Proper fraction

quarters

1. Orderliness. a and b there is a rule that allows you to uniquely identify between them one and only one of the three relations: “< », « >' or ' = '. This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relationship as two integers and ; two non-positive numbers a and b are related by the same relation as two non-negative numbers and ; if suddenly a non-negative, and b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">

   

   summation of fractions

2. addition operation. For any rational numbers a and b there is a so-called summation rule c. However, 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 has next view: .
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. However, the number itself c called work 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 smaller b and b smaller c, then a smaller c, and if a equals b and b equals c, then a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
5. Associativity of addition. Order adding three rational numbers does not affect the result.
6. The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
8. Commutativity of multiplication. By changing the places of rational factors, the product does not change.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
12. Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
13. Connection of the order relation with the operation of addition. to the left and right parts rational inequality you can add the same rational number. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
14. Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum will exceed a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

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

Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">

Set countability

Numbering of rational numbers

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 suffices to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers.

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

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

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

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

Following this algorithm, one can enumerate all positive rational numbers. 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, simply by assigning to each rational number its opposite. 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 about the countability of the set of rational numbers may cause some bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.

Insufficiency 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 arbitrarily small quantities can be measured. This fact creates misleading impression that rational numbers can measure any geometric distances in general. It is easy to show that this is not true.

It is known from the Pythagorean theorem that the hypotenuse of a right triangle is expressed as the square root of the sum of the squares of its legs. That. isosceles hypotenuse length right triangle with a single leg is equal to, i.e., 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, i.e., the numbers m and n are coprime.

If , then , i.e. m 2 = 2n 2. Therefore, the number m 2 is even, but the product of two odd numbers odd, which means that the number itself m also clear. So there is a natural number k, such that the number m can be represented as m = 2k. Number square 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, which means that the number n- exactly like m. But then they are not coprime, since both are divisible in half. The resulting contradiction proves that is not a rational number.

At the word "fractions" many goosebumps run. Because I remember the school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. But what if we treat tasks containing the correct and improper fractions how to puzzle? After all, many adults solve digital and Japanese crosswords. Understand the rules and that's it. Same here. One has only to delve into the theory - and everything will fall into place. And examples will turn into a way to train the brain.

What types of fractions are there?

Let's start with what it is. A fraction is a number that has some fraction of one. It can be written in two forms. The first is called ordinary. That is, one that has a horizontal or oblique stroke. It equates to the division sign.

In such a notation, the number above the dash is called the numerator, and below it is called the denominator.

Among ordinary fractions, right and wrong fractions are distinguished. For the former, the modulo numerator is always less than the denominator. The wrong ones are called that because they have the opposite. The value of a proper fraction is always less than one. While the wrong one is always greater than this number.

There are also mixed numbers, that is, those that have an integer and a fractional part.

The second type of record is decimal. About her separate conversation.

What is the difference between improper fractions and mixed numbers?

Basically, nothing. It's just a different notation of the same number. Improper fractions after simple operations easily become mixed numbers. And vice versa.

It all depends on specific situation. Sometimes in tasks it is more convenient to use an improper fraction. And sometimes it is necessary to translate it into mixed number and then the example will be solved very easily. Therefore, what to use: improper fractions, mixed numbers - depends on the observation of the solver of the problem.

The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second is always less than unity.

How to represent a mixed number as an improper fraction?

If you want to perform some action with several numbers that are written in different types, then you need to make them the same. One method is to represent numbers as improper fractions.

For this purpose, you will need to follow the following algorithm:

• multiply the denominator by the integer part;
• add the value of the numerator to the result;
• write the answer above the line;
• leave the denominator the same.

Here are examples of how to write improper fractions from mixed numbers:

• 17 ¼ \u003d (17 x 4 + 1): 4 \u003d 69/4;
• 39 ½ \u003d (39 x 2 + 1): 2 \u003d 79/2.

How to write an improper fraction as a mixed number?

