In this article we will define the set of integers, consider which integers are called positive and which are negative. We will also show how integers are used to describe changes in certain quantities. Let's start with the definition and examples of integers.
Whole numbers. Definition, examples
First, let's remember about natural numbers ℕ. The name itself suggests that these are numbers that have naturally been used for counting since time immemorial. In order to cover the concept of integers, we need to expand the definition of natural numbers.
Definition 1. Integers
Integers are the natural numbers, their opposites, and the number zero.
The set of integers is denoted by the letter ℤ.
The set of natural numbers ℕ is a subset of the integers ℤ. Every natural number is an integer, but not every integer is a natural number.
From the definition it follows that any of the numbers 1, 2, 3 is an integer. . , the number 0, as well as the numbers - 1, - 2, - 3, . .
In accordance with this, we will give examples. The numbers 39, - 589, 10000000, - 1596, 0 are integers.
Let the coordinate line be drawn horizontally and directed to the right. Let's take a look at it to visualize the location of integers on a line.
The origin on the coordinate line corresponds to the number 0, and points lying on both sides of zero correspond to positive and negative integers. Each point corresponds to a single integer.
You can get to any point on a line whose coordinate is an integer by setting aside a certain number of unit segments from the origin.
Positive and negative integers
Of all the integers, it is logical to distinguish positive and negative integers. Let us give their definitions.
Definition 2: Positive integers
Positive integers are integers with a plus sign.
For example, the number 7 is an integer with a plus sign, that is, a positive integer. On the coordinate line, this number lies to the right of the reference point, which is taken to be the number 0. Other examples of positive integers: 12, 502, 42, 33, 100500.
Definition 3: Negative integers
Negative integers are integers with a minus sign.
Examples of negative integers: - 528, - 2568, - 1.
The number 0 separates positive and negative integers and is itself neither positive nor negative.
Any number that is the opposite of a positive integer is, by definition, a negative integer. The opposite is also true. The inverse of any negative integer is a positive integer.
It is possible to give other formulations of the definitions of negative and positive integers using their comparison with zero.
Definition 4: Positive integers
Positive integers are integers that are greater than zero.
Definition 5: Negative integers
Negative integers are integers that are less than zero.
Accordingly, positive numbers lie to the right of the origin on the coordinate line, and negative integers lie to the left of zero.
We said earlier that natural numbers are a subset of integers. Let's clarify this point. The set of natural numbers consists of positive integers. In turn, the set of negative integers is the set of numbers opposite to the natural ones.
Important!
Any natural number can be called an integer, but any integer cannot be called a natural number. Answering the question whether they are negative numbers natural, we must boldly say - no, they are not.
Non-positive and non-negative integers
Let's give some definitions.
Definition 6. Non-negative integers
Non-negative integers are positive integers and the number zero.
Definition 7. Non-positive integers
Non-positive integers are negative integers and the number zero.
As you can see, the number zero is neither positive nor negative.
Examples of non-negative integers: 52, 128, 0.
Examples of non-positive integers: - 52, - 128, 0.
A non-negative number is a number greater than or equal to zero. Accordingly, a non-positive integer is a number less than or equal to zero.
The terms "non-positive number" and "non-negative number" are used for brevity. For example, instead of saying that the number a is an integer that is greater than or equal to zero, you can say: a is a non-negative integer.
Using integers to describe changes in quantities
What are integers used for? First of all, with their help it is convenient to describe and determine changes in the quantity of any objects. Let's give an example.
Let a certain number of crankshafts be stored in a warehouse. If 500 more crankshafts are brought to the warehouse, their number will increase. The number 500 precisely expresses the change (increase) in the number of parts. If 200 parts are then taken from the warehouse, then this number will also characterize the change in the number of crankshafts. This time, downward.
If nothing is taken from the warehouse and nothing is delivered, then the number 0 will indicate that the number of parts remains unchanged.
The obvious convenience of using integers, as opposed to natural numbers, is that their sign clearly indicates the direction of change in the value (increase or decrease).
A decrease in temperature by 30 degrees can be characterized by a negative integer - 30, and an increase by 2 degrees - by a positive integer 2.
Let's give another example using integers. This time, let's imagine that we have to give 5 coins to someone. Then, we can say that we have - 5 coins. The number 5 describes the size of the debt, and the minus sign indicates that we must give away the coins.
