If we add the number 0 to the left of a series of natural numbers, we get series of positive integers:
0, 1, 2, 3, 4, 5, 6, 7, ...
Negative integers
Let's consider small example. The picture on the left shows a thermometer that shows a temperature of 7 °C. If the temperature drops by 4 °C, the thermometer will show 3 °C of heat. A decrease in temperature corresponds to the action of subtraction:
Note: all degrees are written with the letter C (Celsius), the degree sign is separated from the number by a space. For example, 7 °C.
If the temperature drops by 7 °C, the thermometer will show 0 °C. A decrease in temperature corresponds to the action of subtraction:
If the temperature drops by 8 °C, the thermometer will show -1 °C (1 °C below zero). But the result of subtracting 7 - 8 cannot be written using natural numbers and zero.
Let's illustrate subtraction using a series of positive integers:
1) From the number 7, count 4 numbers to the left and get 3:
2) From the number 7, count 7 numbers to the left and get 0:
It is impossible to count 8 numbers from the number 7 to the left in a series of positive integers. To make actions 7 - 8 feasible, we expand the range of positive integers. To do this, to the left of zero, we write (from right to left) in order all the natural numbers, adding to each of them the sign - , indicating that this number is to the left of zero.
The entries -1, -2, -3, ... read minus 1, minus 2, minus 3, etc.:
5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...
The resulting series of numbers is called series of integers. The dots to the left and right in this entry mean that the series can be continued indefinitely to the right and left.
To the right of the number 0 in this row are numbers called natural or positive integers(briefly - positive).
To the left of the number 0 in this row are numbers called integer negative(briefly - negative).
The number 0 is an integer, but is neither a positive nor a negative number. It separates positive and negative numbers.
Hence, the series of integers consists of negative integers, zero and positive integers.
Integer Comparison
Compare two integers- means to find out which one is greater, which one is smaller, or determine that the numbers are equal.
You can compare integers using a row of integers, since the numbers in it are arranged from smallest to largest if you move along the row from left to right. Therefore, in a series of integers, you can replace commas with a less than sign:
5 < -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < ...
Hence, of two integers, the greater is the number that is to the right in the series, and the smaller is the one that is to the left, Means:
1) Any positive number is greater than zero and greater than any negative number:
1 > 0; 15 > -16
2) Any negative number less than zero:
7 < 0; -357 < 0
3) Of two negative numbers, the one that is to the right in the series of integers is greater.
Teacher of the highest category
What numbers are called integers?
Lesson objectives:
-Expand the concept of number by introducing negative numbers:
-Develop the skill of writing positive and negative numbers.
Lesson objectives.
Educational – promote the development of the ability to generalize and systematize, promote the development of mathematical horizons, thinking and speech, attention and memory.
Educational – fostering an attitude towards self-education, self-education, precise performance, a creative attitude to activity, critical thinking.
Developmental – develop in schoolchildren the ability to compare and generalize, logically express thoughts, develop mathematical horizons, thinking and speech, attention and memory.
During the classes:
1. Introductory conversation.
So far in mathematics lessons we have looked at what numbers?
-Natural and fractional.
What numbers are called natural numbers?
- These are numbers used when counting objects.
How many can you say?
- infinitely many.
Is zero a natural number? Why?
-What are fractional numbers used for?
-We not only count objects, but parts of certain quantities.
What fractions do you know?
- Ordinary and decimal.
Task No. 1.
Among the numbers, what are the natural numbers? Common fractions? Decimals?
10; 1,1; https://pandia.ru/text/77/504/images/image002_2.png" width="16" height="35 src="> ; https://pandia.ru/text/77/504/images/image004_0.png" width="24" height="35 src="> .
2. Explanation of new material:
However, in your life you have probably already encountered other numbers, which ones? Where?
-Negative. For example, in a weather report.
Before moving on to learning a new topic, let's discuss signs that will help in expanding the set of numbers. These are plus and minus signs. Think about what these signs are associated with in life. It can be anything: white - black, good - bad. We will write your examples in the form of a table.
