Fraction - what is it? Types of fractions. The concept of a common fraction

In mathematics, a fraction is a number consisting of one or more parts (fractions) of a unit. According to the form of recording, fractions are divided into ordinary (example \frac(5)(8)) and decimal (for example 123.45).

Definition. Common fraction (or simple fraction)

Ordinary (simple) fraction is called a number of the form \pm\frac(m)(n) where m and n are natural numbers. The number m is called numerator this fraction, and the number n is its denominator.

A horizontal or slash indicates a division sign, that is, \frac(m)(n)=()^m/n=m:n

Common fractions are divided into two types: proper and improper.

Definition. Proper and improper fractions

Correct A fraction whose numerator is less than its denominator is called a fraction. For example, \frac(9)(11) , because 9

Wrong A fraction is called in which the modulus of the numerator is greater than or equal to the modulus of the denominator. This fraction is rational number, modulo greater than or equal to one. An example would be the fractions \frac(11)(2) , \frac(2)(1) , -\frac(7)(5) , \frac(1)(1)

Along with the improper fraction, there is another representation of the number, which is called a mixed fraction (mixed number). This is not an ordinary fraction.

Definition. Mixed fraction (mixed number)

Mixed fraction is a fraction written as a whole number and a proper fraction and is understood as the sum of this number and the fraction. For example, 2\frac(5)(7)

(record in the form mixed number) 2\frac(5)(7)=2+\frac(5)(7)=\frac(14)(7)+\frac(5)(7)=\frac(19)(7) (record in the form improper fraction)

A fraction is just a representation of a number. The same number can correspond to different fractions, both ordinary and decimal. Let us form a sign for the equality of two ordinary fractions.

Definition. Sign of equality of fractions

The two fractions \frac(a)(b) and \frac(c)(d) are equal, if a\cdot d=b\cdot c . For example, \frac(2)(3)=\frac(8)(12) since 2\cdot12=3\cdot8

From this attribute follows the main property of a fraction.

Property. The main property of a fraction

If the numerator and denominator of a given fraction are multiplied or divided by the same number, not equal to zero, you get a fraction equal to the given one.

\frac(A)(B)=\frac(A\cdot C)(B\cdot C)=\frac(A:K)(B:K);\quad C \ne 0,\quad K \ne 0

Using the basic property of a fraction, you can replace a given fraction with another fraction that is equal to the given one, but with a smaller numerator and denominator. This replacement is called fraction reduction. For example, \frac(12)(16)=\frac(6)(8)=\frac(3)(4) (here the numerator and denominator were divided first by 2, and then by 2 more). A fraction can be reduced if and only if its numerator and denominator are not mutually exclusive. prime numbers. If the numerator and denominator of a given fraction are mutually prime, then the fraction cannot be reduced, for example, \frac(3)(4) is an irreducible fraction.

Rules for positive fractions:

From two fractions with the same denominators The fraction whose numerator is greater is greater. For example, \frac(3)(15)

From two fractions with the same numerators The larger is the fraction whose denominator is smaller. For example, \frac(4)(11)>\frac(4)(13) .

To compare two fractions with different numerators and denominators, you must convert both fractions so that their denominators are the same. This transformation is called reducing fractions to a common denominator.

When talking about mathematics, one cannot help but remember fractions. A lot of attention and time is devoted to their study. Remember how many examples you had to solve in order to learn certain rules for working with fractions, how you memorized and applied the main property of a fraction. How much nerve was spent finding the common denominator, especially if the examples had more than two terms!

Let's remember what it is and a little refresher on the basic information and rules of working with fractions.

Definition of fractions

Let's start, perhaps, with the most important thing - the definition. A fraction is a number that is made up of one or more parts of a unit. A fractional number is written as two numbers separated by a horizontal or slash. In this case, the top (or first) is called the numerator, and the bottom (second) is called the denominator.

It is worth noting that the denominator shows how many parts the unit is divided into, and the numerator shows the number of shares or parts taken. Often fractions, if proper, are less than one.

Now let's look at the properties of these numbers and the basic rules that are used when working with them. But before we examine such a concept as the “basic property rational fraction", let's talk about the types of fractions and their features.

What are fractions?