The next method is the opposite of the one discussed above. That is, when all mixed numbers are replaced with improper fractions. The algorithm of actions will be as follows:

• divide the numerator by the denominator to get the remainder;
• write the quotient in place of the integer part of the mixed;
• the remainder should be placed above the line;
• the divisor will be the denominator.

Examples of such a transformation:

76/14; 76:14 = 5 with a remainder of 6; the answer is 5 integers and 6/14; the fractional part in this example needs to be reduced by 2, you get 3/7; the final answer is 5 whole 3/7.

108/54; after division, the quotient 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer is an integer - 2.

How do you turn an integer into an improper fraction?

There are situations when such action is necessary. To get improper fractions with a predetermined denominator, you will need to perform the following algorithm:

• multiply an integer by the desired denominator;
• write this value above the line;
• place a denominator below it.

The simplest option is when the denominator equal to one. Then there is no need to multiply. It is enough just to write an integer, which is given in the example, and place a unit under the line.

Example: Make 5 an improper fraction with a denominator of 3. After multiplying 5 by 3, you get 15. This number will be the denominator. The answer to the task is a fraction: 15/3.

Two approaches to solving tasks with different numbers

In the example, it is required to calculate the sum and difference, as well as the product and quotient of two numbers: 2 integers 3/5 and 14/11.

In the first approach the mixed number will be represented as an improper fraction.

After performing the steps described above, you get the following value: 13/5.

To find the sum, you need to convert the fractions to same denominator. 13/5 multiplied by 11 becomes 143/55. And 14/11 after multiplying by 5 will take the form: 70/55. To calculate the sum, you only need to add the numerators: 143 and 70, and then write down the answer with one denominator. 213/55 - this improper fraction is the answer to the problem.

When finding the difference, these same numbers are subtracted: 143 - 70 = 73. The answer is a fraction: 73/55.

When multiplying 13/5 and 14/11, you do not need to lead to common denominator. Just multiply the numerators and denominators in pairs. The answer will be: 182/55.

Likewise with division. For right decision you need to replace division with multiplication and flip the divisor: 13/5: 14/11 \u003d 13/5 x 11/14 \u003d 143/70.

In the second approach An improper fraction becomes a mixed number.

After performing the actions of the algorithm, 14/11 will turn into a mixed number with whole part 1 and fractional 3/11.

When calculating the sum, you need to add the integer and fractional parts separately. 2 + 1 = 3, 3/5 + 3/11 = 33/55 + 15/55 = 48/55. The final answer is 3 whole 48/55. In the first approach there was a fraction 213/55. You can check the correctness by converting it to a mixed number. After dividing 213 by 55, the quotient is 3 and the remainder is 48. It is easy to see that the answer is correct.

When subtracting, the "+" sign is replaced by "-". 2 - 1 = 1, 33/55 - 15/55 = 18/55. To check the answer from the previous approach, you need to convert it to a mixed number: 73 is divided by 55 and you get a quotient of 1 and a remainder of 18.

To find the product and the quotient, it is inconvenient to use mixed numbers. Here it is always recommended to switch to improper fractions.

Fraction in mathematics, a number consisting of one or more parts (fractions) of a unit. Fractions are part of the field of rational numbers. Fractions are divided into 2 formats according to the way they are written: ordinary kind and decimal .

The numerator of a fraction- a number showing the number of shares taken (located at the top of the fraction - above the line). Fraction denominator- a number showing how many parts the unit is divided into (located under the line - in the lower part). , in turn, are divided into: correct and wrong, mixed and composite closely related to units of measurement. 1 meter contains 100 cm. Which means that 1 m is divided into 100 equal parts. Thus, 1 cm = 1/100 m (one centimeter is equal to one hundredth of a meter).

or 3/5 (three fifths), here 3 is the numerator, 5 is the denominator. If the numerator is less than the denominator, then the fraction is less than one and is called correct:

If the numerator equal to the denominator, the fraction is equal to one. If the numerator is greater than the denominator, the fraction is greater than one. In both recent cases the fraction is called wrong:

To isolate the largest integer contained in an improper fraction, you need to divide the numerator by the denominator. If the division is performed without a remainder, then the improper fraction taken is equal to the quotient:

If the division is performed with a remainder, then the (incomplete) quotient gives the desired integer, the remainder becomes the numerator of the fractional part; the denominator of the fractional part remains the same.

