Teaching without coercion

(A guide to the fascinating world of mathematics)

Mathematics already then needs to be taught, that it puts the mind in order. (M.V. Lomonosov)

So how do you learn math?

This question interests many.

The first step is to close the gaps from the past. If you missed (didn’t understand, didn’t study in principle, etc.) any topic, sooner or later you will definitely step on this rake. With a classic result... That's the way mathematics works.

Whether you're learning a new topic or revisiting an old one, master the math definitions and terms! Pay attention, I do not say - "learn", but I say "master". These are different things. You must understand, for example, what is the denominator, discriminant, or arcsine at a simple, even primitive level. What is it, why is it needed and how to deal with it. Life will become easier.

If I ask you how to use the dense restricted environment transition device, you will feel uncomfortable answering, right? And if you understand that this very device is an ordinary door? It's actually kind of more fun.

And, of course, you need to decide. If you don't know how to decide, no big deal. You have to try and try. All once did not know how. But those who tried and tried, albeit incorrectly, with mistakes, now know how to solve. And who did not try, did not study - he never learned.

Here are the three components of the answer to the question: "How to teach mathematics?" Eliminate gaps, master the terms at an understandable level and meaningfully solve tasks.

If mathematics seems to you a jungle of some rules, formulas, expressions in which it is impossible to navigate, then I will console you. There are paths and guiding stars there! You will settle in, get used to it, and you will also begin to admire these wilds ...

Mathematics of the school course does not solve complex examples, because it does not know how. She can well solve something like 5x \u003d 10, a quadratic equation through the discriminant, and the same simple one from trigonometry, logarithms, etc. And all the power of mathematics is aimed at simplifying complex expressions. It is for this that rules and formulas for various transformations are needed. They allow us to write the original expression in a different form convenient for us without changing its essence.

"Mathematics is the art of calling different things by the same name." (A. Poincare)

For example, 8 = 6 + 2 = 2 = = log 6561 = 32: 4. It's still the same number 8! Only recorded in a variety of forms. Which type to choose - we decide! Consistent with the task and common sense.

The main guiding star in mathematics is the ability to transform expressions. Almost any solution starts with a transformation of the original expression. With the help of rules and formulas, which are not at all such an insane amount as you think.

We often say "All formulas work from left to right and from right to left." Let's say (a + b) almost everyone writes it down as a + 2ab + b . But not everyone (unfortunately) realizes that x + 2x + 1 can be written as (x + 1) . And here's what you need to know! Formulas need to know in person! To be able to recognize them in expressions encrypted by cunning teachers, to identify parts of the formulas, to bring them, if necessary, to complete ones.

Expression conversions are troublesome at first. Requires labor. At the initial stage, it is necessary to check, where possible, the correctness of the transformation by inverse transformation. Factored out - multiply back and bring similar ones. It turned out the original expression - hurray! Found the roots of the equation - substitute in the original expression. See what happened. And so on.

So, I invite you to the wonderful world of mathematics. And let's start our journey by getting to know fractions, as this is perhaps the most vulnerable spot for most schoolchildren.

Good luck!

Lesson 1.

Types of fractions. Transformations.

Who knows fractions, he is strong, he is brave in mathematics!

Fractions are of three types.

1. Common fractions , for example: , , , .

Sometimes, instead of a horizontal line, they put a slash: 1/2, 3/7, 19/5. A line, both horizontal (vinculium) and oblique (solidus) means the same operation: dividing the top number (numerator) by the bottom number (denominator). And that's it! Instead of a line, it is quite possible to put a division sign - two dots. 1/2 = 1:2.

When the division is possible entirely, it must be done. So, instead of the fraction 32/8, it is much more pleasant to write the number 4. That is. 32 is simply divided by 8. 32/8 = 32: 8 = 4. I'm not talking about the fraction 4/1, which is also equal to 4. And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.

2. Decimals , for example: 0.5; 3.28; 0.543; 23.32.

3. mixed numbers , for example: , , , .

Mixed numbers are practically not used in high school. In order to work with them, they must be converted to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the task and hang ... From scratch. But we remember this procedure!

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

So go ahead! The whole variety of fraction transformations is provided by a single property! That's what it's called basic property of a fraction. Remember: if the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:

And we need it, all these transformations? - you ask. And how! Now you will see for yourself. First, let's use the basic property of a fraction to reduce fractions. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction of the form 5/10, but a fractional rational expression.

