Shares of a unit and is represented as \frac(a)(b).

Fraction numerator (a)- the number above the line of the fraction and showing the number of shares into which the unit was divided.

Fraction denominator (b)- the number under the line of the fraction and showing how many shares the unit was divided.

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Basic property of a fraction

If ad=bc , then two fractions \frac(a)(b) and \frac(c)(d) are considered equal. For example, fractions will be equal \frac35 and \frac(9)(15), since 3 \cdot 15 = 15 \cdot 9 , \frac(12)(7) and \frac(24)(14), since 12 \cdot 14 = 7 \cdot 24 .

From the definition of the equality of fractions it follows that the fractions will be equal \frac(a)(b) and \frac(am)(bm), since a(bm)=b(am) is a clear example of the use of the associative and commutative properties of multiplication of natural numbers in action.

Means \frac(a)(b) = \frac(am)(bm)- looks like this basic property of a fraction.

In other words, we get a fraction equal to the given one by multiplying or dividing the numerator and denominator of the original fraction by the same natural number.

Fraction reduction is the process of replacing a fraction, in which the new fraction is equal to the original, but with a smaller numerator and denominator.

It is customary to reduce fractions based on the main property of a fraction.

For example, \frac(45)(60)=\frac(15)(20)(the numerator and denominator are divisible by the number 3); the resulting fraction can again be reduced by dividing by 5, i.e. \frac(15)(20)=\frac 34.

irreducible fraction is a fraction of the form \frac 34, where the numerator and denominator are relatively prime numbers. The main purpose of fraction reduction is to make the fraction irreducible.

Bringing fractions to a common denominator

Let's take two fractions as an example: \frac(2)(3) and \frac(5)(8) with different denominators 3 and 8 . In order to bring these fractions to a common denominator and first multiply the numerator and denominator of the fraction \frac(2)(3) by 8 . We get the following result: \frac(2 \cdot 8)(3 \cdot 8) = \frac(16)(24). Then multiply the numerator and denominator of the fraction \frac(5)(8) by 3 . We get as a result: \frac(5 \cdot 3)(8 \cdot 3) = \frac(15)(24). So, the original fractions are reduced to a common denominator 24.

Arithmetic operations on ordinary fractions

Addition of ordinary fractions

a) With the same denominators, the numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As seen in the example:

\frac(a)(b)+\frac(c)(b)=\frac(a+c)(b);

b) With different denominators, the fractions are first reduced to a common denominator, and then the numerators are added according to the rule a):

\frac(7)(3)+\frac(1)(4)=\frac(7 \cdot 4)(3)+\frac(1 \cdot 3)(4)=\frac(28)(12) +\frac(3)(12)=\frac(31)(12).

Subtraction of ordinary fractions

a) With the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same:

\frac(a)(b)-\frac(c)(b)=\frac(a-c)(b);

b) If the denominators of the fractions are different, then first the fractions are reduced to a common denominator, and then repeat the steps as in paragraph a).

Multiplication of ordinary fractions

Multiplication of fractions obeys the following rule:

\frac(a)(b) \cdot \frac(c)(d)=\frac(a \cdot c)(b \cdot d),

that is, multiply the numerators and denominators separately.

For example:

\frac(3)(5) \cdot \frac(4)(8) = \frac(3 \cdot 4)(5 \cdot 8)=\frac(12)(40).

Division of ordinary fractions

Fractions are divided in the following way:

\frac(a)(b) : \frac(c)(d)= \frac(ad)(bc),

that is a fraction \frac(a)(b) multiplied by a fraction \frac(d)(c).

Example: \frac(7)(2) : \frac(1)(8)=\frac(7)(2) \cdot \frac(8)(1)=\frac(7 \cdot 8)(2 \cdot 1 )=\frac(56)(2).

Reciprocal numbers

If ab=1 , then the number b is reverse number for number a .

Example: for the number 9, the reverse is \frac(1)(9), as 9 \cdot \frac(1)(9)=1, for the number 5 - \frac(1)(5), as 5 \cdot \frac(1)(5)=1.

Decimals

Decimal is a proper fraction whose denominator is 10, 1000, 10\,000, ..., 10^n .

For example: \frac(6)(10)=0.6;\enspace \frac(44)(1000)=0.044.

In the same way, incorrect numbers with a denominator 10 ^ n or mixed numbers are written.

