Ordinary and decimal fractions and operations on them. Decimals, definitions, notation, examples, operations with decimals

That if they know the theory of series, then without it no metamatic concepts can be introduced. Moreover, these people believe that anyone who does not use it widely is ignorant. Let us leave the views of these people to their conscience. Let's better understand what an infinite periodic fraction is and how we, uneducated people who know no limits, should deal with it.

Let's divide 237 by 5. No, you don't need to launch the Calculator. Let's better remember secondary (or even primary?) school and simply divide it into a column:

Well, did you remember? Then you can get down to business.

The concept of “fraction” in mathematics has two meanings:

1. Non-integer number.
2. Non-integer form.

There are two types of fractions - in the sense, two forms of writing non-integer numbers:

1. Simple (or vertical) fractions, like 1/2 or 237/5.
2. Decimal fractions, such as 0.5 or 47.4.

Note that in general the very use of a fraction-notation does not mean that what is written is a fraction-number, for example 3/3 or 7.0 - not fractions in the first sense of the word, but in the second, of course, fractions.

In mathematics, in general, decimal counting has always been accepted, and therefore decimals more convenient than simple ones, i.e. a fraction with a decimal denominator (Vladimir Dal. Dictionary living Great Russian language. "Ten").

And if so, then I want to make every vertical fraction a decimal (“horizontal”). And to do this you simply need to divide the numerator by the denominator. Let's take, for example, the fraction 1/3 and try to make a decimal out of it.

Even a completely uneducated person will notice: no matter how long it takes, it will not separate: triplets will continue to appear ad infinitum. So let’s write it down: 0.33... We mean “the number that is obtained when you divide 1 by 3,” or, in short, “one third.” Naturally, one third is a fraction in the first sense of the word, and “1/3” and “0.33...” are fractions in the second sense of the word, that is entry forms a number that is located on the number line at such a distance from zero that if you put it aside three times, you get one.

Now let's try to divide 5 by 6:

Let's write it down again: 0.833... We mean “the number that you get when you divide 5 by 6,” or, in short, “five-sixths.” However, confusion arises here: does this mean 0.83333 (and then the triplets are repeated), or 0.833833 (and then 833 is repeated). Therefore, notation with an ellipsis does not suit us: it is not clear where the repeating part begins (it is called a “period”). Therefore, we will put the period in brackets, like this: 0,(3); 0.8(3).

0,(3) not easy equals one third, that's There is one third, because we specially invented this notation to represent this number as a decimal fraction.

This entry is called infinite periodic fraction, or simply a periodic fraction.

Whenever we divide one number by another, if we don’t get a finite fraction, we get an infinite periodic fraction, that is, someday the sequences of numbers will definitely begin to repeat. Why this is so can be understood purely speculatively by looking carefully at the column division algorithm:

In the places marked with checkmarks, different pairs of numbers cannot always be obtained (because, in principle, there are a finite number of such pairs). And as soon as such a pair appears there, which already existed, the difference will also be the same - and then the whole process will begin to repeat itself. There is no need to check this, because it is quite obvious that if you repeat the same actions, the results will be the same.

Now that we understand well essence periodic fraction, let's try multiplying one third by three. Yes, of course, you will get one, but let's write this fraction in decimal form and multiply it in a column (ambiguity does not arise here due to the ellipsis, since all the numbers after the decimal point are the same):

And again we notice that nines, nines and nines will appear after the decimal point all the time. That is, using the reverse bracket notation, we get 0,(9). Since we know that the product of one third and three is one, then 0.(9) is such a fancy way of writing one. However, it is inappropriate to use this form of recording, because a unit can be written perfectly without using a period, like this: 1.

As you can see, 0,(9) is one of those cases where the whole number is written in fraction form, like 3/3 or 7.0. That is, 0,(9) is a fraction only in the second sense of the word, but not in the first.

So, without any limits or series, we figured out what 0.(9) is and how to deal with it.

But let us still remember that in fact we are smart and studied analysis. Indeed, it is difficult to deny that:

But, perhaps, no one will argue with the fact that:

All this is, of course, true. Indeed, 0,(9) is both the sum of the reduced series and the double sine of the indicated angle, and natural logarithm Euler numbers.

