Teacher of the highest category
What numbers are called integers?
Lesson Objectives:
-Expand the concept of number by introducing negative numbers:
-To form the skill of writing positive and negative numbers.
Lesson objectives.
Educational - to promote the development of the ability to generalize and systematize, to promote the development of mathematical horizons, thinking and speech, attention and memory.
Educational - education of the attitude towards self-education, self-education, precise diligence, creative attitude to activity, critical thinking.
Educational - to 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.
Until now, in mathematics lessons, we have considered what numbers?
-Natural and fractional.
What numbers are called natural?
- These are the numbers used in counting objects.
How many can you say?
- infinitely many.
Is zero a natural number? Why?
What are fractional numbers for?
-We not only count objects, but parts of certain quantities.
What fractions do you know?
- Ordinary and decimal.
Task number 1.
Can you name 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 the new material:
However, in life you have probably already met with other numbers, which ones? Where?
-Negative. For example, in the weather report.
Before moving on to 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.
How many thoughts are caused by just two signs. In fact, these two signs make it possible to go to different sides. Such numbers, "similar" to natural ones, but with a minus sign, are needed in cases where the value can change in two opposite directions. To express a value as a negative number, some initial, zero mark is introduced. Let's look at examples that others have made, and at home think and make your presentation. Slide number 2-7.
The use of the sign is very convenient. Its use is accepted all over 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 assigning 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 -15 and 15. O is opposite to 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, we learn it by heart at home.
A natural number is also called a positive integer, that is, it is the same thing. Before him, in order to emphasize outward difference from negative, sometimes a plus sign is put. +5=5.
3. Formation of skills and abilities:
1) № 000.
2) Write these numbers in two groups: positive and negative:
-15, 7, 28, -41, 0, 382, -591, -999, 2000.
3) The game "my mood".
Now you will evaluate 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 say out loud in turn: “I have good mood for 4 points"
4) Clapperboard game
I will call 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 of studying the addition of integers:
No. 000 (a).
We look at the solution with the help of the presentation. Slide number 8.
4. Lesson summary:
What are positive numbers? Negative?
-What did you find out about?
What are negative numbers for?
How are positive and negative numbers written?
5. D/Z: 8.1, No. 000, 721(b), 715(b). Creative task: compose a poem about integers, a drawing, a presentation, a fairy tale.
We subtract another from the number,
We make a straight line.
We recognize this sign
"Minus" we call him.
1.
Worth a unit
Looks like a match.
She's just a dash
With a little bang.
2.
Barely glides on water
Like a swan, number two.
Arched neck,
Chasing the waves.
3.
Two hooks, look
Got the number three.
But these two hooks
Don't plant a worm.
4.
Somehow the fork was dropped
One tooth was broken off.
This fork in the whole world
It's called "four".
5.
Number five - with a big belly,
He 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 curled up?
You try to eat it
This cherry is the number six.
7.
I am such a poker
I can't put it in the oven.
Everyone knows about her
That it's called "seven".
8.
The rope twisted, twisted,
Weaved into two loops.
"What's the number?" - Let's ask mom.
Mom will answer us: "Eight."
9.
The wind blew strong and blew,
Flip the cherry.
Number six, pray tell
Turned into the number nine.
10.
Like an older sister
Zero one leads.
We just walked together
Immediately the number ten became.
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, |
· A lot of mathematics does not remain in memory, but when you understand it, then it is easy to recall forgotten things on occasion.
Integers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. Here are the numbers:
This is a natural series of numbers.
Zero is a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite set of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It cannot be specified, because there is an infinite set of natural numbers.
The sum of natural numbers is a natural number. So, the addition of natural numbers a and b:
The product of natural numbers is a natural number. So, the product of natural numbers a and b:
c is always a natural number.
Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers There is not always a natural number. If for natural numbers a and b
where c is a natural number, it means that a is evenly divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.
The divisor of a natural number is the natural number by which the first number is evenly divisible.
Every natural number is divisible by 1 and itself.
Simple natural numbers are only divisible by 1 and themselves. Here we mean divided completely. Example, numbers 2; 3; 5; 7 is only divisible by 1 and itself. These are simple natural numbers.