There are several types of such numbers. First of all, these are ordinary and decimal. The first represent the type of recording we have already indicated using a horizontal or slash. The second type of fractions is indicated using the so-called positional notation, when the integer part of the number is indicated first, and then, after the decimal point, the fractional part is indicated.

It is worth noting here that in mathematics both decimal and ordinary fractions are used equally. The main property of the fraction is valid only for the second option. In addition, in ordinary fractions there are proper and wrong numbers. For the former, the numerator is always less than the denominator. Note also that such a fraction is less than one. In an improper fraction, on the contrary, the numerator is greater than the denominator, and the fraction itself is greater than one. In this case, an integer can be extracted from it. In this article we will consider only ordinary fractions.

Properties of Fractions

Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers were no exception. They have one important feature, with the help of which certain operations can be carried out on them. What is the main property of a fraction? The rule states that if its numerator and denominator are multiplied or divided by the same rational number, we get a new fraction, the value of which will be equal to the value of the original one. That is, by multiplying two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, and they will be equal.

Based on this property, you can reduce fractions, as well as select common denominators for a particular pair of numbers.

Operations

Although fractions seem more complex, they can also be used to perform basic math operations, such as addition and subtraction, multiplication, and division. In addition, there is such a specific action as reducing fractions. Naturally, each of these actions is performed according to certain rules. Knowing these laws makes working with fractions easier, easier and more interesting. That is why next we will look at the basic rules and algorithm of actions when working with such numbers.

But before we talk about mathematical operations such as addition and subtraction, let's look at an operation such as reduction to a common denominator. This is where knowledge of what basic property of a fraction exists comes in handy.

Common denominator

In order to reduce a number to a common denominator, you first need to find the least common multiple of the two denominators. That is smallest number, which is simultaneously divisible by both denominators without a remainder. The easiest way to find the LCM (least common multiple) is to write down on a line for one denominator, then for the second, and find the matching number among them. If the LCM is not found, that is, these numbers do not have a common multiple, you should multiply them, and the resulting value is considered the LCM.

So, we have found the LCM, now we need to find an additional factor. To do this, you need to alternately divide the LCM into the denominators of the fractions and write the resulting number over each of them. Next, you should multiply the numerator and denominator by the resulting additional factor and write the results as a new fraction. If you doubt that the number you received is equal to the previous one, remember the basic property of a fraction.

Addition

Now let's move directly to mathematical operations on fractional numbers. Let's start with the simplest one. There are several options for adding fractions. In the first case, both numbers have same denominator. In this case, all that remains is to add the numerators together. But the denominator does not change. For example, 1/5 + 3/5 = 4/5.

If fractions have different denominators, you should reduce them to a common denominator and only then perform addition. We discussed how to do this a little higher. In this situation, the basic property of a fraction will come in handy. The rule will allow you to bring numbers to a common denominator. The value will not change in any way.

Alternatively, it may happen that the fraction is mixed. Then you should first add together the whole parts, and then the fractional ones.

Multiplication

Does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of a fraction. It is enough to first multiply the numerators and denominators together. In this case, the product of the numerators will become the new numerator, and the denominators will become the new denominator. As you can see, nothing complicated.

The only thing that is required of you is knowledge of the multiplication tables, as well as attentiveness. In addition, after receiving the result, you should definitely check whether this number can be reduced or not. We'll talk about how to reduce fractions a little later.

Subtraction

When performing, you should be guided by the same rules as when adding. So, in numbers with the same denominator, it is enough to subtract the numerator of the subtrahend from the numerator of the minuend. If the fractions have different denominators, you should bring them to a common denominator and then perform this operation. As with addition, you will need to use the basic property algebraic fraction, as well as skills in finding LOC and common divisors for fractions.

Division

And the last one, the most interesting operation when working with such numbers - division. It is quite simple and does not cause any particular difficulties even for those who have little understanding of how to work with fractions, especially addition and subtraction. When dividing, the same rule applies as multiplying by a reciprocal fraction. The main property of a fraction, as in the case of multiplication, will not be used for this operation. Let's take a closer look.

When dividing numbers, the dividend remains unchanged. The divisor fraction turns into its inverse, that is, the numerator and denominator change places. After this, the numbers are multiplied with each other.

