The sum of opposites. Opposite numbers

Let's consider such an example. It is necessary to sequentially calculate: .

You can rearrange the numbers to be added, and then subtract the remaining ones: .

But this is not always convenient. For example, we can calculate the balance of things in some warehouse and we need to know the intermediate result.

You can perform actions in a row: .

We know that , which means that the result will be a subtraction from the number . This means that it is necessary to subtract, but not yet from anything. When there is something to subtract from, subtract:

But we can "cheat" and designate . Thus, we will introduce a new object - negative numbers .

We have already performed such an operation - in nature, for example, the number "" also did not exist, but we introduced such an object in order to facilitate the recording of actions.

Imagine that we were instructed to issue and receive balls in a sports warehouse. We need to keep records. You can write in words:

Issued , Accepted , Issued , Accepted , ... (See Fig. 1.)

Rice. 1. Accounting

Agree, if you need to issue and receive many times a day, then the recording is not very convenient.

You can divide the sheet into two columns, one - Accepted, the other - Issued. (See Figure 2.)

Rice. 2. Simplified notation

The entry got shorter. But here's the problem: how to understand how many balls were taken (or given away) at any particular moment in time?

The following consideration can be used for recording: when we issue balls from the warehouse, their number in the warehouse decreases, and when we receive, it increases.

But how to write "gave out the ball"? You can enter such an object: .

This object allows us to mathematically record the movement of the balls in the order in which they happened:

Let's consider one more example.

On the account of your phone rubles. You went online, and it cost roubles. It turned out a debt of rubles. The operator could write down like this: "the client owes rubles." You have put rubles. The operator deducted the debt. It turned out on the account of rubles.

But it is convenient to record both transactions and money in the account using the signs "" and "". (See Figure 3.)

Rice. 3. Convenient recording

We enter a negative number to write down the result of subtraction from fewer more: .

Adding a negative number is the same as subtracting: .

In order to distinguish negative numbers from the positive numbers that we dealt with earlier, we agreed to put a minus sign in front of it: .

Could you do without them? Yes, you can. In each specific situation, we would use the words “back”, “in debt”, and so on. But they, these words, would be different.

And so we have a universal convenient tool. One for all such cases.

We can draw an analogy with a car. It consists of a large number parts, many of which are not needed individually, but together they allow you to ride. So are negative numbers - a tool that, together with other mathematical tools, makes it easier to calculate and simplify the solution and recording of many problems.

So, we have introduced a new object - negative numbers. What are they used for in life?

First, let's recall the roles of positive numbers:

Quantity: e.g. wood, liters of milk. (See Figure 4.)

Rice. 4. Quantity

Ordering: for example, houses are numbered positive numbers. (See Figure 5.)

Rice. 5. Ordering

Name: e.g. player number. (See Figure 6.)

Rice. 6. Number as a name

Now let's look at the functions of negative numbers:

Designation of the missing quantity. The number is not negative. But a negative number is used to show that the amount is being subtracted. For example, we can pour out of a bottle and write it as . (See Figure 7.)

Rice. 7. Designation of the missing quantity

Ordering. Sometimes zero is selected during numbering and you need to number objects on both sides of zero. For example, the floors located below the -th, in the basement. (See Figure 8.) Or a temperature that is below the selected zero. (See Figure 9.)

Rice. 8. Floor below th, in the basement

Rice. 9. Negative numbers on the thermometer scale

But still, the main purpose of negative numbers is a tool for simplifying mathematical calculations.

But for negative numbers to become such a handy tool, you need to:

A negative temperature is one that is below zero, below zero temperature. But what is zero temperature? To measure, record the temperature, you need to select the unit of measurement and the reference point. Both are an agreement. We use the Celsius scale named after the scientist who proposed it. (See Figure 10.)

Rice. 10. Anders Celsius

Here, the freezing point of water is chosen as the reference point. Anything below is marked negative value. (See Figure 11.)

Rice. eleven.

But it is clear that if we take another reference point, another zero, then the negative temperature in Celsius can be positive in this other scale. And so it happens. In physics, the Kelvin scale is widely used. It is similar to the Celsius scale, only the value of the lowest possible temperature is chosen as zero (there is no lower). This value is called absolute zero". In Celsius, this is approximately. (See Figure 12.)

Rice. 12. Two scales

That is, there are no negative values ​​​​in the Kelvin scale at all.

Yes, our summer .

And frosty .

That is, a negative temperature is a convention, an agreement of people to call it that.