One is not considered a prime number.
Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:
One is not considered a composite number.
The set of natural numbers consists of one, prime numbers and composite numbers.
The set of natural numbers is denoted Latin letter N.
Properties of addition and multiplication of natural numbers:
commutative property of addition
associative property of addition
(a + b) + c = a + (b + c);
commutative property of multiplication
associative property of multiplication
(ab)c = a(bc);
distributive property multiplication
A (b + c) = ab + ac;
Whole numbers
Integers are natural numbers, zero and the opposite of natural numbers.
Numbers opposite to natural numbers are negative integers, for example:
1; -2; -3; -4;...
The set of integers is denoted by the Latin letter Z.
Rational numbers
Rational numbers are integers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
It can be seen from the examples that any integer is a periodic fraction with a period of zero.
Any rational number can be represented as a fraction m/n, where m is an integer number, n natural number. Let's represent the number 3,(6) from the previous example as such a fraction.
Number is an abstraction used for quantitative characteristics objects. Numbers arose in primitive society in connection with the need for people to count objects. Over time, with the development of science, the number has become the most important mathematical concept.
For problem solving and proof various theorems you need to understand what types of numbers are. The main types of numbers include: natural numbers, integers, rational numbers, real numbers.
Integers- these are the numbers obtained with the natural counting of objects, or rather, with their numbering ("first", "second", "third" ...). The set of natural numbers is denoted by the Latin letter N (can be remembered based on English word natural). It can be said that N ={1,2,3,....}
Whole numbers are numbers from the set (0, 1, -1, 2, -2, ....). This set consists of three parts - natural numbers, negative integers (opposite natural numbers) and the number 0 (zero). Integers are denoted by a Latin letter Z . It can be said that Z ={1,2,3,....}.
Rational numbers are numbers that can be represented as a fraction, where m is an integer and n is a natural number. The Latin letter is used to denote rational numbers Q . All natural and integer numbers are rational. Also, as examples of rational numbers, you can give: ,,.
Real (real) numbers are the numbers used to measure continuous quantities. The set of real numbers is denoted by the Latin letter R. Real numbers include rational numbers and irrational numbers. Irrational numbers are numbers that are obtained as a result of performing various operations on rational numbers (for example, extracting a root, calculating logarithms), but are not rational at the same time. Examples of irrational numbers are ,,.
Any real number can be displayed on the number line:
For the sets of numbers listed above, the following statement is true:
That is, the set of natural numbers is included in the set of integers. The set of integers is included in the set of rational numbers. And the set of rational numbers is included in the set of real numbers. This statement can be illustrated using Euler circles.
Important notes!
1. If instead of formulas you see abracadabra, clear your cache. How to do it in your browser is written here:
2. Before you start reading the article, pay attention to our navigator for the most useful resource For
To MUCH simplify your life when you need to calculate something, to win precious time at the OGE or the USE, to make fewer stupid mistakes - read this section!
Here's what you'll learn:
- how to calculate faster, easier and more accurately usinggrouping of numberswhen adding and subtracting,
- how to quickly multiply and divide without errors using multiplication rules and divisibility criteria,
- how to significantly speed up calculations using least common multiple(NOC) and greatest common divisor(GCD).
Possession of the techniques of this section can tip the scales in one direction or another ... whether you enter the university of your dreams or not, you or your parents will have to pay a lot of money for education or you will enter the budget.
Let's dive right in... (Let's go!)
P.S. LAST VALUABLE ADVICE...
A bunch of integers consists of 3 parts:
- integers(we will consider them in more detail below);
- numbers opposite to natural numbers(everything will fall into place as soon as you know what natural numbers are);
- zero - " " (where without it?)
letter Z.
Integers
“God created natural numbers, everything else is the work of human hands” (c) German mathematician Kronecker.
The natural numbers are the numbers that we use to count objects and it is on this that their history of occurrence is based - the need to count arrows, skins, etc.
1, 2, 3, 4...n
letter N.
Accordingly, this definition does not include (can’t you count what is not there?) and, moreover, do not include negative values(is there an apple?).