Reduction

So, we have already examined the definition and structure of fractions, their types, the rules of operations on these numbers, and found out the main property of an algebraic fraction. Now let's talk about such an operation as reduction. Reducing a fraction is the process of converting it - dividing the numerator and denominator by the same number. Thus, the fraction is reduced without changing its properties.

Usually, when performing a mathematical operation, you should carefully look at the resulting result and find out whether it is possible to reduce the resulting fraction or not. Remember that the final result always contains a fractional number that does not require reduction.

Other operations

Finally, we note that we have not listed all operations on fractional numbers, mentioning only the most well-known and necessary. Fractions can also be compared, converted to decimals and vice versa. But in this article we did not consider these operations, since in mathematics they are carried out much less frequently than those we presented above.

Conclusions

We talked about fractional numbers and operations with them. We also examined the main property. But let us note that all these issues were considered by us in passing. We have given only the most well-known and used rules and given the most important, in our opinion, advice.

This article is intended to refresh information about fractions that you have forgotten rather than to give new information and fill your head with endless rules and formulas that, most likely, will never be useful to you.

We hope that the material presented in the article, simply and concisely, was useful to you.

Do you want to feel like a sapper? Then this lesson is for you! Because now we will study fractions - these are such simple and harmless mathematical objects that, in terms of their ability to “blow the mind”, are superior to the rest of the algebra course.

The main danger of fractions is that they occur in real life. This is how they differ, for example, from polynomials and logarithms, which you can study and easily forget after the exam. Therefore, the material presented in this lesson can, without exaggeration, be called explosive.

A number fraction (or just a fraction) is a pair of integers written separated by a slash or a horizontal bar.

Fractions written through a horizontal line:

The same fractions written with a slash:
5/7; 9/(−30); 64/11; (−1)/4; 12/1.

Fractions are usually written through a horizontal line - it’s easier to work with them this way, and they look better. The number written on top is called the numerator of the fraction, and the number written below is called the denominator.

Any integer can be represented as a fraction with a denominator of 1. For example, 12 = 12/1 is the fraction from the example above.

In general, you can put any whole number into the numerator and denominator of a fraction. The only limitation is that the denominator must be different from zero. Remember the good old rule: “You can’t divide by zero!”

If the denominator still has a zero, the fraction is called an indefinite fraction. Such a record does not make sense and cannot be used in calculations.

The main property of a fraction

Fractions a /b and c /d are said to be equal if ad = bc.

From this definition it follows that the same fraction can be written in different ways. For example, 1/2 = 2/4, since 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4, since 1 4 ≠ 3 5.

A reasonable question arises: how to find all fractions equal to a given one? We give the answer in the form of a definition:

The main property of a fraction is that the numerator and denominator can be multiplied by the same number other than zero. This will result in a fraction equal to the given one.

This is very important property- remember it. Using the basic property of a fraction, you can simplify and shorten many expressions. In the future, it will constantly “pop up” in the form of various properties and theorems.

Improper fractions. Selecting a whole part

If the numerator is less than the denominator, it is called a proper fraction. Otherwise (i.e., when the numerator is greater than or at least equal to the denominator), the fraction is called improper, and an integer part can be distinguished in it.

The whole part is written with a large number in front of the fraction and looks like this (marked in red):

To isolate the whole part of an improper fraction, you need to follow three simple steps:

1. Find how many times the denominator fits in the numerator. In other words, find the maximum integer that, when multiplied by the denominator, will still be less than the numerator (in as a last resort- equals). This number will be the integer part, so we write it in front;
2. Multiply the denominator by the integer part found in the previous step, and subtract the result from the numerator. The resulting “stub” is called the remainder of the division; it will always be positive (in extreme cases, zero). We write it in the numerator of the new fraction;
3. We rewrite the denominator without changes.

Well, is it difficult? At first glance, it may be difficult. But with a little practice, you will be able to do it almost orally. In the meantime, take a look at the examples:

Task. Select the whole part in the indicated fractions:

In all examples, the whole part is highlighted in red, and the remainder of the division is highlighted in green.

Pay attention to the last fraction, where the remainder of the division turns out to be zero. It turns out that the numerator is completely divided by the denominator. This is quite logical, because 24: 6 = 4 is a hard fact from the multiplication table.

