Numbers. Whole numbers. Properties of integers. Greatest common multiple and least common divisor. Divisibility criteria and grouping methods (2019)

§ 77. On the fractions of a unit.

We studied the properties of integers and actions on them. In addition to integers, there are fractional numbers, which we will now familiarize ourselves with. When a student says that it takes him half an hour to walk from home to school, he expresses time not in whole hours, but in parts of an hour. When the doctor advises the patient to dissolve the powder in a quarter of a glass of hot water, then the water is measured not in whole glasses, but in parts of a glass. If one watermelon is to be divided equally among three boys, then each of them can only get a third of the watermelon, or a third of it.

In all cases, we were not talking about whole units, but about parts, or fractions of a unit. Shares can be very diverse, for example, a gram is a thousandth of a kilogram, a millimeter is a millionth of a kilometer. First we will talk about the most simple shares (half, third, quarter, etc.).

For greater clarity, we will depict these shares as straight line segments.

If we take the segment AB as a unit (Fig. 9), then, dividing it into two equal parts, we can say that the resulting segments AC and CB will be halves of the segment AB.

Further, if we take the segment DE (Fig. 10) as a unit and divide it into 3 equal parts, then each of the obtained segments DF, FH, HE will be equal to one third of the segment DE, and the segment DH will be equal to two thirds of the segment DE. Similarly, segment FE will be equal to two-thirds of segment DE.

Let's take another segment MN (Fig. 11), take it as a unit and divide it into four equal parts; then each of the segments MP, PQ, QR, RN will be equal to one quarter of the segment MN; each of the segments MQ, PR, QN will be equal to two quarters of it, and each of the segments MR and PN will be equal to three quarters of MN.

In the examples considered, we got acquainted with a half, a third, a quarter, two thirds, two quarters, three quarters, that is, either with one share of a unit, or with two, or with three equal parts of a unit.

A number made up of one or more equal parts of one is called shot.

We have already said that instead of the word "share" you can say the word "part"; therefore, a fraction can be called a number expressing one or more identical parts of a unit.

Thus, the numbers mentioned in this paragraph: half, or one second, one third, one quarter, two thirds and others, will be fractions.

Often it is necessary to consider not only parts of objects, but together with them whole objects. For example, two boys decide to share equally their five apples. Obviously, each of them will first take two apples, and they will cut the remaining last apple into two equal parts. Then each will have two and a half apples. Here the number of apples for each boy is expressed as a whole number (two) with some fraction (half).

Numbers that include a whole number and a fraction are called mixed numbers.

§ 78. Image of fractions.

Consider the last drawing of the previous paragraph (Fig. 11). We said that segment MR is three quarters of segment MN. Now the question arises how this fraction, i.e. three quarters, can be written using numbers. Recall how the fraction three-quarters arose. We took the segment MN as a unit, divided it into 4 equal parts and took 3 from these parts. It is this process of the emergence of a fraction that should be reflected in its record, i.e. from this record it should be seen that the unit is divided into 4 equal parts and the resulting parts are taken 3. Because of this, the fraction is depicted using two numbers separated by a horizontal line. A number is written under the line, indicating how many equal parts the unit is divided into, from which the fraction is taken, and another number is written above the line, showing how many shares are contained

in this fraction. A fraction of three quarters will be written like this: 3 / 4.

The number above the line is called numerator fractions; this number indicates the number of parts contained in the given fraction.

The number below the line is called denominator fractions; it shows how many equal parts the unit is divided.

3 - numerator,
_
4 is the denominator.

The dash that separates the numerator from the denominator is called the fractional bar. The numerator and denominator are both collectively called terms of a fraction. Let's write a fraction as an example:

two thirds - 2/3; five twelfths - 5/12.

Mixed numbers are written as follows: first they write an integer and next to it a fraction is attributed to the right. For example, a mixed number of two and four fifths should be written like this: 2 4 / 5.

§ 79. The emergence of fractions.

Consider the question of how and where fractions arise, why and under what circumstances they appear.

Take, for example, this fact. You need to measure the length of the blackboard with a meter. We take a meter long wooden ruler and apply it along the bottom edge of the board, moving from left to right. Let it fit in twice, but there is still some part of the board where the ruler will not fit in the third time, because the length of the remaining part is less than the length of the ruler.

If the rest of the board contains, for example, half a meter, then the length of the board is two and a half (2 1/2) meters.

We will now measure the width of the board with the same ruler. Let's say that she did it once, but after this single delay, a small part of the board remained, less than a meter long. Applying a meter to this part of the board, let's say, it was possible to find that it is equal to one quarter (1/4) of a meter.

