The opposite of this. Opposite numbers. Complete lessons – Knowledge Hypermarket
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.
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 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 numerical expressions 2 + 1 and - 2 + 1.
The opposite irrational numbers will also be 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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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.
Page navigation.
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 the 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
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 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 direct 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; -3:0; -6; -8
920. Find the numbers that on the coordinate line are at a distance of: 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 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 everything down natural numbers, less than 5, and numbers opposite to them.
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
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
