Already in elementary school, students are exposed to fractions. And then they appear in every topic. You cannot forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are not complicated, the main thing is to understand everything in order.
Why are fractions needed?
The world around us consists of entire objects. Therefore, there is no need for shares. But everyday life constantly pushes people to work with parts of objects and things.
For example, chocolate consists of several pieces. Consider a situation where his tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It can easily be divided into three. But it will not be possible to give five people a whole number of chocolate slices.
By the way, these slices are already fractions. And their further division leads to the appearance of more complex numbers.
What is a "fraction"?
This is a number made up of parts of a unit. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written at the top (left) is called the numerator. What is at the bottom (right) is the denominator.
Essentially, the slash turns out to be a division sign. That is, the numerator can be called the dividend, and the denominator can be called the divisor.
What fractions are there?
In mathematics there are only two types: ordinary and decimal fractions. Schoolchildren become acquainted with the first ones in elementary school, calling them simply “fractions.” The latter will be learned in 5th grade. That's when these names appear.
Common fractions are all those that are written as two numbers separated by a line. For example, 4/7. A decimal is a number in which the fractional part has a positional notation and is separated from the whole number by a comma. For example, 4.7. Students need to clearly understand that the two examples given are completely different numbers.
Every simple fraction can be written as a decimal. This statement is almost always true in reverse. There are rules that allow you to write a decimal fraction as a common fraction.
What subtypes do these types of fractions have?
It is better to start in chronological order, as they are studied. Common fractions come first. Among them, 5 subspecies can be distinguished.
Correct. Its numerator is always less than its denominator.
Wrong. Its numerator is greater than or equal to its denominator.
Reducible/irreducible. It may turn out to be either right or wrong. Another important thing is whether the numerator and denominator have common factors. If there are, then it is necessary to divide both parts of the fraction by them, that is, reduce it.
Mixed. An integer is assigned to its usual regular (irregular) fractional part. Moreover, it is always on the left.
Composite. It is formed from two fractions divided by each other. That is, it contains three fractional lines at once.
Decimal fractions have only two subtypes:
finite, that is, one whose fractional part is limited (has an end);
infinite - a number whose digits after the decimal point do not end (they can be written endlessly).
How to convert a decimal fraction to a common fraction?
If this is a finite number, then an association is applied based on the rule - as I hear, so I write. That is, you need to read it correctly and write it down, but without a comma, but with a fractional bar.
As a hint about the required denominator, you need to remember that it is always one and several zeros. You need to write as many of the latter as there are digits in the fractional part of the number in question.
How to convert decimal fractions into ordinary fractions if their integer part is missing, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. All that remains is to write down the fractional parts. The first number will have a denominator of 10, the second will have a denominator of 100. That is, the given examples will have the following numbers as answers: 9/10, 5/100. Moreover, it turns out that the latter can be reduced by 5. Therefore, the result for it needs to be written as 1/20.
How can you convert a decimal fraction into an ordinary fraction if its integer part is different from zero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its value is written. In the first case it is 5, in the second it is 13. Then you need to move on to the fractional part. The same operation is supposed to be carried out with them. The first number appears 23/100, the second - 108/100000. The second value needs to be reduced again. The answer gives the following mixed fractions: 5 23/100 and 13 27/25000.
How to convert an infinite decimal fraction to an ordinary fraction?
If it is non-periodic, then such an operation will not be possible. This fact is due to the fact that each decimal fraction is always converted to either a finite or a periodic fraction.
The only thing you can do with such a fraction is round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal will never give the initial value. That is, infinite non-periodic fractions are not converted into ordinary fractions. This needs to be remembered.
How to write an infinite periodic fraction as an ordinary fraction?
In these numbers, there are always one or more digits after the decimal point that are repeated. They are called a period. For example, 0.3(3). Here "3" is in the period. They are classified as rational because they can be converted into ordinary fractions.
Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second, the fractional part begins with some numbers, and then the repetition begins.
The rule by which you need to write an infinite decimal as an ordinary fraction will be different for the two types of numbers indicated. It is quite easy to write pure periodic fractions as ordinary fractions. As with finite ones, they need to be converted: write down the period in the numerator, and the denominator will be the number 9, repeated as many times as the number of digits the period contains.
For example, 0,(5). The number does not have an integer part, so you need to immediately start with the fractional part. Write 5 as the numerator and 9 as the denominator. That is, the answer will be the fraction 5/9.
The rule on how to write an ordinary decimal periodic fraction that is mixed.
Look at the length of the period. That's how many 9s the denominator will have.
Write down the denominator: first nines, then zeros.
To determine the numerator, you need to write down the difference of two numbers. All numbers after the decimal point will be minified, along with the period. Deductible - it is without a period.
For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period contains one digit. So there will be one zero. There is also only one number in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.
To determine the numerator, you need to subtract 5 from 58. It turns out 53. For example, the answer would have to be written as 53/90.
How are fractions converted to decimals?
The simplest option is a number whose denominator is the number 10, 100, etc. Then the denominator is simply discarded, and a comma is placed between the fractional and integer parts.
There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. You just need to multiply not only the denominator, but also the numerator by the same number.
For all other cases, a simple rule is useful: divide the numerator by the denominator. In this case, you may get two possible answers: a finite or a periodic decimal fraction.
Operations with ordinary fractions
Addition and subtraction
Students become acquainted with them earlier than others. Moreover, at first the fractions have the same denominators, and then they have different ones. General rules can be reduced to this plan.
Find the least common multiple of the denominators.
Write additional factors for all ordinary fractions.