So the entire width of the board is 1 1/4 m.

Thus, when measuring the length and width of the board, we got the numbers 2 1/2 m and 1 1/4 m (ie, fractional numbers).

Not only the length and width of objects, but also many other quantities are often expressed in fractional numbers.

We measure time not only in hours, minutes and seconds, but often in parts of an hour, in parts of a minute and even in parts of a second.

Very often, fractional numbers express weight, for example, they say: 1/2 kg, l 1/2 kg, 1/2 g, 3/4 g, 1/2 t, etc.

So far, we have been talking about the origin of fractions from measurement, but there is another source of fractions - this is the action of division. Let's stop there. Let 3 apples be required to be divided among 4 boys; obviously, in this case, each boy will not get a whole apple, because there are fewer apples than children. First, take 2 apples and cut each in half. It will turn out 4 halves, and since there are four boys, each can be given half an apple. We will cut the remaining third apple into 4 parts and then add each boy to what he has, another quarter. Then all the apples will be distributed and each boy will receive one half and one quarter of an apple. But since each half contains 2 quarters, it can finally be said that each boy will have two quarters and plus one quarter each, that is, a total of three quarters (3/4) of an apple.

§ 80. Comparison of fractions in size.

If we compare any quantities with each other, for example, two segments, then it may turn out that one of them is exactly equal to the other, or it is greater than the other, or less than the other.

In Figure 12, segment AB is equal to segment CD; segment EF is greater than segment QH; segment KL is less than segment MN.

We will meet the same three cases when comparing fractions. Let's try to compare some fractions with each other.

1. Two fractions are considered equal if the quantities corresponding to these fractions are equal to each other (with the same unit of measurement). Let's take the segment SC and take it as a unit.

We divide the segment SK in half by point D (Fig. 13). Then we will denote the part of this segment CD by the fraction 1 / 2 . If we divide the same segment SK into 4 equal parts, then the segment CD will be expressed as a fraction 2 / 4; if we divide the segment SK into 8 equal parts, then the segment CD will correspond to the fraction 4/8. Since we took the same segment three times, the fractions 1/2, 2/4 and 4/8 are equal to each other.

2. Let's take two fractions with equal numerators: 1/4 and 1/8, and see what values ​​correspond to them. In the first case, some value is divided into 4 equal parts, and in the second case, it is also divided into 8 equal parts.

Figure 14 shows that 1/4 is greater than 1/8. Therefore, of two fractions with the same numerator, the larger fraction is the one with the smaller denominator.

3. Take two fractions with equal denominators: 5/8 and 3/8. If we mark each of these fractions in the previous drawing, we will see that the segment corresponding to the first fraction is larger than the segment corresponding to the second. So, of two fractions with the same denominator, the larger fraction is the one with the larger numerator.

4. If two fractions are given with different numerators and denominators, then their value can be judged by comparing each of them with one. For example, 2 / 3 is less than 4 / 5, because the first fraction differs from unity by 1 / 3, and the second by 1 / 5, i.e. the second fraction is less short of unity than the first.

However, it is easiest to compare such fractions by reducing them to a common denominator, which will be discussed below.

§ 81. Fractions are regular and improper. Mixed numbers.

Let's take the segment AB equal to two linear units (Fig. 15). We divide each unit into 10 equal parts, then each part will be equal to 1 / 10, i.e.

AD = DE = EF = FH = ... = 1/10 AC.

Consider other segments and think about what fractions they are expressed in. For example, AF - 3/10, AK - 5/10, AM - 7/10; AO - 9 / 10 , AS - 10 / 10 , AR - 11 / 10 , AR - 13 / 10 . We expressed all the segments taken as fractional numbers with a denominator of 10. The first four fractions (3/10, 5/10, 7/10; 9/10) have numerators less than denominators, each of them is less than 1.

The fifth fraction (10 / 10) has the numerator equal to the denominator, and the fraction itself is equal to 1, it corresponds to the segment AC, taken as a unit.

The last two fractions (11/10, 13/10) have numerators greater than denominators, and each fraction is greater than 1.

A fraction whose numerator is less than the denominator is called a proper fraction. As stated above, a proper fraction is less than one. This means that the first four fractions are correct and therefore we can write: 3 / 10<1, 5 / 10 <1, 7 / 10 <1, 9 / 10 <1.