Usually the student does not think about dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where a typical mistake lurks, a blunder, if you like.

For example, you need to simplify the expression: .

What are we doing? We cross out the factor a above and the degree below! We get: .

Everything is correct. But really you shared the whole numerator and the whole denominator on the multiplier a. If you are used to just crossing out, then, in a hurry, you can cross out the letter a in the expression and get again. Which would be categorically wrong: an unforgivable mistake. Because here the whole numerator on a already not shared! This fraction cannot be reduced.

When reducing, you need to divide the entire numerator and the entire denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example, 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced. We 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! It's important in CT, right?

It's easy with decimals. As it is heard, so it is written! Let's say 0.25. It's zero point, twenty-five hundredths. So we write: 25/100. We reduce (divide the numerator and denominator by 25), we get an ordinary fraction: 1/4. Everything. It happens, and nothing is reduced. For example, 0.3. This is three tenths, i.e. 3/10.

What if integers are non-zero? It's OK. We write the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three whole, 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. From all of the above, a useful conclusion: Any decimal fraction can be converted to a common fraction.

But the reverse conversion, ordinary to decimal, some cannot do without a calculator. But you must! How are you going to write down the answer? We carefully read and master this process.

What is a decimal fraction? Her denominator is always 10, or 100, or 1000, or 10,000, and so on. If your common fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the result is 1/2? And the answer must be written in decimal ...

We remember basic property of a fraction! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature 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...)? 5, obviously. Feel free to multiply the denominator by 5. But then the numerator must also be multiplied by 5. We get 1/2 = 0.5. That's all.

However, the denominators may be different. For example, the fraction 3/16. Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide by a corner, as they taught in elementary grades. We get 0.1875.

And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. And on the calculator, and when dividing by a corner, we get 0.3333333 ... Hence one more useful conclusion. Not every common fraction converts to a decimal!

So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they need to be converted to ordinary fractions. How to do it? You can catch a fifth grader and ask him. But not always a fifth-grader will be nearby ... You will have to do it yourself. This is not difficult. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator of a common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.

Suppose that in the task you saw a number with horror:

Calmly, without panic, we argue. The whole part is 1. One. 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. Consider: 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 an ordinary fraction. That's all. It looks even simpler in mathematical notation:

Easily? Then secure your success! Convert these mixed numbers , , to common fractions. You should get 10/3, 23/10 and 21/4.

Well, almost everything. You remembered the types of fractions and understood how to translate them from one type to another. The question remains: why do this? Where and when to apply this deep knowledge?

Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed into a bunch, we translate everything into ordinary fractions. It can always be done. Well, if it is written, for example, 0.8 + 0.3, then we think so, without any translation. Why do we need extra work? We choose the way to solve which is convenient for us!

If the task is full of decimal fractions, but um ... some scary ones, go to ordinary ones, try it! Maybe everything will work out. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction? 0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Still shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. Everything!

Let's summarize our lesson.

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

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

3. The choice of the type of fractions for working with the task depends on this very task. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.

Practical Tips:

1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not common words, not good wishes! This is a severe need! It is better to write two extra lines in a draft than to make a mistake when calculating in your head.

2. In examples with different types of fractions - go to ordinary fractions.

3. We reduce all fractions to the stop.

4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).

5. We divide the unit into a fraction in our mind, simply by turning the fraction over.

Now try to put the theory into practice.

So, let's solve it in the exam mode! We solve an example, we check, we solve the following. We decided everything - we checked again from the first to the last example. And then we look at the answers.

Decided? Looking for answers that match yours. The answers are written in disorder, away from temptation, so to speak...

0; 17/22; 3; 1; 3/4; 14; -5/4; 17/12; 1/3; 5; 2/5; 25.

And now we draw conclusions. If everything worked out - happy for you! Elementary calculations with fractions are not your problem! You can do more serious things. If not... Patience and work will grind everything.

The material of this article is a general look at the transformation of expressions containing fractions. Here we will consider the basic transformations that are characteristic of expressions with fractions.

Page navigation.

Fractional expressions and fractional expressions

To begin with, let's clarify what kind of expression transformation we are going to deal with.

The title of the article contains the self-explanatory phrase " expressions with fractions". That is, below we will talk about the transformation of numeric expressions and expressions with variables, in the record of which there is at least one fraction.