If we owe 2 coins to one person and 3 to another, then the total debt (5 coins) can be calculated using the rule of adding negative numbers:
2 + (- 3) = - 5
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Natural numbers are the numbers with which it all began. And today these are the first numbers that a person encounters in his life, when in childhood he learns to count on his fingers or counting sticks.Definition: Natural numbers are numbers that are used to count objects (1, 2, 3, 4, 5, ...) [The number 0 is not natural. It has its own separate history in the history of mathematics and appeared much later than natural numbers.]
The set of all natural numbers (1, 2, 3, 4, 5, ...) is denoted by the letter N.
Integers
Having learned to count, the next thing we do is learn to perform arithmetic operations on numbers. Usually, addition and subtraction are taught first (using counting sticks).With addition, everything is clear: adding any two natural numbers, the result will always be the same natural number. But in subtraction we discover that we cannot subtract the larger from the smaller so that the result is a natural number. (3 − 5 = what?) This is where the idea of negative numbers comes into play. (Negative numbers are no longer natural numbers)
At the stage of occurrence of negative numbers (and they appeared later than fractional ones) there were also their opponents, who considered them nonsense. (Three objects can be shown on your fingers, ten can be shown, a thousand objects can be represented by analogy. And what is “minus three bags”? - At that time, numbers were already used on their own, in isolation from specific objects, the number of which they denote were still in the minds of people much closer to these specific subjects than today.) But, like the objections, the main argument in favor of negative numbers came from practice: negative numbers made it possible to conveniently count debts. 3 − 5 = −2 - I had 3 coins, I spent 5. This means that I not only ran out of coins, but I also owed someone 2 coins. If I return one, the debt will change −2+1=−1, but can also be represented by a negative number.
As a result, negative numbers appeared in mathematics, and now we have an infinite number of natural numbers (1, 2, 3, 4, ...) and there is the same number of their opposites (−1, −2, −3, −4 , ...). Let's add another 0 to them. And we will call the set of all these numbers integers.
Definition: The natural numbers, their opposites, and zero make up the set of integers. It is designated by the letter Z.
Any two integers can be subtracted from each other or added to form a whole number.
The idea of adding integers already suggests the possibility of multiplication as simply a faster way of doing addition. If we have 7 bags of 6 kilograms each, we can add 6+6+6+6+6+6+6 (add 6 to the current total seven times), or we can simply remember that such an operation will always result in 42. Just like adding six sevens, 7+7+7+7+7+7 will also always give 42.
Results of the addition operation certain numbers with yourself certain the number of times for all pairs of numbers from 2 to 9 are written out and a multiplication table is made up. To multiply integers greater than 9, the column multiplication rule is invented. (Which also applies to decimal fractions, and which will be discussed in one of the following articles.) When multiplying any two integers by each other, the result will always be an integer.
Rational numbers
Now division. Just as subtraction is the inverse operation of addition, we come to the idea of division as the inverse operation of multiplication.When we had 7 bags of 6 kilograms each, using multiplication we easily calculated that total weight the contents of the bags are 42 kilograms. Let's imagine that we poured the entire contents of all the bags into one common pile weighing 42 kilograms. And then they changed their minds and wanted to distribute the contents back into 7 bags. How many kilograms will end up in one bag if we distribute it equally? – Obviously, 6.
What if we want to distribute 42 kilograms into 6 bags? Here we will think that the same total 42 kilograms could be obtained if we poured 6 bags of 7 kilograms into a pile. And this means that when dividing 42 kilograms into 6 bags equally, we get 7 kilograms in one bag.
What if you divide 42 kilograms equally into 3 bags? And here, too, we begin to select a number that, when multiplied by 3, would give 42. For “tabular” values, as in the case of 6 · 7 = 42 => 42: 6 = 7, we perform the division operation simply by recalling the multiplication table. For more complex cases, column division is used, which will be discussed in one of the following articles. In the case of 3 and 42, you can “select” to remember that 3 · 14 = 42. This means 42:3 = 14. Each bag will contain 14 kilograms.
Now let's try to divide 42 kilograms equally into 5 bags. 42:5=?
We notice that 5 · 8 = 40 (few), and 5 · 9 = 45 (many). That is, we will not get 42 kilograms from 5 bags, neither 8 kilograms in a bag, nor 9 kilograms. At the same time, it is clear that in reality nothing prevents us from dividing any quantity (cereals, for example) into 5 equal parts.