A fraction whose numerator is equal to or greater than the denominator is called an improper fraction. Thus, an improper fraction is either equal to one or greater than it. So the last three fractions are improper and you can write:

10 / 10 =1 ; 11 / 10 >1 ; 13 / 10 >1 ;

Let's focus on the last two (improper) fractions. The fraction 11/10 consists of one whole unit and the correct fraction 1/10, which means that it can be written like this: 1 1/10. The result was a number that is a combination of an integer and a proper fraction, that is, a mixed number. The same can be repeated for the improper fraction 13/10. We can represent it as 1 3/10. This will also be a mixed number.

You need to learn how to replace an improper fraction with a mixed number. We easily replaced the previous two improper fractions with mixed numbers. But if we met a fraction, for example 545/32, then it is more difficult to extract the integer part from it, and without extracting the integer part it is difficult to judge the value of this number.

On the other hand, when performing various calculations, it is sometimes more convenient to use not mixed numbers, but improper fractions. This means that, if necessary, you need to be able to do the inverse transformation, that is, replace the mixed number with an improper fraction.

§ 82. Conversion of an improper fraction to a mixed number and inverse transformation.

Let's take an improper fraction 9/4 and try to replace it with a mixed number. We will argue as follows: if 4 quarters are contained in one unit, then as many whole units are contained in 9 quarters as many times 4 quarters are contained in 9 quarters. To answer this question, it is enough to divide 9 by 4. The resulting quotient will indicate the number of integers, and the remainder will give the number of quarters that do not constitute a whole unit. 4 is contained in 9 twice with a remainder of 1. So 9 / 4 = 2 1 / 4, since 9: 4 = 2 and 1 in the remainder.

Let's turn the improper fraction 545/32 mentioned above into a mixed number.

545; 32 \u003d 17 and 1 in the remainder, so 545 / 32 \u003d 17 1 / 32.

To convert an improper fraction to a mixed number, you need to divide the numerator of the fraction by the denominator and find the remainder; the quotient will show the number of whole units, and the remainder will show the number of fractions of a unit.

Since, by converting an improper fraction into a mixed number, we each time select an integer part, this transformation is usually called the elimination of an integer from an improper fraction.

Consider the case when an improper fraction is equal to an integer. Let it be required to exclude an integer from an incorrect

fractions 36/12 According to the rule, we get 36: 12 = 3 and 0 in the remainder, i.e. the numerator is divided by the denominator without a remainder, which means 36/12 = 3.

Let us now turn to the inverse transformation, i.e., to the conversion of a mixed number into an improper fraction.

Let's take the mixed number 3 3/4 and turn it into an improper fraction. Let's reason like this: each whole unit contains 4 quarters, and 3 units will contain 3 times more fourths, i.e. 4 x 3 \u003d 12 fourths. This means that 3 whole units contain 12 quarters, and even in the fractional part of the mixed number there are 3 quarters, and there will be 15 quarters in total, or 15 / 4. Therefore, 3 3 / 4 = 15 / 4 .

Example. Convert the mixed number 8 4 / 9 to an improper fraction:

To turn a mixed number into an improper fraction, you need to multiply the denominator by an integer, add the numerator to the resulting product and make this sum the numerator of the desired fraction, and leave the denominator the same.

§ 83. Converting an integer to an improper fraction.

Any whole number can be expressed in any number of fractions of one. This is sometimes useful in calculations. Let, for example, the number 5 be expressed in sixths of a unit.

We will argue as follows: since there are six sixths in one unit, then in 5 units of these shares there will be not six, but 5 times more, i.e. 6 x 5 \u003d 30 sixths. The action is arranged like this:

In the same way, we can turn any whole number into an improper fraction with any denominator. Let's take the number 10 and represent it as an improper fraction with different denominators:

denominator 2, then

denominator 3, then

denominator 5, then

Thus, in order to express an integer as an improper fraction with a given denominator, you need to multiply this denominator by a given number, make the resulting product a numerator and sign this denominator.

The smallest possible denominator is one (1). Therefore, when they want to represent an integer as a fraction, they often take one as the denominator (l2 = 12 / 1). This thought is sometimes expressed as follows: any whole number can be considered as a fraction with a denominator equal to one (2 = 2 / 1; 3 = 3 / 1; 4 = 4 / 1; 5 = 5 / 1, etc.)

§ 84. Change in the value of a fraction with a change in its members.

In this section, we will consider how the value of a fraction will change when its members change.

1st question. What happens to the value of a fraction as its numerator increases several times? Let's take the fraction 1/12 and we will gradually increase its numerator by two, three, four, etc. times. Then you get the following fractions:

If we begin to compare these fractions with each other, we will see that they gradually increase: the second fraction is twice as large as the first, because it has twice as many parts, the third fraction is three times as large as the first, etc.

From this we can conclude: If the numerator of a fraction is increased several times, then the fraction will increase by the same amount.