Let's start from scratch. Zero takes special position among the numbers.

As we have already discussed, for our convenience, we can designate the subtraction of seven as a negative number. Since it means subtraction, we leave the sign "" as its sign. Let's call a new number.

That is, "" is a number that adds up to zero: . And in any order. This is the definition of a negative (or opposite) number.

For each number that we studied before, we introduce a new number, negative, whose sign is a minus sign in front of it. That is, for each previous number, its negative twin appeared. Such twins are called opposite numbers. (See Figure 13.)

Rice. 13. Opposite numbers

So, definition: two numbers are called opposite numbers, the sum of which is equal to zero.

Outwardly, they differ only in the sign "".

If a variable is preceded by the sign "", for example, what does this mean? It doesn't mean that given value negative. The minus sign means that this value is opposite to the number: . Which of these numbers is positive, which is negative, we do not know.

If , then .

If (negative number), then (positive number).

What is the opposite of zero? We already know this.

If zero is added to any number, including zero, then the original number will not change. That is, the sum of two zeros is equal to zero: . But numbers whose sum is zero are opposite. Thus, zero is the opposite of itself.

So, we have given the definition of negative numbers, found out why they are needed.

Now let's spend some time on technology. For now, we need to learn how to find its opposite for any number:

In the last part of the lesson, we will talk about the new names and designations of sets that appear after the introduction of negative numbers.

Subject

Lesson type

• study and primary assimilation of new material

Lesson Objectives

Get to know the definitions of positive and negative, opposite numbers

Find opposite numbers when solving exercises, when solving equations

Developing - to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.

Educational - through a lesson, to cultivate an attentive attitude towards each other, to instill the ability to listen to comrades, mutual assistance, independence.

Lesson objectives

Learn what opposite numbers are

Learn to use this concept when solving problems

Check students' ability to solve problems.

Lesson Plan

1. Introduction.

2. Theoretical part

3. Practical part.

4. Homework.

5. Interesting Facts

Introduction

Look at the pictures and describe in one word what is the difference in them.

The pictures show opposites.

are two numbers that are equal in absolute value, but having different signs, eg. 5 and -5.

Theoretical part

First, let's remember what is negative numbers. Look video:

Points with coordinates 5 and -5 are equally distant from point O and are located along different sides from her. To get from point O to these points, one must travel the same distances, but in opposite directions. The numbers 5 and -5 are called opposite numbers: 5 is the opposite of -5 and -5 is the opposite of 5.

Two numbers that differ from each other only in signs are called opposite numbers.

For example, 35 and -35 will be opposite numbers, since the number 35 \u003d +35, which means that the numbers 35 and -35 differ only in signs. The opposite numbers will also be 0.8 and -0.8, ¾ and -¾.

Properties of opposite numbers

one). For every number, there is only one opposite number.

2). The number 0 is the opposite of itself.

3). The opposite of a is called -a. If a = -7.8, then -a = 7.8; if a = 8.3, then -a = -8.3; if a = 0, then -a = 0.

4). The entry "-(-15)" means the opposite of -15. Since the opposite of -15 is 15, then -(-15) = 15. In general -(-a) = a.

The natural numbers, their opposite numbers and zero are called whole numbers.

opposite number n" in relation to the number n is the number that, when added to n, gives zero.

n + n" = 0

This equality can be rewritten as follows:

n + n" - n = 0 - n or n" = − n

Thus, opposite numbers have the same modules but opposite signs.

In accordance with this, the number opposite to the number n is denoted − n. When a number is positive, then its opposite number will be negative, and vice versa.

1. Give examples of opposite numbers.

2. Draw them on the coordinate line.

3. What is the opposite of -3.6; 7; 0; 8/9; -1/2

Practical part

Example

1) Mark points A(2), B(-2), C(+4), D(-3), E(-5.2), F(5.2), G(-6) on the coordinate line , H(7). 2) Among these points, find and indicate those that are symmetrical with respect to the point O (0). What can be said about the coordinates of symmetrical points?

Points symmetric with respect to point O(0): A(2) and B(-2), E(-5.2) and F(5.2)

Symmetric point coordinates are numbers that differ only in sign. Such numbers are called opposite.

Mark on the coordinate line points A (-3), B (+6), C (+4.2), D (+3), E (-4.2), F (-6) What can be said about these numbers ?

From the numbers 15; 2.5; - 2.5; - eighteen; 0; 45; - 45 choose: a) natural numbers; b) whole numbers; c) negative numbers; d) positive numbers; e) opposite numbers.

