The sign of the definition is called the opposite. Opposite numbers: definition, notation, examples
In this article we will explore opposite numbers. Here we will answer the question, what numbers are called opposites, we will show how to designate the opposite number given number, and give examples. We will also list the main results characteristic of opposite numbers.
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Determining opposite numbers
It will help us to get an idea of opposite numbers.
Let us mark some point M on the coordinate line, different from the origin. We can get to point M by sequentially laying off a unit segment from the origin in the direction of point M, as well as its tenth, hundredth, and so on parts. If we plot the same number of unit segments and its shares in the opposite direction, then we will get to another point, denoted by the letter N. Let's give an example to illustrate our actions (see figure below). To get to point M on the coordinate line, we laid off two unit segments and 4 segments, constituting a tenth of a unit, in the negative direction. Now let's put two unit segments and 4 segments, constituting a tenth of a unit, in the positive direction. This will give us point N.
We are almost ready to understand the definition of opposite numbers; all that remains is to discuss a couple of nuances.
We know that each point on the coordinate line corresponds to a single real number, therefore, both point M and point N correspond to some real numbers. So the numbers corresponding to points M and N are called opposite.
Separately, it is necessary to say about point O - the origin. Point O corresponds to the number 0. The number zero is considered to be the opposite of itself.
Now we can voice determining opposite numbers.
Definition.
Two numbers are called opposite if the points on the coordinate line corresponding to these numbers can be reached by laying off the same number of unit segments from the origin in opposite directions, as well as fractions of a unit segment, the number 0 is opposite to itself.
Notation of opposite numbers and examples
It's time to enter symbols of opposite numbers.
To indicate the opposite of a given number, use the minus sign, which is written in front of the given number. That is, the number opposite to the number a is written as −a. For example, the opposite number 0.24 is −0.24, and the opposite number −25 is −(−25).
Let's give examples of opposite numbers. The pair of numbers 17 and −17 (or −17 and 17) is an example of opposite integers. The numbers and are opposite rational numbers. Other examples of opposite rational numbers are the pairs of numbers 5.126 and −5.126. as well as 0,(1201) and −0,(1201) . It remains to give a few examples of the opposite
In this article we will try to figure out what opposite numbers are. We will explain what they are in general, show what specific designations are used for them, and look at a few examples. In the last part of the material we will list the main properties of opposite numbers.
To explain the very concept of opposites, we first need to depict a coordinate line. Let's take point M on it (but not at the very beginning of the countdown). 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 the one in which M is located, then we can get to another similar point. Let's call it N. For example, from M to zero is a distance of 2.4 unit segments, and from N to zero is the same. Take a look at the picture:
Let us remember that each point on a 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. Every number has an opposite number, except zero. Since this is the beginning of the countdown, it is considered the opposite of itself.
Let's write down the definition of what opposite numbers are:
Definition 1
Opposite numbers are called that 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 section we will introduce basic notation for such numbers. If we have a certain number and we need to write down the opposite of it, then we use a minus for this.
Example 1
Let's say our number is a, therefore its opposite is a (minus a). In exactly the same way, for 0.26 the opposite is - 0.26, and for 145 it will be - 145. If the original number itself is negative, for example, - 9, then we write the opposite as – (- 9).
What other examples of opposite numbers can you give? Let's take the integers: 12 and - 12. Opposite rational numbers– these 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, the values of numerical expressions 2 + 1 and - 2 + 1 .
Opposite irrational numbers so will e and - e .
Basic properties of opposite numbers
Such numbers are inherent certain properties. Below we will 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 located on opposite sides of the reference 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 looks like this: for any positive a it will be true − (− a) = a. Let's show with an example why this is important.
Let's take the number 5. Using the coordinate line, you can see that the opposite number is 5, and vice versa. Using the notation that we indicated above, we write the number opposite - 5 as – (- 5) . It turns out that – (- 5) = 5. Hence the conclusion: opposite numbers differ from each other only by the presence of a minus sign.
2. Next property is usually 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 statement does not need additional evidence.
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 points on a coordinate line cannot correspond to many numbers at once.
Definition 5
4. The moduli of opposite numbers are equal.
This follows from the module definition. It is logical that points on a 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.
