Numbers are called whole numbers. Numbers
For the first time negative numbers began to be used in ancient China and India, in Europe they were introduced into mathematical use by Nicolas Shuquet (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 integers. a, b And c.
| addition | multiplication | |
| closure : | 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 |
| the existence of a neutral element: | a + 0 = a | a× 1 = a |
| the existence of an opposite element: | a + (−a) = 0 | a≠ ±1 ⇒ 1/ a is not whole |
| distributivity of multiplication with respect to addition: | a × ( b + c) = (a × b) + (a × c) | |
number systems |heading4= Hierarchy of numbers |list4=
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number systems
|list5=Cardinal numbers - You should definitely transfer to the bed, it will not 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 go out of sight for a moment during the entire service. Pierre guessed from the cautious movement of the people surrounding the chair that the dying man was being lifted and carried.
“Hold on to my hand, you’ll drop it like that,” he heard the frightened whisper of one of the servants, “from below ... another one,” voices said, and the heavy breathing and stepping of people’s feet became more hasty, as if the burden they were carrying was beyond their strength. .
The bearers, 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, a high, fat, open chest, the fat shoulders of the patient, raised upwards by the people holding him under the armpits, and a gray-haired curly, lion head. This head, with an unusually wide forehead and cheekbones, a beautiful sensual mouth and a majestic cold look, 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 bearers, and the cold, indifferent look did not know where to stop.
A few minutes of fuss passed by the high bed; the people carrying the sick man dispersed. Anna Mikhailovna touched Pierre's hand and said to him: "Venez." [Go.] Pierre, together with her, went up to the bed, on which, in a festive pose, apparently related to the sacrament that had just been performed, the sick man was laid. He lay with his head propped high on the pillows. His hands were symmetrically laid out on a green silk blanket, palms down. When Pierre approached, the count looked directly at him, but looked with that look, the meaning and meaning of which cannot be understood by a person. Either this glance said absolutely nothing, only that, as long as there are eyes, one must look somewhere, or it said too much. Pierre stopped, not knowing what to do, and looked inquiringly at his leader, Anna Mikhailovna. Anna Mikhailovna made a hurried gesture to him with her eyes, pointing to the patient's hand and kissing it with her lips. Pierre, diligently stretching his neck so as not to catch on the blanket, carried out her advice and kissed her broad-boned and fleshy hand. Not a hand, not a single muscle of the count's face trembled. Pierre again looked inquiringly at Anna Mikhailovna, now asking what he should do. Anna Mikhaylovna pointed out to him with her eyes a chair that stood beside the bed. Pierre obediently began to sit down on an armchair, continuing to ask with his eyes whether he had done what was needed. 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 large space, and using all his mental strength to seem 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 the touching importance of this last minute of meeting between father and son. This lasted two minutes, which seemed to Pierre an hour. Suddenly a shudder appeared in the large muscles and wrinkles of the count's face. The shudder intensified, the beautiful mouth twisted (it was only then that Pierre realized to what extent his father was close to death), an indistinct hoarse sound was heard from the twisted mouth. Anna Mikhailovna diligently looked into the patient's eyes and, trying to guess what he needed, she pointed either to Pierre, then to the drink, then in a whisper she called Prince Vasily inquiringly, then she pointed to the blanket. The patient's eyes and face showed impatience. He made an effort to look at the servant, who was standing at the head of the bed without leaving.
“They want to roll over onto the other side,” the servant whispered and rose to turn the count’s heavy body facing the wall.
Pierre got up to help the servant.
While the count was being turned over, one of his arms fell back helplessly, and he made a vain effort to drag it. Did the count notice that 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 he had a weak, suffering smile that did not suit his features, expressing, as it were, mockery at his own impotence. Suddenly, at the sight of this smile, Pierre felt a shudder in his chest, a pinching in his nose, and tears clouded his vision. The patient was turned over on his side against the wall. He sighed.
- Il est assoupi, [He dozed off,] - said Anna Mikhailovna, noticing the princess who came to replace. - Allons. [Let's go to.]
Pierre left.
1) I divide immediately by, since both numbers are 100% divisible by:
2) I will divide by the remaining large numbers (s), since they are divided by without a remainder (at the same time, I will not decompose - it is already a common divisor):
6 2 4 0 = 1 0 ⋅ 4 ⋅ 1 5 6
6 8 0 0 = 1 0 ⋅ 4 ⋅ 1 7 0
3) I will leave and alone and begin to consider the numbers and. Both numbers are exactly divisible by (end in even numbers(in this case, we present as, but can be divided into)):
4) We work with numbers and. Do they have common divisors? It’s as easy as in the previous steps, and you can’t say, so then we’ll just decompose them into prime factors:
5) As we can see, we were right: and have no common divisors, and now we need to multiply.
GCD
Task number 2. Find GCD of numbers 345 and 324
I can’t quickly find at least one common divisor here, so I just decompose into prime factors (as few as possible):
Exactly, GCD, and I did not initially check the divisibility criterion for, and, perhaps, I would not have to do so many actions.
But you checked, right?
As you can see, it's quite easy.
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 is the smallest number that fits 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)
Let's break down each number:
Why did I just write?
Remember the signs of divisibility by: divisible by (the last digit is even) and the sum of the digits is divisible by.
Accordingly, we can immediately divide by, writing it as.
Now we write out the longest expansion in a line - the second:
Let's add to it the numbers from the first expansion, which are not in what we wrote out:
Note: we wrote out everything except for, since we already have it.
Now we need to multiply all these numbers!
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:
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 of 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 - shortage).
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)).
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 does not mean anything, since fractions do not apply to many things either (it does not happen that a digger digs a hole, you cannot buy a ticket to the cinema, 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 zero) as a 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 was often hated, feared for a long time, and even banned”— 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!
