Decimal numbers such as 0.2; 1.05; 3.017, etc. as they are heard, so they are written. Zero point two, we get a fraction. One point five hundredths, we get a fraction. Three point seventeen thousandths, we get the fraction. The numbers before the decimal point are the whole part of the fraction. The number after the decimal point is the numerator of the future fraction. If there is a single-digit number after the decimal point, the denominator will be 10, if there is a two-digit number - 100, a three-digit number - 1000, etc. Some resulting fractions can be reduced. In our examples

Converting a fraction to a decimal

This is the reverse of the previous transformation. What is the characteristic of a decimal fraction? Its denominator is always 10, or 100, or 1000, or 10000, and so on. If your common fraction has a denominator like this, there's no problem. For example, or

If the fraction is, for example . In this case, it is necessary to use the basic property of a fraction and convert the denominator to 10 or 100, or 1000... In our example, if we multiply the numerator and denominator by 4, we get a fraction that can be written as a decimal number 0.12.

Some fractions are easier to divide than to convert the denominator. For example,

Some fractions cannot be converted to decimals!
For example,

Converting a mixed fraction to an improper fraction

A mixed fraction, for example, can be easily converted to an improper fraction. To do this, you need to multiply the whole part by the denominator (bottom) and add it with the numerator (top), leaving the denominator (bottom) unchanged. That is

When converting a mixed fraction to an improper fraction, you can remember that you can use fraction addition

Converting an improper fraction to a mixed fraction (highlighting the whole part)

An improper fraction can be converted to a mixed fraction by highlighting the whole part. Let's look at an example. We determine how many integer times “3” fits into “23”. Or divide 23 by 3 on a calculator, the whole number to the decimal point is the desired one. This is "7". Next, we determine the numerator of the future fraction: we multiply the resulting “7” by the denominator “3” and subtract the result from the numerator “23”. It’s as if we find the extra that remains from the numerator “23” if we remove the maximum amount of “3”. We leave the denominator unchanged. Everything is done, write down the result

From the school algebra course we move on to specifics. In this article we will study in detail a special type of rational expressions - rational fractions, and also consider what characteristic identical rational fraction conversions take place.

Let us immediately note 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 rational and algebraic fractions to mean the same thing.

As usual, let's start with a definition and examples. Next we’ll talk about bringing a rational fraction to a new denominator and changing the signs of the members of the fraction. After this, we will look at how to reduce fractions. Finally, let's look at representing a rational fraction as a sum of several fractions. We will provide all information with examples and detailed descriptions of solutions.

Page navigation.

Definition and examples of rational fractions

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

This definition does not specify whether the polynomials in the numerator and denominator of a rational fraction must be polynomials of the standard form or not. Therefore, we will assume that the notations for 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 stated definition of a rational fraction, since in the first of them the numerator does not contain 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, these are polynomials; in a particular case, monomials and numbers. Therefore, identical transformations can be carried out with the numerator and denominator of a rational fraction, as with any expression. In other words, the expression in the numerator of a rational fraction can be replaced by an identically equal expression, just like the denominator.

You can perform identical transformations in the numerator and denominator of a rational fraction. For example, in the numerator you can group and reduce similar terms, and in the denominator you can replace the product of several numbers with 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 in the form of a product.

For clarity, let's consider solutions to several examples.

Example.

Convert rational fraction so that the numerator contains a polynomial of standard form, and the denominator contains the product of polynomials.

Solution.

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

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

The main property of a fraction can be used to change the signs of the members of a fraction. Indeed, multiplying the numerator and denominator of a rational fraction by -1 is equivalent to changing their signs, and the result is a fraction identically equal to the given one. This 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 is answered by equality.

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

With fractions, you can carry out another identical transformation, in which the sign of either the numerator or the denominator changes. Let us state the corresponding 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 one. The written statement corresponds to the equalities and .

Proving these equalities is not difficult. The proof is based on the properties of multiplication of numbers. Let's prove the first of them: . Using similar transformations, the equality is proved.

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

To conclude this point, we present two more useful equalities and . That is, if you change the sign of only the numerator or only the denominator, 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 fractional rational expressions.

Reducing rational fractions

The following transformation of rational fractions, called 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 reducing a rational fraction implies getting rid of the common factor in its numerator and denominator.

Example.

Cancel a rational fraction.

Solution.

The common factor 2 is immediately visible, let’s perform a reduction by it (when writing, it is convenient to cross out the common factors that are being reduced by). We have . Since x 2 =x·x and y 7 =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, as is y 3. Let's reduce by these factors: . This completes the reduction.

