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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; in) 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.
AT example 1 when a) denominator 25=5 5; when in) 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) by 5 so that “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 in)(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.
Decimal. Whole part. Decimal point.
Decimals. Properties of decimal fractions.
Periodic decimal. Period .
Decimal is the result of dividing one by ten, one hundred, one thousand, etc. parts. These fractions are very convenient for calculations, since they are based on the same positional system on which counting and notation of integers are built. Due to this, the notation and rules for decimals are actually the same as for integers. When writing decimal fractions, there is no need to mark the denominator, this is determined by the place that the corresponding figure occupies. First spelled whole part numbers, then put on the right decimal point. The first digit after the decimal point means the number of tenths, the second - the number of hundredths, the third - the number of thousandths, etc. The numbers after the decimal point are called decimal places.
EXAMPLE
One of advantages of decimals- they are easy brought to mindordinary: the number after the decimal point (in our case 5047) is the numerator; the denominator is equaln-th degree 10, wheren- number of decimal places(in our case n= 4):
If the decimal fraction does not contain an integer part, then a zero is placed before the decimal point:
Properties of decimal fractions.
1. Decimal does not change if zeros are added to the right:
13.6 =13.6000.
2. The decimal fraction does not change if you remove the zeros located
in the end decimal fraction:
0.00123000 = 0.00123 .
Attention! Cannot remove non-terminal zeros decimal!
These properties allow you to quickly multiply and divide decimals by 10, 100, 1000, and so on.
Periodic decimal contains an infinitely repeating group of digits called period. The period is written in brackets. For example, 0.12345123451234512345… = 0.(12345).
EXAMPLE If we divide 47 by 11, we get 4.27272727… = 4.(27).
In this article, we will understand what a decimal fraction is, what features and properties it has. Go! 🙂
The decimal fraction is a special case of ordinary fractions (in which the denominator is a multiple of 10).
Definition
Decimals are fractions whose denominators are numbers consisting of one and a certain number of zeros following it. That is, these are fractions with a denominator of 10, 100, 1000, etc. Otherwise, a decimal fraction can be characterized as a fraction with a denominator of 10 or one of the powers of ten.
Fraction examples:
, ,
A decimal fraction is written differently than a common fraction. Operations with these fractions are also different from operations with ordinary ones. The rules for operations on them are to a large extent close to the rules for operations on integers. This, in particular, determines their relevance in solving practical problems.
Representation of a fraction in decimal notation
There is no denominator in the decimal notation, it displays the number of the numerator. In general, decimal fractions are written as follows:
where X is the integer part of the fraction, Y is its fractional part, "," is the decimal point.
For the correct representation of an ordinary fraction as a decimal, it is required that it be correct, that is, with a highlighted integer part (if possible) and a numerator that is less than the denominator. Then, in decimal notation, the integer part is written before the decimal point (X), and the numerator of the ordinary fraction is written after the decimal point (Y).
If the numerator represents a number with a number of digits less than the number of zeros in the denominator, then in the Y part, the missing number of digits in the decimal notation is filled with zeros in front of the numerator digits.
Example: 
If the ordinary fraction is less than 1, i.e. does not have an integer part, then 0 is written in decimal form for X.
In the fractional part (Y), after the last significant (other than zero) digit, an arbitrary number of zeros can be entered. It does not affect the value of the fraction. And vice versa: all zeros at the end of the fractional part of the decimal fraction can be omitted.
Reading decimals
Part X is read in the general case as follows: "X integers."
The Y part is read according to the number in the denominator. For the denominator 10, you should read: "Y tenths", for the denominator 100: "Y hundredths", for the denominator 1000: "Y thousandths" and so on ... 😉
Another approach to reading is considered more correct, based on counting the number of digits of the fractional part. To do this, you need to understand that the fractional digits are located in a mirror image with respect to the digits of the integer part of the fraction.
Names for correct reading are given in the table:
Based on this, the reading should be based on the correspondence to the name of the category of the last digit of the fractional part.
- 3.5 reads "three point five"
- 0.016 reads like "zero point sixteen thousandths"
Converting an arbitrary ordinary fraction to a decimal
If the denominator of an ordinary fraction is 10 or some power of ten, then the fraction is converted as described above. In other situations, additional transformations are required.
There are 2 ways to translate.
The first way of translation
The numerator and denominator must be multiplied by such an integer that the denominator is 10 or one of the powers of ten. And then the fraction is represented in decimal notation.
This method is applicable for fractions, the denominator of which is decomposed only into 2 and 5. So, in the previous example 
. If there are other prime factors in the expansion (for example, ), then you will have to resort to the 2nd method.
The second way of translation
The 2nd method is to divide the numerator by the denominator in a column or on a calculator. The integer part, if any, is not involved in the transformation.