We note right away that after the publication of the article " Transformation of fractions: a general view"We are no longer interested in individual fractions. Thus, further we will consider sums, differences, products, partial and more complex expressions with roots, powers, logarithms, which are united only by the presence of at least one fraction.

And let's talk about fractional expressions. This is not the same as expressions with fractions. Fraction expressions are a more general concept. Not every expression with fractions is a fractional expression. For example, the expression is not a fractional expression, although it contains a fraction, it is an integer rational expression. So don't call an expression with fractions a fractional expression without being completely sure that it is.

Basic identical transformations of expressions with fractions

Example.

Simplify the expression .

Solution.

In this case, you can open the brackets, which will give the expression , which contains like terms and , as well as −3 and 3 . After their reduction, we get a fraction.

Let's show a short form of writing the solution:

Answer:

.

Working with individual fractions

The expressions we are talking about transforming differ from other expressions mainly in the presence of fractions. And the presence of fractions requires tools to work with them. In this paragraph, we will discuss the transformation of individual fractions included in the record of this expression, and in the next paragraph we will proceed to perform operations with the fractions that make up the original expression.

With any fraction that is a component of the original expression, you can perform any of the transformations indicated in the article Fraction conversion. That is, you can take a separate fraction, work with its numerator and denominator, reduce it, bring it to a new denominator, etc. It is clear that with this transformation, the selected fraction will be replaced by a fraction identically equal to it, and the original expression will be replaced by an expression identically equal to it. Let's look at an example.

Example.

Convert expression with fraction to a simpler form.

Solution.

Let's start the transformation by working with a fraction. First, open the brackets and give similar terms in the numerator of the fraction: . Now it begs the bracketing of the common factor x in the numerator and the subsequent reduction of the algebraic fraction: . It remains only to substitute the result obtained instead of a fraction in the original expression, which gives .

Answer:

.

Performing actions with fractions

Part of the process of converting expressions with fractions is often to do actions with fractions. They are carried out in accordance with the accepted procedure for performing actions. It is also worth keeping in mind that any number or expression can always be represented as a fraction with a denominator of 1.

Example.

Simplify the expression .

Solution.

The problem can be approached from different angles. In the context of the topic under consideration, we will go by performing actions with fractions. Let's start by multiplying fractions:

Now we write the product as a fraction with a denominator 1, after which we subtract the fractions:

If desired and necessary, one can still get rid of irrationality in the denominator , on which you can finish the transformation.

Answer:

Application of properties of roots, powers, logarithms, etc.

The class of expressions with fractions is very wide. Such expressions, in addition to fractions themselves, may contain roots, degrees with different exponents, modules, logarithms, trigonometric functions, etc. Naturally, when they are converted, the corresponding properties are applied.

Applicable to fractions, it is worth highlighting the property of the root of the fraction, the property of the fraction to the degree, the property of the modulus of the quotient and the property of the logarithm of the difference .

For clarity, we give a few examples. For example, in the expression It may be useful, based on the properties of the degree, to replace the first fraction with a degree, which further allows us to represent the expression as a squared difference. When converting a logarithmic expression it is possible to replace the logarithm of a fraction with the difference of logarithms, which further allows us to bring similar terms and thereby simplify the expression: . Converting trigonometric expressions may require replacing the ratio of the sine to the cosine of the same angle with a tangent. It may also be necessary to move from a half argument using the appropriate formulas to a whole argument, thereby getting rid of the fraction argument, for example, .

Applying properties of roots, degrees, etc. to the transformation of expressions is covered in more detail in the articles:

• Transformation of irrational expressions using properties of roots,
• Transformation of expressions using the properties of powers,
• Converting logarithmic expressions using the properties of logarithms,
• Converting trigonometric expressions.

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 whole five hundredths, we get a fraction. Three whole seventeen thousandths, we get a fraction. The digits before the decimal point in a decimal number are the integer part of the fraction. The number after the decimal point is the numerator of the future fraction. If there is a one-digit number after the decimal point, the denominator will be 10, if two-digit - 100, three-digit - 1000, etc. Some of the resulting fractions can be reduced. In our examples

Converting a fraction to a decimal number

This is the reverse of the previous transformation. What is a decimal fraction? Her denominator is always 10, or 100, or 1000, or 10,000, and so on. If your usual fraction has such a denominator, there is no problem. For example, or

If a fraction, for example . In this case, you need to use the basic property of the 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 decimal numbers!
For example,