For example: 5\frac(1)(10)=5.1;\enspace \frac(763)(100)=7\frac(63)(100)=7.63.

In the form of a decimal fraction, any ordinary fraction with a denominator that is a divisor of a certain power of the number 10 is represented.

Example: 5 is a divisor of 100 so the fraction \frac(1)(5)=\frac(1 \cdot 20)(5 \cdot 20)=\frac(20)(100)=0.2.

Arithmetic operations on decimal fractions

Adding decimals

To add two decimal fractions, you need to arrange them so that the same digits and a comma under a comma appear under each other, and then add the fractions as ordinary numbers.

Subtraction of decimals

It works in the same way as addition.

Decimal multiplication

When multiplying decimal numbers, it is enough to multiply the given numbers, ignoring the commas (as natural numbers), and in the received answer, the comma on the right separates as many digits as there are after the decimal point in both factors in total.

Let's do the multiplication of 2.7 by 1.3. We have 27 \cdot 13=351 . We separate two digits from the right with a comma (the first and second numbers have one digit after the decimal point; 1+1=2). As a result, we get 2.7 \cdot 1.3=3.51 .

If the result is fewer digits than it is necessary to separate with a comma, then the missing zeros are written in front, for example:

To multiply by 10, 100, 1000, in a decimal fraction, move the comma 1, 2, 3 digits to the right (if necessary, a certain number of zeros are assigned to the right).

For example: 1.47 \cdot 10\,000 = 14,700 .

Decimal division

Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number. A comma in the private is placed after the division of the integer part is completed.

If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:

Consider dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. First of all, we multiply the dividend and the divisor of the fraction by 100, that is, we move the comma to the right in the dividend and divisor by as many characters as there are in the divisor after the decimal point (in this example, two). Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:

It happens that the final decimal fraction is not always obtained when dividing one number by another. The result is an infinite decimal. In such cases, go to ordinary fractions.

2.8: 0.09= \frac(28)(10) : \frac (9)(100)= \frac(28 \cdot 100)(10 \cdot 9)=\frac(280)(9)= 31 \frac(1)(9).

Possess basic property of a fraction:

Remark 1

If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then as a result we get a fraction equal to the original one:

$\frac(a\cdot n)(b\cdot n)=\frac(a)(b)$

$\frac(a\div n)(b\div n)=\frac(a)(b)$

Example 1

Let a square divided into $4$ equal parts be given. If $2$ of $4$ parts are shaded, we get the shaded $\frac(2)(4)$ of the entire square. If you look at this square, it is obvious that exactly half of it is shaded, i.e. $(1)(2)$. Thus, we get $\frac(2)(4)=\frac(1)(2)$. Let's factorize the numbers $2$ and $4$:

Substitute these expansions into equality:

$\frac(1)(2)=\frac(2)(4)$,

$\frac(1)(2)=\frac(1\cdot 2)(2\cdot 2)$,

$\frac(1)(2)=\frac(2\div 2)(4\div 2)$.

Example 2

Is it possible to get an equal fraction if both the numerator and denominator of the given fraction are multiplied by $18$ and then divided by $3$?

Decision.

Let some ordinary fraction $\frac(a)(b)$ be given. By condition, the numerator and denominator of this fraction were multiplied by $ 18 $, we got:

$\frac(a\cdot 18)(b\cdot 18)$

$\frac(a\cdot 18)(b\cdot 18)=\frac(a)(b)$

$\frac(a\div 3)(b\div 3)$

According to the basic property of a fraction:

$\frac(a\div 3)(b\div 3)=\frac(a)(b)$

Thus, the resulting fraction is equal to the original.

Answer: You can get a fraction equal to the original.

Application of the basic property of a fraction

The main property of a fraction is most often used for:

• converting fractions to a new denominator:
• fraction abbreviations.

Bringing a fraction to a new denominator- replacement of a given fraction with a fraction that will be equal to it, but have a larger numerator and a larger denominator. To do this, the numerator and denominator of the fraction are multiplied by the same natural number, as a result of which, according to the main property of the fraction, a fraction is obtained that is equal to the original one, but with a larger numerator and denominator.

Fraction reduction- replacement of a given fraction with a fraction that will be equal to it, but have a smaller numerator and a smaller denominator. To do this, the numerator and denominator of the fraction are divided by a positive common divisor of the numerator and denominator, which is different from zero, as a result of which, according to the main property of the fraction, a fraction is obtained that is equal to the original one, but with a smaller numerator and denominator.