The long division rule that results in a decimal fraction is described below (see Dividing Decimals).
Convert decimal to ordinary
To do this, its fractional part (to the right of the comma) should be written as a numerator, and the result of reading the fractional part should be written as the corresponding number in the denominator. Further, if possible, you need to reduce the resulting fraction.
End and Infinite Decimal
The decimal fraction is called final, the fractional part of which consists of a finite number of digits.
All the above examples contain exactly the final decimal fractions. However, not every ordinary fraction can be represented as a final decimal. If the 1st translation method for a given fraction is not applicable, and the 2nd method demonstrates that the division cannot be completed, then only an infinite decimal fraction can be obtained.
It is impossible to write an infinite fraction in its full form. In an incomplete form, such fractions can be represented:
- as a result of reduction to the desired number of decimal places;
- in the form of a periodic fraction.
A fraction is called periodic, in which, after the decimal point, an infinitely repeating sequence of digits can be distinguished.
The remaining fractions are called non-periodic. For non-periodic fractions, only the 1st representation method (rounding) is allowed.
An example of a periodic fraction: 0.8888888 ... There is a repeating figure 8 here, which, obviously, will be repeated indefinitely, since there is no reason to assume otherwise. This number is called fraction period.
Periodic fractions are pure and mixed. A decimal fraction is pure, in which the period begins immediately after the decimal point. A mixed fraction has 1 or more digits before the decimal point.
54.33333 ... - periodic pure decimal fraction
2.5621212121 ... - periodic mixed fraction
Examples of writing infinite decimals:
The 2nd example shows how to properly form a period in a periodic fraction.
Converting periodic decimals to ordinary
To convert a pure periodic fraction into an ordinary period, write it in the numerator, and write in the denominator a number consisting of nines in an amount equal to the number of digits in the period.
A mixed recurring decimal is translated as follows:
- you need to form a number consisting of the number after the decimal point before the period, and the first period;
- from the resulting number subtract the number after the decimal point before the period. The result will be the numerator of an ordinary fraction;
- in the denominator, you need to enter a number consisting of the number of nines equal to the number of digits of the period, followed by zeros, the number of which is equal to the number of digits of the number after the decimal point before the 1st period.
Decimal Comparison
Decimal fractions are compared initially by their whole parts. The larger is the fraction that has the larger integer part.
If the integer parts are the same, then the digits of the corresponding digits of the fractional part are compared, starting from the first (from the tenths). The same principle applies here: the larger of the fractions, which has a larger rank of tenths; if the tenths digits are equal, the hundredths digits are compared, and so on.
Because the
, since with equal integer parts and equal tenths in the fractional part, the 2nd fraction has more hundredths.
Adding and subtracting decimals
Decimals are added and subtracted in the same way as whole numbers, writing the corresponding digits one under the other. To do this, you need to have decimal points under each other. Then the units (tens, etc.) of the integer part, as well as the tenths (hundredths, etc.) of the fractional part will match. The missing digits of the fractional part are filled with zeros. Directly The process of addition and subtraction is carried out in the same way as for integers.
Decimal multiplication
To multiply decimal fractions, you need to write them one under the other, aligned with the last digit and not paying attention to the location of the decimal points. Then you need to multiply the numbers in the same way as when multiplying integers. After receiving the result, you should recalculate the number of digits after the decimal point in both fractions and separate the total number of fractional digits in the resulting number with a comma. If there are not enough digits, they are replaced by zeros.
Multiplying and dividing decimals by 10 n
These actions are simple and come down to moving the decimal point. P in multiplication, the comma is moved to the right (the fraction increases) by the number of digits equal to the number of zeros in 10 n, where n is an arbitrary integer power. That is, a certain number of digits are transferred from the fractional part to the integer. When dividing, respectively, the comma is transferred to the left (the number decreases), and some of the digits are transferred from the integer part to the fractional part. If there are not enough digits to transfer, then the missing digits are filled with zeros.
Dividing a decimal and an integer by an integer and a decimal
Dividing a decimal by an integer is the same as dividing two integers. Additionally, only the position of the decimal point must be taken into account: when demolishing the digit of the digit followed by a comma, it is necessary to put a comma after the current digit of the generated answer. Then you need to keep dividing until you get zero. If there are not enough signs in the dividend for complete division, zeros should be used as them.
Similarly, 2 integers are divided into a column if all the digits of the dividend have been demolished, and the full division has not yet been completed. In this case, after the demolition of the last digit of the dividend, a decimal point is placed in the resulting answer, and zeros are used as the demolished digits. Those. the dividend here, in fact, is represented as a decimal fraction with a zero fractional part.
To divide a decimal fraction (or an integer) by a decimal number, it is necessary to multiply the dividend and the divisor by the number 10 n, in which the number of zeros is equal to the number of digits after the decimal point in the divisor. In this way, they get rid of the decimal point in the fraction by which you want to divide. Further, the division process is the same as described above.