Just two signs evoke so many thoughts. In fact, these two signs provide the opportunity to go in different directions. Such numbers, “similar” to natural numbers, but with a minus sign, are needed in cases where a quantity can change in two opposite directions. To express a value as a negative number, some initial, zero mark is introduced. Let's look at the examples that others have done, and at home you can think about it and make your own presentation. Slide No. 2-7.
Using the sign is very convenient. Its use is accepted throughout the world. But it was not always so. Slide number 8.
So, along with the natural numbers
1, 2, 3, 4, 5, …100, …, 1000, …
We will consider negative numbers, each of which is obtained by adding a minus sign to the corresponding natural number:
-1,- 2, - 3, - 4, - 5, …-100, …,- 1000, …
A natural number and its corresponding negative number are called opposites. For example, the numbers 15 and -15. You can use -15 and 15. O is the opposite of itself.
Rule: Natural numbers, their negative opposites and the number 0 are called whole numbers. All these numbers together make up the set of integers.
Open the textbook, page 159, find the rule, read it again, and learn it by heart at home.
A natural number is also commonly called a positive integer, that is, it is the same thing. In order to emphasize the external difference from the negative, a plus sign is sometimes placed in front of it. +5=5.
3. Formation of skills and abilities:
1) № 000.
2) Write these numbers into two groups: positive and negative:
-15, 7, 28, -41, 0, 382, -591, -999, 2000.
3) Game “my mood”.
Now you will rate your mood at the moment on the following scale:
Good mood: +1, +2, +3, +4, +5.
Bad mood: -1, -2, -3, -4, -5.
One person will write the results on the board, and everyone else will take turns saying out loud: “I have good mood by 4 points"
4) Game "cracker"
I will name pairs of numbers, if the pair is opposite, then you clap your hands, if not, then there should be silence in the class:
5 and -5; 6 and 0.6; -300 and 300; 3 and 1/3; 8 and 80; 14 and -14; 5/7 and 7/5; -1 and 1.
5) Propaedeutics for learning the addition of integers:
No. 000 (a).
We look at the solution using the presentation. Slide number 8.
4. Lesson summary:
-What numbers are called positive? Negative?
-What did you find out about O?
- What are negative numbers used for?
-How are positive and negative numbers written?
5. D/Z: clause 8.1, No. 000, 721(b), 715(b). Creative task: write a poem about whole numbers, a drawing, a presentation, a fairy tale.
We will subtract another from the number,
We put a straight line.
We recognize this sign
“Minus” we call him.
1.
Worth one
Looks like a match.
She's just a devil
With a small bang.
2.
It barely glides through the water,
Like a swan, number two.
She arched her neck,
Drives the waves behind him.
3.
Two hooks, look
The result was number three.
But these two hooks
You can't get a worm.
4.
Somehow the fork was dropped
One clove was broken off.
This fork is in the whole world
It's called "four".
5.
Number five - with a big belly,
Wears a cap with a visor.
At school this number is five
Children love to receive.
6.
What a cherry, my friend,
Is the stem bent upward?
Try to eat it
This cherry is number six.
7.
I'm such a poker
I can't put it in the oven.
Everyone knows about her
That it's called "seven".
8.
The rope was twisting, twisting,
Braided into two loops.
"What is this number?" - Let's ask mom.
Mom will answer us: “Eight.”
9.
Wind blew strong and blew
He turned the cherry over.
Number six, please tell me
It turned into the number nine.
10.
Like an older sister
The zero is led by one.
We just walked together
They immediately became the number ten.
Poems about mathematics
Mathematics is the basis and queen of all sciences, Arzhnikova Svetlana, Complex science mathematics: Razborov Roman, | Find your speed , One two three four five, Vityutneva Marina, |
· Much of mathematics does not remain in memory, but when you understand it, then it is easy to remember what you have forgotten on occasion.
The information in this article provides a general understanding of integers. First, a definition of integers is given and examples are given. Next, we consider integers on the number line, from where it becomes clear which numbers are called positive integers and which are called negative integers. After this, it is shown how changes in quantities are described using integers, and negative integers are considered in the sense of debt.
Page navigation.
Integers - Definition and Examples
Definition.