If we divide (reduce) the numerator and denominator by their GCD, then the result is irreducible form of the original fraction.

Fraction reduction

As you know, ordinary fractions are divisible by contractible and irreducible.

To reduce a fraction, you need to divide both the numerator and the denominator of the fraction by their positive common divisor, which is not equal to zero. When reducing the fraction, a new fraction is obtained with a smaller numerator and denominator, which, according to the main property of the fraction, is equal to the original one.

Example 3

Reduce the fraction $\frac(15)(25)$.

Decision.

Reduce the fraction by $5$ (divide its numerator and denominator by $5$):

$\frac(15)(25)=\frac(15\div 5)(25\div 5)=\frac(3)(5)$

Answer: $\frac(15)(25)=\frac(3)(5)$.

Getting an irreducible fraction

Most often, a fraction is reduced to obtain an irreducible fraction equal to the original reducible fraction. This result can be achieved by dividing both the numerator and denominator of the original fraction by their GCD.

$\frac(a\div gcd (a,b))(b\div gcd (a,b))$ is an irreducible fraction, because according to the properties of GCD, the numerator and denominator of a given fraction are coprime numbers.

gcd(a,b) is the largest number by which both the numerator and denominator of the fraction $\frac(a)(b)$ can be divided. Thus, to reduce a fraction to an irreducible form, it is necessary to divide its numerator and denominator by their gcd.

Remark 2

Fraction reduction rule: 1. Find the GCD of two numbers that are in the numerator and denominator of the fraction. 2. Perform the division of the numerator and denominator of the fraction by the found GCD.

Example 4

Reduce the fraction $6/36$ to an irreducible form.

Decision.

Let's reduce this fraction by GCD$(6,36)=6$, because $36\div 6=6$. We get:

$\frac(6)(36)=\frac(6\div 6)(36\div 6)=\frac(1)(6)$

Answer: $\frac(6)(36)=\frac(1)(6)$.

In practice, the phrase "reduce a fraction" means that you need to reduce the fraction to an irreducible form.

When studying ordinary fractions, we encounter the concepts of the main property of a fraction. A simplified form is necessary for solving examples with ordinary fractions. This article involves the consideration of algebraic fractions and the application to them of the main property, which will be formulated with examples of its application.

Formulation and rationale

The main property of a fraction has a formulation of the form:

Definition 1

When simultaneously multiplying or dividing the numerator and denominator by the same number, the value of the fraction remains unchanged.

That is, we get that a · m b · m = a b and a: m b: m = a b are equivalent, where a b = a · m b · m and a b = a: m b: m are considered valid. The values ​​a , b , m are some natural numbers.

Dividing the numerator and denominator by a number can be represented as a · m b · m = a b . This is similar to solving example 8 12 = 8: 4 12: 4 = 2 3 . When dividing, an equality of the form a is used: m b: m \u003d a b, then 8 12 \u003d 2 4 2 4 \u003d 2 3. It can also be represented as a m b m \u003d a b, that is, 8 12 \u003d 2 4 3 4 \u003d 2 3.

That is, the main property of the fraction a · m b · m = a b and a b = a · m b · m will be considered in detail in contrast to a: m b: m = a b and a b = a: m b: m .

If the numerator and denominator contain real numbers, then the property applies. We must first prove the validity of the written inequality for all numbers. That is, prove the existence of a · m b · m = a b for all real a , b , m , where b and m are non-zero values ​​to avoid division by zero.

Proof 1

Let a fraction of the form a b be considered part of the record z, in other words, a b = z, then it is necessary to prove that a · m b · m corresponds to z, that is, to prove a · m b · m = z. Then this will allow us to prove the existence of the equality a · m b · m = a b .

The fraction bar means the division sign. Applying the relationship with multiplication and division, we get that from a b = z after transformation we get a = b · z . According to the properties of numerical inequalities, both parts of the inequality should be multiplied by a number other than zero. Then we multiply by the number m, we get that a · m = (b · z) · m . By property, we have the right to write the expression in the form a · m = (b · m) · z . Hence, it follows from the definition that a b = z . That's all the proof of the expression a · m b · m = a b .

