The main property of an algebraic fraction: formulation, proof, application examples. Basic property of a fraction

Dismantled in detail basic property of a fraction, its formulation is given, a proof and an explanatory example are given. The application of the main property of a fraction in the reduction of fractions and reduction of fractions to a new denominator is also considered.

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The main property of a fraction - formulation, proof and explanatory examples

Let's look at an example that illustrates the basic property of a fraction. Let's say we have a square divided into 9 "big" squares, and each of these "big" squares is divided into 4 "small" squares. Thus, we can also say that the original square is divided into 4·9=36 "small" squares. Let's paint over 5 "big" squares. In this case, 4 5 = 20 “small” squares will be filled in. We present a figure corresponding to our example.

The shaded part is 5/9 of the original square, or, which is the same, 20/36 of the original square, that is, the fractions 5/9 and 20/36 are equal: or . From these equalities, as well as from the equalities 20=5 4 , 36=9 4 , 20:4=5 and 36:4=9, it follows that and .

To consolidate the disassembled material, consider the solution of the example.

Example.

The numerator and denominator of some ordinary fraction were multiplied by 62, after which the numerator and denominator of the resulting fraction were divided by 2. Is the resulting fraction equal to the original?

Decision.

Multiplying the numerator and denominator of a fraction by any natural number, in particular by 62, gives a fraction, which, due to the main property of the fraction, is equal to the original one. The main property of a fraction allows us to assert that after dividing the numerator and denominator of the resulting fraction by 2, a fraction will be obtained that will be equal to the original fraction.

Answer:

Yes, the resulting fraction is equal to the original.

Application of the basic property of a fraction

The basic property of a fraction is mainly applied in two cases: firstly, when reducing fractions to a new denominator, and, secondly, when reducing fractions.

The main property of a fraction allows you to reduce fractions, and as a result, move from the original fraction to a fraction equal to it, but with a smaller numerator and denominator. Fraction reduction consists in dividing the numerator and denominator of the original fraction by any positive numerator and denominator other than one (if there are no such common divisors, then the original fraction is irreducible, that is, cannot be reduced). In particular, division by will bring the original fraction to an irreducible form.

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.

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In this article, we will analyze what the main property of a fraction is, formulate it, give a proof and a good example. Then we will consider how to apply the basic property of a fraction when performing the actions of reducing fractions and bringing fractions to a new denominator.

All ordinary fractions have the most important property, which we call the basic property of a fraction, and it sounds like this:

Definition 1

If the numerator and denominator of one fraction are multiplied or divided by the same natural number, then the result will be a fraction equal to the given one.

Let's represent the main property of a fraction in the form of equality. For natural numbers a , b and m the equalities will be valid:

a m b m = a b and a: m b: m = a b

Consider the proof of the main property of a fraction. Based on the properties of multiplication of natural numbers and the properties of division of natural numbers, we write the equalities: (a · m) · b = (b · m) · a and (a: m) · b = (b: m) · a. So the fractions a m b m and a b , as well as a: m b: m and a b are equal by the definition of equality of fractions.

Let's look at an example that graphically illustrates the main property of a fraction.

Example 1

Let's say we have a square divided into 9 "big" parts-squares. Each "big" square is divided into 4 smaller ones. It is possible to say that the given square is divided into 4 9 = 36 "small" squares. Highlight 5 "large" squares with color. In this case, 4 · 5 = 20 "small" squares will be colored. Let's show a picture demonstrating our actions:

The colored part is 59 of the original figure or 2036 which is the same. Thus, the fractions 5 9 and 20 36 are equal: 5 9 = 20 36 or 20 36 = 5 9 .

These equalities, as well as the equalities 20 = 4 5, 36 = 4 9, 20: 4 = 5 and 36: 4 = 9, make it possible to conclude that 5 9 = 5 4 9 4 and 20 36 = 20 4 36 4 .

To consolidate the theory, we will analyze the solution of an example.

Example 2

It is given that the numerator and denominator of some ordinary fraction were multiplied by 47, after which these numerator and denominator were divided by 3. Is the resulting fraction equal to the given one?

Decision

Based on the basic property of a fraction, we can say that multiplying the numerator and denominator of a given fraction by a natural number 47 will result in a fraction equal to the original one. We can assert the same thing by dividing further by 3. Ultimately, we will get a fraction equal to the given one.

Answer: Yes, the resulting fraction will be equal to the original.

Application of the basic property of a fraction

The main property is used when you need to bring fractions to a new denominator and when reducing fractions.

Reducing a fraction to a new denominator is the act of replacing a given fraction with a fraction equal to it, but with a larger numerator and denominator. To bring a fraction to a new denominator, you need to multiply the numerator and denominator of the fraction by the required natural number. Operations with ordinary fractions would be impossible without a way to bring fractions to a new denominator.

Definition 2

Fraction reduction- the action of the transition to a new fraction equal to the given one, but with a smaller numerator and denominator. To reduce a fraction, you need to divide the numerator and denominator of the fraction by the same necessary natural number, which will be called common divisor.