The operation of dividing integers by each other does not necessarily result in an integer. This is how we came to the concept of fractions. 42:5 = 42/5 = 8 whole 2/5 (if counted in fractions) or 42:5 = 8.4 (if counted in decimals).
Common and decimal fractions
We can say that any ordinary fraction m/n (m is any integer, n is any natural number) is simply a special form of writing the result of dividing the number m by the number n. (m is called the numerator of the fraction, n is the denominator) The result of dividing, for example, the number 25 by the number 5 can also be written as an ordinary fraction 25/5. But this is not necessary, since the result of dividing 25 by 5 can simply be written as the integer 5. (And 25/5 = 5). But the result of dividing the number 25 by the number 3 can no longer be represented as an integer, so here the need arises to use a fraction, 25:3 = 25/3. (You can distinguish the whole part 25/3 = 8 whole 1/3. Ordinary fractions and operations with ordinary fractions will be discussed in more detail in the following articles.)The good thing about ordinary fractions is that in order to represent the result of dividing any two integers as such a fraction, you simply need to write the dividend in the numerator of the fraction and the divisor in the denominator. (123:11=123/11, 67:89=67/89, 127:53=127/53, ...) Then, if possible, reduce the fraction and/or highlight the whole part (these actions with ordinary fractions will be discussed in detail in the following articles ). The problem is that performing arithmetic operations (addition, subtraction) with ordinary fractions is no longer as convenient as with whole numbers.
For the convenience of writing (in one line) and for the convenience of calculations (with the possibility of calculations in a column, as for ordinary integers), in addition to ordinary fractions, decimal fractions were also invented. A decimal fraction is a specially written ordinary fraction with a denominator of 10, 100, 1000, etc. For example, the common fraction 7/10 is the same as the decimal fraction 0.7. (8/100 = 0.08; 2 whole 3/10 = 2.3; 7 whole 1/1000 = 7, 001). A separate article will be devoted to converting ordinary fractions to decimals and vice versa. Operations with decimals– other articles.
Any integer can be represented as a common fraction with a denominator of 1. (5=5/1; −765=−765/1).
Definition: All numbers that can be represented as a fraction are called rational numbers. The set of rational numbers is denoted by the letter Q.
When dividing any two integers by each other (except when dividing by 0), the result will always be a rational number. For ordinary fractions, there are rules for addition, subtraction, multiplication and division that allow you to perform the corresponding operation with any two fractions and also obtain a rational number (fraction or integer) as a result.
The set of rational numbers is the first of the sets we have considered in which you can add, subtract, multiply, and divide (except for division by 0), never going beyond the boundaries of this set (that is, always getting a rational number as a result) .
It would seem that there are no other numbers; all numbers are rational. But this is not true either.
Real numbers
There are numbers that cannot be represented as a fraction m/n (where m is an integer, n is a natural number).What are these numbers? We have not yet considered the operation of exponentiation. For example, 4 2 =4 ·4 = 16. 5 3 =5 ·5 ·5=125. Just as multiplication is a more convenient form of writing and calculating addition, so exponentiation is a form of writing the multiplication of the same number by itself a certain number of times.
But now let’s look at the inverse operation of exponentiation—root extraction. The square root of 16 is a number that when squared gives 16, that is, the number 4. The square root of 9 is 3. But the square root of 5 or 2, for example, cannot be represented rational number. (The proof of this statement, other examples of irrational numbers and their history can be found, for example, on Wikipedia)
In the GIA in grade 9 there is a task to determine whether a number containing a root in its notation is rational or irrational. The task is to try to convert this number to a form that does not contain a root (using the properties of roots). If you can’t get rid of the root, then the number is irrational.
Another example of an irrational number is the number π, familiar to everyone from geometry and trigonometry.
Definition: Rational and irrational numbers together are called real (or real) numbers. The set of all real numbers is denoted by the letter R.
In real numbers, as opposed to rational numbers, we can express the distance between any two points on a line or plane.
If you draw a straight line and select two arbitrary points on it or select two arbitrary points on a plane, it may turn out that the exact distance between these points cannot be expressed as a rational number. (Example - the hypotenuse of a right triangle with legs 1 and 1, according to the Pythagorean theorem, will be equal to the root of two - that is, an irrational number. This also includes the exact length of the diagonal of a tetrad cell (the length of the diagonal of any ideal square with integral sides).)