In addition, all fractional numbers are not included (we also cannot say "I have a laptop", or "I sold cars")
Any natural number can be written using 10 digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
So 14 is not a number. This is a number. What numbers does it consist of? That's right, from numbers and.
Addition. Grouping when adding for faster counting and fewer mistakes
What interesting things can you say about this procedure? Of course, you will now answer "the value of the sum does not change from the rearrangement of the terms." It would seem that a primitive rule familiar from the first class, however, when solving large examples, it instantly forgotten!
Don't forget about himuse groupingto facilitate the counting process and reduce the likelihood of errors, because on USE Calculator you won't have.
See for yourself which expression is easier to add?
- 4 + 5 + 3 + 6
- 4 + 6 + 5 + 3
Of course the second! Although the result is the same. But! Considering the second way, you are less likely to make a mistake and you will do everything faster!
So, in your mind, you think like this:
4 + 5 + 3 + 6 = 4 + 6 + 5 + 3 = 10 + 5 + 3 = 18
Subtraction. Grouping when subtracting for faster counting and less error
When subtracting, we can also group subtracted numbers, for example:
32 - 5 - 2 - 6 = (32 - 2) - 5 - 6 = 30 - 5 - 6 = 19
What if subtraction is interleaved with addition in the example? You can also group, you will answer, and rightly so. Just please, do not forget about the signs in front of the numbers, for example: 32 - 5 - 2 - 6 = (32 - 2) - (6 + 5) = 30 - 11 = 19
Remember: incorrectly affixed signs will lead to an erroneous result.
Multiplication. How to multiply in your mind
It is obvious that the value of the product will also not change from changing the places of the factors:
2 ⋅ 4 ⋅ 6 ⋅ 5 = (2 ⋅ 5 ) ⋅ (4 ⋅ 6 ) = 1 0 ⋅ 2 4 = 2 4 0
I won’t tell you to “use this when solving problems” (you got the hint yourself, right?), but rather tell you how to quickly multiply some numbers in your head. So, carefully look at the table:
And a little more about multiplication. Of course, you remember two special occasions… Guess what I mean? Here's about it:
Oh yeah, let's take a look signs of divisibility. In total, there are 7 rules for the signs of divisibility, of which you already know the first 3 for sure!
But the rest is not at all difficult to remember.
7 signs of divisibility of numbers that will help you quickly count in your head!
- You, of course, know the first three rules.
- The fourth and fifth are easy to remember - when dividing by and we look to see if the sum of the digits that make up the number is divisible by this.
- When dividing by, we pay attention to the last two digits of the number - is the number they make up divisible by?
- When dividing by a number, it must be divisible by and by at the same time. That's all wisdom.
Are you now thinking - "why do I need all this"?
First, the exam is without calculator and these rules will help you navigate the examples.
And secondly, you heard the tasks about GCD And NOC? Familiar abbreviation? Let's begin to remember and understand.
Greatest common divisor (gcd) - needed for reducing fractions and fast calculations
Let's say you have two numbers: and. What largest number are both numbers divisible? You will answer without hesitation, because you know that:
12 = 4 * 3 = 2 * 2 * 3
8 = 4 * 2 = 2 * 2 * 2
What numbers in the expansion are common? That's right, 2 * 2 = 4. That was your answer. Keeping this simple example in mind, you will not forget the algorithm for finding GCD. Try to "build" it in your head. Happened?
To find the NOD you need:
- Split the numbers into prime factors(into numbers that cannot be divided by anything other than itself or by, for example, 3, 7, 11, 13, etc.).
- Multiply them.
Do you understand why we needed signs of divisibility? So that you look at the number and you can start dividing without a remainder.
For example, let's find the GCD of numbers 290 and 485
First number - .
Looking at it, you can immediately tell what it is divisible by, let's write:
you can’t divide it into anything else, but you can - and, we get:
290 = 29 * 5 * 2
Let's take another number - 485.
According to the signs of divisibility, it must be divisible by without a remainder, since it ends with. We share:
Let's analyze the original number.
- It cannot be divided by (the last digit is odd),
- - is not divisible by, so the number is also not divisible by,
- is also not divisible by and (the sum of the digits in the number is not divisible by and by)
- is also not divisible, because it is not divisible by and,
- is also not divisible by and, since it is not divisible by and.
