1 is located to the left by three steps, and ended up at the point where the negative number −4 is located.

The minus sign in the expression -1 - 3 tells us that we should move to the left in the direction of decreasing numbers.

Example 5 Find the value of the expression −2 + 2

The value of this expression is 0

This example can be solved using a coordinate line. To do this, from the point where the negative number −2 is located, you need to move two steps to the right. As a result, we will find ourselves at the point where the number 0 is located

It can be seen that we have moved from the point where the negative number −2 is located to the right by two steps and ended up at the point where the number 0 is located.

The plus sign in the expression -2 + 2 tells us that we should move to the right in the direction of increasing numbers.

Rules for adding and subtracting integers

To add or subtract integers, it is not at all necessary to imagine a coordinate line every time, let alone draw it. It is more convenient to use ready-made rules.

When applying the rules, you need to pay attention to the sign of the operation and the signs of the numbers to be added or subtracted. This will determine which rule to apply.

Example 1 Find the value of the expression −2 + 5

Here a positive number is added to a negative number. In other words, the addition of numbers with different signs is carried out. −2 is negative and 5 is positive. For such cases, the following rule applies:

To add numbers with different signs, you need to subtract a smaller module from a larger module, and put the sign of the number whose module is greater in front of the answer.

So, let's see which module is larger:

The modulus of 5 is greater than the modulus of −2. The rule requires subtracting the smaller from the larger module. Therefore, we must subtract 2 from 5, and before the received answer put the sign of the number whose modulus is greater.

The number 5 has a larger modulus, so the sign of this number will be in the answer. That is, the answer will be positive:

−2 + 5 = 5 − 2 = 3

Usually written shorter: −2 + 5 = 3

Example 2 Find the value of the expression 3 + (−2)

Here, as in the previous example, the addition of numbers with different signs is carried out. 3 is positive and -2 is negative. Note that the number -2 is enclosed in parentheses to make the expression clearer. This expression is much easier to understand than the expression 3+−2.

So, we apply the rule of adding numbers with different signs. As in the previous example, we subtract the smaller module from the larger module and put the sign of the number whose module is greater before the answer:

3 + (−2) = |3| − |−2| = 3 − 2 = 1

The modulus of the number 3 is greater than the modulus of the number −2, so we subtracted 2 from 3, and put the sign of the greater modulus number before the answer. The number 3 has a larger module, so the sign of this number is put in the answer. That is, the answer is yes.

Usually written shorter 3 + (−2) = 1

Example 3 Find the value of the expression 3 − 7

In this expression, the larger number is subtracted from the smaller number. In such a case, the following rule applies:

To subtract a larger number from a smaller number, you need to subtract a smaller number from a larger number, and put a minus in front of the received answer.

3 − 7 = 7 − 3 = −4

There is a slight snag in this expression. Recall that the equal sign (=) is placed between values ​​and expressions when they are equal to each other.

The value of the expression 3 − 7, as we learned, is −4. This means that any transformations that we will perform in this expression must be equal to −4

But we see that the expression 7 − 3 is located at the second stage, which is not equal to −4.

To correct this situation, the expression 7 - 3 must be put in brackets and put a minus before this bracket:

3 − 7 = − (7 − 3) = − (4) = −4

In this case, equality will be observed at each stage:

After the expression is evaluated, the brackets can be removed, which we did.

So to be more precise, the solution should look like this:

3 − 7 = − (7 − 3) = − (4) = − 4

This rule can be written using variables. It will look like this:

a − b = − (b − a)

A large number of brackets and operation signs can complicate the solution of a seemingly very simple task, so it is more expedient to learn how to write such examples briefly, for example 3 − 7 = − 4.

In fact, the addition and subtraction of integers is reduced to just addition. This means that if you want to subtract numbers, this operation can be replaced by addition.

So, let's get acquainted with the new rule:

To subtract one number from another means to add to the minuend a number that will be the opposite of the subtracted one.

For example, consider the simplest expression 5 − 3. At the initial stages of studying mathematics, we put an equal sign and wrote down the answer:

But now we are progressing in learning, so we need to adapt to the new rules. The new rule says that to subtract one number from another means to add to the minuend a number that will be subtracted.

Using the expression 5 − 3 as an example, let's try to understand this rule. The minuend in this expression is 5, and the subtrahend is 3. The rule says that in order to subtract 3 from 5, you need to add to 5 such a number that will be opposite to 3. The opposite number for the number 3 is −3. We write a new expression:

And we already know how to find values ​​for such expressions. This is the addition of numbers with different signs, which we considered earlier. To add numbers with different signs, we subtract a smaller module from a larger module, and put the sign of the number whose module is greater before the answer received:

5 + (−3) = |5| − |−3| = 5 − 3 = 2

The modulus of 5 is greater than the modulus of −3. Therefore, we subtracted 3 from 5 and got 2. The number 5 has a larger modulus, so the sign of this number was put in the answer. That is, the answer is positive.

