The concept of opposite numbers. Negative numbers. Opposite numbers (Slupko M.V.)
§ 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 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 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 a set 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
- Handbook for students in secondary school http://shkolo.ru
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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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
Interesting concept from school course learning are opposite numbers, which can be considered both mathematically and geometrically. Understanding this topic simplifies the study of mathematics and allows you to quickly cope with some problems - so we will look at what numbers are called opposites, and what rules work for them.
What is the essence of the term?
To understand the meaning of opposite numbers, let's turn to geometry for a moment. Let's draw a coordinate line and mark the zero point on it, and then put two more marks on the line - for example, “2” with right side and "-2" to the left of zero. Of course, from both points the distance to the origin will be exactly the same - and this is easily verified by measurements. “2” and “-2” are the same distance from zero, but in different directions - accordingly, they are completely opposite to each other.
That's the point. Numbers can be as large or small as desired, whole or fractional. However, each of them has a certain number that is its exact opposite. The definition can be given as follows - if on the coordinate line from two points placed on both sides of zero, an equal distance can be set aside to the origin - these points, or rather, the numbers corresponding to them, will be opposite.
What rules can be derived from the definition?
It is worth remembering a few absolute statements regarding the topic under consideration:
- The principle of opposites for two numbers works both ways. For example, the number 3 is opposite to the number -3 - and therefore only the number 3 is opposite to the number -3, and not any other.
- A number cannot have two opposites - there is always only one.
- Numbers can be opposite to each other different signs. If a number is positive, then its opposite number will have a minus sign - for example, 5 and -5. The same thing works in reverse side- for a number with a minus sign, the opposite will always be that with a plus sign - for example, -6 and 6.
- Two opposite numbers have the same absolute value, or modulus. In other words, if for the number 4
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 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