Equalities of the form a · m b · m = a b and a b = a · m b · m make sense when instead of a , b , m there are polynomials, and instead of b and m they are non-zero.

The main property of an algebraic fraction: when you simultaneously multiply the numerator and denominator by the same number, we get an identically equal to the original expression.

The property is considered fair, since operations with polynomials correspond to operations with numbers.

Example 1

Consider the example of the fraction 3 · x x 2 - x y + 4 · y 3 . It is possible to convert to the form 3 x (x 2 + 2 x y) (x 2 - x y + 4 y 3) (x 2 + 2 x y).

Multiplication by the polynomial x 2 + 2 · x · y was performed. In the same way, the main property helps to get rid of x 2, which is present in the fraction of the form 5 x 2 (x + 1) x 2 (x 3 + 3) given by the condition, to the form 5 x + 5 x 3 + 3. This is called simplification.

The main property can be written as expressions a · m b · m = a b and a b = a · m b · m , when a , b , m are polynomials or ordinary variables, and b and m must be non-zero.

Scope of application of the main property of an algebraic fraction

The use of the main property is relevant for reduction to a new denominator or when reducing a fraction.

Definition 2

Reduction to a common denominator is the multiplication of the numerator and denominator by a similar polynomial to obtain a new one. The resulting fraction is equal to the original.

That is, a fraction of the form x + y x 2 + 1 (x + 1) x 2 + 1 when multiplied by x 2 + 1 and reduced to a common denominator (x + 1) (x 2 + 1) will get the form x 3 + x + x 2 y + y x 3 + x + x 2 + 1 .

After performing operations with polynomials, we get that the algebraic fraction is converted to x 3 + x + x 2 y + y x 3 + x + x 2 + 1.

Reduction to a common denominator is also performed when adding or subtracting fractions. If fractional coefficients are given, then it is first necessary to make a simplification, which will simplify the form and the very finding of the common denominator. For example, 2 5 x y - 2 x + 1 2 = 10 2 5 x y - 2 10 x + 1 2 = 4 x y - 20 10 x + 5.

The application of the property when reducing fractions is performed in 2 stages: decomposing the numerator and denominator into factors to find the common m, then making the transition to the form of the fraction a b , based on the equality of the form a · m b · m = a b .

If a fraction of the form 4 x 3 - x y 16 x 4 - y 2 after decomposition is converted to x (4 x 2 - y) 4 x 2 - y 4 x 2 + y, it is obvious that the general the multiplier is the polynomial 4 · x 2 − y . Then it will be possible to reduce the fraction according to its main property. We get that

x (4 x 2 - y) 4 x 2 - y 4 x 2 + y = x 4 x 2 + y. The fraction is simplified, then when substituting the values, it will be necessary to perform much less actions than when substituting into the original one.

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From the algebra course of the school curriculum, we turn to the specifics. In this article, we will study in detail a special kind of rational expressions − rational fractions, and also analyze what characteristic identical transformations of rational fractions take place.

We note right away that rational fractions in the sense in which we define them below are called algebraic fractions in some algebra textbooks. That is, in this article we will understand the same thing under rational and algebraic fractions.

As usual, we start with a definition and examples. Next, let's talk about bringing a rational fraction to a new denominator and about changing the signs of the members of the fraction. After that, we will analyze how the reduction of fractions is performed. Finally, let us dwell on the representation of a rational fraction as a sum of several fractions. All information will be provided with examples with detailed descriptions of solutions.

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Definition and examples of rational fractions

Rational fractions are studied in algebra lessons in grade 8. We will use the definition of a rational fraction, which is given in the algebra textbook for grades 8 by Yu. N. Makarychev and others.

This definition does not specify whether the polynomials in the numerator and denominator of a rational fraction must be polynomials of standard form or not. Therefore, we will assume that rational fractions can contain both standard and non-standard polynomials.

Here are a few examples of rational fractions. So , x/8 and - rational fractions. And fractions and do not fit the sounded definition of a rational fraction, since in the first of them the numerator is not a polynomial, and in the second both the numerator and the denominator contain expressions that are not polynomials.

Converting the numerator and denominator of a rational fraction

The numerator and denominator of any fraction are self-sufficient mathematical expressions, in the case of rational fractions they are polynomials, in a particular case they are monomials and numbers. Therefore, with the numerator and denominator of a rational fraction, as with any expression, identical transformations can be carried out. In other words, the expression in the numerator of a rational fraction can be replaced by an expression that is identically equal to it, just like the denominator.

