Right fractions. What is a legal fraction? Right and wrong fraction: rules

This article is about common fractions... Here we will get acquainted with the concept of a fraction of a whole, which will lead us to the definition of an ordinary fraction. Further we will dwell on the accepted notation for ordinary fractions and give examples of fractions, say about the numerator and denominator of a fraction. After that, we will give definitions of correct and incorrect, positive and negative fractions, and also consider the position of fractional numbers on the coordinate ray. In conclusion, we list the main actions with fractions.

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Shares of the whole

First we introduce share concept.

Suppose we have some object made up of several absolutely identical (that is, equal) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange, consisting of several equal slices. Each of these equal parts that make up the whole thing is called shares of a whole or simply shares.

Note that the shares are different. Let us explain this. Let's say we have two apples. Let's cut the first apple into two equal parts, and the second into 6 equal parts. It is clear that the share of the first apple will differ from the share of the second apple.

Depending on the number of beats that make up the whole subject, these beats have their own names. Let's analyze share names... If the object consists of two parts, any of them is called one-second part of the whole object; if the object consists of three parts, then any of them is called one third part, and so on.

One second share has a special name - half... One third of the share is called third, and one quarter is a quarter.

For the sake of brevity, the following have been introduced. share designations... One second share is designated as either 1/2, one third share as or 1/3; one fourth is like or 1/4, and so on. Note that notation with a horizontal bar is used more often. To consolidate the material, we will give one more example: the record denotes one hundred and sixty-seventh part of the whole.

The concept of a share naturally extends from objects to quantities. For example, one of the measures for measuring length is the meter. For measuring lengths shorter than a meter, you can use fractions of a meter. So you can use, for example, half a meter or a tenth or thousandth of a meter. The fractions of other quantities are applied in the same way.

Common fractions, definitions and examples of fractions

To describe the number of beats, use common fractions... Let's give an example that will allow us to approach the definition of ordinary fractions.

Let the orange have 12 parts. Each beat in this case represents one twelfth of a whole orange, that is,. Let's designate two shares as, three shares - as, and so on, we'll designate 12 shares as. Each of these entries is called a fraction.

Now let's give a general definition of common fractions.

The sounded definition of ordinary fractions allows us to give examples of common fractions: 5/10,, 21/1, 9/4,. And here are the records do not fit the voiced definition of ordinary fractions, that is, they are not ordinary fractions.

Numerator and Denominator

For convenience, a common fraction is distinguished numerator and denominator.

Definition.

Numerator fraction (m / n) is a natural number m.

Definition.

Denominator fraction (m / n) is a natural number n.

So, the numerator is above the fraction line (to the left of the oblique slash), and the denominator is below the fraction line (to the right of the oblique line). For example, let's give an ordinary fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.

It remains to discuss the meaning of the numerator and denominator of an ordinary fraction. The denominator of a fraction shows how many parts one item consists of, the numerator, in turn, indicates the number of such parts. For example, the denominator 5 of the fraction 12/5 means that one item has five parts, and the numerator 12 means that there are 12 such parts.

Natural number as a fraction with denominator 1

The denominator of an ordinary fraction can be equal to one. In this case, we can assume that the object is indivisible, in other words, it is something whole. The numerator of such a fraction indicates how many whole items were taken. Thus, an ordinary fraction of the form m / 1 has the meaning of a natural number m. This is how we substantiated the validity of the equality m / 1 = m.

We rewrite the last equality as follows: m = m / 1. This equality enables us to represent any natural number m as an ordinary fraction. For example, 4 is a 4/1 fraction, and 103 498 is equal to 103 498/1.

So, any natural number m can be represented as an ordinary fraction with denominator 1 as m / 1, and any ordinary fraction of the form m / 1 can be replaced by a natural number m.

Slash of fraction as a sign of division

The representation of the original item in the form of n shares is nothing more than a division into n equal parts. After the item is divided into n shares, we can divide it equally among n people - each will receive one share.

If we initially have m identical objects, each of which is divided into n shares, then we can equally divide these m objects among n people, giving each person one share of each of the m objects. In this case, each person will have m shares of 1 / n, and m shares of 1 / n gives an ordinary fraction m / n. Thus, the common fraction m / n can be used to denote the division of m objects between n people.

