Addition subtraction multiplication and division negative numbers. Adding numbers with different signs

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 look at several examples of adding numbers with different signs.

Look at this gear (see Fig. 1).

Rice. 1. Clock gear

This is not a hand that directly shows the time and not a dial (see Fig. 2). But without this part the clock does not work.

Rice. 2. Gear inside the clock

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 equivalent operations and can be performed in any order. In direct order, we can calculate: , but we can’t start with subtraction, since we haven’t yet agreed on what .

It is clear that increasing the number by and then decreasing by means ultimately decreasing by three. Why not designate this object and count like that: adding means subtracting. Then .

The number can mean, for example, an apple. 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 make calculations easier.

Let's name new numbers negative. Now we can subtract the larger number from the smaller number. Technically, you still need to subtract the smaller number from the larger number, but put a minus sign in your 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 , we introduce a new number, which we denote , and determine that it has the following property: the sum of the number and is equal to : .

We will call the number negative, and the numbers and - opposite. Thus, we got an infinite number of new numbers, for example:

The opposite of number ;

Subtract the larger number from the smaller number: . Let's add to this expression: . We got zero. However, according to the property: the number that adds zero to five 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 zero: .

2. If you subtract a positive number from zero, 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 already covered adding numbers like these in the previous lesson, but let's make sure we understand what to do with them. For example: .

To find this sum, add the opposite positive numbers and put a minus sign.

3. One number can be positive and the other negative.

If it is convenient for us, we can replace the addition of a negative number with the subtraction of a positive one: .

One more example: . Again we write the amount as the difference. You can subtract a larger number from a smaller number by subtracting a smaller number from a larger one, but using a minus sign.

We can swap the terms: .

Another similar example: .

In all cases, the result is a subtraction.

To briefly formulate these rules, let's remember one more term. Opposite numbers are, of course, not equal to each other. But it would be strange not to notice what they have in common. We called this common modulo number. The modulus of opposite numbers is the same: for a positive number it is equal to the number itself, and for a negative number it is equal to the opposite, positive. For example: , .

To add two negative numbers, you need to add their modules 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, we add their modules and put a minus sign:

Two numbers with different signs, therefore, from the modulus of the number (the larger modulus), we subtract the modulus of the number and put a minus sign (the sign of the number with the larger modulus):

Two numbers with different signs, therefore, from the modulus of the number (the larger modulus), we subtract the modulus of the number and put a plus sign (the sign of the number with the larger modulus): .

Positive and negative numbers have historically had different roles.

First we introduced natural numbers to count objects:

Then we introduced other positive numbers - fractions, for counting non-integer quantities, parts: .

Negative numbers appeared as a tool to simplify calculations. It was not like there were any quantities in life that we could not count, and we invented negative numbers.

That is, negative numbers did not arise from the real world. They just turned out to be so convenient that in some places they found application in life. For example, we often hear about negative temperatures. However, we never encounter a negative number of apples. What's the difference?

The difference is that in life, negative quantities are used only for comparison, but not for quantities. If a hotel has a basement and an elevator is installed there, then in order to maintain the usual numbering of regular floors, a minus first floor may appear. This first minus 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 apples, but six. 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 latter 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. M.: Education, 1989.
Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. M.: ZSh MEPhI, 2011.
Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at 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 secondary school. M.: Education, Mathematics Teacher Library, 1989.

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Allforchildren.ru ().

Homework

Almost the entire mathematics course 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 appear 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 about adding and subtracting numbers with different signs. Let us recall the rules by which these actions occur.

Adding numbers with different signs

If to solve a problem we need to add a negative number “-b” to some number “a”, then we need to act as follows.

Let's take the modules of both numbers - |a| and |b| - and compare these absolute values with each other.
Let us note which module is larger and which is smaller, and subtract the smaller value from the larger value.
Let us put in front of the resulting number the sign of the number whose modulus is greater.

This will be the answer. We can put it 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 turn out to be equal. If so, then we can stop at this point - we are talking about opposite numbers, and their sum will always be equal to zero.

Subtracting numbers with different signs

We've dealt with addition, now let's look at the rule for subtraction. It is also quite simple - and in addition, 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 you need to subtract “c” from “a”, then we write it like this: a – (-c) = a + c.
If “a” is a negative number, and “c” is positive, and “c” needs to be subtracted from “a”, then we write it as follows: (- a)– c = - a+ (-c).