1) Write down the number opposite to number a.

2) Indicate the number opposite to the number a, if:

a=5, a=-3, a=0, a=-2/5;

A \u003d 6, -a \u003d - 2, -a \u003d 3.4.

1) Remember what the entry means: - (- a).

2) Replace * with such a number to get the correct equality: a) - (- 5) = *; b) 3 = - *.

Homework

one). Fill in the table:

2). Find: a) -m,

if m = -8,

if m = -16

if -k = 27

if -k = -35

if c = 41

if c = -3.6

3). How many pairs of opposite numbers are located between the numbers -7.2 and 3.6. Mark on the coordinate line.

4). Find out the name of an outstanding French scientist:

Do you know where in Everyday life do we encounter positive and negative numbers?

List of sources used

1. Mathematical encyclopedia (in 5 volumes). - M.: Soviet Encyclopedia, 2002. - T. 1.
2. " The latest guide schoolchild" "HOUSE XXI century" 2008
3. Summary of the lesson on the topic "Opposite numbers" Author: Petrova V.P., mathematics teacher (grades 5-9), Kyiv
4. N.Ya. Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for Grade 6, Textbook for high school s

In this article, we will try to figure out what opposite numbers are. We will explain what they are in general, show what kind of designations are used for them, and analyze a few examples. In the last part of the material, we list the main properties of opposite numbers.

To explain the very concept of opposites, we first need to draw a coordinate line. Let's take a point M on it (only not at the very beginning of the reference). Its distance to zero will be equal to a certain number of unit segments, which can, in turn, be divided into tenths and hundredths. If we measure the same distance from the origin in the direction opposite to that on which M is located, then we can get to another similar point. Let's call it N. For example, from M to zero - the distance is 2, 4 unit segments, and from N to zero - too. Take a look at the picture:

Recall that each point on the coordinate line can be associated with only one real number. In this case, our points M and N correspond to certain numbers, which are called opposite. Each number has opposite number except for zero. Since this is the origin, it is considered the opposite of itself.

Let's write down the definition of what opposite numbers are:

Definition 1

Opposite the numbers are called, which correspond to such points on the coordinate line that we will get to if we mark the same distance from the origin in different directions (positive and negative). Zero is at the origin and is opposite to itself.

How are opposite numbers indicated?

In this subsection we introduce the basic notation for such numbers. If we have a certain number and we need to write down the opposite of it, then for this we use a minus.

Example 1

Let's say our number is a, therefore, its opposite is a (minus a). In the same way, for 0.26 the opposite is -0.26, and for 145 it will be -145. If the original number is itself negative, for example, - 9, then we write the opposite as - (- 9) .

What other examples of opposite numbers can you give? Let's take integers: 12 and - 12. Opposite rational numbers are 3 2 11 and - 3 2 11, as well as 8, 128 and - 8, 128, 0, (18901) and - 0, (18901), etc. Irrational numbers can also be opposite, for example, values numeric expressions 2 + 1 and - 2 + 1 .

Opposite ir rational numbers will also be e and - e .

Basic properties of opposite numbers

Such numbers have certain properties. Below we give a list of them with explanations.

Definition 2

1. If the original number is positive, then its opposite will be negative.

This statement is obvious and follows from the graph above: such numbers are on opposite sides of the reference on the coordinate line. If you have forgotten the concepts of positive and negative numbers, look at the material that we published earlier.

Another very important statement can be deduced from this rule. In literal form, its notation is as follows: for any positive a, it will be true − (− a) = a . Let's use an example to show why this is important.

Let's take the number 5. With the help of the coordinate line, you can see that the number is opposite to it - 5, and vice versa. Using the notation that we indicated above, we write the number opposite - 5 as - (- 5). It turns out that - (- 5) \u003d 5. Hence the conclusion: opposite numbers differ from each other only by the presence of a minus sign.

2. Next property called the property of symmetry. It can also be derived from the very definition of opposite numbers. It sounds like this:

Definition 3

If some number a is the opposite of b, then b is the opposite of a.

Obviously, this assertion does not need additional proof.

3. The third property of opposite numbers says:

Definition 4

Every real number has only one opposite number.

This statement follows from the fact that the points of the coordinate line cannot correspond to many numbers at once.

Definition 5

4. Modules of opposite numbers are equal.

This follows from the module definition. It is logical that the points on the line corresponding to any opposite numbers are at the same distance from the reference point.

Definition 6

5. If we add opposite numbers, we get 0.

In literal form, this statement looks like a + (− a) = 0 .