But neither one, nor the other, nor the third is a definition.

To say that 0,(9) is the sum of the infinite series 9/(10 n), with n equal to one, is the same as to say that sine is the sum of the infinite Taylor series:

This absolutely right, and this is the most important fact For computational mathematics, but this is not a definition, and, most importantly, it does not bring a person any closer to understanding essentially sinus The essence of the sine of a certain angle is that it just everything the ratio of the leg opposite the angle to the hypotenuse.

So, a periodic fraction is just everything a decimal fraction that is obtained when when dividing by a column the same set of numbers will be repeated. There is no trace of analysis here.

And this is where the question arises: where does it come from? at all did we take the number 0,(9)? What do we divide by with a column to get it? Indeed, there are no numbers such that when divided into a column, we would have endlessly appearing nines. But we managed to get this number by multiplying 0,(3) by 3 with a column? Not really. After all, you need to multiply from right to left in order to correctly take into account the transfers of digits, and we did this from left to right, cunningly taking advantage of the fact that transfers do not occur anywhere anyway. Therefore, the legality of writing 0,(9) depends on whether we recognize the legality of such multiplication by a column or not.

Therefore, we can generally say that the notation 0,(9) is incorrect - and to a certain extent be right. However, since the notation a ,(b ) is accepted, it is simply ugly to abandon it when b = 9; It’s better to decide what such an entry means. So, if we generally accept the notation 0,(9), then this notation, of course, means the number one.

It only remains to add that if we used, say, the ternary number system, then when dividing by a column of one (1 3) by three (10 3) we would get 0.1 3 (read “zero point one third”), and when dividing One by two would be 0,(1) 3.

So the periodicity of a fraction-number is not some objective characteristic of a fraction-number, but just side effect using one or another number system.

It is known that if the denominator n irreducible fraction in its canonical expansion has a prime factor not equal to 2 and 5, then this fraction cannot be represented as a finite decimal fraction. If we try in this case to write down the original irreducible fraction as a decimal, dividing the numerator by the denominator, then the division process cannot be completed, because if it were completed after a finite number of steps, we would get a finite decimal fraction, which contradicts the previously proven theorem. So in this case the decimal notation of a positive rational number is A= appears to be an infinite fraction.

For example, fraction = 0.3636... . It is easy to notice that the remainders when dividing 4 by 11 are periodically repeated, therefore, the decimal places will be periodically repeated, i.e. it turns out infinite periodic decimal fraction, which can be written as 0,(36).

Periodically repeating numbers 3 and 6 form a period. It may turn out that there are several digits between the decimal point and the beginning of the first period. These numbers form the pre-period. For example,

0.1931818... The process of dividing 17 by 88 is endless. The numbers 1, 9, 3 form the pre-period; 1, 8 – period. The examples we have considered reflect a pattern, i.e. any positive rational number representable as either a finite or infinite periodic decimal fraction.

Theorem 1. Let the ordinary fraction be irreducible in the canonical expansion of the denominator n is a prime factor different from 2 and 5. Then the common fraction can be represented as an infinite periodic decimal fraction.

Proof. We already know that the process of dividing a natural number m to a natural number n will be endless. Let us show that it will be periodic. In fact, when dividing m on n the resulting balances will be smaller n, those. numbers of the form 1, 2, ..., ( n– 1), from which it is clear that the number of different remainders is finite and therefore, starting from a certain step, some remainder will be repeated, which will entail the repetition of the decimal places of the quotient, and the infinite decimal fraction becomes periodic.

Two more theorems hold.

Theorem 2. If the expansion of the denominator of an irreducible fraction into prime factors does not include the numbers 2 and 5, then when this fraction is converted into an infinite decimal fraction, a pure periodic fraction will be obtained, i.e. a fraction whose period begins immediately after the decimal point.

Theorem 3. If the expansion of the denominator includes factors 2 (or 5) or both, then the infinite periodic fraction will be mixed, i.e. between the decimal point and the beginning of the period there will be several digits (pre-period), namely as many as the largest of the exponents of the factors 2 and 5.

Theorems 2 and 3 are proposed to the reader to prove independently.