2nd question. What happens to the value of a fraction when decreasing its numerator several times? Let's take the fraction 24/25 and we will gradually decrease its numerator by two times, three times, four times, etc. Then we get the following fractions:

Look at these fractions one by one from left to right and you will see that the second fraction (12 / 25) is half the first 24 / 25, because it has half the parts, that is, half the numerator; the fourth fraction 6/25 is four times less than the first and half the second.

Means, If the numerator of a fraction is reduced several times, then the fraction will decrease by the same amount.

3rd question. What happens to the value of a fraction when increasing its denominator several times? We can answer this question by taking some fraction, for example 1 / 2, and increasing its denominator without changing the numerator. Let's double the denominator, triple it, etc. and see what happens to the fraction:

Gradually increasing the denominator, we finally brought it to 100. The denominator became quite large, but the value of the share greatly decreased, it became equal to one hundredth. From this it is clear that an increase in the denominator of a fraction will inevitably lead to a decrease in the fraction itself.

Means, If the denominator of a fraction is increased several times, then the fraction will decrease by the same amount.

4th question. What happens to the value of a fraction when its denominator is multiplied? We will take those fractions that were recently written and rewrite them from the end; then our first fraction will be the smallest, and the last the largest, but the first will have the largest denominator, and the last fraction will have the smallest denominator:

It is easy to conclude: If the denominator of a fraction is reduced by a factor of 1, then the fraction will increase by the same factor.

5th question. What happens to a fraction when both the numerator and denominator increase or decrease by the same amount?

Let's take the fraction 1/2 and we will sequentially and simultaneously increase its numerator and denominator. A factor is sometimes put next to the fraction, by which the terms of the first fraction are multiplied:

We wrote six fractions, they are different in their appearance, but it is easy to figure out that they are all equal in size. In fact, let's compare at least the first fraction with the second. The first fraction is 1/2; if we double its numerator, then the fraction will double, but if we immediately double its denominator, then it will decrease by half, that is, in other words, it will remain unchanged. So 1/2 = 2/4. The same reasoning can be repeated for other fractions.

Conclusion: if the numerator and denominator of a fraction are multiplied by the same number(increase the same number of times), the value of the fraction will not change.

We write this property in a general form. Let's denote the fraction by a / b , the number by which the numerator and denominator are multiplied - by the letter t ; then the specified property will take the form of equality:

It remains to consider the question of simultaneously reducing the numerator and denominator by the same number of times. Let's write several fractions in a row, where in the first place there will be a fraction 36/48, and in the last 3/4:

All of them will be equal to each other, which can be found by comparing any two adjacent fractions, for example, halving the numerator of the first fraction (36), we reduce the fraction by 2 times, but halving its denominator (48), we increase the fraction by 2 times, i.e. as a result, we leave it unchanged.

Conclusion: if the numerator and denominator of a fraction are divided by the same number (reduced by the same number of times), then the value of the fraction will not change:

The essence of the last two conclusions is that with a simultaneous increase or decrease in the numerator and denominator by the same number of times, the value of the fraction will not change.

This remarkable property of a fraction will be of great importance in what follows, so we will call it basic property of a fraction.

§ 85. Reduction of fractions.

Let's take the segment AB (Fig. 16) and divide it into 20 equal parts, then each of these parts will be equal to 1/20; The segment AC, which contains 15 such parts, will be represented by a fraction 15 / 20.

Now let's try to enlarge the shares, for example, we divide the segment not into 20 parts, but into 4 equal parts. The new shares turned out to be larger than the previous ones, since each new share contains 5 former ones, which is clearly visible in the drawing. Now let's think about what the segment AC is equal to at the new crushing, which at the first crushing was equal to 15/20 of the segment AB. It can be seen from the drawing that if the segment AB is divided into 4 parts, then the segment AC will be equal to 3/4 of the segment AB.

So, segment AC, depending on how many parts the segment AB is divided into, can be represented by both a fraction 15/20 and a fraction 3/4. In magnitude, this is the same fraction, because it measures the same segment in the same units of measurement. So, instead of the fraction 15/20, we can use the fraction 3/4, and vice versa.

The question arises, which fraction is more convenient to use? It is more convenient to use the second fraction, because its numerator and denominator are expressed in smaller numbers than the first, and in this sense it is simpler.

In the process of reasoning, it turned out that one value (segment AC) was expressed in two fractions, different in appearance, but the same in value (15/20, 3/4). Obviously, there can be not two such fractions, but an uncountable set. Based on the basic property of a fraction, we can bring the first of these fractions to such a form that the numerator and denominator will be the smallest. In fact, if the numerator and denominator of the fraction 15/20 are divided by 5, then it will be equal to 3/4, i.e. 15/20 = 3/4.