If everything is done correctly, the numerator of the new fraction will definitely be less than the denominator, i.e. the fraction will become correct. I will also note that it is better to highlight the whole part at the very end of the problem, before writing down the answer. Otherwise, the calculations can be significantly complicated.

Going to an improper fraction

There is also a reverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because working with improper fractions is much easier.

The transition to an improper fraction is also performed in three steps:

1. Multiply the whole part by the denominator. The result can be quite big numbers, but this should not bother us;
2. Add the resulting number to the numerator of the original fraction. Write the result in the numerator of the improper fraction;
3. Rewrite the denominator - again, without changes.

Here are specific examples:

Task. Convert to improper fraction:

For clarity, the integer part is again highlighted in red, and the numerator of the original fraction is highlighted in green.

Consider the case when the numerator or denominator of the fraction contains negative number. For example:

In principle, there is nothing criminal in this. However, working with such fractions can be inconvenient. Therefore, in mathematics it is customary to place minuses as fraction signs.

This is very easy to do if you remember the rules:

1. “Plus for minus gives minus.” Therefore, if the numerator contains a negative number, and the denominator contains a positive number (or vice versa), feel free to cross out the minus and put it in front of the entire fraction;
2. "Two negatives make an affirmative". When there is a minus in both the numerator and the denominator, we simply cross them out - no additional actions not required.

Of course, these rules can also be applied in the opposite direction, i.e. You can enter a minus sign under the fraction sign (most often in the numerator).

We deliberately do not consider the “plus on plus” case - with it, I think, everything is clear. Let's see how these rules work in practice:

Task. Take out the negatives of the four fractions written above.

Pay attention to the last fraction: there is already a minus sign in front of it. However, it is “burned” according to the rule “minus for minus gives a plus.”

Also, do not move minuses in fractions with the whole part highlighted. These fractions are first converted to improper fractions - and only then do calculations begin.

Fraction- a form of representing a number in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.

Correct A fraction whose numerator is greater than its denominator is called a fraction. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.

The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:

The main property of a fraction

If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,

Reducing fractions to a common denominator

To bring two fractions to a common denominator, you need:

1. Multiply the numerator of the first fraction by the denominator of the second
2. Multiply the numerator of the second fraction by the denominator of the first
3. Replace the denominators of both fractions with their product

Operations with fractions

Addition. To add two fractions you need

1. Add the new numerators of both fractions and leave the denominator unchanged

Example:

Subtraction. To subtract one fraction from another, you need

1. Reduce fractions to a common denominator
2. Subtract the numerator of the second from the numerator of the first fraction, and leave the denominator unchanged

Example:

Multiplication. To multiply one fraction by another, multiply their numerators and denominators:

Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:

Examples with fractions are one of the basic elements of mathematics. There are many different types equations with fractions. Below is detailed instructions for solving examples of this type.

How to solve examples with fractions - general rules

To solve examples with fractions of any type, be it addition, subtraction, multiplication or division, you need to know the basic rules:

• In order to add fractional expressions with the same denominator (the denominator is the number at the bottom of the fraction, the numerator at the top), you need to add their numerators and leave the denominator the same.
• In order to subtract a second fractional expression (with the same denominator) from one fraction, you need to subtract their numerators and leave the denominator the same.
• To add or subtract fractional expressions with different denominators, you need to find the smallest common denominator.
• In order to find a fractional product, you need to multiply the numerators and denominators, and, if possible, reduce.
• To divide a fraction by a fraction, you multiply the first fraction by the second fraction reversed.

How to solve examples with fractions - practice

Rule 1, example 1:

Calculate 3/4 +1/4.

According to Rule 1, if two (or more) fractions have the same denominator, you simply add their numerators. We get: 3/4 + 1/4 = 4/4. If a fraction has the same numerator and denominator, the fraction will equal 1.

Answer: 3/4 + 1/4 = 4/4 = 1.

Rule 2, example 1:

Calculate: 3/4 – 1/4

Using rule number 2, to solve this equation you need to subtract 1 from 3 and leave the denominator the same. We get 2/4. Since two 2 and 4 can be reduced, we reduce and get 1/2.