28. Methods of transition from infinite periodic
decimal fractions to common fractions

Let a periodic fraction be given A= 0,(4), i.e. 0.4444... .

Let's multiply A by 10, we get

10A= 4.444…4…Þ 10 A = 4 + 0,444….

Those. 10 A = 4 + A, we obtained an equation for A, solving it, we get: 9 A= 4 Þ A = .

We note that 4 is both the numerator of the resulting fraction and the period of the fraction 0,(4).

Rule converting a pure periodic fraction into an ordinary fraction is formulated as follows: the numerator of the fraction is equal to the period, and the denominator consists of the same number of nines as there are digits in the period of the fraction.

Let us now prove this rule for a fraction whose period consists of n

A= . Let's multiply A by 10 n, we get:

10n × A = = + 0, ;

10n × A = + a;

(10n – 1) A = Þ a = = .

So, the previously formulated rule has been proven for any pure periodic fraction.

Let us now give a fraction A= 0.605(43) – mixed periodic. Let's multiply A by 10 with the same indicator, how many digits are in the pre-period, i.e. by 10 3, we get

10 3 × A= 605 + 0,(43) Þ 10 3 × A = 605 + = 605 + = = ,

those. 10 3 × A= .

Rule converting a mixed periodic fraction into an ordinary fraction is formulated as follows: the numerator of the fraction is equal to the difference between the number written in digits before the beginning of the second period and the number written in digits before the beginning of the first period, the denominator consists of the number of nines equal to the number of digits in the period and such number of zeros how many digits there are before the start of the first period.

Let us now prove this rule for a fraction whose preperiod consists of n numbers, and the period is from To numbers Let a periodic fraction be given

Let's denote V= ; r= ,

With= ; Then With=in × 10k + r.

Let's multiply A by 10 with such an exponent how many digits are in the preperiod, i.e. by 10 n, we get:

A×10 n = + .

Taking into account the notations introduced above, we write:

a× 10n= V+ .

So, the rule formulated above has been proven for any mixed periodic fraction.

Every infinite periodic decimal fraction is a form of writing some rational number.

For the sake of consistency, sometimes a finite decimal is also considered an infinite periodic decimal with period "zero". For example, 0.27 = 0.27000...; 10.567 = 10.567000...; 3 = 3,000... .

Now the following statement becomes true: every rational number can (and in a unique way) be expressed as an infinite periodic decimal fraction, and every infinite periodic decimal fraction expresses exactly one rational number (periodic decimal fractions with a period of 9 are not considered).

The fact that many square roots are irrational numbers, does not at all detract from their significance; in particular, the number $\sqrt2$ is very often used in various engineering and scientific calculations. This number can be calculated with the accuracy required in each specific case. You can get this number to as many decimal places as you have the patience for.

For example, the number $\sqrt2$ can be determined with an accuracy of six decimal places: $\sqrt2=1.414214$. This value is not very different from the true value, since $1.414214 \times 1.414214=2.000001237796$. This answer differs from 2 by barely more than one millionth. Therefore, the value of $\sqrt2$ equal to $1.414214$ is considered quite acceptable for solving most practical problems. In cases where greater accuracy is required, it is not difficult to obtain as much significant figures after the decimal point, as needed in this case.

However, if you show rare stubbornness and try to extract square root from the number $\sqrt2$ until you achieve the exact result, you will never finish your work. It's a never-ending process. No matter how many decimal places you get, there will always be a few more left.

This fact may surprise you just as much as turning $\frac13$ into an infinite decimal $0.333333333…$ and so on indefinitely, or turning $\frac17$ into $0.142857142857142857…$ and so on indefinitely. At first glance it may seem that these infinite and irrational square roots are phenomena of the same order, but this is not at all the case. After all, these infinite fractions have a fractional equivalent, while $\sqrt2$ does not have such an equivalent. Why exactly? The fact is that the decimal equivalent of $\frac13$ and $\frac17$, as well as an infinite number of other fractions, are periodic infinite fractions.

At the same time, the decimal equivalent of $\sqrt2$ is a non-periodic fraction. This statement is also true for any irrational number.

