Subtraction of opposite numbers. Integer addition: general idea, rules, examples

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 ordinary fractions, to do this, we will perform the transition from a mixed number to an improper fraction: , and translate the decimal fraction into an ordinary: .

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 ease of implementation further action, we bring 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 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.

Lesson content

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 right side 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 left side 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, more Subtract the smaller one and put a minus sign in front of the 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. On early stages 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 its 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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Instruction

There are four types of mathematical operations: addition, subtraction, multiplication and division. Therefore, there will be four types of examples with. Negative numbers within the example are highlighted so as not to confuse the mathematical operation. For example, 6-(-7), 5+(-9), -4*(-3) or 34:(-17).

Addition. This action can look like: 1) 3+(-6)=3-6=-3. Replacing the action: first, the brackets are opened, the "+" sign is reversed, then the smaller "3" is subtracted from the larger (modulo) number "6", after which the answer is assigned the larger sign, that is, "-".
2) -3+6=3. This one can be written as - ("6-3") or according to the principle "subtract the smaller from the larger and assign the sign of the larger to the answer."
3) -3+(-6)=-3-6=-9. When opening, the replacement of the action of addition by subtraction, then the modules are summed up and the result is given a minus sign.

Subtraction.1) 8-(-5)=8+5=13. The brackets are opened, the sign of the action is reversed, and an addition example is obtained.
2) -9-3=-12. The elements of the example are added together and get common sign "-".
3) -10-(-5)=-10+5=-5. When opening the brackets, the sign changes again to "+", then the smaller number is subtracted from the larger number and the sign of the larger number is taken from the answer.

Multiplication and division. When performing multiplication or division, the sign does not affect the operation itself. When multiplying or dividing numbers, a minus sign is assigned to the answer, if numbers with the same signs, the result always has a plus sign. 1)-4*9=-36; -6:2=-3.
2)6*(-5)=-30; 45:(-5)=-9.
3)-7*(-8)=56; -44:(-11)=4.

Sources:

table with cons

How to decide examples? This question is often asked by children to their parents if homework needs to be done. How to correctly explain to a child the solution of examples for addition and subtraction of multi-digit numbers? Let's try to figure this out.

You will need

1. Mathematics textbook.
2. Paper.
3. Handle.

Instruction

Read the example. To do this, each multivalued is divided into classes. Starting from the end of the number, count off three digits and put a dot (23.867.567). Recall that the first three digits from the end of the number to units, the next three - to the class, then there are millions. We read the number: twenty-three eight hundred sixty-seven thousand sixty-seven.

Write down an example. Please note that the units of each digit are written strictly under each other: units under units, tens under tens, hundreds under hundreds, etc.

Perform addition or subtraction. Start doing the action with units. Write the result under the category with which the action was performed. If it turned out to be a number (), then we write the units at the place of the answer, and add the number of tens to the units of the discharge. If the number of units of any digit in the reduced is less than in the subtracted, we take 10 units of the next digit, perform the action.

Read the answer.

The order of the multiplication operation

The essence of the multiplication operation is based on a simpler arithmetic operation -. In fact, multiplication is the summation of the first factor, or multiplicand, such a number of times that corresponds to the second factor. For example, in order to multiply 8 by 4, you need to add the number 8 4 times, resulting in 32. This method, in addition to providing an understanding of the essence of the multiplication operation, can be used to check the result obtained by calculating the desired product. It should be borne in mind that the verification necessarily assumes that the terms involved in the summation are the same and correspond to the first factor.

Solving multiplication examples

Thus, in order to solve, associated with the need to carry out multiplication, it may be sufficient to add the required number of first factors a given number of times. Such a method can be convenient for performing almost any calculations associated with this operation. At the same time, in mathematics quite often there are typical ones, in which standard single-digit integers participate. In order to facilitate their calculation, the so-called multiplication was created, which includes complete list products of positive integer single-digit numbers, that is, numbers from 1 to 9. Thus, having once learned, you can significantly simplify the process of solving multiplication examples based on the use of such numbers. However, for more complex options, it will be necessary to carry out this mathematical operation yourself.

Number multiplication operation

There are three main elements involved in the multiplication operation. The first of these, which is commonly referred to as the first factor or multiplicand, is the number that will be subjected to the multiplication operation. The second, which is called the second factor, is the number by which the first factor will be multiplied. Finally, the result of the multiplication operation carried out is most often called the product.

It should be remembered that the essence of the multiplication operation is actually based on addition: for its implementation, it is necessary to add together a certain number of first factors, and the number of terms in this sum must be equal to the second factor. In addition to calculating the product of the two factors under consideration, this algorithm can also be used to check the resulting result.

An example of solving a multiplication task

Consider solutions to the multiplication problem. Suppose, according to the conditions of the assignment, it is necessary to calculate the product of two numbers, among which the first factor is 8, and the second is 4. In accordance with the definition of the multiplication operation, this actually means that you need to add the number 8 4 times. The result is 32 - this is the product considered numbers, that is, the result of their multiplication.

In addition, it must be remembered that the so-called commutative law applies to the multiplication operation, which establishes that changing the places of factors in the original example will not change its result. Thus, you can add the number 4 8 times, resulting in the same product - 32.

Multiplication table

It is clear that to solve in this way a large number of examples of the same type is a rather tedious task. In order to facilitate this task, the so-called multiplication was invented. In fact, it is a list of products of integer positive single-digit numbers. Simply put, a multiplication table is a collection of results of multiplication between each other from 1 to 9. Once you have learned this table, you can no longer resort to multiplication whenever you need to solve an example for such prime numbers, but simply remember its result.

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 greater and which is smaller, and subtract from greater value lesser.
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 module "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.