Converting a mixed fraction to an improper

A mixed fraction, such as , is easily converted to an improper fraction. To do this, you need to multiply the integer part by the denominator (bottom) and add it to the numerator (top), leaving the denominator (bottom) unchanged. That is

When converting a mixed fraction to an improper one, you can remember that you can use the addition of fractions

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

An improper fraction can be converted to a mixed fraction by highlighting the whole part. Consider an example, . Determine how many integer times "3" fit in "23". Or we divide 23 by 3 on the calculator, the whole number up 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". How would we find the excess that remains from the numerator "23", if we remove the maximum number of "3". The denominator is left unchanged. Everything is done, write down the result

Fractions

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

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

Let's deal with fractions, finally! Well, how much can you get confused in them!? Moreover, it is all simple and logical. So, what are fractions?

Types of fractions. Transformations.

Fractions are of three types.

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 ...), tell yourself the phrase with the expression: " Zzzzz remember! Zzzzz denominator - out zzzz u!" Look, everything will be remembered.)

A dash, which is horizontal, which is oblique, means division top number (numerator) to bottom number (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.

When the division is possible entirely, it 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 talking about the fraction "4/1". Which is also just "4". And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.

2. Decimals , for example:

It is in this form that it will be necessary 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 to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the puzzle and hang ... From scratch. But we remember this procedure! A little lower.

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

Basic property of a fraction.

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

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

And we need it, all these transformations? And how! Now you will see for yourself. First, let's use the basic property of a fraction for fraction abbreviations. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to reduce fractions correctly and quickly without doing unnecessary work can be found in special Section 555.

A normal student does not bother dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where a typical mistake lurks, a blunder, if you like.

For example, you need to simplify the expression:

There is nothing to think about, we cross out the letter "a" from above and the deuce from below! We get:

Everything is correct. But really you shared the whole numerator and the whole denominator "a". If you are used to just cross out, then, in a hurry, you can cross out the "a" in the expression

and get again

Which would be categorically wrong. Because here the whole numerator on "a" already not shared! This fraction cannot be reduced. By the way, such an abbreviation is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When reducing, it is necessary to divide the whole numerator and the whole denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?

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

How to convert fractions from one form to another.

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

What if integers are non-zero? It's OK. 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 whole, 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 of the above, a useful conclusion: any decimal fraction can be converted to a common fraction .

But the reverse conversion, ordinary to decimal, some cannot do without a calculator. But you must! How will you write down the answer on the exam!? We carefully read and master this process.

What is a decimal fraction? She has in the denominator always is worth 10 or 100 or 1000 or 10000 and so on. If your usual fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if in the answer to the task of section "B" it turned out 1/2? What will we write in response? Decimals are required...

We remember basic property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature 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...)? 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 maths demands! We get 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.

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

And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 into an exact decimal fraction does not translate. Just like 1/7, 5/6 and so on. Many of them are untranslatable. Hence another useful conclusion. Not every common fraction converts to a decimal. !

By the way, this is useful information for self-examination. In section "B" in response, you need to write down a decimal fraction. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere along the way you made a mistake! Come back, check the solution.

So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they all need to be converted to ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always a sixth grader will be at hand ... We will have to do it ourselves. This is not difficult. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator of a common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.

Let in the problem you saw with horror the number:

Calmly, without panic, we understand. The whole part is 1. One. 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 an ordinary fraction. That's all. It looks even simpler in mathematical notation:

Clearly? Then secure your success! Convert to common fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction into a mixed number - is rarely required in high school. Well, if... And if you - not in high school - you can look into the special Section 555. In the same place, by the way, you will learn about improper fractions.

Well, almost everything. You remembered the types of fractions and understood how convert them from one type to another. The question remains: why do it? 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 into a bunch, we translate everything into ordinary fractions. It can always be done. Well, if something like 0.8 + 0.3 is written, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is full of decimal fractions, but um ... some kind of evil ones, go to ordinary ones, try it! Look, everything will be fine. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction?

0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Oh, it's shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. Everything!

Let's summarize this lesson.

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

2. Decimals and mixed numbers always can be converted to common fractions. Reverse Translation not always available.

3. The choice of the type of fractions for working with the task depends on this very task. 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 ones:

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

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

On this we will finish. In this lesson, we brushed up on the key points on fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not mastered it yet ... Those can go to a special Section 555. All the basics are detailed there. Many suddenly understand everything are starting. And they solve fractions on the fly).

