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 the given number as a decimal.

Solution. Divide the numerator of each fraction 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 unity - the integer part of this mixed number.

Irreducible ordinary fractions whose denominators contain no prime divisors 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 got the final decimals 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a final decimal.

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

Therefore, the denominator of an irreducible ordinary fraction will need to be represented as a 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. So that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which the denominator was multiplied.

Express the following fractions as a decimal:

Solution. Each of these fractions is irreducible. Let us decompose the denominator of each fraction into prime factors.

20=2 2 5. Conclusion: one "five" is missing.

8=2 2 2. Conclusion: there are not enough three "fives".

25=5 5. Conclusion: two "twos" are missing.

Comment. In practice, they often do not use the factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is a unit 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 work, 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) out of 25 you get 100 when multiplied by 4. This means that the numerator 8 must also be multiplied by 4.

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

When b)(example 1 ) the repeated digit is one and equals 6. Therefore, our result 0.66... ​​will be written like this: 0,(6) . They read: zero integers, six in the period.

If there is one or more non-recurring digits between the comma and the first period, then such a periodic fraction is called a mixed periodic fraction.

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

Write the number as a decimal.

For example.$\frac(3)(10), 4 \frac(7)(100), \frac(11)(10000)$

Such fractions are usually written without a denominator, and the value of each digit depends on the place where it stands. For such fractions, the integer part is separated by a comma, and after the comma there should be as many digits as there are zeros in the denominator of an ordinary fraction. The fractional digits are called decimal places.

For example.$\frac(21)(100)=0.21 ; 3 \frac(21)(100)=$3.21

The first decimal place after the decimal point corresponds to tenths, the second to hundredths, the third to thousandths, and so on.

If the number of zeros in the denominator of a decimal fraction is greater than the number of digits in the numerator of the same fraction, then after the decimal point, the required number of zeros is added before the numerator digits.

Since there are four zeros in the denominator, and two digits in the numerator, we add $4-2=2$ zeros in the decimal notation before the numerator.

The main property of a decimal fraction

Property

If you add several zeros to the decimal fraction on the right, then the value of the decimal fraction will not change.

For example.$12.034=12.0340=12.03400=12.034000=\ldots$

Comment

Thus, the zeros at the end of the decimal are not taken into account, so when performing various actions, these zeros can be crossed out / discarded.

Decimal Comparison

To compare two decimals (to find out which of the two decimals is greater), you need to compare their whole parts, then tenths, hundredths, and so on. If the integer part of one of the fractions is greater than the integer part of the other fraction, then the first fraction is considered larger. In the case of equality of integer parts, the fraction with more tenths is greater, etc.

Example

Exercise. Compare fractions $2,432$ ; $2.41 and $1.234

Solution. The fraction $1,234$ is the smallest because its integer part is 1 and $1

Now let's compare the fractions $2,432$ and $1,234$ in size. Their integer parts are equal to each other and equal to 2. Compare tenths: $4=4$ . Compare hundredths: $3>1$ . So $2.432>2.41$ .

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

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

Decimal examples:

Why was it necessary to isolate such fractions at all? Why do they need their own entry 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, bring the fractions to a common denominator. In decimal fractions, none of this is required;
2. Reduction of calculations. Decimals add and multiply according to their own rules, and with a little practice you will be able to work with them much faster than with ordinary ones;
3. Ease of recording. Unlike ordinary fractions, decimals are written in 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 demand change in the amount of 2/3 rubles in a store :)

Rules for writing decimal fractions

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

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

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

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

To write an arbitrary decimal fraction in the specified 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 shifted, and after it there are zeros at the end of the record, 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, any number of zeros can be assigned to the left of any number without harm to health. It's ugly, but sometimes useful.

At first glance, this algorithm may seem rather 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: 73. We shift the decimal point by one sign (because the denominator is 10) - we get 7.3.

The numerator of the second fraction: 9. We shift the decimal point by two digits (because 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 notation like “.09”.

The numerator of the third fraction: 10029. We shift the decimal point by three digits (because 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. We cross them out - 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 is done in the last example. However, in no case should you do this with zeros that are inside the number (which are surrounded by other digits). That is why we got 10.029 and 10.5, and not 1.29 and 1.5.