A number that contains an integer and a fractional part is called mixed. Fraction mixed number maybe improper fraction. Then it is possible to extract the largest integer from the fractional part and represent the mixed number in such a way that the fractional part becomes a proper fraction (or disappears altogether).

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. Next, 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 give definitions of right and wrong, positive and negative fractions, and also consider the position of fractional numbers on coordinate beam. In conclusion, we list the main actions with fractions.

Page navigation.

Shares of the whole

First we introduce share concept.

Let's assume that 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 object is called share of the whole or simply shares.

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

Depending on the number of shares that make up the whole object, these shares 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 beat has a special name - half. One third is called third, and one quadruple - quarter.

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

The concept of a share naturally extends from objects to magnitudes. For example, one of the measures of length is the meter. To measure lengths less than a meter, fractions of a meter can be used. So you can use, for example, half a meter or a tenth or thousandth of a meter. Shares of other quantities are applied similarly.

Common fractions, definition and examples of fractions

To describe the number of shares are used common fractions. Let's give an example that will allow us to approach the definition of ordinary fractions.

Let an orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . Let's denote two beats as , three beats as , and so on, 12 beats as . Each of these entries is called an ordinary fraction.

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

The voiced definition of ordinary fractions allows us to bring 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, in ordinary fractions we distinguish numerator and denominator.

Definition.

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

Definition.

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

So, the numerator is located above the fraction bar (to the left of the slash), and the denominator is below the fraction bar (to the right of the slash). For example, let's take 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 contained in the numerator and denominator of an ordinary fraction. The denominator of the fraction shows how many shares one item consists of, the numerator, in turn, indicates the number of such shares. For example, the denominator 5 of the fraction 12/5 means that one item consists of five parts, and the numerator 12 means that 12 such parts are taken.

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 are taken. Thus, common fraction of the form m/1 has the meaning of a natural number m . This is how we substantiated the equality m/1=m .

Let's rewrite the last equality like this: m=m/1 . This equality allows us to represent any natural number m as an ordinary fraction. For example, the number 4 is the fraction 4/1, and the number 103498 is the fraction 103498/1.

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.

Fraction bar as division sign

The representation of the original object 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 items, each of which is divided into n shares, then we can equally divide these m objects among n people, giving each person one share from each of the m objects. In this case, each person will have m shares 1/n, and m shares 1/n gives an ordinary fraction m/n. Thus, the common fraction m/n can be used to represent the division of m items among 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 as follows: The bar of a fraction can be understood as a division sign, that is, m/n=m:n.

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

Equal and unequal ordinary fractions, comparison of fractions

Enough natural action is an comparison of common fractions, because it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as the other 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 common fractions, and in the second unequal common fractions. Let's give a definition of equal and unequal ordinary fractions.

Definition.

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

Definition.

Two common fractions a/b and c/d not equal, if the equality a d=b c is not satisfied.

Here are some examples of equal fractions. For example, the common fraction 1/2 is equal to the fraction 2/4, since 1 4=2 2 (if necessary, see the rules and examples of multiplication of natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second - into 4 shares. It is obvious that two-fourths of an apple is 1/2 a share. Other examples of equal common fractions are the fractions 4/7 and 36/63, and the pair of fractions 81/50 and 1620/1000.

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

If, when comparing two ordinary fractions, it turns out that they are not equal, then you may need to find out which of these ordinary fractions smaller another, and which more. To find out, the rule for comparing ordinary fractions is used, the essence of which is to bring the compared fractions to a common denominator and then compare 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 all semantic load is contained in a fractional number. However, for brevity and convenience, the concept of a fraction and a fractional number are combined and simply called a fraction. Here it is appropriate to paraphrase the well-known saying: we say a fraction - we mean fractional number, we say a fractional number - we mean a fraction.