Above we carried out the reduction of rational fractions sequentially. Or 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 verify its absence, you need to factor 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, reduction is carried out.

Various nuances can arise in the process of reducing rational fractions. The main subtleties are discussed in the article reducing algebraic fractions using examples and in detail.

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 factoring the 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 entire expression and a fraction.

A rational fraction, the numerator of which contains a polynomial representing the sum of several monomials, can always be written as a sum of fractions with the same denominators, the numerators of which contain the corresponding monomials. For example, . This representation is explained by the rule for adding and subtracting algebraic fractions with like denominators.

In general, any rational fraction can be expressed 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 holds . For example, a rational fraction can be represented as a sum of fractions in various ways: Let's imagine the original fraction as the sum of an integer expression and a fraction. By dividing the numerator by the denominator with 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 .

References.

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

The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation.

That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized).

If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).

To reinforce this, solve a few examples yourself:

Examples:

Solutions:

1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:

The first step should be factorization:

4. Adding and subtracting fractions. Reducing fractions to a common denominator.

Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators.

Let's remember:

Answers:

1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. Here, first of all, we convert mixed fractions into improper ones, and then according to the usual scheme:

It's a completely different matter if the fractions contain letters, for example:

Let's start with something simple:

a) Denominators do not contain letters

Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:

Now in the numerator you can give similar ones, if any, and factor them:

Try it yourself:

Answers:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· first of all, we determine the common factors;

· then we write out all the common factors one at a time;

· and multiply them by all other non-common factors.

To determine the common factors of the denominators, we first factor them into prime factors:

Let us emphasize the common factors:

Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

· factor the denominators;

· determine common (identical) factors;

· write out all common factors once;

· multiply them by all other non-common factors.

So, in order:

1) factor the denominators:

2) determine common (identical) factors:

3) write out all common factors once and multiply them by all other (unemphasized) factors:

So there's a common denominator here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to a degree

to a degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?

So, another unshakable rule:

When you reduce fractions to a common denominator, use only the multiplication operation!

But what do you need to multiply by to get?

So multiply by. And multiply by:

We will call expressions that cannot be factorized “elementary factors.”

For example, - this is an elementary factor. - Same. But no: it can be factorized.

What about the expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic “”).

So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.

We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).

The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:

Great! Then:

Another example:

Solution:

As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:

So let's write:

That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now let's bring it to a common denominator:

Got it? Let's check it now.

Tasks for independent solution:

Answers:

Here we need to remember one more thing - the difference of cubes:

Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .

A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:

What to do if there are already three fractions?

Yes, the same thing! First of all, let’s make sure that the maximum number of factors in the denominators is the same:

Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.

We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:

Hmm... It’s clear what to do with fractions. But what about the two?

It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:

Just what you need!

5. Multiplication and division of fractions.

Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:

Did you count?

It should work.

So, let me remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in brackets is evaluated out of turn!

If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.

What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But this is not the same as an expression with letters?

No, it's the same! Only instead of arithmetic operations, you need to do algebraic ones, that is, the actions described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.

Usually our goal is to represent the expression as a product or quotient.

For example:

Let's simplify the expression.

1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).

2) We get:

Multiplying fractions: what could be simpler.

3) Now you can shorten:

Well, that's all. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

Solution:

First of all, let's determine the order of actions.

First, let's add the fractions in parentheses, so instead of two fractions we get one.

Then we will do division of fractions. Well, let's add the result with the last fraction.

I will number the steps schematically:

Now I’ll show you the process, tinting the current action in red:

1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.

2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And what was promised at the very beginning:

Answers:

Solutions (brief):

If you have coped with at least the first three examples, then consider yourself to have mastered the topic.

Now on to learning!

CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS

Basic simplification operations:

Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
Factorization: putting the common factor out of brackets, applying it, etc.
Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
1) numerator and denominator factorize
2) if the numerator and denominator have common factors, they can be crossed out.
IMPORTANT: only multipliers can be reduced!
Adding and subtracting fractions:
;
Multiplying and dividing fractions:
;

Simplifying algebraic expressions is one of the keys to learning algebra and is an extremely useful skill for all mathematicians. Simplification allows you to reduce a complex or long expression to a simple expression that is easy to work with. Basic skills of simplification are good even for those who are not enthusiastic about mathematics. By following a few simple rules, you can simplify many of the most common types of algebraic expressions without any special mathematical knowledge.