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

Change 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 you get a power of ten below. But before doing so, please read the following:

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

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

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

Task. Check if the specified fractions can be represented as decimals:

We write out and factorize the denominators of these fractions:

20 \u003d 4 5 \u003d 2 2 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.

12 \u003d 4 3 \u003d 2 2 3 - there is a "forbidden" factor 3. The fraction cannot be represented as a decimal.

640 \u003d 8 8 10 \u003d 2 3 2 3 2 5 \u003d 2 7 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction is represented as a decimal.

48 \u003d 6 8 \u003d 2 3 2 3 \u003d 2 4 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.

So, we figured out the denominator - now we will consider the entire algorithm for switching to decimal fractions:

1. Factorize 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 decomposition (there will be no other numbers there, remember?). Choose such an additional multiplier so 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 such factor from all possible ones.

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

Task. Convert these numbers to decimals:

Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, a fraction can be represented as a decimal. There are two twos and no fives in the expansion, so the additional factor is 5 2 = 25. The number of twos and fives will be equal to it. We have:

Now let's deal with the second fraction. To do this, note that 24 \u003d 3 8 \u003d 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 (a prime number) and 20 = 4 5 = 2 2 5 respectively - only twos and fives are present everywhere. At the same time, in the first case, “for complete happiness”, there is not enough multiplier 2, and in the second - 5. We get:

Switching from decimals to ordinary

The reverse conversion - from decimal notation to normal - is much easier. There are no restrictions and special checks, so you can always convert a decimal fraction into a classic "two-story" one.

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 - do not overdo it and do not cross out the internal zeros surrounded by other numbers;
2. Calculate how many digits are in the original decimal fraction after the decimal point. Take the number 1 and add as many zeros to the right as you counted the characters. This will be the denominator;
3. Actually, write down the fraction whose numerator and denominator we just found. Reduce if possible. If there was an integer part in the original fraction, now we will get an improper fraction, which is very convenient for further calculations.

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

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

In the first and second fractions after the decimal point 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. Once again, I note that any decimal fraction can be represented as an ordinary one. The reverse transformation is not always possible.

fractional number.

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

Example 1

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

The leftmost digit in the decimal representation of a number cannot be zero, except 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 by a decimal point. For example, $35.02$; $100.7$; $123 \ $456.5; $54.89.

Decimal definition

Definition 1

Decimals are fractional numbers that are represented in decimal notation.

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

Decimals are used for a more compact representation of regular fractions whose denominators are the numbers $10$, $100$, $1\000$, etc. and mixed numbers whose denominators are $10$, $100$, $1\000$, etc.

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

Reading decimals

Decimals that correspond to regular fractions are read the same as ordinary fractions, only the phrase "zero integers" 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").

Decimals 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 decimal notation, the value of each digit depends on its position. Those. in decimal fractions, the concept also takes place discharge.

The digits in decimal fractions up to the decimal point are called the same as the digits in natural numbers. The digits in decimal fractions after the decimal point are listed in the table:

Picture 1.

Example 3

For example, in the decimal fraction $56,328$, $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenth place, $2$ is in the hundredth place, $8$ is in the thousandth place.

The digits in decimal fractions are distinguished by seniority. When reading a decimal fraction, they move from left to right - from senior discharge to junior.

Example 4

For example, in decimal $56.328$, the most significant (highest) digit is the tens digit, and the least significant (lowest) digit is the thousandths digit.

A decimal fraction can be expanded into digits in the same way as expansion into digits of a natural number.

Example 5

For example, let's expand the decimal fraction $37,851$ into digits:

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

End decimals

Definition 2

End 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 final decimal fraction can be converted to a common 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 fraction $\frac(5)(10)$ (or any fraction, which is equal to it, for example, $\frac(1)(2)$ or $\frac(10)(20)$.

Converting an ordinary fraction to a decimal fraction

Convert common fractions with denominators $10, 100, \dots$ to decimals

Before converting some proper ordinary 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 number of zeros in the denominator.

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

Example 7

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

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

write $0$;

put a decimal point after it;

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

Example 8

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

Solution.