Fractions on the coordinate beam

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

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

For 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 the point O and the point closest to it, marked with a small dash, is 1/10 of the unit segment. The point with coordinate 14/10 is removed from the origin by 14 such segments.

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 written fractions are equal (it is located at a distance of half the unit segment, laid down from the origin in the positive direction).

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

Proper and improper fractions, definitions, examples

Among ordinary fractions, there are proper and improper fractions. This division basically has a comparison of the numerator and denominator.

Let's give a definition of proper and improper 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 improper.

Here are some examples of proper 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 comparison of 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 written ordinary fractions is equal to the denominator, and in the remaining fractions the numerator is greater than the denominator.

There are also definitions of proper and improper fractions based on comparing fractions with one.

Definition.

correct if it is less than one.

Definition.

The common 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 how ordinary fractions with a numerator greater than or equal to the denominator deserve such a name - "wrong".

Let's take the improper fraction 9/9 as an example. This fraction means that nine parts of an object are taken, which consists of nine parts. That is, from the available nine shares, we can make up a whole subject. That is, the improper fraction 9/9 essentially gives a whole object, that is, 9/9=1. In general, improper fractions with a numerator equal to the denominator denote one whole object, and such a fraction can be replaced by a natural number 1.

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

The considered examples 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 the sum of a natural number and a proper fraction, when the numerator is not evenly divisible by the denominator (for example, 7/3=2+1/3 ). Perhaps this is precisely what improper fractions deserve such a name - “wrong”.

Of particular interest is the representation of an improper fraction as the sum of a natural number and a proper fraction (7/3=2+1/3). This process is called the extraction of an integer 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 ordinary fraction corresponds to a positive fractional number (see the article positive and negative numbers). That is, ordinary fractions are positive fractions. For example, ordinary 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 entry will correspond to a negative fractional number. In this case, one can speak of 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 an increase, income, a change in some value upwards, etc. Negative fractions correspond to expense, debt, a change in any value in the direction of decrease. For example, a negative fraction -3/4 can be interpreted as a debt, the value of which is 3/4.

On the horizontal and right-directed negative fractions are located to the left of the reference point. The points of the coordinate line whose coordinates 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 combine to form rational numbers.

Actions with fractions

One action with ordinary fractions - comparing fractions - we have already considered above. Four more arithmetic are defined operations 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 draw an analogy.

Multiplication of fractions can be considered as an action in which a fraction is found from a fraction. To clarify, let's take an example. Suppose 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 a 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 5 cells. educational institutions.
• Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

At the word "fractions" many goosebumps run. Because I remember the school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. But what if we treat tasks containing proper and improper fractions as a puzzle? After all, many adults solve digital and Japanese crosswords. Understand the rules and that's it. Same here. One has only to delve into the theory - and everything will fall into place. And examples will turn into a way to train the brain.

What types of fractions are there?

Let's start with what it is. A fraction is a number that has some fraction of one. It can be written in two forms. The first is called ordinary. That is, one that has a horizontal or oblique stroke. It equates to the division sign.

In such a notation, the number above the dash is called the numerator, and below it is called the denominator.

Among ordinary fractions, right and wrong fractions are distinguished. For the former, the modulo numerator is always less than the denominator. The wrong ones are called that because they have the opposite. The value of a proper fraction is always less than one. While the wrong one is always greater than this number.

There are also mixed numbers, that is, those that have an integer and a fractional part.

The second type of notation is decimal. About her separate conversation.

What is the difference between improper fractions and mixed numbers?

Basically, nothing. It's just a different notation of the same number. Improper fractions after simple operations easily become mixed numbers. And vice versa.

