Negative numbers. A negative number
§ 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 integer 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 required 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 terms.
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 members 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 makes it possible to obtain from a fraction with a large numerator and denominator another in appearance, but equal in size, a fraction 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, that is, 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.
In this article, we will define a set of integers, consider which integers are called positive and which are negative. We will also show how integers are used to describe the change in some quantities. Let's start with the definition and examples of integers.
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Whole numbers. Definition, examples
First, let's recall the 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 integers ℤ. Every natural number is an integer, but not every integer is a natural number.
It follows from the definition that any of the numbers 1 , 2 , 3 is an integer. . , the number 0 , as well as the numbers - 1 , - 2 , - 3 , . .
Accordingly, we give examples. The numbers 39 , - 589 , 10000000 , - 1596 , 0 are whole numbers.
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 straight line.
The reference point on the coordinate line corresponds to the number 0, and the points lying on both sides of zero correspond to positive and negative integers. Each point corresponds to a single integer.
Any point on a straight line whose coordinate is an integer can be reached by setting aside a certain number of unit segments from the origin.
Positive and negative integers
Of all integers, it is logical to distinguish between positive and negative integers. Let's 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, for which the number 0 is taken. 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 reverse is also true. The reciprocal 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.
Earlier we said that natural numbers are a subset of integers. Let's clarify this point. The set of natural numbers are 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 negative numbers are natural, one must boldly say - no, they are not.
Non-positive and non-negative integers
Let's give 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 greater than or equal to zero, you can say: a is a non-negative integer.
Using Integers When Describing Changes in Values
What are integers used for? First of all, with their help it is convenient to describe and determine the change in the number of any objects. Let's take an example.
Let a certain number of crankshafts be stored in the warehouse. If another 500 crankshafts are brought to the warehouse, their number will increase. The number 500 just expresses the change (increase) in the number of parts. If then 200 parts are taken away from the warehouse, then this number will also characterize the change in the number of crankshafts. This time, in the direction of reduction.
If nothing is taken from the warehouse, and nothing is brought, then the number 0 will indicate the invariance of the number of parts.
The obvious convenience of using integers, in contrast to natural numbers, is that their sign clearly indicates the direction of change in magnitude (increase or decrease).
A decrease in temperature by 30 degrees can be characterized by a negative number - 30 , and an increase by 2 degrees - by a positive integer 2 .
Here is 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 amount of the debt, and the minus sign indicates that we must give back the coins.
If we owe 2 coins to one person and 3 to another, then the total debt (5 coins) can be calculated by the rule of adding negative numbers:
2 + (- 3) = - 5
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Integers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. Here are the numbers:
This is a natural series of numbers.
Zero is a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite set of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It cannot be specified, because there is an infinite set of natural numbers.
The sum of natural numbers is a natural number. So, the addition of natural numbers a and b:
The product of natural numbers is a natural number. So, the product of natural numbers a and b:
c is always a natural number.
Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers There is not always a natural number. If for natural numbers a and b
where c is a natural number, it means that a is evenly divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.
The divisor of a natural number is the natural number by which the first number is evenly divisible.
Every natural number is divisible by 1 and itself.
Simple natural numbers are only divisible by 1 and themselves. Here we mean divided completely. Example, numbers 2; 3; 5; 7 is only divisible by 1 and itself. These are simple natural numbers.
One is not considered a prime number.
Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:
One is not considered a composite number.
The set of natural numbers consists of one, prime numbers and composite numbers.
The set of natural numbers is denoted by the Latin letter N.
Properties of addition and multiplication of natural numbers:
commutative property of addition
associative property of addition
(a + b) + c = a + (b + c);
commutative property of multiplication
associative property of multiplication
(ab)c = a(bc);
distributive property of multiplication
A (b + c) = ab + ac;
Whole numbers
Integers are natural numbers, zero and the opposite of natural numbers.
Numbers opposite to natural numbers are negative integers, for example:
1; -2; -3; -4;...
The set of integers is denoted by the Latin letter Z.
Rational numbers
Rational numbers are integers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
It can be seen from the examples that any integer is a periodic fraction with a period of zero.
Any rational number can be represented as a fraction m/n, where m is an integer and n is a natural number. Let's represent the number 3,(6) from the previous example as such a fraction.
To whole numbers include natural numbers, zero, and numbers opposite to natural numbers.
Integers are positive integers.
For example: 1, 3, 7, 19, 23, etc. We use such numbers for counting (there are 5 apples on the table, the car has 4 wheels, etc.)
Latin letter \mathbb(N) - denoted set of natural numbers.
Natural numbers cannot include negative (a chair cannot have a negative number of legs) and fractional numbers (Ivan could not sell 3.5 bicycles).
Numbers opposite to natural numbers are negative integers: -8, -148, -981, ....
