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 encounter degrees with rational indicators yes 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? And how! 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 not shared! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge for 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 all.

However, all sorts of denominators come across. You will 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 does not translate. 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 helpful information for self-test. 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 it? 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 is 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 all. 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 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 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 available.

3. The choice of the type of fractions to work with a task depends on the task itself. In the presence of different types fractions in one task, the most reliable thing is to move on 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).

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. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Decimal. The whole part. Decimal point.

Decimal places. Properties of decimal fractions.

Periodic decimal fraction. Period .

Decimal is the result of dividing one by ten, one hundred, thousand, etc. parts. These fractions are very convenient for calculations, since they are based on the same positional system on which counting and writing integers are based. Thanks to this, the notation and rules for working with decimal fractions are essentially the same as for whole numbers. When writing decimal fractions, there is no need to mark the denominator; this is determined by the place occupied by the corresponding digit. First it is written whole part numbers, then put on the right decimal point. The first digit after the decimal point means the number of tenths, the second – the number of hundredths, the third – the number of thousandths, etc. The numbers located after the decimal point are called decimals.

EXAMPLE

One of advantages of decimals– they are easy brought to mindordinary: the number after the decimal point (in our case 5047) is the numerator; the denominator is equaln-th power of 10, wheren- number of decimal places(in our case n= 4):

If the decimal fraction does not contain an integer part, then a zero is placed before the decimal point:

Properties of decimal fractions.

1. The decimal does not change if you add zeros to the right:

13.6 =13.6000.

2. The decimal fraction does not change if you remove the zeros located

at the end decimal:

0.00123000 = 0.00123 .

Attention! You cannot remove non-terminal zeros. decimal!

These properties allow you to quickly multiply and divide decimals by 10, 100, 1000, etc.

Periodic decimal contains an infinitely repeating group of numbers called period. The period is written in brackets. For example, 0.12345123451234512345… = 0.(12345).

EXAMPLE If we divide 47 by 11, we get 4.27272727… = 4.(27).

Of the many fractions found in arithmetic, those that have 10, 100, 1000 in the denominator - in general, any power of ten - deserve special attention. These fractions have a special name and notation.

A decimal is any number fraction whose denominator is a power of ten.

Examples of decimal fractions:

Why was it necessary to separate out such fractions at all? Why do they need their own recording form? There are at least three reasons for this:

1. Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, reduce the fractions to a common denominator. In decimals nothing like this is required;
2. Reduce computation. Decimals add and multiply according to their own rules, and with a little practice you'll be able to work with them much faster than with regular fractions;
3. Ease of recording. Unlike ordinary fractions, decimals are written on one line without loss of clarity.

Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you ask for change in the store in the amount of 2/3 of a ruble :)

Rules for writing decimal fractions

The main advantage of decimal fractions is convenient and visual notation. Namely:

Decimal notation is a form of writing decimal fractions where the integer part is separated from the fractional part by a regular period or comma. In this case, the separator itself (period or comma) is called a decimal point.

For example, 0.3 (read: “zero pointers, 3 tenths”); 7.25 (7 whole, 25 hundredths); 3.049 (3 whole, 49 thousandths). All examples are taken from the previous definition.

In writing, a comma is usually used as a decimal point. Here and further throughout the site, the comma will also be used.

To write an arbitrary decimal fraction in this form, you need to follow three simple steps:

1. Write out the numerator separately;
2. Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
3. If the decimal point has moved, and after it there are zeros at the end of the entry, they must be crossed out.

It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, to the left of any number you can assign any number of zeros without harm to your health. It's ugly, but sometimes useful.

At first glance, this algorithm may seem quite complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:

Task. For each fraction, indicate its decimal notation:

The numerator of the first fraction is: 73. We shift the decimal point by one place (since the denominator is 10) - we get 7.3.

Numerator of the second fraction: 9. We shift the decimal point by two places (since the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange entry like “.09”.

The numerator of the third fraction is: 10029. We shift the decimal point by three places (since the denominator is 1000) - we get 10.029.

The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10,500. There are extra zeros at the end of the number. Cross them out and we get 10.5.

Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as was done in the last example. However, you should never do this with zeros inside a number (which are surrounded by other numbers). That's why we got 10.029 and 10.5, and not 1.29 and 1.5.

So, we figured out the definition and form of writing decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.

Conversion from fractions to decimals

Consider a simple numerical fraction of the form a /b. You can use the basic property of a fraction and multiply the numerator and denominator by such a number that the bottom turns out to be a power of ten. But before you do, read the following:

There are denominators that cannot be reduced to powers of ten. Learn to recognize such fractions, because they cannot be worked with using the algorithm described below.

That's it. Well, how do you understand whether the denominator is reduced to a power of ten or not?

The answer is simple: factor the denominator into prime factors. If the expansion contains only factors 2 and 5, this number can be reduced to a power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the power of ten.

Task. Check whether the indicated fractions can be represented as decimals:

Let's write out and factor the denominators of these fractions:

20 = 4 · 5 = 2 2 · 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.

12 = 4 · 3 = 2 2 · 3 - there is a “forbidden” factor 3. The fraction cannot be represented as a decimal.