The denominator is the number $100$, which contains $2$ two zeros. The numerator contains the number $23$, which contains $2$.digits. this means that preparation for this fraction for conversion to decimal is not necessary.

Let's write $0$, put a decimal point and write 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 has $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction needs to be prepared for conversion to decimal. To do this, 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 $0$, then put a comma and write the number from the numerator. We get the decimal fraction $0.00351$.

Answer: $0,00351$.

Let's formulate rule for converting improper common fractions with denominators $10$, $100$, $\dots$ to decimals:

write a number from the numerator;

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

Example 10

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

Solution.

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

We will devote this material to such an important topic as decimal fractions. First, let's define the basic definitions, give examples and dwell on the rules of decimal notation, as well as what the digits of decimal fractions are. Next, we highlight the main types: finite and infinite, periodic and non-periodic fractions. In the final part, we will show how the points corresponding to fractional numbers are located on the coordinate axis.

What is decimal notation for fractional numbers

The so-called decimal notation for fractional numbers can be used for both natural and fractional numbers. It looks like a set of two or more numbers with a comma between them.

The decimal point is used to separate the integer part from the fractional part. As a rule, the last digit of a decimal is never a zero, unless the decimal point is immediately after the first zero.

What are some examples of fractional numbers in decimal notation? It can be 34 , 21 , 0 , 35035044 , 0 , 0001 , 11 231 552 , 9 etc.

In some textbooks, you can find the use of a dot instead of a comma (5. 67, 6789. 1011, etc.). This option is considered equivalent, but it is more typical for English-language sources.

Definition of decimals

Based on the above concept of decimal notation, we can formulate the following definition of decimal fractions:

Definition 1

Decimals are fractional numbers in decimal notation.

Why do we need to write fractions in this form? It gives us some advantages over ordinary ones, for example, a more compact notation, especially in cases where the denominator is 1000, 100, 10, etc. or a mixed number. For example, instead of 6 10 we can specify 0 , 6 , instead of 25 10000 - 0 , 0023 , instead of 512 3 100 - 512 , 03 .

How to correctly represent ordinary fractions with tens, hundreds, thousands in the denominator in decimal form will be described in a separate material.

How to read decimals correctly

There are some rules for reading records of decimals. So, those decimal fractions that correspond to their correct ordinary equivalents are read almost the same, but with the addition of the words "zero tenths" at the beginning. So, the entry 0 , 14 , which corresponds to 14 100 , is read as "zero point fourteen hundredths."

If a decimal fraction can be associated with a mixed number, then it is read in the same way as this number. So, if we have a fraction 56, 002, which corresponds to 56 2 1000, we read such an entry as "fifty-six point two thousandths."

The value of a digit in a decimal notation depends on where it is located (just like in the case of natural numbers). So, in decimal fraction 0, 7, seven is tenths, in 0, 0007 it is ten thousandths, and in fraction 70,000, 345 it means seven tens of thousands of whole units. Thus, in decimal fractions, there is also the concept of a number digit.

The names of the digits located before the comma are similar to those that exist in natural numbers. The names of those that are located after are clearly presented in the table:

Let's take an example.

Example 1

We have decimal 43, 098. She has a four in the tens place, a three in the units place, zero in the tenth place, 9 in the hundredth place, and 8 in the thousandth place.

It is customary to distinguish the digits of decimal fractions by seniority. If we move through the numbers from left to right, then we will go from high to low digits. It turns out that hundreds are older than tens, and millionths are younger than hundredths. If we take that final decimal fraction, which we cited as an example above, then in it the highest, or highest, will be the digit of hundreds, and the lowest, or lowest, will be the digit of 10 thousandths.

Any decimal fraction can be decomposed into separate digits, that is, represented as a sum. This operation is performed in the same way as for natural numbers.

Example 2

Let's try to expand the fraction 56, 0455 into digits.

We will be able to:

56 , 0455 = 50 + 6 + 0 , 4 + 0 , 005 + 0 , 0005

If we remember the properties of addition, we can represent this fraction in other forms, for example, as the sum 56 + 0, 0455, or 56, 0055 + 0, 4, etc.