- cannot be completely divided
So the number can only be decomposed into and.
And now let's find GCD these numbers (and). What is this number? Right, .
Shall we practice?
Task number 1. Find GCD of numbers 6240 and 6800
1) I divide immediately by, since both numbers are 100% divisible by:
Task number 2. Find GCD of numbers 345 and 324
I can't find one here common divisor, so I just factor it into prime factors (as few as possible):
Least common multiple (LCM) - saves time, helps to solve problems outside the box
Let's say you have two numbers - and. What is the smallest number that is divisible by without a trace(i.e. completely)? Hard to imagine? Here's a visual clue for you:
Do you remember what the letter means? That's right, just whole numbers. So what smallest number fit x? :
In this case.
From this a simple example several rules follow.
Rules for quickly finding the NOC
Rule 1. If one of two natural numbers is divisible by another number, then the larger of these two numbers is their least common multiple.
Find the following numbers:
- NOC (7;21)
- NOC (6;12)
- NOC (5;15)
- NOC (3;33)
Of course, you easily coped with this task and you got the answers -, and.
Note that in the rule we are talking about TWO numbers, if there are more numbers, then the rule does not work.
For example, LCM (7;14;21) is not equal to 21, since it cannot be divided without a remainder by.
Rule 2. If two (or more than two) numbers are coprime, then the least common multiple is equal to their product.
find NOC for the following numbers:
- NOC (1;3;7)
- NOC (3;7;11)
- NOC (2;3;7)
- NOC (3;5;2)
Did you count? Here are the answers - , ; .
As you understand, it is not always so easy to take and pick up this same x, so for slightly more complex numbers there is the following algorithm:
Shall we practice?
Find the least common multiple - LCM (345; 234)
Find the least common multiple (LCM) yourself
What answers did you get?
Here's what happened to me:
How long did it take you to find NOC? My time is 2 minutes, I really know one trick, which I suggest you open right now!
If you are very attentive, then you probably noticed that for the given numbers we have already searched for GCD and you could take the factorization of these numbers from that example, thereby simplifying your task, but this is far from all.
Look at the picture, maybe some other thoughts will come to you:
Well? I'll give you a hint: try to multiply NOC And GCD among themselves and write down all the factors that will be when multiplying. Did you manage? You should end up with a chain like this:
Take a closer look at it: compare the factors with how and are decomposed.
What conclusion can you draw from this? Right! If we multiply the values NOC And GCD between themselves, then we get the product of these numbers.
Accordingly, having numbers and meaning GCD(or NOC), we can find NOC(or GCD) in the following way:
1. Find the product of numbers:
2. We divide the resulting product by our GCD (6240; 6800) = 80:
That's all.
Let's write the rule in general form:
Try to find GCD if it is known that:
Did you manage? .
Negative numbers - "false numbers" and their recognition by mankind.
As you already understood, these are numbers opposite to natural ones, that is:
Negative numbers can be added, subtracted, multiplied and divided - just like natural numbers. It would seem that they are so special? But the fact is that negative numbers “won” their rightful place in mathematics right up to the 19th century (until that moment there was a huge amount of controversy whether they exist or not).
The negative number itself arose because of such an operation with natural numbers as "subtraction". Indeed, subtract from - that's a negative number. That is why the set of negative numbers is often called "an extension of the set natural numbers».
Negative numbers were not recognized by people for a long time. So, Ancient Egypt, Babylon and Ancient Greece- the lights of their time, did not recognize negative numbers, and in the case of obtaining negative roots in the equation (for example, as we have), the roots were rejected as impossible.
For the first time negative numbers got their right to exist in China, and then in the 7th century in India. What do you think about this confession? That's right, negative numbers began to denote debts (otherwise - shortages). It was believed that negative numbers are a temporary value, which as a result will change to positive (that is, the money will still be returned to the creditor). However, the Indian mathematician Brahmagupta already then considered negative numbers on an equal footing with positive ones.