In the numerator and denominator of a rational fraction, identical transformations can be performed. For example, in the numerator, you can group and reduce similar terms, and in the denominator, the product of several numbers can be replaced by its value. And since the numerator and denominator of a rational fraction are polynomials, it is possible to perform transformations characteristic of polynomials with them, for example, reduction to a standard form or representation as a product.

For clarity, consider the solutions of several examples.

Example.

Convert Rational Fraction so that the numerator is a polynomial of the standard form, and the denominator is the product of polynomials.

Decision.

Reducing rational fractions to a new denominator is mainly used when adding and subtracting rational fractions.

Changing signs in front of a fraction, as well as in its numerator and denominator

The basic property of a fraction can be used to change the signs of the terms of the fraction. Indeed, multiplying the numerator and denominator of a rational fraction by -1 is tantamount to changing their signs, and the result is a fraction that is identically equal to the given one. Such a transformation has to be used quite often when working with rational fractions.

Thus, if you simultaneously change the signs of the numerator and denominator of a fraction, you will get a fraction equal to the original one. This statement corresponds to equality.

Let's take an example. A rational fraction can be replaced by an identically equal fraction with reversed signs of the numerator and denominator of the form.

With fractions, one more identical transformation can be carried out, in which the sign is changed either in the numerator or in the denominator. Let's go over the appropriate rule. If you replace the sign of a fraction together with the sign of the numerator or denominator, you get a fraction that is identically equal to the original. The written statement corresponds to the equalities and .

It is not difficult to prove these equalities. The proof is based on the properties of multiplication of numbers. Let's prove the first of them: . With the help of similar transformations, the equality is also proved.

For example, a fraction can be replaced by an expression or .

To conclude this subsection, we present two more useful equalities and . That is, if you change the sign of only the numerator or only the denominator, then the fraction will change its sign. For example, and .

The considered transformations, which allow changing the sign of the terms of a fraction, are often used when transforming fractionally rational expressions.

Reduction of rational fractions

The following transformation of rational fractions, called the reduction of rational fractions, is based on the same basic property of a fraction. This transformation corresponds to the equality , where a , b and c are some polynomials, and b and c are non-zero.

From the above equality, it becomes clear that the reduction of a rational fraction implies getting rid of the common factor in its numerator and denominator.

Example.

Reduce the rational fraction.

Decision.

The common factor 2 is immediately visible, let's reduce it (when writing, it is convenient to cross out the common factors by which the reduction is made). We have . Since x 2 \u003d x x and y 7 \u003d y 3 y 4 (see if necessary), it is clear that x is a common factor of the numerator and denominator of the resulting fraction, like y 3 . Let's reduce by these factors: . This completes the reduction.

Above, we performed the reduction of a rational fraction sequentially. And it was possible to perform the reduction in one step, immediately reducing the fraction by 2·x·y 3 . In this case, the solution would look like this: .

Answer:

.

When reducing rational fractions, the main problem is that the common factor of the numerator and denominator is not always visible. Moreover, it does not always exist. In order to find a common factor or make sure that it does not exist, you need to factorize the numerator and denominator of a rational fraction. If there is no common factor, then the original rational fraction does not need to be reduced, otherwise, the reduction is carried out.

In the process of reducing rational fractions, various nuances may arise. The main subtleties with examples and details are discussed in the article reduction of algebraic fractions.

Concluding the conversation about the reduction of rational fractions, we note that this transformation is identical, and the main difficulty in its implementation lies in the factorization of polynomials in the numerator and denominator.

Representation of a rational fraction as a sum of fractions

Quite specific, but in some cases very useful, is the transformation of a rational fraction, which consists in its representation as the sum of several fractions, or the sum of an integer expression and a fraction.

A rational fraction, in the numerator of which there is a polynomial, which is the sum of several monomials, can always be written as the sum of fractions with the same denominators, in the numerators of which are the corresponding monomials. For example, . This representation is explained by the rule of addition and subtraction of algebraic fractions with the same denominators.