Steps

Important Definitions

Similar members . These are members with a variable of the same order, members with the same variables, or free members (members that do not contain a variable). In other words, similar terms include the same variable to the same degree, include several of the same variables, or do not include a variable at all. The order of the terms in the expression does not matter.
- For example, 3x 2 and 4x 2 are similar terms because they contain a second-order (to the second power) variable "x". However, x and x2 are not similar terms, since they contain the variable “x” of different orders (first and second). Likewise, -3yx and 5xz are not similar terms because they contain different variables.
Factorization . This is finding numbers whose product leads to the original number. Any original number can have several factors. For example, the number 12 can be factored into the following series of factors: 1 × 12, 2 × 6 and 3 × 4, so we can say that the numbers 1, 2, 3, 4, 6 and 12 are factors of the number 12. The factors are the same as the factors , that is, the numbers by which the original number is divided.
- For example, if you want to factor the number 20, write it like this: 4×5.
- Note that when factoring, the variable is taken into account. For example, 20x = 4(5x).
- Prime numbers cannot be factored because they are only divisible by themselves and 1.
Remember and follow the order of operations to avoid mistakes.
- Parentheses
- Degree
- Multiplication
- Division
- Addition
- Subtraction
Bringing similar members
1. Write down the expression. Simple algebraic expressions (those that don't contain fractions, roots, etc.) can be solved (simplified) in just a few steps.
  - For example, simplify the expression 1 + 2x - 3 + 4x.
2. Define similar terms (terms with the same variable, terms with the same variables, or free terms).
  - Find similar terms in this expression. The terms 2x and 4x contain a variable of the same order (first). Also, 1 and -3 are free terms (do not contain a variable). Thus, in this expression the terms 2x and 4x are similar, and the members 1 and -3 are also similar.
3. Give similar members. This means adding or subtracting them and simplifying the expression.
  - 2x + 4x = 6x
  - 1 - 3 = -2
4. Rewrite the expression taking into account the given terms. You will get a simple expression with fewer terms. The new expression is equal to the original one.
  - In our example: 1 + 2x - 3 + 4x = 6x - 2, that is, the original expression is simplified and easier to work with.
5. Follow the order of operations when bringing similar members. In our example it was easy to provide similar terms. However, in the case of complex expressions in which terms are enclosed in parentheses and fractions and roots are present, it is not so easy to bring such terms. In these cases, follow the order of operations.
  - For example, consider the expression 5(3x - 1) + x((2x)/(2)) + 8 - 3x. Here it would be a mistake to immediately define 3x and 2x as similar terms and present them, because it is necessary to open the parentheses first. Therefore, perform the operations according to their order.
    - 5(3x-1) + x((2x)/(2)) + 8 - 3x
    - 15x - 5 + x(x) + 8 - 3x
    - 15x - 5 + x 2 + 8 - 3x. Now, when the expression contains only addition and subtraction operations, you can bring similar terms.
    - x 2 + (15x - 3x) + (8 - 5)
    - x 2 + 12x + 3
Taking the multiplier out of brackets
1. Find greatest common divisor(GCD) of all coefficients of the expression. GCD is the largest number by which all coefficients of the expression are divided.
  - For example, consider the equation 9x 2 + 27x - 3. In this case, GCD = 3, since any coefficient of this expression is divisible by 3.
2. Divide each term of the expression by gcd. The resulting terms will contain smaller coefficients than in the original expression.
  - In our example, divide each term in the expression by 3.
    - 9x 2 /3 = 3x 2
    - 27x/3 = 9x
    - -3/3 = -1
    - The result was an expression 3x 2 + 9x - 1. It is not equal to the original expression.
3. Write down the original expression as equal to the product of gcd and the resulting expression. That is, enclose the resulting expression in brackets, and take the gcd out of the brackets.
  - In our example: 9x 2 + 27x - 3 = 3(3x 2 + 9x - 1)
4. Simplifying fractional expressions by putting the factor out of brackets. Why simply put the multiplier out of brackets, as was done earlier? Then, to learn how to simplify complex expressions, such as fractional expressions. In this case, putting the factor out of brackets can help get rid of the fraction (from the denominator).
  - For example, consider the fractional expression (9x 2 + 27x - 3)/3. Use factoring out to simplify this expression.
    - Put the factor of 3 out of brackets (as you did earlier): (3(3x 2 + 9x - 1))/3
    - Notice that there is now a 3 in both the numerator and the denominator. This can be reduced to give the expression: (3x 2 + 9x – 1)/1
    - Since any fraction that has the number 1 in the denominator is simply equal to the numerator, the original fractional expression simplifies to: 3x 2 + 9x - 1.
Additional simplification methods
1. Simplifying fractional expressions. As noted above, if both the numerator and denominator contain the same terms (or even the same expressions), then they can be reduced. To do this, you need to take out the common factor of the numerator or the denominator, or both the numerator and the denominator. Or you can divide each term in the numerator by the denominator and thus simplify the expression.
  - For example, consider the fractional expression (5x 2 + 10x + 20)/10. Here, simply divide each numerator term by the denominator (10). But note that the term 5x 2 is not evenly divisible by 10 (since 5 is less than 10).
    - So write a simplified expression like this: ((5x 2)/10) + x + 2 = (1/2)x 2 + x + 2.
2. Simplification of radical expressions. Expressions under the root sign are called radical expressions. They can be simplified through their decomposition into appropriate factors and the subsequent removal of one factor from under the root.
  - Let's look at a simple example: √(90). The number 90 can be factored into the following factors: 9 and 10, and from 9 we can take the square root (3) and take 3 out from under the root.
    - √(90)
    - √(9×10)
    - √(9)×√(10)
    - 3×√(10)
    - 3√(10)
3. Simplifying expressions with powers. Some expressions contain operations of multiplication or division of terms with powers. In the case of multiplying terms with the same base, their powers are added; in the case of dividing terms with the same base, their powers are subtracted.
  - For example, consider the expression 6x 3 × 8x 4 + (x 17 /x 15). In the case of multiplication, add the powers, and in the case of division, subtract them.
    - 6x 3 × 8x 4 + (x 17 /x 15)
    - (6 × 8)x 3 + 4 + (x 17 - 15)
    - 48x 7 + x 2
  - The following is an explanation of the rules for multiplying and dividing terms with powers.
    - Multiplying terms with powers is equivalent to multiplying terms by themselves. For example, since x 3 = x × x × x and x 5 = x × x × x × x × x, then x 3 × x 5 = (x × x × x) × (x × x × x × x × x), or x 8 .
    - Likewise, dividing terms with degrees is equivalent to dividing terms by themselves. x 5 /x 3 = (x × x × x × x × x)/(x × x × x). Since similar terms found in both the numerator and the denominator can be reduced, the product of two “x”, or x 2 , remains in the numerator.