Thus, when subtracting numbers with different signs, we end up returning to the rules of addition, and when adding numbers with different signs, we return to the rules of subtraction. Memorizing these rules allows you to solve problems quickly and easily.

>>Math: Adding numbers with different signs

33. Addition of numbers with different signs

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

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

This means 9+(- 6) = 3. The number 3 has the same sign as the term 9, and its module equal to the difference between the moduli of terms 9 and -6.

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

If the same air temperature of 9 °C changed by -12 °C (i.e., decreased by 12 °C), then it became equal to 9 + (-12) degrees (Fig. 85). Adding the numbers 9 and -12 using the coordinate line (Fig. 86), we get 9 + (-12) = -3. The number -3 has the same sign as the term -12, and its module is equal to the difference between the modules of the terms -12 and 9.

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

To add two numbers with different signs, you need to:

1) subtract the smaller one from the larger module of the terms;

2) put in front of the resulting number the sign of the term whose modulus is greater.

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

For example:

1) 6,1+(- 4,2)= +(6,1 - 4,2)= 1,9,
or shorter 6.1+(- 4.2) = 6.1 - 4.2 = 1.9;

When adding positive and negative numbers you can use micro calculator. To enter a negative number into a microcalculator, you need to enter the modulus of this number, then press the “change sign” key |/-/|. For example, to enter the number -56.81, you must press the keys sequentially: | 5 |, | 6 |, | ¦ |, | 8 |, | 1 |, |/-/|. 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 using program

? The numbers a and b have different signs. What sign will the sum of these numbers have if the larger module is negative?

if the smaller modulus is negative?

if the larger modulus is a positive number?

if the smaller modulus is a positive number?

Formulate a rule for adding numbers with different signs. How to enter a negative number into a microcalculator?

TO 1045. The number 6 was changed to -10. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is it equal to sum 6 and -10?

1046. The number 10 was changed to -6. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of 10 and -6?

1047. The number -10 was changed to 3. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 3?

1048. The number -10 was changed to 15. On which side of the origin is the resulting number located? At what distance from the origin is it located? What is the sum of -10 and 15?

1049. In the first half of the day the temperature changed by - 4 °C, and in the second half - by + 12 °C. By how many degrees did the temperature change during the day?

1050. Perform addition:

1051. 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 -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.

1052. Which number is 8; 7.1; -7.1; -7; -0.5 is the root equations- 6 + x = -13.1?

1053. Guess the root of the equation and check:

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

1054. Find the meaning of the expression:

1055. Follow the steps using a microcalculator:

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

P 1056. Find the value of the sum:

1057. Find the meaning of the expression:

1058. How many integers are located between the numbers:

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

1059. Imagine 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 regular ordinary fraction.

1060. 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?

M 1061. The radii of the geographical parallels of the earth's surface on which the cities of Athens and Moscow are located are respectively equal to 5040 km and 3580 km (Fig. 87). How much shorter is the Moscow parallel than the Athens parallel?

1062. Write an equation to solve the problem: “A field with an area of 2.4 hectares was divided into two sections. Find square each site, if it is known that one of the sites:

a) 0.8 hectares more than another;
b) 0.2 hectares less than another;
c) 3 times more than another;
d) 1.5 times less than another;
e) constitutes another;
e) is 0.2 of the other;
g) constitutes 60% of the other;
h) is 140% of the other.”

1063. 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 travel on the fifth day, if for 5 days they drove an average of 230 km per day?

2) Father’s monthly income is 280 rubles. My daughter's scholarship is 4 times less. How much does a mother earn per month if there are 4 people in the family, the youngest son is a schoolboy and each person receives an average of 135 rubles?

1064. Follow these steps:

1) (2,35 + 4,65) 5,3:(40-2,9);

2) (7,63-5,13) 0,4:(3,17 + 6,83).

1066. Present each of the numbers as a sum of two equal terms:

1067. Find the value of a + b if:

a) a= -1.6, b = 3.2; b) a=- 2.6, b = 1.9; V)

1068. There were 8 apartments on one floor of a residential building. 2 apartments had a living area of 22.8 m2, 3 apartments - 16.2 m2, 2 apartments - 34 m2. What living area did the eighth apartment have if on this floor on average each apartment had 24.7 m2 of living space?