In Europe, the usefulness of negative numbers, as well as the fact that they can denote debt, came much later, that is, a millennium. The first mention was seen in 1202 in the "Book of the Abacus" by Leonard of Pisa (I say right away that the author of the book has nothing to do with the Leaning Tower of Pisa, but the Fibonacci numbers are his work (the nickname of Leonardo of Pisa is Fibonacci)). Further, the Europeans came to the conclusion that negative numbers can mean not only debts, but also a lack of something, however, not everyone recognized this.
So, in the XVII century, Pascal believed that. How do you think he justified it? That's right, "nothing can be less than NOTHING". An echo of those times remains the fact that a negative number and the operation of subtraction are denoted by the same symbol - minus "-". And true: . Is the number " " positive, which is subtracted from, or negative, which is added to? ... Something from the series "which comes first: the chicken or the egg?" Here is such a kind of this mathematical philosophy.
Negative numbers secured their right to exist with the advent of analytic geometry, in other words, when mathematicians introduced such a thing as a real axis. 
It was from this moment that equality came. However, there were still more questions than answers, for example:
proportion
This proportion is called the Arno paradox. Think about it, what is doubtful about it?
Let's talk together " " more than " " right? Thus, according to logic, the left side of the proportion should be greater than the right side, but they are equal ... Here it is the paradox.
As a result, mathematicians agreed that Karl Gauss (yes, yes, this is the one who considered the sum (or) of numbers) in 1831 put an end to it - he said that negative numbers have the same rights as positive ones, and the fact that they do not apply to all things means nothing, since fractions do not apply to many things either (it does not happen that a digger digs a hole, you cannot buy a movie ticket, etc.).
Mathematicians calmed down only in the 19th century, when the theory of negative numbers was created by William Hamilton and Hermann Grassmann.
That's how controversial they are, these negative numbers.
Emergence of "emptiness", or the biography of zero.
In mathematics, a special number. At first glance, this is nothing: add, subtract - nothing will change, but you just have to attribute it to the right to "", and the resulting number will be many times greater than the original one. By multiplying by zero, we turn everything into nothing, but we cannot divide by "nothing". In a word, the magic number)
The history of zero is long and complicated. A trace of zero is found in the writings of the Chinese in 2000 AD. and even earlier with the Maya. The first use of the zero symbol, as it is today, was seen among the Greek astronomers.
There are many versions of why such a designation "nothing" was chosen. Some historians are inclined to believe that this is an omicron, i.e. The first letter of the Greek word for nothing is ouden. According to another version, the word “obol” (a coin of almost no value) gave life to the symbol of zero.
Zero (or null) as mathematical symbol first appears among the Indians (note that negative numbers began to “develop” there). The first reliable evidence of writing zero dates back to 876, and in them "" is a component of the number.
Zero also came to Europe belatedly - only in 1600, and just like negative numbers, it faced resistance (what can you do, they are Europeans).
“Zero has often been hated, feared, or even banned from time immemorial,” writes the American mathematician Charles Seif. So, the Turkish Sultan Abdul-Hamid II at the end of the 19th century. ordered his censors to delete the H2O water formula from all chemistry textbooks, taking the letter "O" for zero and not wanting his initials to be defamed by the proximity to the despicable zero.
On the Internet you can find the phrase: “Zero is the most powerful force in the Universe, it can do anything! Zero creates order in mathematics, and it also brings chaos into it. Absolutely correct point :)
Summary of the section and basic formulas
The set of integers consists of 3 parts:
- natural numbers (we will consider them in more detail below);
- numbers opposite to natural ones;
- zero - " "
The set of integers is denoted letter Z.
1. Natural numbers
Natural numbers are the numbers that we use to count objects.
The set of natural numbers is denoted letter N.
In operations with integers, you will need the ability to find GCD and LCM.
Greatest Common Divisor (GCD)
To find the NOD you need:
- Decompose numbers into prime factors (into numbers that cannot be divided by anything other than itself or by, for example, etc.).
- Write down the factors that are part of both numbers.
- Multiply them.
Least common multiple (LCM)
To find the NOC you need:
- Factorize numbers into prime factors (you already know how to do this very well).
- Write out the factors included in the expansion of one of the numbers (it is better to take the longest chain).