The problem is that any decimal that is an approximation of the square root of 2 is non-periodic fraction. No matter how far we go in our calculations, any fraction we get will be non-periodic.

Imagine a fraction with a huge number of non-periodic digits after the decimal point. If suddenly after the millionth digit the entire sequence of decimal places is repeated, it means decimal- periodic and there is an equivalent for it in the form of a ratio of integers. If a fraction with a huge number (billions or millions) of non-periodic decimal places at some point has an endless series of repeating digits, for example $...55555555555...$, this also means that this fraction is periodic and there is an equivalent in the form of a ratio of integers numbers.

However, in case, their decimal equivalents are completely non-periodic and cannot become periodic.

Of course, you can ask the following question: “Who can know and say for sure what happens to a fraction, say, after the trillion sign? Who can guarantee that a fraction will not become periodic?” There are ways to conclusively prove that irrational numbers are non-periodic, but such proofs require complex mathematics. But if it suddenly turned out that the irrational number becomes periodic fraction, this would mean a complete collapse of the foundations of mathematical sciences. And in fact this is hardly possible. It’s not easy for you to throw it on your knuckles from side to side, there’s a complex mathematical theory here.

Remember how in the very first lesson about decimals I said that there are numerical fractions that cannot be represented as decimals (see lesson “ Decimals”)? We also learned how to factor the denominators of fractions to see if there were any numbers other than 2 and 5.

So: I lied. And today we will learn how to convert absolutely any numerical fraction into a decimal. At the same time, we will get acquainted with a whole class of fractions with an infinite significant part.

A periodic decimal is any decimal that:

1. The significant part consists of an infinite number of digits;
2. At certain intervals, the numbers in the significant part are repeated.

The set of repeating digits that make up the significant part is called the periodic part of a fraction, and the number of digits in this set is called the period of the fraction. The remaining segment of the significant part, which is not repeated, is called the non-periodic part.

Since there are many definitions, it is worth considering in detail several of these fractions:

This fraction appears most often in problems. Non-periodic part: 0; periodic part: 3; period length: 1.

Non-periodic part: 0.58; periodic part: 3; period length: again 1.

Non-periodic part: 1; periodic part: 54; period length: 2.

Non-periodic part: 0; periodic part: 641025; period length: 6. For convenience, repeating parts are separated from each other by a space - this is not necessary in this solution.

Non-periodic part: 3066; periodic part: 6; period length: 1.

As you can see, the definition of a periodic fraction is based on the concept significant part of a number. Therefore, if you have forgotten what it is, I recommend repeating it - see the lesson “”.

Transition to periodic decimal fraction

Consider an ordinary fraction of the form a /b. Let's factorize its denominator into prime factors. There are two options:

1. The expansion contains only factors 2 and 5. These fractions are easily converted to decimals - see the lesson “Decimals”. We are not interested in such people;
2. There is something else in the expansion other than 2 and 5. In this case, the fraction cannot be represented as a decimal, but it can be converted into a periodic decimal.

To define a periodic decimal fraction, you need to find its periodic and non-periodic parts. How? Convert the fraction to an improper fraction, and then divide the numerator by the denominator using a corner.

The following will happen:

1. Will split first whole part , if it exists;
2. There may be several numbers after the decimal point;
3. After a while the numbers will start repeat.

That's it! Repeating numbers after the decimal point are denoted by the periodic part, and those in front are denoted by the non-periodic part.

Task. Convert ordinary fractions to periodic decimals:

All fractions without an integer part, so we simply divide the numerator by the denominator with a “corner”:

As you can see, the remainders are repeated. Let's write the fraction in the “correct” form: 1.733 ... = 1.7(3).

The result is a fraction: 0.5833 ... = 0.58(3).

We write it in normal form: 4.0909 ... = 4,(09).

We get the fraction: 0.4141 ... = 0,(41).

Transition from periodic decimal fraction to ordinary fraction

Consider the periodic decimal fraction X = abc (a 1 b 1 c 1). It is required to convert it into a classic “two-story” one. To do this, follow four simple steps:

1. Find the period of the fraction, i.e. count how many digits are in the periodic part. Let this be the number k;
2. Find the value of the expression X · 10 k. This is equivalent to shifting the decimal point to the right a full period - see the lesson "Multiplying and dividing decimals";
3. The original expression must be subtracted from the resulting number. In this case, the periodic part is “burned” and remains common fraction;
4. Find X in the resulting equation. We convert all decimal fractions to ordinary fractions.