1069. The freight train consisted of 42 cars. There were 1.2 times more covered cars than platforms, and the number of tanks was equal to the number of platforms. How many cars of each type were on the train?

1070. Find the meaning of the expression

N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school

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In this article we will look in detail at how it is done addition of integers. First, let's form a general idea of the addition of integers, and see what the addition of integers on a coordinate line is. This knowledge will help us formulate rules for adding positive, negative, and integers with different signs. Here we will examine in detail the application of addition rules when solving examples and learn how to check the results obtained. To conclude the article, we will talk about adding three or more integers.

Page navigation.

Understanding addition of integers

Here are examples of adding integer opposite numbers. The sum of the numbers −5 and 5 is zero, the sum of 901+(−901) is zero, and the result of adding the opposite integers 1,567,893 and −1,567,893 is also zero.

Addition of an arbitrary integer and zero

Let's use the coordinate line to understand what the result of adding two integers, one of which is zero, is.

Adding an arbitrary integer a to zero means moving unit segments from the origin to a distance a. Thus, we find ourselves at the point with coordinate a. Therefore, the result of adding zero and an arbitrary integer is the added integer.

On the other hand, adding zero to an arbitrary integer means moving from the point whose coordinate is specified by a given integer to a distance of zero. In other words, we will remain at the starting point. Therefore, the result of adding an arbitrary integer and zero is the given integer.

So, the sum of two integers, one of which is zero, is equal to the other integer. In particular, zero plus zero is zero.

Let's give a few examples. The sum of the integers 78 and 0 is 78; the result of adding zero and −903 is −903 ; also 0+0=0 .

Checking the result of addition

After adding two integers, it is useful to check the result. We already know that to check the result of adding two natural numbers, we need to subtract any of the terms from the resulting sum, and this should result in another term. Checking the result of adding integers performed similarly. But subtracting integers comes down to adding to the minuend the number opposite to the one being subtracted. Thus, to check the result of adding two integers, you need to add to the resulting sum the number opposite to any of the terms, which should result in another term.

Let's look at examples of checking the result of adding two integers.

Example.

When adding two integers 13 and −9, the number 4 was obtained, check the result.

Solution.

Let's add to the resulting sum 4 the number −13, opposite to the term 13, and see if we get another term −9.

So, let's calculate the sum 4+(−13) . This is the sum of integers with opposite signs. The modules of the terms are 4 and 13, respectively. The term whose modulus is greater has a minus sign, which we remember. Now subtract from the larger module and subtract the smaller one: 13−4=9. All that remains is to put the remembered minus sign in front of the resulting number, we have −9.

When checking, we received a number equal to another term, therefore, the original sum was calculated correctly.−19. Since we received a number equal to another term, the addition of the numbers −35 and −19 was performed correctly.

Adding three or more integers

Up to this point we have talked about adding two integers. In other words, we considered sums consisting of two terms. However, the combinative property of adding integers allows us to uniquely determine the sum of three, four, or more integers.

Based on the properties of addition of integers, we can assert that the sum of three, four, and so on numbers does not depend on the way the parentheses are placed indicating the order in which actions are performed, as well as on the order of the terms in the sum. We substantiated these statements when we talked about the addition of three or more natural numbers. For integers, all reasoning is completely the same, and we will not repeat ourselves.0+(−101) +(−17)+5 . After this, placing the parentheses in any acceptable way, we will still get the number −113.

Answer:

5+(−17)+0+(−101)=−113 .

Bibliography.

Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.

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 figure out why the sum of negative numbers will also be a negative number. The coordinate line will help us with this, on which we will add the numbers -3 and -5. Let us 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, left! For 5 unit segments. We mark a point and write the number corresponding to it. This number is -8.

So, when adding negative numbers using the coordinate line, we are always to the left of the origin, 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. This notation is called an algebraic sum. Apply (in our example) the entry: -3-5=-8.

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

Let's decide According to the rule for adding negative numbers: we add the modules of the terms: 23+42+54=119. The result will have a minus sign.

They usually write it like this: -23-42-54=-119.

Addition of numbers with different signs.

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

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

1) -4+6. You need to add the number 6 to the number -4. Let's mark the number -4 with a dot 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 found ourselves 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 module. The result has the same sign as the term with a large modulus.

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

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

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