And in the set of real numbers, any distances on a line, in a plane or in space can be expressed by the corresponding real number.
1) I divide by immediately, since both numbers are 100% divisible by:
2) I will divide by the remaining large numbers (and), since they are divisible by without a remainder (at the same time, I will not expand - 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’ll leave and alone and start looking at the numbers and. Both numbers are exactly divisible by (end with even digits (in this case, we imagine how, or you can divide by)):
4) We work with numbers and. Do they have common divisors? It’s not as easy as in the previous steps, so we’ll simply decompose them into simple factors:
5) As we see, we were right: and have no common divisors, and now we need to multiply.
GCD
Task No. 2. Find the gcd of numbers 345 and 324
I can’t quickly find at least one common divisor here, so I just break it down into prime factors (as small as possible):
Exactly, gcd, but I initially did not check the test of divisibility by, and perhaps I would not have had to do so many actions.
But you checked, right?
As you can see, it's not difficult at all.
Least common multiple (LCM) - saves time, helps solve problems in a non-standard way
Let's say you have two numbers - and. What is the smallest number that can be divided by without a trace(that is, completely)? Is it hard to imagine? Here's a visual hint for you:
Do you remember what the letter stands for? That's right, just whole numbers. So what smallest number fits in place x? :
In this case.
Several rules emerge from this simple example.
Rules for quickly finding NOCs
Rule 1: If one of two natural numbers is divisible by another number, then the larger of the 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 coped with this task without difficulty and you got the answers - , and.
Please 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 is not divisible by.
Rule 2. If two (or more than two) numbers are coprime, then the least common multiple is equal to their product.
Find NOC 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’s not always possible to pick up this same x so easily, so for slightly more complex numbers there is the following algorithm:
Shall we practice?
Let's find the least common multiple - LCM (345; 234)
Let's break down each number:
Why did I write right away?
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 down the longest decomposition on 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 because 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 I got:
How much time did you spend finding 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 we have already searched for the given numbers GCD and you could take the factorization of these numbers from that example, thereby simplifying your task, but that’s not all.
Look at the picture, maybe some other thoughts will come to you:
Well? I'll give you a hint: try multiplying NOC And GCD among themselves and write down all the factors that will appear when multiplying. Did you manage? You should end up with a chain like this:
Take a closer look at it: compare the multipliers with how and are laid out.
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) according to this scheme:
1. Find the product of numbers:
2. Divide the resulting product by ours GCD (6240; 6800) = 80:
That's it.
Let's write the rule in general form:
Try to find GCD, if it is known that:
Did you manage? .
Negative numbers are “false numbers” and their recognition by humanity.
As you already understand, these are numbers opposite to natural ones, that is:
It would seem, what is so special about them?
But the fact is that negative numbers “won” their rightful place in mathematics right up to the 19th century (until that moment it was huge amount disputes whether they exist or not).
The negative number itself arose due to such an operation with natural numbers as “subtraction”.
Indeed, subtract from it and you get a negative number. That is why the set of negative numbers is often called "an expansion 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 luminaries of their time, did not recognize negative numbers, and in the case of obtaining negative roots in an equation (for example, like ours), the roots were rejected as impossible.
Negative numbers first gained their right to exist in China, and then in the 7th century in India.
What do you think is the reason for this recognition?
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 considered negative numbers on an equal basis with positive ones.
In Europe, the usefulness of negative numbers, as well as the fact that they can denote debts, was discovered much later, perhaps a millennium.
The first mention was noticed in 1202 in the “Book of Abacus” by Leonard of Pisa (I’ll 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 17th century, Pascal believed that.
How do you think he justified this?
It’s true, “nothing can be less than NOTHING.”
An echo of those times remains the fact that a negative number and the subtraction operation are denoted by the same symbol - the minus “-”. And the truth: . Is the number “ ” positive, which is subtracted from, or negative, which is summed to?... Something from the series “what comes first: the chicken or the egg?” This is such a peculiar mathematical philosophy.
Negative numbers secured their right to exist with the advent of analytical geometry, in other words, when mathematicians introduced such a concept as the number axis. 
It was from this moment that equality came. However, there were still more questions than answers, for example:
proportion
This proportion is called “Arnaud’s paradox”. Think about it, what's dubious about it?
Let's argue together "" is more than "" right? Thus, according to logic, the left side of the proportion should be greater than the right, but they are equal... This is the paradox.