Task. Convert the number to an ordinary improper fraction:

• 9,(6);
• 32,(39);
• 0,30(5);
• 0,(2475).

We work with the first fraction: X = 9,(6) = 9.666 ...

The parentheses contain only one digit, so the period is k = 1. Next, we multiply this fraction by 10 k = 10 1 = 10. We have:

10X = 10 9.6666... ​​= 96.666...

Subtract the original fraction and solve the equation:

10X − X = 96.666 ... − 9.666 ... = 96 − 9 = 87;
9X = 87;
X = 87/9 = 29/3.

Now let's look at the second fraction. So X = 32,(39) = 32.393939...

Period k = 2, so multiply everything by 10 k = 10 2 = 100:

100X = 100 · 32.393939 ... = 3239.3939 ...

Subtract the original fraction again and solve the equation:

100X − X = 3239.3939 ... − 32.3939 ... = 3239 − 32 = 3207;
99X = 3207;
X = 3207/99 = 1069/33.

Let's move on to the third fraction: X = 0.30(5) = 0.30555 ... The diagram is the same, so I’ll just give the calculations:

Period k = 1 ⇒ multiply everything by 10 k = 10 1 = 10;

10X = 10 0.30555... = 3.05555...
10X − X = 3.0555 ... − 0.305555 ... = 2.75 = 11/4;
9X = 11/4;
X = (11/4) : 9 = 11/36.

Finally, the last fraction: X = 0,(2475) = 0.2475 2475... Again, for convenience, the periodic parts are separated from each other by spaces. We have:

k = 4 ⇒ 10 k = 10 4 = 10,000;
10,000X = 10,000 0.2475 2475 = 2475.2475 ...
10,000X − X = 2475.2475 ... − 0.2475 2475 ... = 2475;
9999X = 2475;
X = 2475: 9999 = 25/101.

This article is about decimals. Here we will understand the decimal notation of fractional numbers, introduce the concept of a decimal fraction and give examples of decimal fractions. Next we’ll talk about the digits of decimal fractions and give the names of the digits. After this, we will focus on infinite decimal fractions, let's talk about periodic and non-periodic fractions. Next we list the basic operations with decimal fractions. In conclusion, let us establish the position of decimal fractions on the coordinate beam.

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Decimal notation of a fractional number

Reading Decimals

Let's say a few words about the rules for reading decimal fractions.

Decimal fractions, which correspond to proper ordinary fractions, are read in the same way as these ordinary fractions, only “zero integer” is first added. For example, the decimal fraction 0.12 corresponds to the common fraction 12/100 (read “twelve hundredths”), therefore, 0.12 is read as “zero point twelve hundredths”.

Decimal fractions that correspond to mixed numbers are read exactly the same as these mixed numbers. For example, the decimal fraction 56.002 corresponds to mixed number, therefore, the decimal fraction 56.002 is read as "fifty-six point two thousandths."

Places in decimals

In writing decimal fractions, as well as in writing natural numbers, the meaning of each digit depends on its position. Indeed, the number 3 in the decimal fraction 0.3 means three tenths, in the decimal fraction 0.0003 - three ten thousandths, and in the decimal fraction 30,000.152 - three ten thousandths. So we can talk about decimal places, as well as about the digits in natural numbers.

The names of the digits in the decimal fraction up to the decimal point completely coincide with the names of the digits in natural numbers. And the names of the decimal places after the decimal point can be seen from the following table.

For example, in the decimal fraction 37.051, the digit 3 is in the tens place, 7 is in the units place, 0 is in the tenths place, 5 is in the hundredths place, and 1 is in the thousandths place.

Places in decimal fractions also differ in precedence. If in writing a decimal fraction we move from digit to digit from left to right, then we will move from seniors To junior ranks. For example, the hundreds place is older than the tenths place, and the millions place is lower than the hundredths place. In a given final decimal fraction, we can talk about the major and minor digits. For example, in decimal fraction 604.9387 senior (highest) the place is the hundreds place, and junior (lowest)- ten-thousandths digit.

