The formula for adding numbers with different signs. Addition of numbers with different signs: rule, examples

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 quantity, 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 the record 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)

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. "Gymnasium", 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. M.: Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. M.: ZSh MEPhI, 2011.
5. 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.
6. 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.

1. Math-prosto.ru ().
2. youtube().
3. School-assistant.ru ().
4. Allforchildren.ru ().

Homework

Addition of negative numbers.

The sum of negative numbers is a negative number. The module of the sum is equal to the sum of the modules of the terms.

Let's see why the sum of negative numbers will also be a negative number. The coordinate line will help us with this, on which we will perform the addition of the numbers -3 and -5. Let's mark a point on the coordinate line corresponding to the number -3.

To the number -3 we need to add the number -5. Where do we go from the point corresponding to the number -3? That's right, to the left! For 5 single segments. We mark the point and write the number corresponding to it. This number is -8.

So, when adding negative numbers using a coordinate line, we are always to the left of the reference point, therefore, it is clear that the result of adding negative numbers is also a negative number.

Note. We added the numbers -3 and -5, i.e. found the value of the expression -3+(-5). Usually, when adding rational numbers, they simply write down these numbers with their signs, as if listing all the numbers that need to be added. Such a notation is called an algebraic sum. Apply (in our example) record: -3-5=-8.

Example. Find the sum of negative numbers: -23-42-54. (Agree that this entry is shorter and more convenient like this: -23+(-42)+(-54))?

We decide according to the rule of adding negative numbers: we add the modules of the terms: 23+42+54=119. The result will be with a minus sign.

They usually write it down like this: -23-42-54 \u003d -119.

Addition of numbers with different signs.

The sum of two numbers with different signs has the sign of the addend with a large modulus. To find the modulus of the sum, you need to subtract the smaller modulus from the larger modulus.

Let's perform the addition of numbers with different signs using the coordinate line.

1) -4+6. It is required to add the number -4 to the number 6. We mark the number -4 with a point on the coordinate line. The number 6 is positive, which means that from the point with coordinate -4 we need to go to the right by 6 unit segments. We ended up to the right of the origin (from zero) by 2 unit segments.

The result of the sum of the numbers -4 and 6 is the positive number 2:

— 4+6=2. How could you get the number 2? Subtract 4 from 6, i.e. subtract the smaller one from the larger one. The result has the same sign as the term with a large modulus.

2) Let's calculate: -7+3 using the coordinate line. We mark the point corresponding to the number -7. We go to the right by 3 unit segments and get a point with coordinate -4. We were and remained to the left of the origin: the answer is a negative number.

— 7+3=-4. We could get this result as follows: we subtracted the smaller one from the larger module, i.e. 7-3=4. As a result, the sign of the term with a larger module was set: |-7|>|3|.

Examples. Calculate: a) -4+5-9+2-6-3; b) -10-20+15-25.

the formation of knowledge about the rule for adding numbers with different signs, the ability to apply it in the simplest cases;

development of skills to compare, identify patterns, generalize;

education of a responsible attitude to educational work.

Equipment: multimedia projector, screen.

Lesson type: lesson learning new material.

DURING THE CLASSES

1. Organizational moment.

Stand up straight

They sat down quietly.

Now the bell has rung

Let's start our lesson.

Guys! Today we have guests at our lesson. Let's turn to them and smile at each other. So we start our lesson.

slide 2- The epigraph of the lesson: “He who does not notice anything does not study anything.

Whoever studies nothing is always whining and bored.

Roman Sef (children's writer)

Sweet 3 - I suggest you play the reverse game. Rules of the game: you need to divide the words into two groups: gain, lie, warmth, gave, truth, good, loss, took, evil, cold, positive, negative.

There are many contradictions in life. With their help, we define the surrounding reality. For our lesson, I need the latter: positive - negative.

What are we talking about in mathematics when we use these words? (About numbers.)

The great Pythagoras said: "Numbers rule the world." I propose to talk about the most mysterious numbers in science - numbers with different signs. - Negative numbers appeared in science as the opposite of positive ones. Their path to science was difficult, because even many scientists did not support the idea of ​​their existence.

What concepts and quantities do people measure with positive and negative numbers? (charges of elementary particles, temperature, losses, height and depth, etc.)

slide 4- Words opposite in meaning - antonyms (table).

2. Setting the topic of the lesson.

Slide 5 (work with the table) What numbers did you learn in previous lessons?
– What tasks related to positive and negative numbers can you perform?
- Attention to the screen. (Slide 5)
What numbers are in the table?
- Name the modules of numbers written horizontally.
– Specify the largest number, specify the number with the largest modulus.
- Answer the same questions for numbers written vertically.
– Do the largest number and the number with the largest modulus always coincide?
Find the sum of positive numbers, the sum of negative numbers.
- Formulate the rule for adding positive numbers and the rule for adding negative numbers.
What numbers are left to add?
- Can you put them together?
Do you know the rule for adding numbers with different signs?
- Formulate the topic of the lesson.
- What is your goal? .Think what we will do today? (Answers of children). Today we continue to get acquainted with positive and negative numbers. The topic of our lesson is “Addition of numbers with different signs.” And our goal: to learn without errors, to add numbers with different signs. Write down the number and topic of the lesson in your notebook.