This transformation (simultaneous reduction of the numerator and denominator by the same number of times), which allows you to get a fraction with a large numerator and denominator from a fraction with a large numerator and denominator, but equal in size with smaller members, is called the reduction of fractions.

Therefore, the reduction of a fraction is the replacement of it with another fraction equal to it with smaller terms, by dividing the numerator and denominator by the same number.

We reduced the fraction 15 / 20 and came to the fraction 3 / 4, which can no longer be reduced, because its terms 3 and 4 do not have a common divisor (except for one). Such a fraction is called irreducible. There are two paths you can take when reducing fractions. The first way is that the fraction is reduced gradually, and not immediately, i.e. after the first reduction, a reducible fraction is obtained again, which is then reduced again, and this process can be lengthy if the numerator and denominator are expressed in large numbers and have many common dividers.

Let's take the fraction 60/120 and reduce it sequentially, first by 2, we get 60/120 = 30/60 The new fraction (30/60) can also be reduced by 2, we get 30/60 = 15/30. The terms of the new fraction 15/30 have common divisors, so you can reduce this fraction by 3, you get 15/30 = 5/10. Finally, the last fraction can be reduced by 5, i.e. 5/10 = 1/2. This is the successive reduction of fractions.

It is easy to figure out that this fraction (60 / 120) could be reduced immediately by 60, and we would get the same result. What is 60 for the numbers 60 and 120? Greatest common divisor. This means that reducing a fraction by the greatest common divisor of its members makes it possible to immediately bring it to the form of an irreducible fraction, bypassing intermediate divisions. This is the second way to reduce fractions.

§ 86. Reduction of fractions to the smallest common denominator.

Let's take some fractions:

If we begin to compare the first fraction with the second (1/2 and 1/3), we will feel some difficulty. Of course, we understand that half is more than one third, since in the first case the value is divided into two equal parts, and in the second case into three equal parts; but what is the difference between them, it is still difficult to answer. Another thing is the second fraction and the third (1/3 and 2/3), it is easy to compare them, since it is immediately clear that the second fraction is less than the third by one third. It is easy to understand that in those cases when we compare fractions with the same denominators, there are no difficulties, in the same cases when the denominators of the compared fractions are different, some inconvenience arises. Verify this by comparing the rest of the fraction data.

Therefore, the question arises: is it possible, when comparing two fractions, to ensure that the denominators are the same? This can be done based on the basic property of a fraction, i.e. if we increase the denominator several times, then in order for the value of the fraction not to change, its numerator must be increased by the same amount.

This way we can reduce fractions with different denominators to a common denominator.

If you want to reduce some fractions to a common denominator, then you first need to find a number that would be divisible by the denominator of each of these fractions. Therefore, the first step in the process of reducing fractions to a common denominator is finding the least common multiple for given denominators. After the least common multiple has been found, it is necessary, by dividing it by each denominator, to obtain for each fraction the so-called additional multiplier. These will be numbers indicating how many times the numerator and denominator of each fraction must be increased so that their denominators become equal. Consider examples.

1. Let's reduce the fractions 7/30 and 8/15 to a common denominator. Find the least common multiple for the denominators 30 and 15. In this case, this will be the denominator of the first fraction, i.e. 30. This will be the lowest common denominator for the fractions 7/30 and 8/15. Now let's find additional factors: 30: 30 = 1, 30: 15 = 2. So, for the first fraction, the additional factor will be 1, and for the second, 2. The first fraction will remain unchanged. Multiplying the terms of the second fraction by an additional factor, we bring it to the denominator 30:

2. Let's bring three fractions to a common denominator: 7/30, 11/60 and 3/70.

Let's find for the denominators 30, 60 and 70 the least common multiple:

The least common multiple will be 2 2 3 5 7 = 420.

This will be the least common denominator of these fractions.

Now let's find additional factors: 420: 30 = 14; 420: 60 = 7; 420: 70 = 6. So, for the first fraction, the additional factor will be 14, for the second 7 and for the third 6. Multiplying the terms of the fractions by the corresponding additional factors, we get fractions with equal denominators:

3. Let's reduce the fraction to a common denominator: 8/25 and 5/12. The denominators of these fractions (25 and 12) are coprime numbers. Therefore, the least common multiple will be obtained from their multiplication: 25 x 12 \u003d 300. An additional factor for the first fraction will be 12, and for the second 25. These fractions will take the form:

To reduce fractions to the least common denominator, you must first find the least common multiple of all denominators and determine an additional factor for each denominator, and then multiply both terms of each fraction by the corresponding additional factor.