Whole numbers– these are natural numbers, the number zero, as well as numbers opposite to the natural ones.
The definition of integers states that any of the numbers 1, 2, 3, …, the number 0, as well as any of the numbers −1, −2, −3, … is an integer. Now we can easily bring examples of integers. For example, the number 38 is an integer, the number 70,040 is also an integer, zero is an integer (remember that zero is NOT a natural number, zero is an integer), the numbers −999, −1, −8,934,832 are also examples of integers numbers.
It is convenient to represent all integers as a sequence of integers, which has the following form: 0, ±1, ±2, ±3, ... A sequence of integers can be written like this: …, −3, −2, −1, 0, 1, 2, 3, …
From the definition of integers it follows that the set of natural numbers is a subset of the set of integers. Therefore, every natural number is an integer, but not every integer is a natural number.
Integers on a coordinate line
Definition.
Positive integers are integers greater than zero.
Definition.
Negative integers are integers that are less than zero.
Positive and negative integers can also be determined by their position on the coordinate line. On a horizontal coordinate line, points whose coordinates are positive integers lie to the right of the origin. In turn, points with negative integer coordinates are located to the left of point O.
It is clear that the set of all positive integers is the set of natural numbers. In turn, the set of all negative integers is the set of all numbers opposite natural numbers.
Separately, let us draw your attention to the fact that we can safely call any natural number an integer, but we cannot call any integer a natural number. We can only call any positive integer a natural number, since negative integers and zero are not natural numbers.
Non-positive and non-negative integers
Let us give definitions of non-positive integers and non-negative integers.
Definition.
All positive integers, together with the number zero, are called non-negative integers.
Definition.
Non-positive integers– these are all negative integers together with the number 0.
In other words, a non-negative integer is an integer that is greater than zero or equal to zero, and a non-positive integer is an integer that is less than zero or equal to zero.
Examples of non-positive integers are the numbers −511, −10,030, 0, −2, and as examples of non-negative integers we give the numbers 45, 506, 0, 900,321.
Most often, the terms “non-positive integers” and “non-negative integers” are used for brevity. For example, instead of the phrase “the number a is an integer, and a is greater than zero or equal to zero,” you can say “a is a non-negative integer.”
Describing changes in quantities using integers
It's time to talk about why integers are needed in the first place.
The main purpose of integers is that with their help it is convenient to describe changes in the quantity of any objects. Let's understand this with examples.
Let there be a certain number of parts in the warehouse. If, for example, 400 more parts are brought to the warehouse, then the number of parts in the warehouse will increase, and the number 400 expresses this change in quantity in positive side(increasing). If, for example, 100 parts are taken from the warehouse, then the number of parts in the warehouse will decrease, and the number 100 will express a change in quantity in the negative direction (downward). Parts will not be brought to the warehouse, and parts will not be taken away from the warehouse, then we can talk about the constant quantity of parts (that is, we can talk about zero change in quantity).
In the examples given, the change in the number of parts can be described using the integers 400, −100 and 0, respectively. A positive integer 400 indicates a change in quantity in a positive direction (increase). A negative integer −100 expresses a change in quantity in a negative direction (decrease). The integer 0 indicates that the quantity remains unchanged.
The convenience of using integers compared to using natural numbers is that you do not have to explicitly indicate whether the quantity is increasing or decreasing - the integer quantifies the change, and the sign of the integer indicates the direction of the change.
Integers can also express not only a change in quantity, but also a change in some quantity. Let's understand this using the example of temperature changes.
A rise in temperature of, say, 4 degrees is expressed as a positive integer 4. A decrease in temperature, for example, by 12 degrees can be described by a negative integer −12. And the invariance of temperature is its change, determined by the integer 0.
Separately, it is necessary to say about the interpretation of negative integers as the amount of debt. For example, if we have 3 apples, then the positive integer 3 represents the number of apples we own. On the other hand, if we have to give 5 apples to someone, but we don’t have them in stock, then this situation can be described using a negative integer −5. In this case, we “own” −5 apples, the minus sign indicates debt, and the number 5 quantifies debt.