3. Work on the topic of the lesson.

slide 6.– Using these concepts, find the results of adding numbers with different signs on the screen.
What numbers are the result of adding positive numbers, negative numbers?
What numbers are the result of adding numbers with different signs?
What determines the sign of the sum of numbers with different signs? (Slide 5)
– From the term with the largest modulus.
“It's like pulling a rope. The strongest wins.

Slide 7- Let's play. Imagine that you are pulling a rope. . Teacher. Rivals usually meet in competitions. And today we will visit several tournaments with you. The first thing that awaits us is the final of the tug-of-war contest. There are Ivan Minusov at number -7 and Petr Plusov at number +5. Who do you think will win? Why? So, Ivan Minusov won, he really turned out to be stronger than his opponent, and was able to drag him to his negative side by exactly two steps.

Slide 8.- . And now we will visit other competitions. Here is the final of the shooting competition. The best in this form were Minus Troikin with three balloons and Plus Chetverikov, who had four balloons in stock. And here guys, what do you think, who will be the winner?

Slide 9- Competitions have shown that the strongest wins. So when adding numbers with different signs: -7 + 5 = -2 and -3 + 4 = +1. Guys, how do numbers with different signs add up? Students offer their own options.

The teacher formulates the rule, gives examples.

10 + 12 = +(12 – 10) = +2

4 + 3,6 = -(4 – 3,6) = -0,4

Students during the demonstration can comment on the solution that appears on the slide.

slide 10- Teacher, let's play one more game "Sea battle". An enemy ship is approaching our coast, it must be knocked out and sunk. For this we have a gun. But to hit the target, you need to make accurate calculations. What will you see now. Ready? Then go ahead! Please do not be distracted, the examples change exactly after 3 seconds. Is everyone ready?

Students take turns going to the board and calculating the examples that appear on the slide. - List the steps to complete the task.

slide 11- Textbook work: p.180 p.33, read the rule for adding numbers with different signs. Comments on a rule.
- What is the difference between the rule proposed in the textbook and the algorithm you compiled? Consider examples in the textbook with commentary.

slide 12- Teacher-Now guys, let's have a experiment. But not chemical, but mathematical! Take the numbers 6 and 8, the plus and minus signs, and mix everything well. Let's get four examples-experience. Do them in your notebook. (two students decide on the wings of the board, then the answers are checked). What conclusions can be drawn from this experiment?(The role of signs). Let's do 2 more experiments. , but with your numbers (one person goes out to the board). Let's invent numbers for each other and check the results of the experiment (mutual verification).

slide 13 .- The rule is displayed on the screen in verse form. .

4. Fixing the topic of the lesson.

Slide 14 - Teacher - “All kinds of signs are needed, all kinds of signs are important!” Now, guys, we will divide with you into two teams. The boys will be in the team of Santa Claus, and the girls will be in the team of the Sun. Your task, without calculating the examples, is to determine in which of them negative answers will be obtained, and in which positive ones, and write out the letters of these examples in a notebook. Boys, respectively, are negative, and girls are positive (cards are issued from the application). A self-check is in progress.

Well done! You have an excellent sense for signs. This will help you complete the following task

Slide 15 - Physical education. -10, 0,15,18, -5,14,0, -8, -5, etc. (negative numbers - squat, positive numbers - pull up, jump up)

slide 16-Solve 9 examples on your own (task on cards in the application). 1 person at the board. Do a self test. Answers are displayed on the screen, students correct errors in their notebooks. Raise your hands who's right. (Marks are given only for good and excellent results)

Slide 17- Rules help us to solve examples correctly. Let's repeat them On the screen, the algorithm for adding numbers with different signs.

5. Organization of independent work.

Slide 18-FRontal work through the game "Guess the word"(task on cards in the application).

Slide 19 - You should get a score for the game - "five"

Slide 20-A now, attention. Homework. Homework shouldn't be difficult for you.

Slide 21 - The laws of addition in physical phenomena. Come up with examples of adding numbers with different signs and ask them to each other. What new did you learn? Have we achieved our goal?

Slide 22 - So the lesson is over, let's summarize now. Reflection. The teacher comments and grades the lesson.

Slide 23 - Thank you for your attention!

I wish you to have more positive and less negative in your life, I want to tell you guys, thank you for your active work. I think that you can easily apply what you have learned in subsequent lessons. The lesson is over. Thank you all very much. Goodbye!

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.

Page navigation.

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 quantity, 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 the record 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)

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. "Gymnasium", 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. M.: Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. M.: ZSh MEPhI, 2011.
5. 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.
6. 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.

1. Math-prosto.ru ().
2. youtube().
3. School-assistant.ru ().
4. Allforchildren.ru ().

Homework