Literally, 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 - integers, rational, irrational, etc.
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Let's consider this example. You need to count sequentially: .
You can rearrange the numbers that need 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, therefore, the result will be a subtraction from the number. This means that we need to subtract , but not from anything yet. When we have something to subtract from, we subtract:
But we can “cheat” and designate . So 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 to make it easier to record actions.
Imagine that at a sports warehouse we were tasked with issuing and receiving balls. 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 recording is not very convenient.
You can divide the sheet into two columns, one - Accepted, the other - Issued. (See Figure 2.)
Rice. 2. Simplified recording
The recording has become shorter. But here's the problem: how to understand how many balls were taken (or given away) at any particular moment in time?
You can use the following consideration for recording: when we issue balls from the warehouse, their quantity in the warehouse decreases, and when we accept them, it increases.
But how to write “gave the ball out”? You can enter the following object: .
This object allows us to make a mathematical record of the movement of the balls in the order in which it happened: 
Let's look at another example.
There are rubles in your phone account. You went online and it cost rubles. The result was a debt of rubles. The operator could have written down: “the client owes rubles.” You 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 the result of subtracting a larger number from a smaller number: .
Adding a negative number is equivalent to subtracting: .
In order to distinguish negative numbers from the positive numbers with which we dealt earlier, we agreed to put a minus sign in front of it: .
Could you do without them? Yes you can. In any given situation, we would use the words “back”, “borrow” 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 large quantity parts, many of which are not needed individually, but all together allow you to drive. Likewise, negative numbers are a tool that, together with other mathematical tools, makes it possible to simplify calculations and simplify the solution and writing of many problems.
So, we have introduced a new object - negative numbers. What are they used for in life?
First, let's remember the roles of positive numbers:
Quantity: for example wood, liter of milk. (See Figure 4.)
Rice. 4. Quantity
Ordering: For example, houses are numbered with positive numbers. (See Figure 5.)
Rice. 5. Organize
Name: for example, football player number. (See Figure 6.)
Rice. 6. Number as a name
Now let's look at the functions of negative numbers:
Indication of the missing quantity. Quantity is never negative. But a negative number is used to show that a quantity is being subtracted. For example, we can pour from a bottle and write it as . (See Figure 7.)
Rice. 7. Indication of missing quantity
Arranging. Sometimes, when numbering, zero is selected 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 located below the th, in the basement
Rice. 9. Negative numbers on the thermometer scale
But still, the main purpose of negative numbers is as a tool for simplifying mathematical calculations.
But for negative numbers to become like this convenient tool, need to:
A negative temperature is one that is below zero, below zero temperature. But what is zero temperature? To measure and record temperature, you need to select a unit of measurement and a reference point. Both are agreements. We use the Celsius scale after the scientist who proposed it. (See Fig. 10.)
Rice. 10. Anders Celsius
The freezing point of water is chosen as the reference point here. Everything below is indicated negative value. (See Figure 11.)
Rice. 11.
But it is clear that if we take another reference point, another zero, then negative temperature Celsius can be positive on this other scale. This is what happens. The Kelvin scale is widely used in physics. It is similar to the Celsius scale, only the value of the lowest possible temperature is selected as zero (it cannot be 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.
So, our summer 
.
And the frosty ones 
.
That is, negative temperature is a convention, an agreement among people to call it that.
Let's start from scratch. Zero occupies a special position among numbers.
As we have already discussed, for our convenience we can denote the subtraction of seven as a negative number. Since it means subtraction, we leave the “” sign as its sign. Let's name 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 earlier, we will introduce a new number, negative, the sign of which is the minus sign in front of it. That is, for each previous number its negative twin appeared. We call such twins opposite numbers. (See Figure 13.)
Rice. 13. Opposite numbers
So, the definition: opposite numbers are two numbers whose sum is equal to zero.
Externally, they differ only in the “” sign.
If a variable is preceded by a "" sign, for example, what does that mean? This does not mean that this value is negative. The minus sign means that this value is the opposite of the number: . We don’t know which of these numbers is positive and which is negative.
If, then.
If (negative number), then (positive number).