Whole numbers - these are natural numbers, as well as their opposite numbers and zero.
Whole numbers— extension of the set of natural numbers N, which is obtained by adding to N 0 and negative numbers like − n. The set of integers denotes Z.
The sum, difference and product of integers again give integers, i.e. the integers form a ring with respect to the operations of addition and multiplication.
Integers on the number line:
How many integers? How many integers? There is no largest or smallest integer. This series is endless. The largest and smallest integer do not exist.
The natural numbers are also called positive whole numbers, i.e. phrase " natural number" and "positive integer" are the same.
Neither common nor decimal fractions are whole numbers. But there are fractions with whole numbers.
Integer examples: -8, 111, 0, 1285642, -20051 and so on.
In simple terms, integers are (∞... -4,-3,-2,-1,0,1,2,3,4...+ ∞) is a sequence of integers. That is, those whose fractional part (()) is equal to zero. They don't have shares.
Natural numbers are whole, positive numbers. Whole numbers, examples: (1,2,3,4...+ ∞).
Operations on integers.
1. The sum of integers.
To add two integers with the same sign, you need to add the modules of these numbers and put the final sign in front of the sum.
Example:
(+2) + (+5) = +7.
2. Subtraction of whole numbers.
To add two integers with different signs, it is necessary to subtract the modulus of a number that is less from the modulus of a number that is greater and put a sign before the answer more modulo.
Example:
(-2) + (+5) = +3.
3. Multiplication of integers.
To multiply two integers, it is necessary to multiply the modules of these numbers and put a plus sign (+) in front of the product if the original numbers were of the same sign, and minus (-) if they were different.
Example:
(+2) ∙ (-3) = -6.
When multiple numbers are multiplied, the sign of the product will be positive if the number of non-positive factors is even, and negative if it is odd.
Example:
(-2) ∙ (+3) ∙ (-5) ∙ (-3) ∙ (+4) = -360 (3 non-positive factors).
4. Division of integers.
To divide integers, it is necessary to divide the modulus of one by the modulus of the other and put a “+” sign in front of the result if the signs of the numbers are the same, and minus if they are different.
Example:
(-12) : (+6) = -2.
Properties of integers.
Z is not closed under division of 2 integers ( e.g. 1/2). The table below shows some of the basic properties of addition and multiplication for any integers. a, b And c.
|
Property |
addition |
multiplication |
|
isolation |
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 neutral element |
a + 0 = a |
a × 1 = a |
|
existence opposite element |
a + (−a) = 0 |
a ≠ ± 1 ⇒ 1/a is not whole |
|
distributivity multiplication with respect to additions |
a × ( b + c) = (a × b) + (a × c) |
|
From the table it can be concluded that Z is a commutative ring with unity under addition and multiplication.
Standard division does not exist on the set of integers, but there is a so-called division with remainder: for any integers a And b, b≠0, there is one set of integers q And r, What a = bq + r And 0≤r<|b| , Where |b| is the absolute value (module) of the number b. Here a- divisible b- divider, q- private, r- remainder.
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: the modulus of the smaller number is subtracted from the modulus of the larger number, and the sign of the larger modulo number is put in front of the 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 a negative sign is 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.
What does integer mean
So, consider what numbers are called integers.
Thus, integers will denote such numbers: $0$, $±1$, $±2$, $±3$, $±4$, etc.
The set of natural numbers is a subset of the set of integers, i.e. any natural will be an integer, but not any integer is a natural number.
Integer positive and integer negative numbers
Definition 2
plus.
The numbers $3, 78, 569, 10450$ are positive integers.
Definition 3
are signed integers minus.
Numbers $−3, −78, −569, -10450$ are negative integers.
Remark 1
The number zero does not refer to either positive integers or negative integers.
Whole positive numbers are integers greater than zero.
Whole negative numbers are integers less than zero.
The set of natural integers is the set of all positive integers, and the set of all opposites of natural numbers is the set of all negative integers.
Integer non-positive and integer non-negative numbers
All positive integers and the number zero are called integer non-negative numbers.
Integer non-positive numbers are all negative integers and the number $0$.
Remark 2
Thus, whole non-negative number are integers greater than zero or equal to zero, and non-positive integer are integers less than zero or equal to zero.
For example, non-positive integers: $-32, -123, 0, -5$, and non-negative integers: $54, 123, 0.856 342.$
Description of changing values using integers
Integers are used to describe changes in the number of any items.
Consider examples.
Example 1
Suppose a store sells a certain number of items. When the store receives $520$ of items, the number of items in the store will increase, and the number $520$ shows a positive change in the number. When the store sells $50$ items, the number of items in the store will decrease, and the number $50$ will express a negative change in the number. If the store will neither bring nor sell the 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. A positive value of the integer $520$ indicates a positive change in the number. A negative value of the integer $−50$ indicates a negative change in the number. The integer $0$ indicates the immutability of the number.
Integers are convenient to use, because no explicit indication of an increase in number or decrease is needed - the sign of the integer indicates the direction of the change, and the value indicates a quantitative change.
Using integers, you can express not only a change in quantity, but also a change in any value.
Consider an example of a change in the cost of a product.
Example 2
An increase in cost, for example, by $20$ rubles is expressed using a positive integer $20$. Decreasing the cost, for example, by $5$ rubles is described using a negative integer $−5$. If there are no cost changes, then such a change is determined using the integer $0$.
Separately, consider the value of negative integers as the size of the debt.
Example 3
For example, a person has $5,000 rubles. Then, using a positive integer $5,000$, you can show the number of rubles that he has. A person has to pay a rent in the amount of $7,000 rubles, but he does not have that kind of money; in this 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 $7,000$ shows the amount of debt.