It all depends on the specific situation. Sometimes in tasks it is more convenient to use an improper fraction. And sometimes it is necessary to translate it into a mixed number, and then the example will be solved very easily. Therefore, what to use: improper fractions, mixed numbers - depends on the observation of the solver of the problem.

The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second is always less than unity.

How to represent a mixed number as an improper fraction?

If you want to perform some action with several numbers that are written in different forms, then you need to make them the same. One method is to represent numbers as improper fractions.

For this purpose, you will need to follow the following algorithm:

• multiply the denominator by the integer part;
• add the value of the numerator to the result;
• write the answer above the line;
• leave the denominator the same.

Here are examples of how to write improper fractions from mixed numbers:

• 17 ¼ \u003d (17 x 4 + 1): 4 \u003d 69/4;
• 39 ½ \u003d (39 x 2 + 1): 2 \u003d 79/2.

How to write an improper fraction as a mixed number?

The next method is the opposite of the one discussed above. That is, when all mixed numbers are replaced with improper fractions. The algorithm of actions will be as follows:

• divide the numerator by the denominator to get the remainder;
• write the quotient in place of the integer part of the mixed;
• the remainder should be placed above the line;
• the divisor will be the denominator.

Examples of such a transformation:

76/14; 76:14 = 5 with a remainder of 6; the answer is 5 integers and 6/14; the fractional part in this example needs to be reduced by 2, you get 3/7; the final answer is 5 whole 3/7.

108/54; after division, the quotient 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer is an integer - 2.

How do you turn an integer into an improper fraction?

There are situations when such action is necessary. To get improper fractions with a predetermined denominator, you will need to perform the following algorithm:

• multiply an integer by the desired denominator;
• write this value above the line;
• place a denominator below it.

The simplest option is when the denominator is equal to one. Then there is no need to multiply. It is enough just to write an integer, which is given in the example, and place a unit under the line.

Example: Make 5 an improper fraction with a denominator of 3. After multiplying 5 by 3, you get 15. This number will be the denominator. The answer to the task is a fraction: 15/3.

Two approaches to solving tasks with different numbers

In the example, it is required to calculate the sum and difference, as well as the product and quotient of two numbers: 2 integers 3/5 and 14/11.

In the first approach the mixed number will be represented as an improper fraction.

After performing the steps described above, you get the following value: 13/5.

In order to find out the sum, you need to reduce the fractions to the same denominator. 13/5 multiplied by 11 becomes 143/55. And 14/11 after multiplying by 5 will take the form: 70/55. To calculate the sum, you only need to add the numerators: 143 and 70, and then write down the answer with one denominator. 213/55 - this improper fraction is the answer to the problem.

When finding the difference, these same numbers are subtracted: 143 - 70 = 73. The answer is a fraction: 73/55.

When multiplying 13/5 and 14/11, you do not need to reduce to a common denominator. Just multiply the numerators and denominators in pairs. The answer will be: 182/55.

Likewise with division. For the correct solution, you need to replace division with multiplication and flip the divisor: 13/5: 14/11 \u003d 13/5 x 11/14 \u003d 143/70.

In the second approach An improper fraction becomes a mixed number.

After performing the actions of the algorithm, 14/11 will turn into a mixed number with an integer part of 1 and a fractional part of 3/11.

When calculating the sum, you need to add the integer and fractional parts separately. 2 + 1 = 3, 3/5 + 3/11 = 33/55 + 15/55 = 48/55. The final answer is 3 whole 48/55. In the first approach there was a fraction 213/55. You can check the correctness by converting it to a mixed number. After dividing 213 by 55, the quotient is 3 and the remainder is 48. It is easy to see that the answer is correct.

When subtracting, the "+" sign is replaced by "-". 2 - 1 = 1, 33/55 - 15/55 = 18/55. To check the answer from the previous approach, you need to convert it to a mixed number: 73 is divided by 55 and you get a quotient of 1 and a remainder of 18.

To find the product and the quotient, it is inconvenient to use mixed numbers. Here it is always recommended to switch to improper fractions.