What are trailing decimals

All the fractions we talked about above are trailing decimals. This means that the number of digits after the decimal point is finite. Let's get the definition:

Definition 1

Trailing decimals are a type of decimal that has a finite number of digits after the comma.

Examples of such fractions can be 0, 367, 3, 7, 55, 102567958, 231032, 49, etc.

Any of these fractions can be converted either into a mixed number (if the value of their fractional part is different from zero), or into an ordinary fraction (if the integer part is zero). We have devoted a separate material to how this is done. Let's just point out a couple of examples here: for example, we can bring the final decimal fraction 5, 63 to the form 5 63 100, and 0, 2 corresponds to 2 10 (or any other fraction equal to it, for example, 4 20 or 1 5 .)

But the reverse process, i.e. writing an ordinary fraction in decimal form may not always be performed. So, 5 13 cannot be replaced by an equal fraction with a denominator of 100, 10, etc., which means that the final decimal fraction will not work out of it.

The main types of infinite decimal fractions: periodic and non-periodic fractions

We indicated above that finite fractions are called so because they have a finite number of digits after the decimal point. However, it may well be infinite, in which case the fractions themselves will also be called infinite.

Definition 2

Infinite decimals are those that have an infinite number of digits after the decimal point.

Obviously, such numbers simply cannot be written completely, so we indicate only a part of them and then put ellipsis. This sign indicates an infinite continuation of the sequence of decimal places. Examples of infinite decimals would be 0 , 143346732 ... , 3 , 1415989032 ... , 153 , 0245005 ... , 2 , 66666666666 ... , 69 , 748768152 ... . etc.

In the "tail" of such a fraction, there can be not only seemingly random sequences of numbers, but a constant repetition of the same character or group of characters. Fractions with alternation after the decimal point are called periodic.

Definition 3

Periodic decimal fractions are such infinite decimal fractions in which one digit or a group of several digits is repeated after the decimal point. The repeating part is called the period of the fraction.

For example, for the fraction 3, 444444 ... . the period will be the number 4, and for 76, 134134134134 ... - the group 134.

What is the minimum number of characters allowed in a periodic fraction? For periodic fractions, it will be sufficient to write the entire period once in parentheses. So, the fraction is 3, 444444 ... . it will be correct to write as 3, (4) , and 76, 134134134134 ... - as 76, (134) .

In general, entries with multiple periods in brackets will have exactly the same meaning: for example, the periodic fraction 0.677777 is the same as 0.6 (7) and 0.6 (77), etc. Entries like 0 , 67777 (7) , 0 , 67 (7777) and others are also allowed.

In order to avoid errors, we introduce the uniformity of notation. Let's agree to write only one period (the shortest possible sequence of digits), which is closest to the decimal point, and enclose it in parentheses.

That is, for the above fraction, we will consider the entry 0, 6 (7) as the main one, and, for example, in the case of the fraction 8, 9134343434, we will write 8, 91 (34) .

If the denominator of an ordinary fraction contains prime factors that are not equal to 5 and 2, then when converted to decimal notation, infinite fractions will be obtained from them.

In principle, we can write any finite fraction as a periodic one. To do this, we just need to add an infinite number of zeros to the right. How does it look on the record? Let's say we have a final fraction 45, 32. In periodic form, it will look like 45 , 32 (0) . This action is possible because adding zeros to the right of any decimal fraction gives us a fraction equal to it as a result.

Separately, one should dwell on periodic fractions with a period of 9, for example, 4, 89 (9), 31, 6 (9) . They are an alternative notation for similar fractions with a period of 0, so they are often replaced when writing with fractions with a zero period. At the same time, one is added to the value of the next digit, and (0) is indicated in parentheses. The equality of the resulting numbers is easy to check by presenting them as ordinary fractions.

For example, the fraction 8, 31 (9) can be replaced by the corresponding fraction 8, 32 (0) . Or 4 , (9) = 5 , (0) = 5 .

Infinite decimal periodic fractions are rational numbers. In other words, any periodic fraction can be represented as an ordinary fraction, and vice versa.