For decimal fractions, expansion into digits takes place. It is similar to expansion by digits of natural numbers. For example, the expansion into decimal places of 45.6072 is as follows: 45.6072=40+5+0.6+0.007+0.0002. And the properties of addition from the decomposition of a decimal fraction into digits allow you to move on to other representations of this decimal fraction, for example, 45.6072=45+0.6072, or 45.6072=40.6+5.007+0.0002, or 45.6072= 45.0072+0.6.

Ending decimals

Up to this point, we have only talked about decimal fractions, in the notation of which there is a finite number of digits after the decimal point. Such fractions are called finite decimals.

Definition.

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

Here are some examples of final decimal fractions: 0.317, 3.5, 51.1020304958, 230,032.45.

However, not every fraction can be represented as a final decimal. For example, the fraction 5/13 cannot be replaced by an equal fraction with one of the denominators 10, 100, ..., therefore, cannot be converted into a final decimal fraction. We will talk more about this in the theory section, converting ordinary fractions to decimals.

Infinite Decimals: Periodic Fractions and Non-Periodic Fractions

In writing a decimal fraction after the decimal point, it is possible to allow for the possibility of an infinite number of digits. In this case, we will come to consider the so-called infinite decimal fractions.

Definition.

Infinite decimals- These are decimal fractions, which contain an infinite number of digits.

It is clear that we cannot write down infinite decimal fractions in full form, so in their writing we limit ourselves to only a certain finite number of digits after the decimal point and put an ellipsis indicating an infinitely continuing sequence of digits. Here are some examples of infinite decimal fractions: 0.143940932…, 3.1415935432…, 153.02003004005…, 2.111111111…, 69.74152152152….

If you look closely at the last two infinite decimal fractions, then in the fraction 2.111111111... the endlessly repeating number 1 is clearly visible, and in the fraction 69.74152152152..., starting from the third decimal place, a repeating group of numbers 1, 5 and 2 is clearly visible. Such infinite decimal fractions are called periodic.

Definition.

Periodic decimals(or just periodic fractions) are endless decimal fractions, in the recording of which, starting from a certain decimal place, some number or group of numbers is endlessly repeated, which is called period of the fraction.

For example, the period of the periodic fraction 2.111111111... is the digit 1, and the period of the fraction 69.74152152152... is a group of digits of the form 152.

For infinite periodic decimal fractions, a special form of notation is adopted. For brevity, we agreed to write down the period once, enclosing it in parentheses. For example, the periodic fraction 2.111111111... is written as 2,(1) , and the periodic fraction 69.74152152152... is written as 69.74(152) .

It is worth noting that different periods can be specified for the same periodic decimal fraction. For example, the periodic decimal fraction 0.73333... can be considered as a fraction 0.7(3) with a period of 3, and also as a fraction 0.7(33) with a period of 33, and so on 0.7(333), 0.7 (3333), ... You can also look at the periodic fraction 0.73333 ... like this: 0.733(3), or like this 0.73(333), etc. Here, in order to avoid ambiguity and discrepancies, we agree to consider as the period of a decimal fraction the shortest of all possible sequences of repeating digits, and starting from the closest position to the decimal point. That is, the period of the decimal fraction 0.73333... will be considered a sequence of one digit 3, and the periodicity starts from the second position after the decimal point, that is, 0.73333...=0.7(3). Another example: the periodic fraction 4.7412121212... has a period of 12, the periodicity starts from the third digit after the decimal point, that is, 4.7412121212...=4.74(12).

Infinite decimal periodic fractions are obtained by converting into decimal fractions ordinary fractions whose denominators contain prime factors other than 2 and 5.

Here it is worth mentioning periodic fractions with a period of 9. Let us give examples of such fractions: 6.43(9) , 27,(9) . These fractions are another notation for periodic fractions with period 0, and are commonly replaced by periodic fractions with period 0. To do this, period 9 is replaced by period 0, and the value of the next highest digit is increased by one. For example, a fraction with period 9 of the form 7.24(9) is replaced by a periodic fraction with period 0 of the form 7.25(0) or an equal final decimal fraction 7.25. Another example: 4,(9)=5,(0)=5. The equality of a fraction with period 9 and its corresponding fraction with period 0 is easily established after replacing these decimal fractions with equal ordinary fractions.