Arithmetic operations with integers
What can you do with integers? They can be multiplied, added and subtracted from each other. Let's analyze each operation on a specific example.
Integer addition
Two integers with the same signs are added as follows: the modules of these numbers are added and the resulting sum is preceded by the final sign:
(+11) + (+9) = +20
Subtraction of integers
Two integers with different signs are added as follows: the modulus of the smaller number is subtracted from the modulus of the larger number, and the sign of the larger modulo number is put in front of the answer:
(-7) + (+8) = +1
Integer multiplication
To multiply one integer by another, you need to multiply the modules of these numbers and put the “+” sign in front of the received answer if the original numbers were with the same signs, and the “-” sign if the original numbers were with different signs:
(-5) \cdot (+3) = -15
(-3) \cdot (-4) = +12
You should remember the following whole number multiplication rule:
+ \cdot + = +
+\cdot-=-
- \cdot += -
-\cdot-=+
There is a rule for multiplying several integers. Let's remember it:
The sign of the product will be “+” if the number of factors with a negative sign is even and “-” if the number of factors with a negative sign is odd.
(-5) \cdot (-4) \cdot (+1) \cdot (+6) \cdot (+1) = +120
Division of integers
The division of two integers is carried out as follows: the modulus of one number is divided by the modulus of the other, and if the signs of the numbers are the same, then the “+” sign is placed in front of the resulting quotient, and if the signs of the original numbers are different, then the “−” sign is put.
(-25) : (+5) = -5
Properties of addition and multiplication of integers
Let's analyze the basic properties of addition and multiplication for any integers a , b and c :
- a + b = b + a - commutative property of addition;
- (a + b) + c \u003d a + (b + c) - the associative property of addition;
- a \cdot b = b \cdot a - commutative property of multiplication;
- (a \cdot c) \cdot b = a \cdot (b \cdot c)- associative properties of multiplication;
- a \cdot (b \cdot c) = a \cdot b + a \cdot c is the distributive property of multiplication.
What does integer mean
So, consider what numbers are called integers.
Thus, integers will denote such numbers: $0$, $±1$, $±2$, $±3$, $±4$, etc.
The set of natural numbers is a subset of the set of integers, i.e. any natural will be an integer, but not any integer is a natural number.
Integer positive and integer negative numbers
Definition 2
a plus.
The numbers $3, 78, 569, 10450$ are positive integers.
Definition 3
are signed integers minus.
Numbers $−3, −78, −569, -10450$ are negative integers.
Remark 1
The number zero does not refer to either positive integers or negative integers.
Whole positive numbers are integers greater than zero.
Whole negative numbers are integers less than zero.
The set of natural integers is the set of all positive integers, and the set of all opposites of natural numbers is the set of all negative integers.
Integer non-positive and integer non-negative numbers
All positive integers and the number zero are called integer non-negative numbers.
Integer non-positive numbers are all negative integers and the number $0$.
Remark 2
In this way, whole non-negative number are integers greater than zero or equal to zero, and non-positive integer are integers less than zero or equal to zero.
For example, non-positive integers: $−32, −123, 0, −5$, and non-negative integers: $54, 123, 0.856 342.$
Description of changing values using integers
Integers are used to describe changes in the number of any items.
Consider examples.
Example 1
Suppose a store sells a certain number of items. When the store receives $520$ of items, the number of items in the store will increase, and the number of $520$ shows a positive change in the number. When the store sells $50$ items, the number of items in the store will decrease, and the number $50$ will express a negative change in the number. If the store will neither bring nor sell the goods, then the number of goods will remain unchanged (i.e., we can talk about a zero change in the number).
In the above example, the change in the number of goods is described using the integers $520$, $−50$, and $0$, respectively. A positive value of the integer $520$ indicates a positive change in the number. A negative value of the integer $−50$ indicates a negative change in the number. The integer $0$ indicates the immutability of the number.
Integers are convenient to use, because no explicit indication of an increase in number or decrease is needed - the sign of the integer indicates the direction of the change, and the value indicates a quantitative change.
Using integers, you can express not only a change in quantity, but also a change in any value.
Consider an example of a change in the cost of a product.
Example 2
An increase in cost, for example, by $20$ rubles is expressed using a positive integer $20$. Decreasing the cost, for example, by $5$ rubles is described using a negative integer $−5$. If there are no cost changes, then such a change is determined using the integer $0$.
Separately, consider the value of negative integers as the size of the debt.
Example 3
For example, a person has $5,000 rubles. Then, using a positive integer $5,000$, you can show the number of rubles that he has. A person has to pay a rent in the amount of $7,000 rubles, but he does not have that kind of money; in this case, such a situation is described by a negative integer $−7,000$. In this case, the person has $−7,000$ rubles, where "-" indicates debt, and the number $7,000$ shows the amount of debt.