Fractions

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Fractions are not much of a nuisance in high school. For the time being. Until you come across powers with rational exponents and logarithms. And there... You press and press the calculator, and it shows a full display of some numbers. You have to think with your head like in the third grade.

Let's finally figure out fractions! Well, how much can you get confused in them!? Moreover, it’s all simple and logical. So, what are the types of fractions?

Types of fractions. Transformations.

There are three types of fractions.

1. Common fractions , For example:

Sometimes instead of a horizontal line they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens...), say to yourself the phrase: " Zzzzz remember! Zzzzz denominator - look zzzzz uh!" Look, everything will be zzzz remembered.)

The dash, either horizontal or inclined, means division the top number (numerator) to the bottom (denominator). That's all! Instead of a dash, it is quite possible to put a division sign - two dots.

When complete division is possible, this must be done. So, instead of the fraction “32/8” it is much more pleasant to write the number “4”. Those. 32 is simply divided by 8.

32/8 = 32: 8 = 4

I'm not even talking about the fraction "4/1". Which is also just "4". And if it’s not completely divisible, we leave it as a fraction. Sometimes you have to do the opposite operation. Convert a whole number into a fraction. But more on that later.

2. Decimals , For example:

It is in this form that you will need to write down the answers to tasks “B”.

3. Mixed numbers , For example:

Mixed numbers are practically not used in high school. In order to work with them, they must be converted into ordinary fractions. But you definitely need to be able to do this! Otherwise you will come across such a number in a problem and freeze... Out of nowhere. But we will remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if a fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

The main property of a fraction.

So, let's go! To begin with, I will surprise you. The whole variety of fraction transformations is provided by one single property! That's what it's called main property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction does not change. Those:

It is clear that you can continue to write until you are blue in the face. Don’t let sines and logarithms confuse you, we’ll deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.

Do we need it, all these transformations? Yes! Now you will see for yourself. To begin with, let's use the basic property of a fraction for reducing fractions. It would seem like an elementary thing. Divide the numerator and denominator by the same number and that's it! It's impossible to make a mistake! But... man is a creative being. You can make a mistake anywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to correctly and quickly reduce fractions without doing extra work can be read in the special Section 555.

A normal student doesn't bother dividing the numerator and denominator by the same number (or expression)! He simply crosses out everything that is the same above and below! This is where a typical mistake, a blunder, if you will, lurks.