Finally, let's take a closer look at infinite decimal fractions, which do not contain an endlessly repeating sequence of digits. They are called non-periodic.

Definition.

Non-recurring decimals(or just non-periodic fractions) are infinite decimal fractions that have no period.

Sometimes non-periodic fractions have a form similar to that of periodic fractions, for example, 8.02002000200002... is a non-periodic fraction. In these cases, you should be especially careful to notice the difference.

Note that non-periodic fractions do not convert to ordinary fractions; infinite non-periodic decimal fractions represent irrational numbers.

Operations with decimals

One of the operations with decimal fractions is comparison, and the four basic arithmetic functions are also defined operations with decimals: addition, subtraction, multiplication and division. Let's consider separately each of the actions with decimal fractions.

Comparison of decimals essentially based on comparison of ordinary fractions corresponding to the decimal fractions being compared. However, converting decimal fractions into ordinary fractions is a rather labor-intensive process, and infinite non-periodic fractions cannot be represented as an ordinary fraction, so it is convenient to use a place-by-digit comparison of decimal fractions. Place-wise comparison of decimal fractions is similar to comparison of natural numbers. For more detailed information, we recommend studying the material in the article: comparison of decimal fractions, rules, examples, solutions.

Let's move on to the next step - multiplying decimals. Multiplication of finite decimal fractions is carried out similarly to subtraction of decimal fractions, rules, examples, solutions to multiplication by a column of natural numbers. In the case of periodic fractions, multiplication can be reduced to multiplication of ordinary fractions. In turn, the multiplication of infinite non-periodic decimal fractions after rounding them is reduced to the multiplication of finite decimal fractions. We recommend for further study the material in the article: multiplication of decimal fractions, rules, examples, solutions.

Decimals on a coordinate ray

There is a one-to-one correspondence between points and decimals.

Let's figure out how points on the coordinate ray are constructed that correspond to a given decimal fraction.

We can replace finite decimal fractions and infinite periodic decimal fractions with equal ordinary fractions, and then construct the corresponding ordinary fractions on the coordinate ray. For example, the decimal fraction 1.4 corresponds to the common fraction 14/10, therefore the point with coordinate 1.4 is removed from the origin in the positive direction by 14 segments equal to a tenth of a unit segment.

Decimal fractions can be marked on a coordinate ray, starting from the decomposition of a given decimal fraction into digits. For example, let us need to build a point with coordinate 16.3007, since 16.3007=16+0.3+0.0007, then in this point you can get there by sequentially laying off from the origin 16 unit segments, 3 segments whose length is equal to a tenth of a unit segment, and 7 segments whose length is equal to a ten-thousandth of a unit segment.

This way of building decimal numbers on the coordinate ray allows you to get as close as you like to the point corresponding to the infinite decimal fraction.

Sometimes it is possible to accurately plot the point corresponding to an infinite decimal fraction. For example, , then this infinite decimal fraction 1.41421... corresponds to a point coordinate ray, removed from the origin by the length of the diagonal of a square with a side of 1 unit segment.

The reverse process of obtaining the decimal fraction corresponding to a given point on a coordinate ray is the so-called decimal measurement of a segment. Let's figure out how it is done.

Let our task be to get from the origin to a given point on the coordinate line (or to infinitely approach it if we can’t get to it). With the decimal measurement of a segment, we can sequentially lay off from the origin any number of unit segments, then segments whose length is equal to a tenth of a unit, then segments whose length is equal to a hundredth of a unit, etc. By recording the number of segments of each length laid aside, we obtain the decimal fraction corresponding to a given point on the coordinate ray.

For example, to get to point M in the above figure, you need to set aside 1 unit segment and 4 segments, the length of which is equal to a tenth of a unit. Thus, point M corresponds to the decimal fraction 1.4.

It is clear that the points of the coordinate ray, which cannot be reached in the process of decimal measurement, correspond to infinite decimal fractions.

References.

• Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
• Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
• Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.