Multiply the numerators and denominators by the factors specified for them.
Add (subtract) the numerators of the fractions and leave the common denominator unchanged.
If the numerator of the minuend is less than the subtrahend, then we need to find out whether we have a mixed number or a proper fraction.
In the first case, you need to borrow one from the whole part. Add the denominator to the numerator of the fraction. And then do the subtraction.
In the second, it is necessary to apply the rule of subtracting a larger number from a smaller number. That is, from the module of the subtrahend, subtract the module of the minuend, and in response put a “-” sign.
Look carefully at the result of addition (subtraction). If you get an improper fraction, then you need to select the whole part. That is, divide the numerator by the denominator.
Multiplication and division
To perform them, fractions do not need to be reduced to a common denominator. This makes it easier to perform actions. But they still require you to follow the rules.
When multiplying fractions, you need to look at the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.
Multiply the numerators.
Multiply the denominators.
If the result is a reducible fraction, then it must be simplified again.
When dividing, you must first replace division with multiplication, and the divisor (second fraction) with the reciprocal fraction (swap the numerator and denominator).
Then proceed as with multiplication (starting from point 1).
In tasks where you need to multiply (divide) by a whole number, the latter should be written as an improper fraction. That is, with a denominator of 1. Then act as described above.
Operations with decimals
Addition and subtraction
Of course, you can always convert a decimal into a fraction. And act according to the plan already described. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.
Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Add the missing number of zeros to it.
Write the fractions so that the comma is below the comma.
Add (subtract) like natural numbers.
Remove the comma.
Multiplication and division
It is important that you do not need to add zeros here. Fractions should be left as they are given in the example. And then go according to plan.
To multiply, you need to write the fractions one below the other, ignoring the commas.
Multiply like natural numbers.
Place a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.
To divide, you must first transform the divisor: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.
Multiply the dividend by the same number.
Divide a decimal fraction by a natural number.
Place a comma in your answer at the moment when the division of the whole part ends.
What if one example contains both types of fractions?
Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. In such tasks there are two possible solutions. You need to objectively weigh the numbers and choose the optimal one.
First way: represent ordinary decimals
It is suitable if division or translation results in finite fractions. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you don’t like working with ordinary fractions, you will have to count them.
Second way: write decimal fractions as ordinary
This technique turns out to be convenient if the part after the decimal point contains 1-2 digits. If there are more of them, you may end up with a very large common fraction and decimal notation will make the task faster and easier to calculate. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.
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 can be represented as either a finite or an 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 precision is required, it is not difficult to obtain as many significant digits 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 for it 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.
As is known, the set of rational numbers (Q) includes the set of integers (Z), which in turn includes the set of natural numbers (N). In addition to whole numbers, rational numbers include fractions.
Why then is the entire set of rational numbers sometimes considered as infinite periodic decimal fractions? Indeed, in addition to fractions, they also include integers, as well as non-periodic fractions.
The fact is that all integers, as well as any fraction, can be represented as an infinite periodic decimal fraction. That is, for all rational numbers you can use the same recording method.
How is an infinite periodic decimal represented? In it, a repeating group of numbers after the decimal point is placed in brackets. For example, 1.56(12) is a fraction in which the group of digits 12 is repeated, i.e. the fraction has the value 1.561212121212... and so on endlessly. A repeating group of numbers is called a period.
However, we can represent any number in this form if we consider its period to be the number 0, which also repeats endlessly. For example, the number 2 is the same as 2.00000.... Therefore, it can be written as an infinite periodic fraction, i.e. 2,(0).
The same can be done with any finite fraction. For example:
0,125 = 0,1250000... = 0,125(0)
However, in practice they do not use the transformation of a finite fraction into an infinite periodic one. Therefore, they separate finite fractions and infinite periodic ones. Thus, it is more correct to say that the rational numbers include
- all integers
- final fractions,
- infinite periodic fractions.
At the same time, simply remember that integers and finite fractions are representable in theory in the form of infinite periodic fractions.
On the other hand, the concepts of finite and infinite fractions are applicable to decimal fractions. When it comes to fractions, both finite and infinite decimals can be uniquely represented as a fraction. This means that from the point of view of ordinary fractions, periodic and finite fractions are the same thing. Additionally, whole numbers can also be represented as a fraction by imagining that we are dividing the number by 1.
How to represent a decimal infinite periodic fraction as an ordinary fraction? The most commonly used algorithm is something like this:
- Reduce the fraction so that after the decimal point there is only a period.
- Multiply an infinite periodic fraction by 10 or 100 or ... so that the decimal point moves to the right by one period (i.e., one period ends up in the whole part).
- Equate the original fraction (a) to the variable x, and the fraction (b) obtained by multiplying by the number N to Nx.
- Subtract x from Nx. From b I subtract a. That is, they make up the equation Nx – x = b – a.
- When solving an equation, the result is an ordinary fraction.
An example of converting an infinite periodic decimal fraction into an ordinary fraction:
x = 1.13333...
10x = 11.3333...
10x * 10 = 11.33333... * 10
100x = 113.3333...
100x – 10x = 113.3333... – 11.3333...
90x = 102
x =
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:
- Non-integer number.
- Non-integer form.
- Simple (or vertical) fractions, like 1/2 or 237/5.
- Decimal fractions, such as 0.5 or 47.4.
In mathematics, in general, decimal counting has always been accepted, and therefore decimal fractions are more convenient than simple ones, that is, a fraction with a decimal denominator (Vladimir Dal. Explanatory dictionary of the 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 the natural logarithm of the Euler number.
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 it 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 a side effect of using one or another number system.