For example, you need to simplify the expression:

There’s nothing to think about here, cross out the letter “a” on top and the two on the bottom! We get:

Everything is correct. But really you divided all numerator and all the denominator is "a". If you are used to just crossing out, then, in a hurry, you can cross out the “a” in the expression

and get it again

Which would be categorically untrue. Because here all the numerator on "a" is already doesn't share! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge to the teacher. This is not forgiven! Do you remember? When reducing, you need to divide all numerator and all denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. How can I continue to work with her now? Without a calculator? Multiply, say, add, square!? And if you’re not too lazy, and carefully cut it down by five, and by another five, and even... while it’s being shortened, in short. Let's get 3/8! Much nicer, right?

The main property of a fraction allows you to convert ordinary fractions to decimals and vice versa without a calculator! This is important for the Unified State Exam, right?

How to convert fractions from one type to another.

With decimal fractions everything is simple. As it is heard, so it is written! Let's say 0.25. This is zero point twenty-five hundredths. So we write: 25/100. We reduce (we divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

What if the integers are not zero? It's OK. We write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three point seventeen hundredths. We write 317 in the numerator and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary, Watson! From all that has been said, a useful conclusion: any decimal fraction can be converted to a common fraction .

But some people cannot do the reverse conversion from ordinary to decimal without a calculator. And it is necessary! How will you write down the answer on the Unified State Exam!? Read carefully and master this process.

What is the characteristic of a decimal fraction? Her denominator is Always costs 10, or 100, or 1000, or 10000 and so on. If your common fraction has a denominator like this, there's no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the answer to the task in section “B” turned out to be 1/2? What will we write in response? Decimals are required...

Let's remember main property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. Anything, by the way! Except zero, of course. So let’s use this property to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? At 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 = 1x5/2x5 = 5/10 = 0.5. That's it.

However, all sorts of denominators come across. You might come across, for example, the fraction 3/16. Try and figure out what to multiply 16 by to make 100, or 1000... Doesn’t it work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as they taught in elementary school. We get 0.1875.

And there are also very bad denominators. For example, there is no way to turn the fraction 1/3 into a good decimal. Both on the calculator and on a piece of paper, we get 0.3333333... This means that 1/3 is an exact decimal fraction not translated. Same as 1/7, 5/6 and so on. There are many of them, untranslatable. This brings us to another useful conclusion. Not every fraction can be converted to a decimal !

By the way, this is useful information for self-testing. In section "B" you must write down a decimal fraction in your answer. And you got, for example, 4/3. This fraction does not convert to a decimal. This means you made a mistake somewhere along the way! Go back and check the solution.

So, we figured out ordinary and decimal fractions. All that remains is to deal with mixed numbers. To work with them, they must be converted into ordinary fractions. How to do this? You can catch a sixth grader and ask him. But a sixth grader won’t always be at hand... You’ll have to do it yourself. It's not difficult. You need to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator of the common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is simple. Let's look at an example.

Suppose you were horrified to see the number in the problem:

Calmly, without panic, we think. The whole part is 1. Unit. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of a common fraction. That's it. It looks even simpler in mathematical notation:

Is it clear? Then secure your success! Convert to ordinary fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction to a mixed number - is rarely required in high school. Well, if so... And if you are not in high school, you can look into the special Section 555. By the way, you will also learn about improper fractions there.

Well, that's practically all. You remembered the types of fractions and understood How transfer them from one type to another. The question remains: For what do this? Where and when to apply this deep knowledge?

I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed together, we convert everything into ordinary fractions. It can always be done. Well, if it says something like 0.8 + 0.3, then we count it that way, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is all decimal fractions, but um... some kind of evil ones, go to ordinary ones and try it! Look, everything will work out. For example, you will have to square the number 0.125. It’s not so easy if you haven’t gotten used to using a calculator! Not only do you have to multiply numbers in a column, you also have to think about where to insert the comma! It definitely won’t work in your head! What if we move on to an ordinary fraction?

0.125 = 125/1000. We reduce it by 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, it's still shrinking! Back to 5! We get 1/8. We easily square it (in our minds!) and get 1/64. All!

Let's summarize this lesson.

1. There are three types of fractions. Common, decimal and mixed numbers.

2. Decimals and mixed numbers Always can be converted to ordinary fractions. Reverse transfer not always possible

3. The choice of the type of fractions to work with a task depends on the task itself. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary fractions:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

You should get answers like this (in a mess!):

Let's finish here. In this lesson we refreshed our memory on key points about fractions. It happens, however, that there is nothing special to refresh...) If someone has completely forgotten, or has not yet mastered it... Then you can go to a special Section 555. All the basics are covered in detail there. Many suddenly understand everything are starting. And they solve fractions on the fly).