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Simplifying algebraic expressions is one of the keys to learning algebra and 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 simplification skills are good even for those who are not enthusiastic about mathematics. By following a few simple rules, many of the most common types of algebraic expressions can be simplified without any special mathematical knowledge.

Steps

Important definitions

1. 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, like terms include one variable to the same extent, include several identical 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 like terms because they contain the variable "x" of the second order (in the second power). However, x and x 2 are not similar members, since they contain the variable "x" of different orders (first and second). Similarly, -3yx and 5xz are not similar members because they contain different variables.

2. Factorization . This is finding such numbers, the product of which leads to the original number. Any original number can have several factors. For example, the number 12 can be decomposed 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 divisors , that is, the numbers by which the original number is divisible.

   • 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.

3. Remember and follow the order of operations to avoid mistakes.

   • Parentheses
   • Degree
   • Multiplication
   • Division
   • Addition
   • Subtraction

   Casting Like Members

   1. Write down the expression. The simplest algebraic expressions (which do not contain fractions, roots, and so on) can be solved (simplified) in just a few steps.

      • For example, simplify the expression 1 + 2x - 3 + 4x.

   2. Define similar members (members with a variable of the same order, members with the same variables, or free members).

      • 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 members (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 members. You will get a simple expression with fewer terms. The new expression is equal to the original.

      • In our example: 1 + 2x - 3 + 4x = 6x - 2, that is, the original expression is simplified and easier to work with.

   5. Observe the order in which operations are performed when casting like terms. In our example, it was easy to bring similar terms. However, in the case of complex expressions in which members are enclosed in brackets 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 like terms and quote them, because first you need to expand the parentheses. Therefore, perform the operations in 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 cast like terms.
        • x 2 + (15x - 3x) + (8 - 5)
        • x 2 + 12x + 3

   Parenthesizing the multiplier

   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 divisible.

      • 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 expression term by 3.
        • 9x2/3=3x2
        • 27x/3=9x
        • -3/3 = -1
        • It turned out the expression 3x2 + 9x-1. It is not equal to the original expression.

   3. Write the original expression as equal to the product of gcd times the resulting expression. That is, enclose the resulting expression in brackets, and put the GCD out of brackets.

      • In our example: 9x 2 + 27x - 3 = 3(3x 2 + 9x - 1)

   4. Simplifying fractional expressions by taking the multiplier out of brackets. Why just take 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 the brackets can help get rid of the fraction (from the denominator).

      • For example, consider the fractional expression (9x 2 + 27x - 3)/3. Use parentheses to simplify this expression.
        • Factor out the factor 3 (as you did before): (3(3x 2 + 9x - 1))/3
        • Note that both the numerator and denominator now have the number 3. This can be reduced, and you get the expression: (3x 2 + 9x - 1) / 1
        • Since any fraction that has the number 1 in the denominator is just equal to the numerator, the original fractional expression is simplified to: 3x2 + 9x-1.

   Additional Simplification Techniques

   1. Simplifying fractional expressions. As noted above, if both the numerator and the 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 of 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 term of the numerator by the denominator (10). But note that the 5x2 term is not even divisible by 10 (because 5 is less than 10).
        • So write the simplified expression like this: ((5x 2)/10) + x + 2 = (1/2)x 2 + x + 2.

   2. Simplification of radical expressions. Expressions under the radical sign are called radical expressions. They can be simplified through their decomposition into the appropriate factors and the subsequent removal of one factor from under the root.

      • Consider a simple example: √(90). The number 90 can be decomposed into the following factors: 9 and 10, and from 9, 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. In some expressions, there are operations of multiplication or division of terms with a degree. In the case of multiplication of terms with one base, their degrees are added; in the case of dividing terms with the same base, their degrees are subtracted.

      • For example, consider the expression 6x 3 × 8x 4 + (x 17 / x 15). In the case of multiplication, add the exponents, and in the case of division, subtract them.
        • 6x 3 × 8x 4 + (x 17 / x 15)
        • (6 × 8)x 3 + 4 + (x 17 - 15)
        • 48x7+x2
      • The following is an explanation of the rule for multiplying and dividing terms with a degree.
        • 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 .
        • Similarly, dividing terms with powers is equivalent to dividing terms by themselves. x 5 /x 3 \u003d (x × x × x × x × x) / (x × x × x). Since similar terms that are in both the numerator and the denominator can be reduced, the product of two "x", or x 2, remains in the numerator.