Answer: 3/4 – 1/4 = 2/4 = 1/2.

Rule 3, Example 1

Calculate: 3/4 + 1/6

Solution: Using the 3rd rule, we find the lowest common denominator. The least common denominator is the number that is divisible by the denominators of all fractional expressions in the example. Thus, we need to find the minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. Divide 12 by the denominator of the first fraction, we get 3, multiply by 3, write 3 in the numerator *3 and + sign. Divide 12 by the denominator of the second fraction, we get 2, multiply 2 by 1, write 2*1 in the numerator. So, we get a new fraction with a denominator equal to 12 and a numerator equal to 3*3+2*1=11. 11/12.

Answer: 11/12

Rule 3, Example 2:

Calculate 3/4 – 1/6. This example is very similar to the previous one. We do all the same steps, but in the numerator instead of the + sign, we write a minus sign. We get: 3*3-2*1/12 = 9-2/12 = 7/12.

Answer: 7/12

Rule 4, Example 1:

Calculate: 3/4 * 1/4

Using the fourth rule, we multiply the denominator of the first fraction by the denominator of the second and the numerator of the first fraction by the numerator of the second. 3*1/4*4 = 3/16.

Answer: 3/16

Rule 4, Example 2:

Calculate 2/5 * 10/4.

This fraction can be reduced. In the case of a product, the numerator of the first fraction and the denominator of the second and the numerator of the second fraction and the denominator of the first are canceled.

2 cancels from 4. 10 cancels from 5. We get 1 * 2/2 = 1*1 = 1.

Answer: 2/5 * 10/4 = 1

Rule 5, Example 1:

Calculate: 3/4: 5/6

Using the 5th rule, we get: 3/4: 5/6 = 3/4 * 6/5. We reduce the fraction according to the principle of the previous example and get 9/10.

Answer: 9/10.

How to solve examples with fractions - fractional equations

Fractional equations are examples where the denominator contains an unknown. In order to solve such an equation, you need to use certain rules.

Let's look at an example:

Solve the equation 15/3x+5 = 3

Let us remember that you cannot divide by zero, i.e. the denominator value must not be zero. When solving such examples, this must be indicated. For this purpose, there is an OA (permissible value range).

So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3

At x = 5/3 the equation simply has no solution.

Having indicated the ODZ, in the best possible way Solving this equation will get rid of the fractions. To do this, we first represent all non-fractional values ​​in the form of a fraction, in in this case number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions you need to multiply each of them by the lowest common denominator. In this case it will be (3x+5)*1. Sequence of actions:

1. Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
2. Open the brackets: 15*(3x+5) = 45x + 75.
3. We do the same with right side equations: 3*(3x+5) = 9x + 15.
4. We equate left and right side: 45x + 75 = 9x +15
5. Move the X's to the left, numbers to the right: 36x = – 50
6. Find x: x = -50/36.
7. We reduce: -50/36 = -25/18

Answer: ODZ x ≠ 5/3. x = -25/18.

How to solve examples with fractions - fractional inequalities

Fractional inequalities of the type (3x-5)/(2-x)≥0 are solved using the number axis. Let's look at this example.

Sequence of actions:

• We equate the numerator and denominator to zero: 1. 3x-5=0 => 3x=5 => x=5/3
  2. 2-x=0 => x=2
• We draw a number axis, writing the resulting values ​​on it.
• Draw a circle under the value. There are two types of circles - filled and empty. A filled circle means that given value is included in the range of solutions. An empty circle indicates that this value is not included in the solution area.
• Since the denominator cannot be equal to zero, there will be an empty circle under the 2nd.

• To determine the signs, we substitute any number greater than two into the equation, for example 3. (3*3-5)/(2-3)= -4. the value is negative, which means we write a minus above the area after the two. Then substitute for X any value of the interval from 5/3 to 2, for example 1. The value is again negative. We write a minus. We repeat the same with the area located up to 5/3. We substitute any number less than 5/3, for example 1. Again, minus.

• Since we are interested in the values ​​of x at which the expression will be greater than or equal to 0, and there are no such values ​​(there are minuses everywhere), this inequality has no solution, that is, x = Ø (an empty set).

Answer: x = Ø