640 = 8 · 8 · 10 = 2 3 · 2 3 · 2 · 5 = 2 7 · 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction can be represented as a decimal.

48 = 6 · 8 = 2 · 3 · 2 3 = 2 4 · 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.

So, we’ve sorted out the denominator - now let’s look at the entire algorithm for moving to decimal fractions:

1. Factor the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
2. Count how many twos and fives are present in the expansion (there will be no other numbers there, remember?). Choose an additional factor such that the number of twos and fives is equal.
3. Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.

Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest multiplier of all possible.

And one more thing: if the original fraction contains an integer part, be sure to convert this fraction to an improper fraction - and only then apply the described algorithm.

Task. Convert these numerical fractions to decimals:

Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, the fraction can be represented as a decimal. The expansion contains two twos and not a single five, so the additional factor is 5 2 = 25. With it, the number of twos and fives will be equal. We have:

Now let's look at the second fraction. To do this, note that 24 = 3 8 = 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.

The last two fractions have denominators 5 (prime number) and 20 = 4 · 5 = 2 2 · 5 respectively - only twos and fives are present everywhere. Moreover, in the first case, “for complete happiness” a factor of 2 is not enough, and in the second - 5. We get:

Conversion from decimals to common fractions

The reverse conversion - from decimal to regular notation - is much simpler. There are no restrictions or special checks here, so you can always convert a decimal fraction to the classic “two-story” fraction.

The translation algorithm is as follows:

1. Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing is not to overdo it and do not cross out the inner zeros surrounded by other numbers;
2. Count how many decimal places there are after the decimal point. Take the number 1 and add as many zeros to the right as there are characters you count. This will be the denominator;
3. Actually, write down the fraction whose numerator and denominator we just found. If possible, reduce it. If the original fraction contained an integer part, now we get not correct fraction, which is very convenient for further calculations.

Task. Convert decimal fractions to ordinary fractions: 0.008; 3.107; 2.25; 7,2008.

Cross out the zeros on the left and the commas - we get the following numbers (these will be the numerators): 8; 3107; 225; 72008.

In the first and second fractions there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.

Finally, let's combine the numerators and denominators into ordinary fractions:

As can be seen from the examples, the resulting fraction can very often be reduced. Let me note once again that any decimal fraction can be represented as an ordinary fraction. The reverse conversion may not always be possible.

fractional number.

Decimal notation of a fractional number is a set of two or more digits from $0$ to $9$, between which there is a so-called \textit (decimal point).

Example 1

For example, $35.02$; $100.7$; $123\456.5$; $54.89$.

The leftmost digit in the decimal notation of a number cannot be zero, the only exception being when the decimal point is immediately after the first digit $0$.

Example 2

For example, $0.357$; $0.064$.

Often the decimal point is replaced with a decimal point. For example, $35.02$; $100.7$; $123\456.5$; $54.89$.

Decimal definition

Definition 1

Decimals-- these are fractional numbers that are represented in decimal notation.

For example, $121.05; $67.9$; $345.6700$.

Decimals are used to more compactly write proper fractions, the denominators of which are the numbers $10$, $100$, $1\000$, etc. and mixed numbers, the denominators of the fractional part of which are the numbers $10$, $100$, $1\000$, etc.

For example, the common fraction $\frac(8)(10)$ can be written as a decimal $0.8$, and the mixed number $405\frac(8)(100)$ can be written as a decimal $405.08$.

Reading Decimals

Decimal fractions, which correspond to regular fractions, are read the same as ordinary fractions, only the phrase “zero integer” is added in front. For example, the common fraction $\frac(25)(100)$ (read “twenty-five hundredths”) corresponds to the decimal fraction $0.25$ (read “zero point twenty-five hundredths”).

Decimal fractions that correspond to mixed numbers are read the same way as mixed numbers. For example, the mixed number $43\frac(15)(1000)$ corresponds to the decimal fraction $43.015$ (read “forty-three point fifteen thousandths”).

Places in decimals

In writing a decimal fraction, the meaning of each digit depends on its position. Those. in decimal fractions the concept also applies category.

The places in decimal fractions before the decimal point are called the same as the places in natural numbers. The decimal places after the decimal point are listed in the table:

Picture 1.

Example 3

For example, in the decimal fraction $56.328$, the digit $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenths place, $2$ is in the hundredths place, $8$ is in the thousandths place.

Places in decimal fractions are distinguished by precedence. When reading a decimal fraction, move from left to right - from senior rank to younger.

Example 4

For example, in the decimal fraction $56.328$, the most significant (highest) place is the tens place, and the low (lowest) place is the thousandths place.

A decimal fraction can be expanded into digits similar to the digit decomposition of a natural number.

Example 5

For example, let's break down the decimal fraction $37.851$ into digits:

$37,851=30+7+0,8+0,05+0,001$

Ending decimals

Definition 2

Ending decimals are called decimal fractions, the records of which contain a finite number of characters (digits).

For example, $0.138$; $5.34$; $56.123456$; $350,972.54.

Any finite decimal fraction can be converted to a fraction or a mixed number.