After we have learned how to reduce fractions to a common denominator, comparing fractions in size will no longer present any difficulties. We can now compare the value of any two fractions, bringing them first to a common denominator.

There are many types of numbers, one of them is integers. Integers appeared in order to make it easier to count not only in a positive direction, but also in a negative one.

Consider an example:
During the day it was 3 degrees outside. By evening the temperature dropped by 3 degrees.
3-3=0
It was 0 degrees outside. And at night the temperature dropped by 4 degrees and began to show on the thermometer -4 degrees.
0-4=-4

A series of integers.

We cannot describe such a problem with natural numbers; we will consider this problem on a coordinate line.

We have a series of numbers:
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …

This series of numbers is called next to whole numbers.

Integer positive numbers. Whole negative numbers.

A series of integers consists of positive and negative numbers. To the right of zero are natural numbers, or they are also called whole positive numbers. And to the left of zero go whole negative numbers.

Zero is neither positive nor negative. It is the boundary between positive and negative numbers.

is a set of numbers consisting of natural numbers, negative integers and zero.

A series of integers in positive and negative directions is endless multitude.

If we take any two integers, then the numbers between these integers will be called end set.

For example:
Let's take integers from -2 to 4. All numbers between these numbers are included in the finite set. Our finite set of numbers looks like this:
-2, -1, 0, 1, 2, 3, 4.

Natural numbers are denoted by the Latin letter N.
Integers are denoted by the Latin letter Z. The whole set of natural numbers and integers can be depicted in the figure.


Nonpositive integers in other words, they are negative integers.
Non-negative integers are positive integers.

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 a “+” 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 :

1. a + b = b + a - commutative property of addition;
2. (a + b) + c \u003d a + (b + c) - the associative property of addition;
3. a \cdot b = b \cdot a - commutative property of multiplication;
4. (a \cdot c) \cdot b = a \cdot (b \cdot c)- associative properties of multiplication;
5. a \cdot (b \cdot c) = a \cdot b + a \cdot c is the distributive property of multiplication.

First level

Greatest common multiple and least common divisor. Divisibility criteria and grouping methods (2019)

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).

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Let's dive right in... (Let's go!)

Important note!If instead of formulas you see gibberish, clear your cache. To do this, press CTRL+F5 (on Windows) or Cmd+R (on Mac)

Lots of integers consists of 3 parts:

1. integers(we will consider them in more detail below);
2. numbers opposite to natural numbers(everything will fall into place as soon as you know what natural numbers are);
3. 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 even more so does 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 grouping, in order to facilitate the process of counting and reduce the likelihood of errors, because you will not have a calculator for the exam.

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 is the largest number divisible by both of these numbers? 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:

1. Decompose numbers into prime factors (into numbers that cannot be divided by anything other than itself or by, for example, 3, 7, 11, 13, etc.).
2. 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? Correctly, .

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:

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 digits (in this case, we present as, but can be divided by)):

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 simple 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, but I did not initially check the divisibility criterion for, and, perhaps, I would not have to do so many actions. But you checked, right? Well done! 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)? It's difficult 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.

Several rules follow from this simple example.

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? Correctly! 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 it was great amount disputes 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 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 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:

1. Decompose numbers into prime factors (into numbers that cannot be divided by anything other than itself or by, for example, etc.).
2. Write down the factors that are part of both numbers.
3. Multiply them.

Least common multiple (LCM)

To find the NOC you need:

1. Factorize numbers into prime factors (you already know how to do this very well).
2. Write out the factors included in the expansion of one of the numbers (it is better to take the longest chain).
3. Add to them the missing factors from the expansions of the remaining numbers.
4. 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!

The content of the article

The concept of a number in mathematics can refer to objects of a different nature: natural numbers used in counting (positive integers 1, 2, 3, etc.), numbers that are possible results of (idealized) measurements (these are numbers such as 2/ 3, - they are called real numbers), negative numbers, imaginary numbers (say, k), and other more abstract classes of numbers used in higher sections of mathematics (for example, hypercomplex and transfinite numbers). A number must be distinguished from its symbol, or the notation that represents it. We will consider the logical relationships between different classes of numbers.

Such riddles are easily solved, if we consider that different classes of numbers have quite different meanings; although they have enough in common that they can all be called numbers, it should not be thought that they will all satisfy the same rules.

positive integers.