As a result, mathematicians agreed to the point that Karl Gauss (yes, yes, this is the same one who calculated the sum (or) numbers) put an end to it in 1831.
He said that negative numbers have the same rights as positive numbers, and the fact that they do not apply to all things does not mean anything, since fractions also do not apply to many things (it does not happen that a digger digs a hole, you can’t 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.
They are so controversial, these negative numbers.
The emergence of “emptiness”, or the biography of zero.
In mathematics it is a special number.
At first glance, this is nothing: add or subtract - nothing will change, but you just have to add it to the right to “ ”, and the resulting number will be several times larger than the original one.
By multiplying by zero we turn everything into nothing, but dividing by “nothing”, that is, we cannot. In a word, the magic number)
The history of zero is long and complicated.
A trace of zero was found in the writings of the Chinese in the 2nd millennium AD. and even earlier among the Mayans. The first use of the zero symbol, as it is today, was seen among Greek astronomers.
There are many versions of why this designation “nothing” was chosen.
Some historians are inclined to believe that this is an omicron, i.e. first letter Greek word nothing - ouden. According to another version, the word “obol” (a coin with almost no value) gave life to the symbol of zero.
Zero (or null) as a mathematical symbol first appears among Indians(note that negative numbers began to “develop” there).
The first reliable evidence of the recording of zero dates back to 876, and in them “ ” is a component of the number.
Zero also came to Europe late - only in 1600, and just like negative numbers, it encountered resistance (what can you do, that's how Europeans are).
“Zero has often been hated, long feared, or even banned.”- writes American mathematician Charles Safe.
Thus, the Turkish Sultan Abdul Hamid II at the end of the 19th century. ordered his censors to erase the formula of water H2O from all chemistry textbooks, taking the letter “O” for zero and not wanting his initials to be discredited by the proximity of the despised zero.”
On the Internet you can find the phrase: “Zero is the most powerful force in the Universe, he can do anything! Zero creates order in mathematics, and it also introduces chaos into it.” Absolutely correct point:)
Summary of the section and basic formulas
The set of integers consists of 3 parts:
- natural numbers (we'll look at them in more detail below);
- numbers opposite to natural numbers;
- zero - " "
The set of integers is denoted letter Z.
1. Natural numbers
Natural numbers are 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 a GCD you need to:
- Decompose numbers into prime factors (those numbers that cannot be divided by anything else except themselves 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:
- Divide numbers into prime factors (you already know how to do this very well).
- Write down 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 opposite to natural ones, that is:
Now I want to hear you...
I hope you appreciated the super-useful “tricks” in this section and understood how they will help you in the exam.
And more importantly - in life. I don’t talk about it, but believe me, this one is true. The ability to count quickly and without errors saves you in many life situations.
Now it's your turn!
Write, will you use grouping methods, divisibility tests, 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 on your exams!
TO integers include natural numbers, zero, and numbers opposite to natural numbers.
Natural numbers are positive integers.
For example: 1, 3, 7, 19, 23, etc. We use such numbers for counting (there are 5 apples on the table, a car has 4 wheels, etc.)
Latin letter \mathbb(N) - denoted set of natural numbers.
TO natural numbers it is impossible to include negative numbers (a chair cannot have a negative number of legs) and fractional numbers (Ivan could not sell 3.5 bicycles).
The opposite of 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 look at each operation using a specific example.
Addition of integers
Two integers with the same signs are added as follows: the modules of these numbers are added and the resulting sum is preceded by a final sign:
(+11) + (+9) = +20
Subtracting Integers
Two integers with different signs are added up as follows: from the module more the modulus of the smaller one is subtracted and the sign of the larger modulo number is placed in front of the resulting answer:
(-7) + (+8) = +1
Multiplying Integers
To multiply one integer by another, you need to multiply the moduli of these numbers and put a “+” sign in front of the resulting answer if the original numbers had the same signs, and a “−” sign if the original numbers had different signs:
(-5)\cdot (+3) = -15
(-3)\cdot (-4) = +12
The following should be remembered rule for multiplying integers:
+ \cdot + = +
+ \cdot - = -
- \cdot + = -
- \cdot - = +
There is a rule for multiplying multiple 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
Integer division
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 placed.
(-25) : (+5) = -5
Properties of addition and multiplication of integers
Let's look at 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 = a + (b + c) - combinative 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- distributive property of multiplication.
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