At first, not everyone succeeds in quickly replacing subtraction with addition. This is due to the fact that positive numbers are written without a plus sign.

For example, in the expression 3 − 1, the minus sign indicating subtraction is the sign of the operation and does not refer to one. The unit in this case is a positive number, and it has its own plus sign, but we don’t see it, because plus is not written before positive numbers.

And so, for clarity, this expression can be written as follows:

(+3) − (+1)

For convenience, numbers with their signs are enclosed in brackets. In this case, replacing subtraction with addition is much easier.

In the expression (+3) − (+1), this number is subtracted (+1), and the opposite number is (−1).

Let's replace subtraction with addition and instead of subtrahend (+1) we write down the opposite number (−1)

(+3) − (+1) = (+3) + (−1)

Further calculation will not be difficult.

(+3) − (+1) = (+3) + (−1) = |3| − |−1| = 3 − 1 = 2

At first glance, it would seem what is the point in these extra gestures, if you can use the good old method to put an equal sign and immediately write down the answer 2. In fact, this rule will help us out more than once.

Let's solve the previous example 3 − 7 using the subtraction rule. First, let's bring the expression to a clear form, placing each number with its signs.

Three has a plus sign because it is a positive number. The minus indicating subtraction does not apply to the seven. Seven has a plus sign because it is a positive number:

Let's replace subtraction with addition:

(+3) − (+7) = (+3) + (−7)

Further calculation is not difficult:

(+3) − (−7) = (+3) + (-7) = −(|−7| − |+3|) = −(7 − 3) = −(4) = −4

Example 7 Find the value of the expression −4 − 5

Before us is the operation of subtraction again. This operation must be replaced by addition. To the minuend (−4) we add the number opposite to the subtrahend (+5). The opposite number for the subtrahend (+5) is the number (−5).

(−4) − (+5) = (−4) + (−5)

We have come to a situation where we need to add negative numbers. For such cases, the following rule applies:

To add negative numbers, you need to add their modules, and put a minus in front of the received answer.

So, let's add the modules of numbers, as the rule requires us to, and put a minus in front of the received answer:

(−4) − (+5) = (−4) + (−5) = |−4| + |−5| = 4 + 5 = −9

The entry with modules must be enclosed in brackets and put a minus before these brackets. So we provide a minus, which should come before the answer:

(−4) − (+5) = (−4) + (−5) = −(|−4| + |−5|) = −(4 + 5) = −(9) = −9

The solution for this example can be written shorter:

−4 − 5 = −(4 + 5) = −9

or even shorter:

−4 − 5 = −9

Example 8 Find the value of the expression −3 − 5 − 7 − 9

Let's bring the expression to a clear form. Here, all numbers except the number −3 are positive, so they will have plus signs:

(−3) − (+5) − (+7) − (+9)

Let's replace subtractions with additions. All minuses, except for the minus in front of the triple, will change to pluses, and all positive numbers will change to the opposite:

(−3) − (+5) − (+7) − (+9) = (−3) + (−5) + (−7) + (−9)

Now apply the rule for adding negative numbers. To add negative numbers, you need to add their modules and put a minus in front of the received answer:

(−3) − (+5) − (+7) − (+9) = (−3) + (−5) + (−7) + (−9) =

= −(|−3| + |−5| + |−7| + |−9|) = −(3 + 5 + 7 + 9) = −(24) = −24

The solution to this example can be written shorter:

−3 − 5 − 7 − 9 = −(3 + 5 + 7 + 9) = −24

or even shorter:

−3 − 5 − 7 − 9 = −24

Example 9 Find the value of the expression −10 + 6 − 15 + 11 − 7

Let's bring the expression to a clear form:

(−10) + (+6) − (+15) + (+11) − (+7)

There are two operations here: addition and subtraction. Addition is left unchanged, and subtraction is replaced by addition:

(−10) + (+6) − (+15) + (+11) − (+7) = (−10) + (+6) + (−15) + (+11) + (−7)

Observing, we will perform each action in turn, based on the previously studied rules. Entries with modules can be skipped:

First action:

(−10) + (+6) = − (10 − 6) = − (4) = − 4

Second action:

(−4) + (−15) = − (4 + 15) = − (19) = − 19

Third action:

(−19) + (+11) = − (19 − 11) = − (8) = −8

Fourth action:

(−8) + (−7) = − (8 + 7) = − (15) = − 15

Thus, the value of the expression −10 + 6 − 15 + 11 − 7 is −15

Note. It is not necessary to bring the expression to a clear form by enclosing numbers in brackets. When getting used to negative numbers, this action can be skipped, as it takes time and can be confusing.

So, for adding and subtracting integers, you need to remember the following rules:

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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.

"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.