Example 6

For example, the final decimal fraction $7.39$ corresponds to the fractional number $7\frac(39)(100)$, and the final decimal fraction $0.5$ corresponds to the proper common fraction $\frac(5)(10)$ (or any fraction which is equal to it, for example, $\frac(1)(2)$ or $\frac(10)(20)$.

Converting a fraction to a decimal

Converting fractions with denominators $10, 100, \dots$ to decimals

Before converting some proper fractions to decimals, they must first be “prepared.” The result of such preparation should be the same number of digits in the numerator and the same number of zeros in the denominator.

The essence of “preliminary preparation” of proper ordinary fractions for conversion to decimal fractions is adding such a number of zeros to the left in the numerator that the total number of digits becomes equal to the number of zeros in the denominator.

Example 7

For example, let's prepare the fraction $\frac(43)(1000)$ for conversion to a decimal and get $\frac(043)(1000)$. And the ordinary fraction $\frac(83)(100)$ does not need any preparation.

Let's formulate rule for converting a proper common fraction with a denominator of $10$, or $100$, or $1\000$, $\dots$ into a decimal fraction:

write $0$;

after it put a decimal point;

write down the number from the numerator (along with added zeros after preparation, if necessary).

Example 8

Convert the proper fraction $\frac(23)(100)$ to a decimal.

Solution.

The denominator contains the number $100$, which contains $2$ and two zeros. The numerator contains the number $23$, which is written with $2$.digits. This means that there is no need to prepare this fraction for conversion to a decimal.

Let's write $0$, put a decimal point and write down the number $23$ from the numerator. We get the decimal fraction $0.23$.

Answer: $0,23$.

Example 9

Write the proper fraction $\frac(351)(100000)$ as a decimal.

Solution.

The numerator of this fraction contains $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction must be prepared for conversion to a decimal. To do this, you need to add $5-3=2$ zeros to the left in the numerator: $\frac(00351)(100000)$.

Now we can form the desired decimal fraction. To do this, write down $0$, then add a comma and write down the number from the numerator. We get the decimal fraction $0.00351$.

Answer: $0,00351$.

Let's formulate rule for converting improper fractions with denominators $10$, $100$, $\dots$ into decimal fractions:

write down the number from the numerator;

Use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.

Example 10

Convert the improper fraction $\frac(12756)(100)$ to a decimal.

Solution.

Let's write down the number from the numerator $12756$, then separate the $2$ digits on the right with a decimal point, because the denominator of the original fraction $2$ is zero. We get the decimal fraction $127.56$.

To write a rational number m/n as a decimal fraction, you need to divide the numerator by the denominator. In this case, the quotient is written as a finite or infinite decimal fraction.

Write down given number as a decimal fraction.

Solution. Divide the numerator of each fraction into a column by its denominator: A) divide 6 by 25; b) divide 2 by 3; V) divide 1 by 2, and then add the resulting fraction to one - the integer part of this mixed number.

Irreducible ordinary fractions whose denominators do not contain prime factors other than 2 And 5 , are written as a final decimal fraction.

IN example 1 when A) denominator 25=5·5; when V) the denominator is 2, so we get the final decimals 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a finite decimal.

Is it possible, without long division, to convert into a decimal fraction such an ordinary fraction, the denominator of which does not contain other divisors other than 2 and 5? Let's figure it out! What fraction is called a decimal and is written without a fraction bar? Answer: fraction with denominator 10; 100; 1000, etc. And each of these numbers is a product equal number of twos and fives. In fact: 10=2 ·5 ; 100=2 ·5 ·2 ·5 ; 1000=2 ·5 ·2 ·5 ·2 ·5 etc.

Consequently, the denominator of an irreducible ordinary fraction will need to be represented as the product of “twos” and “fives”, and then multiplied by 2 and (or) 5 so that the “twos” and “fives” become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. To ensure that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which we multiplied the denominator.

Express the following common fractions as decimals:

Solution. Each of these fractions is irreducible. Let's factor the denominator of each fraction into prime factors.

20=2·2·5. Conclusion: one “A” is missing.

8=2·2·2. Conclusion: three “A”s are missing.

25=5·5. Conclusion: two “twos” are missing.

Comment. In practice, they often do not use factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is one with zeros (10 or 100 or 1000, etc.). And then the numerator is multiplied by the same number.

So, in case A)(example 2) from the number 20 you can get 100 by multiplying by 5, therefore, you need to multiply the numerator and denominator by 5.

When b)(example 2) from the number 8 the number 100 will not be obtained, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.

When V)(example 2) from 25 you get 100 if you multiply by 4. This means that the numerator 8 must be multiplied by 4.

An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodic as a decimal. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosed in parentheses.

When b)(example 1) there is only one repeating digit and is equal to 6. Therefore, our result 0.66... ​​will be written like this: 0,(6) . They read: zero point, six in period.

If there are one or more non-repeating digits between the decimal point and the first period, then such a periodic fraction is called a mixed periodic fraction.

An irreducible common fraction whose denominator is together with others multiplier contains multiplier 2 or 5 , becomes mixed periodic fraction.

Write numbers as decimals.