- Add to them the missing factors from the expansions of the remaining numbers.
- Find the product of the resulting factors.
2. Negative numbers
These are numbers that are opposite to natural numbers, that is:
Now I want to hear from you...
I hope you appreciated the super-useful "tricks" of this section and understood how they will help you in the exam.
And more importantly, in life. I'm not talking about it, but believe me, this one is. The ability to count quickly and without errors saves in many life situations.
Now it's your turn!
Write, will you use grouping methods, divisibility criteria, GCD and LCM in calculations?
Maybe you have used them before? Where and how?
Perhaps you have questions. Or suggestions.
Write in the comments how you like the article.
And good luck with your exams!
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough ...
For what?
For successful passing the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will just say one thing ...
People who received a good education, earn much more than those who did not receive it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
On the exam, you will not be asked theory.
You will need solve problems on time.
And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.
It's like in sports - you need to repeat many times to win for sure.
Find a collection anywhere you want necessarily with solutions detailed analysis and decide, decide, decide!
You can use our tasks (not necessary) and we certainly recommend them.
In order to get a hand with the help of our tasks, you need to help extend the life of the YouClever textbook that you are currently reading.
How? There are two options:
- Unlock access to all hidden tasks in this article -
- Unlock access to all hidden tasks in all 99 articles of the tutorial - Buy a textbook - 499 rubles
Yes, we have 99 such articles in the textbook and access to all tasks and all hidden texts in them can be opened immediately.
Access to all hidden tasks is provided for the entire lifetime of the site.
In conclusion...
If you don't like our tasks, find others. Just don't stop with theory.
“Understood” and “I know how to solve” are completely different skills. You need both.
Find problems and solve!
TO whole numbers include natural numbers, zero, and numbers opposite to natural numbers.
Integers are positive integers.
For example: 1, 3, 7, 19, 23, etc. We use such numbers for counting (there are 5 apples on the table, the car has 4 wheels, etc.)
Latin letter \mathbb(N) - denoted set of natural numbers.
Natural numbers cannot include negative (a chair cannot have a negative number of legs) and fractional numbers (Ivan could not sell 3.5 bicycles).
Numbers opposite to natural numbers are negative integers: -8, -148, -981, ....
Arithmetic operations with integers
What can you do with integers? They can be multiplied, added and subtracted from each other. Let's analyze each operation on a specific example.
Integer addition
Two integers with the same signs are added as follows: the modules of these numbers are added and the resulting sum is preceded by the final sign:
(+11) + (+9) = +20
Subtraction of integers
Two integers with different signs are added as follows: from the module more the modulus of the smaller one is subtracted and the sign of the larger modulo number is put in front of the received answer:
(-7) + (+8) = +1
Integer multiplication
To multiply one integer by another, you need to multiply the modules of these numbers and put the “+” sign in front of the received answer if the original numbers were with the same signs, and the “-” sign if the original numbers were with different signs:
(-5) \cdot (+3) = -15
(-3) \cdot (-4) = +12
You should remember the following whole number multiplication rule:
+ \cdot + = +
+\cdot-=-
- \cdot += -
-\cdot-=+
There is a rule for multiplying several integers. Let's remember it:
The sign of the product will be "+", if the number of factors with negative sign even and "−" if the number of factors with a negative sign is odd.
(-5) \cdot (-4) \cdot (+1) \cdot (+6) \cdot (+1) = +120
Division of integers
The division of two integers is carried out as follows: the modulus of one number is divided by the modulus of the other, and if the signs of the numbers are the same, then the “+” sign is placed in front of the resulting quotient, and if the signs of the original numbers are different, then the “−” sign is put.
(-25) : (+5) = -5
Properties of addition and multiplication of integers
Let's analyze the basic properties of addition and multiplication for any integers a , b and c :
- a + b = b + a - commutative property of addition;
- (a + b) + c \u003d a + (b + c) - the associative property of addition;
- a \cdot b = b \cdot a - commutative property of multiplication;
- (a \cdot c) \cdot b = a \cdot (b \cdot c)- associative properties of multiplication;
- a \cdot (b \cdot c) = a \cdot b + a \cdot c is the distributive property of multiplication.
