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.
Integers, their opposite numbers 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) The positional decimal number system 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
In this article we will explore opposite numbers. Here we will answer the question of what numbers are called opposites, show how the opposite of a given number is designated, 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, as well as its tenth, hundredth, and so on, from the origin in the direction of point M. 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
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.
§ 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 of a given number, use the minus sign, which is written in front of the given 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 a common attribute 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.
Fractional numbers can also be positive or negative. 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 for the textbook by I.I. Zubareva, A.G. Mordkovich //author-compiler L.A. Topilina. Mnemosyne 2009
- Mathematics. 6th grade: textbook for students of general education 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