So we got an explicit connection between ordinary fractions and division (see the general idea of ​​the division of natural numbers). This relationship is expressed in the following: a slash of a fraction can be understood as a division sign, that is, m / n = m: n.

Using an ordinary fraction, you can write the result of dividing two natural numbers for which integer division is not performed. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, everyone will get five eighths of an apple: 5: 8 = 5/8.

Equal and unequal ordinary fractions, comparison of fractions

A fairly natural action is comparison of ordinary fractions, after all, it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as another 1/6 of this apple.

As a result of comparing two ordinary fractions, one of the results is obtained: the fractions are either equal or not equal. In the first case, we have equal fractions, and in the second - unequal fractions... Let us give a definition of equal and unequal ordinary fractions.

Definition.

are equal if the equality a d = b c is true.

Definition.

Two fractions a / b and c / d not equal if the equality a d = b c does not hold.

Here are some examples of equal fractions. For example, an ordinary fraction 1/2 is equal to 2/4, since 1 4 = 2 2 (if necessary, see the rules and examples for multiplying natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second is cut into 4 parts. Moreover, it is obvious that two-fourths of an apple is 1/2 share. Other examples of equal fractions are 4/7 and 36/63, and a pair of fractions 81/50 and 1,620/1000.

And the ordinary fractions 4/13 and 5/14 are not equal, since 4 · 14 = 56, and 13 · 5 = 65, that is, 4 · 14 ≠ 13 · 5. The fractions 17/7 and 6/4 are another example of unequal ordinary fractions.

If, when comparing two ordinary fractions, it turned out that they are not equal, then you may need to find out which of these ordinary fractions less another, and which - more... To find out, the rule for comparing ordinary fractions is used, the essence of which boils down to bringing the compared fractions to a common denominator and then comparing the numerators. Detailed information on this topic is collected in the article comparison of fractions: rules, examples, solutions.

Fractional numbers

Each fraction is a record fractional number... That is, a fraction is just a "shell" of a fractional number, its appearance, and the entire semantic load is contained in the fractional number. However, for brevity and convenience, the concepts of a fraction and a fractional number are combined and simply said as a fraction. It is appropriate here to rephrase a well-known saying: we say a fraction - we mean a fractional number, we say a fractional number - we mean a fraction.

Fractions on the coordinate ray

All fractional numbers corresponding to ordinary fractions have their unique place on, that is, there is a one-to-one correspondence between fractions and points of the coordinate ray.

To get on the coordinate ray to the point corresponding to the fraction m / n, you need to postpone m segments from the origin in the positive direction, the length of which is 1 / n fraction of a unit segment. Such segments can be obtained by dividing a unit segment into n equal parts, which can always be done using a compass and a ruler.

As an example, let's show the point M on the coordinate ray, corresponding to the fraction 14/10. The length of the segment with ends at point O and the point nearest to it, marked with a small stroke, is 1/10 of a unit segment. The point with the coordinate 14/10 is located at a distance of 14 such segments from the origin.

Equal fractions correspond to the same fractional number, that is, equal fractions are the coordinates of the same point on the coordinate ray. For example, one point corresponds to the coordinates 1/2, 2/4, 16/32, 55/110 on the coordinate ray, since all the written fractions are equal (it is located at a distance of half a unit segment, set aside from the origin in the positive direction).

On the horizontal and right-directed coordinate ray, the point whose coordinate is the major fraction is located to the right of the point whose coordinate is the minor fraction. Similarly, the point with the smaller coordinate lies to the left of the point with the larger coordinate.

Correct and improper fractions, definitions, examples

Among ordinary fractions, there are right and wrong fractions... This division is based on the comparison of the numerator and denominator.

Let us give a definition of regular and irregular ordinary fractions.

Definition.

Proper fraction Is an ordinary fraction, the numerator of which is less than the denominator, that is, if m

Definition.

Improper fraction Is an ordinary fraction in which the numerator is greater than or equal to the denominator, that is, if m≥n, then the ordinary fraction is incorrect.

Here are some examples of regular fractions: 1/4,, 32 765/909 003. Indeed, in each of the written ordinary fractions, the numerator is less than the denominator (if necessary, see the article comparing natural numbers), so they are correct by definition.