There are also fractions in which there is no infinitely repeating sequence after the decimal point. In this case, they are called non-periodic fractions.

Definition 4

Non-periodic decimal fractions include those infinite decimal fractions that do not contain a period after the decimal point, i.e. repeating group of numbers.

Sometimes non-periodic fractions look very similar to periodic ones. For example, 9 , 03003000300003 ... at first glance seems to have a period, but a detailed analysis of the decimal places confirms that this is still a non-periodic fraction. You have to be very careful with numbers like this.

Non-periodic fractions are irrational numbers. They are not converted to ordinary fractions.

Basic operations with decimals

The following operations can be performed with decimal fractions: comparison, subtraction, addition, division and multiplication. Let's analyze each of them separately.

Comparing decimals can be reduced to comparing ordinary fractions that correspond to the original decimals. But infinite non-periodic fractions cannot be reduced to this form, and converting decimal fractions to ordinary ones is often a laborious task. How to quickly perform a comparison action if we need to do it in the course of solving the problem? It is convenient to compare decimal fractions by digits in the same way as we compare natural numbers. We will devote a separate article to this method.

To add one decimal fraction to another, it is convenient to use the column addition method, as for natural numbers. To add periodic decimal fractions, you must first replace them with ordinary ones and count according to the standard scheme. If, according to the conditions of the problem, we need to add infinite non-periodic fractions, then we must first round them up to a certain digit, and then add them. The smaller the digit to which we round, the higher the accuracy of the calculation will be. For subtraction, multiplication and division of infinite fractions, preliminary rounding is also necessary.

Finding the difference of decimal fractions is the opposite of addition. In fact, with the help of subtraction, we can find a number whose sum with the subtracted fraction will give us the reduced one. We will talk about this in more detail in a separate article.

Multiplication of decimal fractions is done in the same way as for natural numbers. The method of calculation by a column is also suitable for this. We again reduce this action with periodic fractions to the multiplication of ordinary fractions according to the rules already studied. Infinite fractions, as we remember, must be rounded before counting.

The process of dividing decimals is the reverse of the multiplication process. When solving problems, we also use column counts.

You can set an exact correspondence between the end decimal and a point on the coordinate axis. Let's figure out how to mark a point on the axis that will exactly correspond to the required decimal fraction.

We have already studied how to construct points corresponding to ordinary fractions, and decimal fractions can be reduced to this form. For example, an ordinary fraction 14 10 is the same as 1 , 4 , so the point corresponding to it will be removed from the origin in the positive direction by exactly the same distance:

You can do without replacing the decimal fraction with an ordinary one, and take the digit expansion method as a basis. So, if we need to mark a point whose coordinate will be equal to 15 , 4008 , then we will first represent this number as a sum 15 + 0 , 4 + , 0008 . To begin with, we set aside 15 whole unit segments in the positive direction from the origin, then 4 tenths of one segment, and then 8 ten-thousandths of one segment. As a result, we will get a coordinate point, which corresponds to the fraction 15, 4008.

For an infinite decimal fraction, it is better to use this particular method, since it allows you to approach the desired point as close as you like. In some cases, it is possible to build an exact correspondence of an infinite fraction on the coordinate axis: for example, 2 = 1, 41421. . . , and this fraction can be associated with a point on the coordinate ray, remote from 0 by the length of the diagonal of the square, the side of which will be equal to one unit segment.

If we find not a point on the axis, but a decimal fraction corresponding to it, then this action is called the decimal measurement of the segment. Let's see how to do it right.

Suppose we need to get from zero to a given point on the coordinate axis (or get as close as possible in the case of an infinite fraction). To do this, we gradually set aside unit segments from the origin of coordinates until we get to the desired point. After whole segments, if necessary, we measure tenths, hundredths and smaller parts so that the correspondence is as accurate as possible. As a result, we got a decimal fraction that corresponds to a given point on the coordinate axis.

Above we gave a picture with a point M. Look at it again: to get to this point, you need to measure one unit segment from zero and four tenths of it, since this point corresponds to the decimal fraction 1, 4.

If we cannot hit a point in the process of decimal measurement, then it means that an infinite decimal fraction corresponds to it.

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