Although we all learn positive integers (1, 2, 3, etc.) in early childhood, when it hardly occurs to think about definitions, nevertheless such numbers can be defined by all the rules of formal logic. A strict definition of the number 1 would take more than a dozen pages, and a formula like 1 + 1 = 2, if written in full detail without any abbreviations, would stretch for several kilometers. However, any mathematical theory is forced to begin with some undefined concepts and axioms or postulates about them. Since the positive integers are well known and it is difficult to define them using something simpler, we will take them as the original undefined concepts and assume that the basic properties of these numbers are known.

Negative integers and zero.

Negative numbers are common these days: they are used, for example, to represent temperatures below zero. Therefore, it seems surprising that a few centuries ago there was no specific interpretation of negative numbers, and negative numbers that appeared in the course of calculations were called "imaginary". Although the intuitive interpretation of negative numbers is useful in itself, when trying to understand "rules" such as (-4)ґ(-3) = +12, we must define negative numbers in terms of positive numbers. To do this, we need to build a set of such mathematical objects that will behave in arithmetic and algebra exactly as one would expect from negative numbers. One way to construct such a set is to consider ordered pairs of positive numbers ( a,b). "Ordered" means that, for example, the pair (2,3) is different from the pair (3,2). Such ordered pairs can be considered as a new class of numbers. Now we must say when two such new numbers are equal and what their addition and multiplication means. Our choice of definitions is driven by the desire that the pair ( a,b) acted as the difference ( a – b), which has so far been defined only when a more b. Since in algebra ( a-b) + (c-d) = (a+c) – (b+d), we come to the need to define the addition of new numbers as ( a,b) + (c,d) = (a+c, b+d); because ( a – b)ґ(c – d) = ac + bd – (bc + ad), we define multiplication by the equality ( a,b)ґ(c,d) = (ac+bd, bc + ad); and since ( a-b) = (c-d), if a + d = b + c, we define the equality of new numbers by the relation ( a,b) = (c,d), if a + d = b + c. In this way,

Using the definitions of equality of pairs, we can write the sum and product of pairs in a simpler form:

All couples ( a,a) are equal (by the definition of equality of pairs) and act as we expect zero to act. For example, (2.3) + (1.1) = (3.4) = (2.3); (2.3)ґ(1.1) = (2 + 3, 2 + 3) = (5.5) = (1.1). Couples ( a,a) we can symbolize 0 (which has not yet been used).

Couples ( a,b), where b more a, behave as negative numbers should, and we can denote the pair ( a,b) symbol –( b – a). For example, -4 is (1.5) and -3 is (1.4); (–4)ґ(–3) = (21.9), or (13.1). We would like to denote the last number as 12, but this is certainly not the same as the positive integer 12, since it denotes a pair of positive integers, not a single positive integer. It must be emphasized that since the pairs ( a,b), where b less a, act as positive integers ( a – b), we will write numbers like ( a – b). At the same time, we must forget about the positive integers with which we started, and henceforth use only our new numbers, which we will call whole numbers. The fact that we intend to use the old names for some of the new numbers should not be misleading that the new numbers are actually objects of a different kind.

Fractions.

Intuitively, we think of the fraction 2/3 as the result of breaking 1 into three equal parts and taking two of them. However, the mathematician strives to rely as little as possible on intuition and to define rational numbers in terms of simpler objects - integers. This can be done by treating 2/3 as an ordered pair of (2,3) integers. To complete the definition, it is necessary to formulate the rules for the equality of fractions, as well as addition and multiplication. Of course, these rules must be equivalent to the rules of arithmetic and, of course, different from the rules for those ordered pairs that we have defined as integers. Here are the rules:

It is easy to see that the pairs ( a,1) act as integers a; Continuing to reason in the same way as in the case of negative numbers, we denote by 2 the fraction (2.1), or (4.2), or any other fraction equal to (2.1). Let us now forget about whole numbers and keep them only as a means of writing certain fractions.

Rational and irrational numbers.

Fractions are also called rational numbers, since they can be represented in the form relations(from lat. ratio ratio) of two integers. But if we need a number whose square is 2, then we cannot get by with rational numbers, because there is no rational number whose square is equal to 2. The same becomes clear if we ask about the number expressing the ratio of the circumference of a circle to its diameter. Therefore, if we want to get the square roots of all positive numbers, then we need to extend the class of rational numbers. New numbers, called irrational (i.e. not rational), can be defined in various ways. Ordered pairs are no good for this; one of the simplest ways is to define irrational numbers as infinite non-recurring decimals.

Real numbers.

Rational and irrational numbers together are called real or real numbers. Geometrically, they can be represented by points on a straight line, with fractions in between integers, and irrational numbers in between fractions, as shown in Fig. 1. It can be shown that the system of real numbers has a property known as "completeness" which means that every point on the line corresponds to some real number.