And here are examples of improper fractions: 9/9, 23/4,. Indeed, the numerator of the first of the recorded ordinary fractions is equal to the denominator, and in the remaining fractions the numerator is greater than the denominator.

There are also definitions of right and wrong fractions based on the comparison of fractions with one.

Definition.

correct if it is less than one.

Definition.

An ordinary fraction is called wrong if it is either equal to one or greater than 1.

So the ordinary fraction 7/11 is correct, since 7/11<1 , а обыкновенные дроби 14/3 и 27/27 – неправильные, так как 14/3>1, and 27/27 = 1.

Let's think about why ordinary fractions with a numerator greater than or equal to the denominator have earned such a name - "incorrect".

Let's take the improper fraction 9/9 as an example. This fraction means that you have taken nine parts of an object that consists of nine parts. That is, out of the available nine parts, we can compose a whole object. That is, the improper fraction 9/9 essentially gives the whole item, that is, 9/9 = 1. In general, irregular fractions with a numerator equal to the denominator denote one whole object, and such a fraction can be replaced by the natural number 1.

Now consider the improper fractions 7/3 and 12/4. It is quite obvious that we can make two whole objects out of these seven-third parts (one whole object is 3 parts, then to compose two whole objects we need 3 + 3 = 6 parts) and one third part will remain. That is, the incorrect fraction 7/3 essentially means 2 items and even 1/3 the share of such an item. And from twelve-fourths, we can make three whole objects (three objects with four parts in each). That is, the fraction 12/4 essentially means 3 whole objects.

The examples considered lead us to the following conclusion: improper fractions can be replaced either by natural numbers, when the numerator is divided entirely by the denominator (for example, 9/9 = 1 and 12/4 = 3), or by the sum of a natural number and a regular fraction, when the numerator is not evenly divisible by the denominator (for example, 7/3 = 2 + 1/3). Perhaps this is precisely what the wrong fractions have earned such a name - "wrong".

Of particular interest is the representation of an improper fraction as a sum of a natural number and a regular fraction (7/3 = 2 + 1/3). This process is called the separation of the whole part from an improper fraction, and deserves a separate and more careful consideration.

It is also worth noting that there is a very close relationship between improper fractions and mixed numbers.

Positive and negative fractions

Each fraction corresponds to a positive fractional number (see the article positive and negative numbers). That is, the common fractions are positive fractions... For example, common fractions 1/5, 56/18, 35/144 are positive fractions. When it is necessary to emphasize the positiveness of a fraction, then a plus sign is placed in front of it, for example, +3/4, +72/34.

If you put a minus sign in front of an ordinary fraction, then this record will correspond to a negative fractional number. In this case, we can talk about negative fractions... Here are some examples of negative fractions: −6/10, −65/13, −1/18.

The positive and negative fractions m / n and -m / n are opposite numbers. For example, the fractions 5/7 and −5/7 are opposite fractions.

Positive fractions, like positive numbers in general, denote addition, income, change in any value upward, etc. Negative fractions correspond to expense, debt, change in any value downward. For example, a negative fraction −3/4 can be interpreted as a debt of 3/4.

Negative fractions on horizontal and right-directed ones are located to the left of the origin. The points of the coordinate line, the coordinates of which are the positive fraction m / n and the negative fraction −m / n, are located at the same distance from the origin, but on opposite sides of the point O.

Here it is worth mentioning fractions of the form 0 / n. These fractions are equal to the number zero, that is, 0 / n = 0.

Positive fractions, negative fractions, and 0 / n fractions are combined to form rational numbers.

Actions with fractions

One action with common fractions - comparing fractions - we have already discussed above. Four more arithmetic actions with fractions- addition, subtraction, multiplication and division of fractions. Let's dwell on each of them.

The general essence of actions with fractions is similar to the essence of the corresponding actions with natural numbers. Let's make an analogy.

Multiplication of fractions can be considered as an action in which there is a fraction of a fraction. Let's give an example for clarification. Let's say we have 1/6 of an apple and we need to take 2/3 of it. The part we need is the result of multiplying the fractions 1/6 and 2/3. The result of multiplying two ordinary fractions is an ordinary fraction (which in the particular case is equal to a natural number). Further we recommend to study the information of the article multiplication of fractions - rules, examples and solutions.