In general, any rational fraction can be represented as a sum of fractions in many different ways. For example, the fraction a/b can be represented as the sum of two fractions - an arbitrary fraction c/d and a fraction equal to the difference between the fractions a/b and c/d. This statement is true, since the equality . For example, a rational fraction can be represented as a sum of fractions in various ways: We represent the original fraction as the sum of an integer expression and a fraction. After dividing the numerator by the denominator by a column, we get the equality . The value of the expression n 3 +4 for any integer n is an integer. And the value of a fraction is an integer if and only if its denominator is 1, −1, 3, or −3. These values ​​correspond to the values ​​n=3 , n=1 , n=5 and n=−1 respectively.

Answer:

−1 , 1 , 3 , 5 .

Bibliography.

• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. 7th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 13th ed., Rev. - M.: Mnemosyne, 2009. - 160 p.: ill. ISBN 978-5-346-01198-9.
• Mordkovich A. G. Algebra. 8th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemozina, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Speaking of mathematics, one cannot help but remember fractions. Their study is given a lot of attention and time. 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 many nerves were spent to find a common denominator, especially if there were more than two terms in the examples!

Let's remember what it is, and refresh our memory a little about the basic information and rules for working with fractions.

Definition of fractions

Let's start with the most important thing - definitions. A fraction is a number that consists of one or more unit parts. A fractional number is written as two numbers separated by a horizontal or slash. In this case, the upper (or first) is called the numerator, and the lower (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 they are correct, 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 analyze such a concept as "the main property of a 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 are the type of record already indicated by us 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, right and wrong numbers are distinguished. For the former, the numerator is always less than the denominator. Note also that such a fraction is less than unity. In an improper fraction, on the contrary, the numerator is greater than the denominator, and it itself is greater than one. In this case, an integer can be extracted from it. In this article, we will consider only ordinary fractions.

Fraction properties

Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers are no exception. They have one important feature, with the help of which it is possible to carry out certain operations on them. What is the main property of a fraction? The rule says that if its numerator and denominator are multiplied or divided by the same rational number, we will get a new fraction, the value of which will be equal to the original value. That is, multiplying the two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, while 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 to us, they can also perform basic mathematical operations, such as addition and subtraction, multiplication and division. In addition, there is such a specific action as the reduction of fractions. Naturally, each of these actions is performed according to certain rules. Knowing these laws makes it easier to work with fractions, making it easier and more interesting. That is why further we will consider the basic rules and the algorithm of actions when working with such numbers.

But before we talk about such mathematical operations as addition and subtraction, we will analyze such an operation as reduction to a common denominator. This is where the knowledge of what basic property of a fraction exists will come 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, the smallest number that is simultaneously divisible by both denominators without a remainder. The easiest way to find the LCM (least common multiple) is to write in a line for one denominator, then for the second and find a matching number among them. In the event that the LCM is not found, that is, these numbers do not have a common multiple, they should be multiplied, and the resulting value should be considered as the LCM.

So, we have found the LCM, now we need to find an additional multiplier. To do this, you need to alternately divide the LCM into denominators of fractions and write down the resulting number over each of them. Next, 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 main property of the fraction.

Addition

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

If the fractions have different denominators, they should be reduced to a common one and only then the addition should be performed. How to do this, we have discussed with you a little higher. In this situation, the main property of the fraction will come in handy. The rule will allow you to bring the 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

It does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of the 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 product of 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 table, as well as attentiveness. In addition, after receiving the result, you should definitely check whether this number can be reduced or not. We will talk about how to reduce fractions a little later.

Subtraction

Performing 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. In the event that the fractions have different denominators, you should bring them to a common one and then perform this operation. As with the analogous addition case, you will need to use the basic property of an algebraic fraction, as well as skills in finding the LCM and common factors for fractions.

Division

And the last, most interesting operation when working with such numbers is division. It is quite simple and does not cause any particular difficulties even for those who do not understand how to work with fractions, especially to perform addition and subtraction operations. When dividing, such a rule applies as multiplication 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 is reversed, i.e. the numerator and denominator are reversed. After that, 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 given 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 result obtained in the end and find out whether it is possible to reduce the resulting fraction or not. Remember that the final result is always written as a fractional number that does not require reduction.

Other operations

Finally, we note that we have listed far from all operations on fractional numbers, mentioning only the most famous 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 that we have given above.

findings

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

This article is intended to refresh the information you have forgotten about fractions, rather than give new information and "fill" your head with endless rules and formulas, which, most likely, will not be useful to you.

We hope that the material presented in the article simply and concisely has become useful to you.