Complex numbers.

Since the squares of positive and negative real numbers are positive, there is no point on the line of real numbers that corresponds to a number whose square is -1. But if we tried to solve quadratic equations like x 2 + 1 = 0, then it would be necessary to act as if there were some number i, whose square would be -1. But since there is no such number, we have no choice but to use an "imaginary" or "imaginary" number. Accordingly, "number" i and its combinations with ordinary numbers (like 2 + 3 i) became known as imaginary. Modern mathematicians prefer to call such numbers "complex" because, as we shall see, they are just as "real" as those we have encountered before. For a long time, mathematicians freely used imaginary numbers and got useful results, although they did not fully understand what they were doing. Until the beginning of the 19th century it never occurred to anyone to "revive" the imaginary numbers with the help of their explicit definition. To do this, you need to build some set of mathematical objects that, from the point of view of algebra, would behave like expressions a+bi, if we agree that i 2 = -1. Such objects can be defined as follows. Consider as our new numbers ordered pairs of real numbers, the addition and multiplication of which is determined by the formulas:

We call such ordered pairs complex numbers. Pairs of private form ( a,0) with the second term equal to zero behave like real numbers, so we will agree to denote them in the same way: for example, 2 means (2,0). On the other hand, the complex number (0, b) by the definition of multiplication has the property (0, b)ґ(0,b) = (0 – b 2 , 0 + 0) = (–b 2 ,0) = –b 2. For example, in the case of (0.1)ґ(0.1) we find the product (-1.0); therefore, (0.1) 2 = (–1.0). We have already agreed to write the complex number (-1.0) as -1, so if the number (0.1) is denoted by the symbol i, then we get a complex number i, such that i 2 = -1. In addition, the complex number (2,3) can now be written as 2 + 3 i.

An important difference between this approach to complex numbers and the traditional one is that in this case the number i does not contain anything mysterious or imaginary: it is something well defined by means of numbers that already existed before, although, of course, it does not coincide with any of them. Similarly, the real number 2 is not complex, although we use the symbol 2 to represent a complex number. Since there is actually nothing “imaginary” about imaginary numbers, it is not surprising that they are widely used in real situations, for example in electrical engineering (where instead of a letter i usually use the letter j, as in electrical engineering i- symbol for the current value of the current).

The algebra of complex numbers in many ways resembles the algebra of real numbers, although there are significant differences. For example, the rule for complex numbers does not hold: , therefore , while .

The addition of complex numbers allows for a simple geometric interpretation. For example, the sum of numbers 2 + 3 i and 3 - i there is a number 5 + 2 i, which corresponds to the fourth vertex of the parallelogram with three vertices at points 0, 2 + 3 i and 3 - i.

A point on a plane can be specified not only by rectangular (Cartesian) coordinates ( x,y), but also by its polar coordinates ( r,q) specifying the distance from the point to the origin and the angle. Therefore, the complex number x+iy can also be written in polar coordinates (Fig. 2, b). Length of the radius vector r equal to the distance from the origin to the point corresponding to the complex number; magnitude r is called the modulus of a complex number and is determined by the formula . Often a module is written as . Corner q is called the "angle", "argument", or "phase" of a complex number. Such a number has infinitely many angles that differ by a multiple of 360°; for example, i has an angle of 90°, 450°, -270°, ј Since the Cartesian and polar coordinates of the same point are related by the relations x = r cos q, y = r sin q, the equality x + iy = r(cos q + i sin q).

If a z = x + iy, then the number x-iy is called the complex conjugate of z and denoted n z = re iq . Logarithm of a complex number re iq, by definition, is equal to ln r + iq, where ln means base logarithm e, a q takes on all possible values ​​measured in radians. Thus, a complex number has infinitely many logarithms. For example, ln (–2) = ln 2 + ip+ any integer multiple of 2 p. In general, the degrees can now be defined using the relation a b = e b ln a. For example, i –2i = e–2ln i. Since the values ​​of the number argument i equal p/2 (90° expressed in radians) plus an integer multiple, then the number i –2i matter ep, e 3 p, e -p etc., which are all valid.

hypercomplex numbers.

Complex numbers were invented to be able to solve all quadratic equations with real coefficients. It can be shown that, in fact, complex numbers allow one to do much more: with their introduction, algebraic equations of any degree become solvable, even with complex coefficients. Consequently, if we were only interested in solving algebraic equations, then the need to introduce new numbers would disappear. However, for other purposes, numbers are needed that are arranged somewhat similarly to complex ones, but with more components. Sometimes such numbers are called hypercomplex. Examples are quaternions and matrices.