Example 2

Here are examples of such calculations:

890 + (- 890) = 0 - 45 + 45 = 0 7 + (- 7) = 0

As you can see, this rule works for all numbers - integer, rational, irrational, etc.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

§ 1 The concept of a positive number

In this lesson, you will learn what numbers are called opposites, how to find the opposite number, and what are integer and rational numbers.

Let's start with practical work. On the coordinate line, mark the points A(2) and B(-2). They are symmetrical and the center of symmetry of these points is the origin O(0), since the distance OA=OB.

We see that the coordinates of points that are symmetrical about the origin are numbers that differ only in sign. Such numbers are called opposites.

There is another definition of opposite numbers. What are the modules of numbers 2 and -2? Equal to 2. Therefore, opposite numbers are numbers that have the same modules, but differ in sign.

To indicate the number opposite given number, use the minus sign, which is written in front of the given number. That is, the opposite of a is written as −a. For example, the number 0.24 is opposite to the number −0.24, the number -25 is opposite to the number −(−25), but the number -25 on the coordinate line is opposite to 25, which means -(-25) = 25. It follows from this that -( -a) = a and a = -(-a).

§ 2 Properties of opposite numbers

Let's single out some properties of opposite numbers.

The number opposite to a positive number is negative, and the number opposite to a negative number is positive. This is understandable, since the points of the coordinate line corresponding to opposite numbers are on opposite sides of the origin.

If the number a is opposite to the number b, then b is opposite to a - this follows from the symmetry property of points on the coordinate line.

Let's look at the coordinate line. How many points can be marked on a coordinate line that are symmetrical to the given one with respect to the origin? Only one. This means that for each number there is only one opposite number.

Only one number is opposite to itself - this is the number 0, since 0 \u003d -0 (therefore, it is not customary to write -0).

Numbers with common feature form a set (or group), each set has its own name.

Recall that the numbers that we use in counting are called natural numbers, they form a set of natural numbers.

Every natural number has its opposite number. Natural numbers, their opposite numbers, and the number 0 are called integers.

Can be positive or negative fractional numbers. All integers and all fractions are called rational numbers. They also say that together they form the set of rational numbers.

Let's single out two more groups of numbers. Let's take a coordinate line. If we remove the part of the straight line on which the negative numbers are located, there will be a ray with positive numbers and the reference number 0. The remaining numbers are called non-negative, that is, numbers that are greater than or equal to 0. Therefore, non-positive numbers are all negative numbers and the number 0, that is, numbers that are less than or equal to 0.

Today we learned what opposite, integer, rational, non-negative, non-positive numbers are, we learned how to find the number opposite to a given one.

List of used literature:

1. Mathematics.6th grade: lesson plans to the textbook by I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilin. Mnemosyne 2009
2. Mathematics. Grade 6: student textbook educational institutions. I.I. Zubareva, A.G. Mordkovich.- M.: Mnemozina, 2013
3. Mathematics. Grade 6: a textbook for students of educational institutions. /N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. – M.: Mnemosyne, 2013
4. Mathematics Handbook - http://lyudmilanik.com.ua
5. Handbook for students in high school http://shkolo.ru

Opposite numbers definition

Opposite numbers definition:

Two numbers are said to be opposite if they differ only in signs.

Examples of opposite numbers

Examples of opposite numbers.

1 -1;
2 -2;
99 -99;
-12 12;
-45 45

From here it is clear how to find the number opposite to the given one: just change the sign of the number.

The opposite of 3 is the number minus three.

Example. The numbers are the opposite of the data.

Given: numbers 1; 5; eight; nine.

Find numbers opposite to given.

To solve this task, simply change the signs of the given numbers:

Let's make a table of opposite numbers:

1	5	8	9
-1	-5	-8	-9

The number opposite to zero

The opposite of zero is zero itself.

So the opposite of 0 is 0.

Opposite integers

Opposite integers differ only in signs.

Examples of opposite integers.

10 -10
20 -20
125 -125

A pair of opposite numbers

When people talk about opposite numbers, they always mean a pair of opposite numbers.

A number is the opposite of another number. And each number has only one opposite number.

Numbers opposite to natural numbers

Numbers opposite to natural numbers are negative integers.

Let's make a table of opposite numbers for the first five natural numbers:

1	2	3	4	5
-1	-2	-3	-4	-5

Sum of opposite numbers

The sum of opposite numbers is zero. After all, opposite numbers differ only in sign.