What number is opposite to 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 zero: . But numbers whose sum is zero are opposites. Thus, zero is opposite to itself.
So, we have given the definition of negative numbers and found out why they are needed.
Now let's spend a little 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 new names and notations for sets that appear after the introduction of negative numbers.
Subject
Lesson type
- studying and primary assimilation of new material
Lesson Objectives
Learn the definitions of positive, negative and opposite numbers.
Find opposite numbers when solving exercises, when solving equations
Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.
Lesson Objectives
Find out what opposite numbers are
Learn to use this concept when solving problems
Test students' problem-solving skills.
Lesson Plan
1. Introduction.
2. Theoretical part
3. Practical part.
4. Homework.
Introduction
Look at the pictures and describe in one word what is different about them.
The pictures show opposites.
- these are two numbers equal in absolute value, but having different signs, eg. 5 and -5.
Theoretical part
First, let's remember what it is negative numbers. Look video:
Points with coordinates 5 and -5 are equally distant from point O and are located on opposite sides of it. To get from point O to these points you need to 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 by signs are called opposite numbers.
For example, opposite numbers would be 35 and -35, since the number 35 = +35, which means that the numbers 35 and -35 differ only in signs. Opposite numbers will also be 0.8 and -0.8, ¾ and -¾.
Properties of opposite numbers
1). For every number there is only one opposite number.
2). The number 0 is the opposite of itself.
3). The opposite number of a is denoted -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 notation "-(-15)" means the opposite number of -15. Since the opposite of -15 is 15, then -(-15) = 15. In general -(-a) = a.
The natural numbers, their opposites and zero are called integers.
Opposite number n" in relation to the number n is a 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.
Accordingly, the opposite number of n is denoted − n. When a number is positive, its opposite number will be negative, and vice versa.
1. Give examples of opposite numbers.
2. Draw them on a coordinate line.
3. Name the number opposite -3.6; 7; 0; 8/9; -1/2
Practical part
Example
1) Mark on the coordinate line points A(2), B(-2), C(+4), D(-3), E(-5.2), F(5.2), G(-6) , 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 symmetrical with respect to point O(0): A(2) and B(-2), E(- 5.2) and F(5.2)
Coordinates of symmetrical points are numbers that differ only in sign. Such numbers are called opposite.
Mark the points A(-3), B(+6), C(+4.2), D(+3), E(-4.2), F(-6) on the coordinate line. What can you say about these numbers? ?
Of the numbers 15; 2.5; – 2.5; – 18; 0; 45; – 45 choose: a) natural numbers; b) integers; c) negative numbers; G) positive numbers; e) opposite numbers.
1) Write down the opposite number of a.
2) Indicate the number opposite to number a if:
a=5, a=-3, a=0, a=-2/5;
A = 6, -a = - 2, -a = 3.4.
1) Remember what the entry means: - (- a).
2) Place a number instead of * to obtain the correct equality: a) - (- 5) = *; b) 3 = – *.
Homework
1). Fill out 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 the outstanding French scientist:
Do you know where in everyday life we encounter positive and negative numbers?
List of sources used
1. Mathematical encyclopedia (in 5 volumes). - M.: Soviet Encyclopedia, 2002. - T. 1.
2. " Latest Directory schoolchild" "HOUSE XXI century" 2008
3. Lesson summary on the topic “Opposite numbers” Author: Petrova V.P., mathematics teacher (grades 5-9), Kiev
4. N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
Opposite numbers definition
Opposite numbers definition:
Two numbers are called 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 opposite of a given number: just change the sign of the number.
The opposite number to 3 is the number minus three.
Example. Numbers are opposite to data.
Given: numbers 1; 5; 8; 9.
Find the opposite numbers of the data.
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 opposite of zero
The opposite of zero is the number zero itself.
So the opposite number to 0 is 0.
Opposite Integers
Opposite integers differ only in sign.
Examples of opposite integers.
10 -10
20 -20
125 -125
Pair of opposite numbers
When they talk about opposite numbers, they always mean a pair of opposite numbers.
A number is the opposite of another number. And every number has only one opposite number.
Numbers opposite to natural numbers
The opposite of 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.