Understanding a negative integer as a debt allows, for example, to justify the rule for adding negative integers. Let's give an example. If someone owes 2 apples to one person and 1 apple to another, then the total debt is 2+1=3 apples, so −2+(−1)=−3.
Bibliography.
- Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
Negative numbers were first used in ancient China and India; in Europe they were introduced into mathematical use by Nicolas Chuquet (1484) and Michael Stiefel (1544).
Algebraic properties
is not closed under division of two integers (for example, 1/2). The following table illustrates several basic properties of addition and multiplication for any integer a, b And c.
| addition | multiplication | |
| closedness: | a + b- whole | a × b- whole |
| associativity: | a + (b + c) = (a + b) + c | a × ( b × c) = (a × b) × c |
| commutativity: | a + b = b + a | a × b = b × a |
| existence of a neutral element: | a + 0 = a | a× 1 = a |
| existence of the opposite element: | a + (−a) = 0 | a≠ ±1 ⇒ 1/ a is not integer |
| distributivity of multiplication relative to addition: | a × ( b + c) = (a × b) + (a × c) | |
number systems |heading4= Hierarchy of numbers |list4=
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|list5=Cardinal numbers – You definitely need to move it to the bed, it won’t be possible here...
The patient was so surrounded by doctors, princesses and servants that Pierre no longer saw that red-yellow head with a gray mane, which, despite the fact that he saw other faces, did not leave his sight for a moment during the entire service. Pierre guessed from the careful movement of the people surrounding the chair that the dying man was being lifted and carried.
“Hold on to my hand, you’ll drop me like this,” he heard the frightened whisper of one of the servants, “from below... there’s another one,” said the voices, and the heavy breathing and stepping of the people’s feet became more hasty, as if the weight they were carrying was beyond their strength .
The carriers, among whom was Anna Mikhailovna, drew level with the young man, and for a moment, from behind the backs and backs of the people’s heads, he saw a high, fat, open chest, the fat shoulders of the patient, raised upward by the people holding him under the arms, and a gray-haired, curly, lion's head. This head, with an unusually wide forehead and cheekbones, a beautiful sensual mouth and a majestic cold gaze, was not disfigured by the proximity of death. She was the same as Pierre knew her three months ago, when the count let him go to Petersburg. But this head swayed helplessly from the uneven steps of the carriers, and the cold, indifferent gaze did not know where to stop.
Several minutes of fussing around the high bed passed; the people carrying the sick man dispersed. Anna Mikhailovna touched Pierre's hand and told him: “Venez.” [Go.] Pierre walked with her to the bed on which the sick man was laid in a festive pose, apparently related to the sacrament that had just been performed. He lay with his head high on the pillows. His hands were laid out symmetrically on the green silk blanket, palms down. When Pierre approached, the count looked straight at him, but he looked with a look whose meaning and meaning cannot be understood by a person. Either this look said absolutely nothing except that as long as you have eyes, you must look somewhere, or it said too much. Pierre stopped, not knowing what to do, and looked questioningly at his leader Anna Mikhailovna. Anna Mikhailovna made a hasty gesture to him with her eyes, pointing to the patient’s hand and blowing her a kiss with her lips. Pierre, diligently craning his neck so as not to get caught in the blanket, followed her advice and kissed the big-boned and fleshy hand. Not a hand, not a single muscle of the count’s face trembled. Pierre again looked questioningly at Anna Mikhailovna, now asking what he should do. Anna Mikhailovna pointed him with her eyes to the chair that stood next to the bed. Pierre obediently began to sit down on the chair, his eyes continuing to ask whether he had done what was necessary. Anna Mikhailovna nodded her head approvingly. Pierre again assumed the symmetrically naive position of the Egyptian statue, apparently condoling that his clumsy and fat body occupied such a position. large space, and using all mental strength to appear as small as possible. He looked at the count. The Count looked at the place where Pierre's face was while he stood. Anna Mikhailovna in her position showed an awareness of the touching importance of this last minute of the meeting between father and son. This lasted two minutes, which seemed like an hour to Pierre. Suddenly a tremor appeared in the large muscles and wrinkles of the count’s face. The shuddering intensified, the beautiful mouth became contorted (only then Pierre realized how close his father was to death), and an indistinct hoarse sound was heard from the contorted mouth. Anna Mikhailovna carefully looked into the patient’s eyes and, trying to guess what he needed, pointed first to Pierre, then to the drink, then in a questioning whisper called Prince Vasily, then pointed to the blanket. The patient's eyes and face showed impatience. He made an effort to look at the servant, who stood relentlessly at the head of the bed.
