Opposite numbers solution with opposite signs. Negative numbers
5 and -5 (Fig. 61) 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 in signs are called opposite numbers.
For example, opposite numbers would be 8 and -8, since the number 8 = + 8, which means numbers 8 and - 8 differ only in signs. The opposite numbers will also be
For every number there is only one opposite number.
The number 0 is the opposite of itself.
The opposite number o 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. The entry “- (-15)” means the number opposite to the number -15. Since the opposite number of -15 is 15, then -(- 15) = 15. In general - (- a) = a.
The natural numbers, their opposites and zero are called integers.
? What numbers are called opposites?
Number b is opposite to number a. What number is the opposite of b?
What number is opposite to zero?
Is there a number that has two opposite numbers?
What numbers are called integers?
TO 910. Find the opposite numbers:
911. Substitute a number to get the correct equality:
912. Find the meaning of the expression:
913. Find the coordinates of points A, B and C (Fig. 62).
914. What number is - x, if x:
a) negative; b) zero; c) positive?
915. Fill in the blanks in the table and mark on the coordinate straight points that have as their coordinates the numbers of the resulting table.
916. Solve the equation:
a) - x = 607; b) - a = 30.4; c) - y= -3
917. What integers are located on the coordinate line between the numbers:
P 918. Calculate conventionally:
919. Between what integers on the coordinate line is the number located: 2.6; -thirty; -6; -8
920. Find the numbers that are at a distance on the coordinate line: a) 6 units from the number -9; b) 10 units from the number 4; c) 10 units from the number -4; d) 100 units from the number 0.
921. Draw a coordinate line, taking as unit line segment the length of 4 notebook cells, and mark the point on this straight line, F (2,25).
A 922. Mark on the “time line” the following events from the history of mathematics:
a) The book “Elements” was written by Euclid in the 3rd century. BC e.
b) Number theory originated in Ancient Greece in the 6th century BC e.
V) Decimals appeared in China in the 3rd century.
d) The theory of relations and proportions was developed in Ancient Greece in the 4th century. BC e.
e) Positional decimal system Notation spread to the countries of the East in the 9th century. How many centuries ago did these events take place? Compare the “time line” and the coordinate line.
923. Specify pairs of mutually inverse numbers:
924. Vitya bought 2.4 kg of carrots. How many carrots bought Kolya, if you know what he bought:
a) 0.7 kg more than Viti; f) what Vitya bought;
b) 0.9 kg less than Viti; g) 0.5 of what Vitya bought;
c) 3 times more than Viti; h) 20% of what Vitya bought;
d) 1.2 times less than Viti; i) 120% of what Vitya bought;
e) what Vitya bought; j) 20% more than what Vitya bought?
925. Solve the problem:
1) The brick factory had to produce 270 thousand bricks for the construction of the Palace of Culture. First
week he produced the tasks, in the second week he produced 10% more than in the first week. How many thousand bricks does the plant have left to produce?
2) The collective farm sold 434 tons of grain to the state in three days. On the first day he sold this amount, on the second day - 10% less than on the first day, and on the third day - the rest of the grain. How many tons of grain did the collective farm sell on the third day?
926. Notes differ in the duration of their sound. The sign denotes a whole note, a note half as long - a half note, a sixteenth note.
Check for equality of durations:
D 927. What numbers are opposite numbers:
928. Write down all the natural numbers less than 5 and their opposites.
929. Find the value:
930. On the second day, 2 times more wire was released from the warehouse than on the first day, and on the third day 3 times more than on the first. How many kilograms of wire were issued in these three days, if on the first day they were issued 30 kg less than on the third?
931. On the collective farm, on irrigated lands, 60.8 centners of wheat were collected per hectare. Replacing an old wheat variety with a new one gives a 25% increase in yield. How much wheat does the collective farm now collect from 23 hectares of irrigated field?
932. Make up an equation for each diagram and solve it:
933. 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
§ 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 also what integers and rational numbers are.
Let's start with practical work. On the coordinate line, mark points A(2) and B(-2). They are symmetrical and the center of symmetry of these points is the origin of coordinates O(0), since the distance OA=OB.
We see that the coordinates of points 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 absolute values 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 opposite number given number, use a minus sign, which is written in front of this number. That is, the opposite number of a is written as −a. For example, the number 0.24 is opposite the number −0.24, the number -25 is the opposite number −(−25), but the number -25 on the coordinate line is opposite 25, which means -(-25) = 25. It follows from this that -( -a) = a and a = -(-a).
§ 2 Properties of opposite numbers
Let us highlight some properties of opposite numbers.
The opposite of a positive number is negative, and the opposite of a negative number is positive. This is understandable, since the points of the coordinate line corresponding to opposite numbers are located 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 property of symmetry of points on the coordinate line.
Let's turn to the coordinate line. How many points can be marked on a coordinate line that are symmetrical to the given one relative 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 = -0 (therefore, it is not customary to write -0).
Numbers with common feature form a set (or group), each set has its own name.
Let us remember that the numbers we use when counting are called natural numbers; they form the set of natural numbers.
For every natural number you can find its opposite number. Natural numbers, their opposites, and the number 0 are called integers.
Can be positive or negative fractional numbers. All whole numbers and all fractions are called rational numbers. They also say that together they form the set of rational numbers.
Let's highlight two more groups of numbers. Let's take a coordinate line. If you remove the part of the line on which the negative numbers are located, you will be left with 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, and learned to find the opposite number of a given one.
List of used literature:
- Mathematics. 6th grade: lesson plans to the textbook I.I. Zubareva, A.G. Mordkovich //author-compiler L.A. Topilina. Mnemosyne 2009
- Mathematics. 6th grade: textbook for students educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
- Mathematics. 6th grade: textbook for students of general education institutions. /N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwartzburd. – M.: Mnemosyne, 2013.
- Handbook of mathematics - http://lyudmilanik.com.ua
- Student's Guide to high school http://shkolo.ru
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 do mathematical notation the movements of the balls in the order they 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 in 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 easier to calculate 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 to simplify mathematical calculations.
But for negative numbers to become such a convenient tool, you 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. eleven.
But it is clear that if we take another reference point, another zero, then a negative temperature in 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 lowest value is selected as zero possible temperature(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.
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.