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for grade 5 educational institutions.
• Vilenkin N. Ya. and other Mathematics. Grade 6: textbook for educational institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a guide for applicants to technical schools).

Proper fraction

Quarters

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   Summation of fractions

2. Addition operation. For any rational numbers a and b there is a so-called summation rule c... Moreover, the number itself c called sum numbers a and b and is denoted, and the process of finding such a number is called summation... The summation rule is as follows: .
3. Multiplication operation. For any rational numbers a and b there is a so-called multiplication rule, which puts them in correspondence with some rational number c... Moreover, the number itself c called product numbers a and b and is denoted, and the process of finding such a number is also called multiplication... The multiplication rule is as follows: .
4. Transitivity of the order relation. For any triple of rational numbers a , b and c if a less b and b less c, then a less c, and if a equals b and b equals c, then a equals c... 6435 "> Commutativity of addition. The sum does not change from the change of places of rational terms.
5. Addition associativity. The order of addition of the three rational numbers does not affect the result.
6. The presence of zero. There is a rational number 0 that preserves any other rational number when summed.
7. The presence of opposite numbers. Any rational number has an opposite rational number, when summed with it gives 0.
8. Commutativity of multiplication. The product does not change from a change in the places of the rational factors.
9. Associativity of multiplication. The order in which the three rational numbers are multiplied does not affect the result.
10. Unit availability. There is a rational number 1 that preserves any other rational number when multiplied.
11. The presence of reciprocal numbers. Any rational number has an inverse rational number, which, when multiplied by, gives 1.
12. Distributivity of multiplication relative to addition. The operation of multiplication is consistent with the operation of addition by means of the distribution law:
13. The relationship of the order relation with the addition operation. The same rational number can be added to the left and right sides of a rational inequality. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png "border =" 0 ">
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Additional properties

All other properties inherent in rational numbers are not singled out as the main ones, because, generally speaking, they no longer rely directly on the properties of integers, but can be proved based on the given basic properties or directly by the definition of a certain mathematical object. There are a lot of such additional properties. It makes sense to cite only a few of them here.

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Countability of a set

Rational numbering

To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it is enough to give an algorithm that numbers rational numbers, that is, it establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms is as follows. An endless table of common fractions is compiled, for each i-th line in each j-th column of which the fraction is located. For definiteness, it is assumed that the rows and columns of this table are numbered starting from one. Table cells are designated, where i is the row number of the table in which the cell is located, and j- column number.

The resulting table is bypassed by the "snake" according to the following formal algorithm.

These rules are scanned from top to bottom and the next position is selected on the first match.

In the process of such a traversal, each new rational number is associated with the next natural number. That is, the fraction 1/1 is assigned the number 1, the fraction 2/1 - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to one of the greatest common divisor of the numerator and denominator of the fraction.

Following this algorithm, all positive rational numbers can be enumerated. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers by simply assigning the opposite to each rational number. That. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.

The statement that the set of rational numbers is countable may cause some bewilderment, since at first glance one gets the impression that it is much more extensive than the set of natural numbers. In fact, this is not so, and there are enough natural numbers to enumerate all rational ones.

Lack of rational numbers

The hypotenuse of such a triangle is not expressed by any rational number

Rational numbers of the form 1 / n at large n you can measure arbitrarily small quantities. This fact creates the deceptive impression that any geometric distance can be measured with rational numbers. It is easy to show that this is not true.

It is known from the Pythagorean theorem that the hypotenuse of a right-angled triangle is expressed as the square root of the sum of the squares of its legs. That. the length of the hypotenuse of an isosceles right-angled triangle with a unit leg is, that is, a number whose square is 2.

If we assume that the number is represented by some rational number, then there is such an integer m and such a natural number n, which, moreover, the fraction is irreducible, that is, the numbers m and n- mutually simple.

If, then , i.e. m 2 = 2n 2. Therefore, the number m 2 is even, but the product of two odd numbers is odd, which means that the number itself m also even. So there is a natural number k such that the number m can be represented as m = 2k... Square number m In this sense m 2 = 4k 2, but on the other hand m 2 = 2n 2 means 4 k 2 = 2n 2, or n 2 = 2k 2. As shown earlier for the number m, this means that the number n- even, like m... But then they are not mutually simple, since both are halved. The resulting contradiction proves that it is not a rational number.