There are cases when there is no such common divisor, then they say that the original fraction is irreducible or cannot be reduced. In particular, reducing a fraction by using the greatest common factor will make the fraction irreducible.

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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.

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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This topic is quite important on the basic properties of fractions, all further mathematics and algebra are based. The considered properties of fractions, despite their importance, are very simple.

To understand basic properties of fractions consider a circle.

It can be seen on the circle that 4 parts or are shaded out of eight possible. Write the resulting fraction \(\frac(4)(8)\)

The next circle shows that one of the two possible parts is shaded. Write the resulting fraction \(\frac(1)(2)\)

If we look closely, we will see that in the first case, that in the second case, half of the circle is shaded, so the resulting fractions are equal to \(\frac(4)(8) = \frac(1)(2)\), that is it's the same number.

How can this be proved mathematically? Very simply, remember the multiplication table and write the first fraction into factors.

\(\frac(4)(8) = \frac(1 \cdot \color(red) (4))(2 \cdot \color(red) (4)) = \frac(1)(2) \cdot \color(red) (\frac(4)(4)) =\frac(1)(2) \cdot \color(red)(1) = \frac(1)(2)\)

What have we done? We factored the numerator and denominator \(\frac(1 \cdot \color(red) (4))(2 \cdot \color(red) (4))\), and then divided the fractions \(\frac(1) (2) \cdot \color(red) (\frac(4)(4))\). Four divided by four is 1, and one multiplied by any number is the number itself. What we have done in the above example is called reduction of fractions.

Let's look at another example and reduce the fraction.

\(\frac(6)(10) = \frac(3 \cdot \color(red) (2))(5 \cdot \color(red) (2)) = \frac(3)(5) \cdot \color(red) (\frac(2)(2)) =\frac(3)(5) \cdot \color(red)(1) = \frac(3)(5)\)

We again painted the numerator and denominator into factors and reduced the same numbers into numerators and denominators. That is, two divided by two gave one, and one multiplied by any number gives the same number.

Basic property of a fraction.

This implies the main property of a fraction:

If both the numerator and the denominator of a fraction are multiplied by the same number (except zero), then the value of the fraction will not change.

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

You can also divide the numerator and denominator by the same number at the same time.
Consider an example:

\(\frac(6)(8) = \frac(6 \div \color(red) (2))(8 \div \color(red) (2)) = \frac(3)(4)\)

If both the numerator and the denominator of a fraction are divided by the same number (except zero), then the value of the fraction will not change.

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

Fractions that have common prime divisors in both numerators and denominators are called cancellable fractions.

Cancellative example: \(\frac(2)(4), \frac(6)(10), \frac(9)(15), \frac(10)(5), …\)

There is also irreducible fractions.

irreducible fraction is a fraction that does not have common prime divisors in the numerators and denominators.

An irreducible fraction example: \(\frac(1)(2), \frac(3)(5), \frac(5)(7), \frac(13)(5), …\)

Any number can be represented as a fraction, because any number is divisible by one, For example:

\(7 = \frac(7)(1)\)

Questions to the topic:
Do you think any fraction can be reduced or not?
Answer: No, there are reducible fractions and irreducible fractions.

Check if the equality is true: \(\frac(7)(11) = \frac(14)(22)\)?
Answer: write a fraction \(\frac(14)(22) = \frac(7 \cdot 2)(11 \cdot 2) = \frac(7)(11)\) yes fair.

Example #1:
a) Find a fraction with a denominator of 15 that is equal to the fraction \(\frac(2)(3)\).
b) Find a fraction with a numerator of 8, equal to the fraction \(\frac(1)(5)\).

Decision:
a) We need the denominator to be the number 15. Now the denominator is the number 3. By what number should the number 3 be multiplied to get 15? Recall the multiplication table 3⋅5. We need to use the basic property of fractions and multiply both the numerator and the denominator of the fraction \(\frac(2)(3)\) by 5.

\(\frac(2)(3) = \frac(2 \cdot 5)(3 \cdot 5) = \frac(10)(15)\)

b) We need the number 8 in the numerator. Now the number 1 is in the numerator. By what number should the number 1 be multiplied to get 8? Of course, 1⋅8. We need to use the basic property of fractions and multiply both the numerator and the denominator of the fraction \(\frac(1)(5)\) by 8. We get:

\(\frac(1)(5) = \frac(1 \cdot 8)(5 \cdot 8) = \frac(8)(40)\)

Example #2:
Find an irreducible fraction equal to a fraction: a) \(\frac(16)(36)\), b) \(\frac(10)(25)\).

Decision:
a) \(\frac(16)(36) = \frac(4 \cdot 4)(9 \cdot 4) = \frac(4)(9)\)

b) \(\frac(10)(25) = \frac(2 \cdot 5)(5 \cdot 5) = \frac(2)(5)\)

Example #3:
Write the number as a fraction: a) 13 b) 123

Decision:
a) \(13 = \frac(13) (1)\)

b) \(123 = \frac(123) (1)\)