“They want to turn over on the other side,” the servant whispered and stood up to turn the count’s heavy body over to face the wall.
Pierre stood up to help the servant.
While the count was being turned over, one of his arms fell helplessly back, and he made a vain effort to drag it. Did the count notice the look of horror with which Pierre looked at this lifeless hand, or what other thought flashed through his dying head at that moment, but he looked at the disobedient hand, at the expression of horror in Pierre’s face, again at the hand, and on the face a weak, suffering smile that did not suit his features appeared, expressing a kind of mockery of his own powerlessness. Suddenly, at the sight of this smile, Pierre felt a shudder in his chest, a pinch in his nose, and tears blurred his vision. The patient was turned on his side against the wall. He sighed.
“Il est assoupi, [He dozed off," said Anna Mikhailovna, noticing the princess coming to replace her. – Аllons. [Let's go to.]
Pierre left.
What does a whole number mean?
So, let's look at what numbers are called integers.
Thus, the following numbers will be denoted by integers: $0$, $±1$, $±2$, $±3$, $±4$, etc.
The set of natural numbers is a subset of the set of integers, i.e. Any natural number will be an integer, but not every integer is a natural number.
Positive integers and negative integers
Definition 2
plus.
The numbers $3, 78, 569, 10450$ are positive integers.
Definition 3
are signed integers minus.
The numbers $−3, −78, −569, -10450$ are negative integers.
Note 1
The number zero is neither a positive nor a negative integer.
Positive integers are integers greater than zero.
Negative integers are integers less than zero.
The set of natural integers is the set of all positive integers, and the set of all opposite natural numbers is the set of all negative integers.
Non-positive and non-negative integers
All positive integers and zero are called non-negative integers.
Non-positive integers are all negative integers and the number $0$.
Note 2
Thus, non-negative integer are integers greater than zero or equal to zero, A non-positive integer– integers less than zero or equal to zero.
For example, non-positive integers: $−32, −123, 0, −5$, and non-negative integers: $54, 123, 0, 856,342.$
Describing changes in quantities using integers
Integers are used to describe changes in the number of objects.
Let's look at examples.
Example 1
Let a store sell a certain number of product names. When the store receives $520$ of items, the number of items in the store will increase, and the number $520$ shows a change in the number in a positive direction. When the store sells $50$ of product items, the number of product items in the store will decrease, and the number $50$ will express a change in the number in the negative direction. If the store neither delivers nor sells goods, then the number of goods will remain unchanged (i.e., we can talk about a zero change in the number).
In the above example, the change in the number of goods is described using the integers $520$, $−50$ and $0$, respectively. Positive value the integer $520$ indicates a change in the number in a positive direction. A negative value of the integer $−50$ indicates a change in the number in a negative direction. The integer $0$ indicates that the number is immutable.
Integers are convenient to use because... there is no need for an explicit indication of an increase or decrease in the number - the sign of the integer indicates the direction of the change, and the value indicates the quantitative change.
Using integers you can express not only a change in quantity, but also a change in any quantity.
Let's consider an example of a change in the cost of a product.
Example 2
An increase in value, for example, by $20$ rubles is expressed using a positive integer $20$. A decrease in price, for example, by $5$ rubles is described using a negative integer $−5$. If there is no change in value, then such change is determined using the integer $0$.
Let us separately consider the meaning of negative integers as the amount of debt.
Example 3
For example, a person has $5,000$ rubles. Then, using the positive integer $5,000$, you can show the number of rubles he has. A person must pay rent in the amount of $7,000$ rubles, but he does not have that kind of money, in which case such a situation is described by a negative integer $−7,000$. In this case, the person has $−7,000$ rubles, where “–” indicates debt, and the number $7,000$ indicates the amount of debt.