Graphical representation of decimals
Graphically, decimal fractions are represented by means of a coordinate line. For this, single segments are additionally divided into 10 equal parts, just as centimeters and millimeters are deposited on a ruler at the same time. This ensures that decimals are displayed accurately and can be compared objectively.
In order for the longitudinal divisions on single segments to be the same, one should carefully consider the length of the single segment itself. It should be such that the convenience of additional division can be ensured.
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:
- 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;
- 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;
- 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:
- Write out the numerator separately;
- 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;
- 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:
A 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.
A 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:
- 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;
- 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.
- 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.
A 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:
- 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;
- 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;
- 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.
A 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.
Already in elementary school, students are faced with fractions. And then they appear in every topic. It is impossible to forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are simple, the main thing is to understand everything in order.
Why are fractions needed?
The world around us consists of whole 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 slices. Consider the situation where its tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It will be well divided into three. But the five will not be able to give a whole number of slices of chocolate.
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 consisting of parts of one. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written on the top (left) is called the numerator. The one on the bottom (right) is the denominator.
In fact, the fractional bar turns out to be a division sign. That is, the numerator can be called a dividend, and the denominator can be called a divisor.
What are the fractions?
In mathematics, there are only two types of them: ordinary and decimal fractions. Schoolchildren get acquainted with the first ones in the elementary grades, calling them simply “fractions”. The second learn in the 5th grade. That's when these names appear.
Common fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the integer with a comma. For example, 4.7. Students need to be clear 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 as well. There are rules that allow you to write a decimal fraction as an ordinary fraction.
What subspecies do these types of fractions have?
It is better to start in chronological order, as they are being studied. Common fractions come first. Among them, 5 subspecies can be distinguished.
Correct. Its numerator is always less than the denominator.
Wrong. Its numerator is greater than or equal to the denominator.
Reducible / irreducible. It can be either right or wrong. Another thing is important, whether the numerator and denominator have common factors. If there are, then they are supposed to divide both parts of the fraction, that is, to reduce it.
Mixed. An integer is assigned to its usual correct (incorrect) fractional part. And it always stands on the left.
Composite. It is formed from two fractions divided into each other. That is, it has three fractional features at once.
Decimals have only two subspecies:
final, that is, one in which the 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 decimal to ordinary?
If this is a finite number, then an association based on the rule is applied - 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 line.
As a hint about the required denominator, remember that it is always a one and a few zeros. The latter need to be written as many as the digits in the fractional part of the number in question.
How to convert decimal fractions to ordinary ones if their whole 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. It remains to write down only the fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter turns out to be possible to reduce by 5. Therefore, the result for it must be written 1/20.
How to make an ordinary fraction from a decimal if its integer part is different from zero? For example, 5.23 or 13.00108. Both examples read the integer part and write its value. In the first case, this is 5, in the second, 13. Then you need to move on to the fractional part. With them it is necessary to carry out the same operation. The first number has 23/100, the second has 108/100000. The second value needs to be reduced again. The answer is mixed fractions: 5 23/100 and 13 27/25000.
How to convert an infinite decimal to a common fraction?
If it is non-periodic, then such an operation cannot be carried out. This fact is due to the fact that each decimal fraction is always converted to either final or periodic.
The only thing that is allowed to be done with such a fraction is to 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 to ordinary fractions. This must be remembered.
How to write an infinite periodic fraction in the form of an ordinary?
In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called periods. For example, 0.3(3). Here "3" in the period. They are classified as rational, as 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 any numbers, and then the repetition begins.
The rule by which you need to write an infinite decimal in the form of an ordinary fraction will be different for these two types of numbers. It is quite easy to write pure periodic fractions as ordinary fractions. As with the final ones, they need to be converted: write the period into the numerator, and the number 9 will be the denominator, repeating as many times as there are digits in the period.
For example, 0,(5). The number does not have an integer part, so you need to immediately proceed to the fractional part. Write 5 in the numerator, and write 9 in the denominator. That is, the answer will be the fraction 5/9.
A rule on how to write a common decimal fraction that is a mixed fraction.
Look at the length of the period. So much 9 will have a denominator.
Write down the denominator: first nines, then zeros.
To determine the numerator, you need to write the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtractable - 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 is one digit. So zero will be one. There is also only one digit 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 from 58, you need to subtract 5. It turns out 53. For example, you will have to write 53/90 as an answer.
How are common fractions converted to decimals?
The simplest option is a number whose denominator is the number 10, 100, and so on. 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. Only it is necessary to multiply not only the denominator, but also the numerator by the same number.
For all other cases, a simple rule will come in handy: divide the numerator by the denominator. In this case, you may get two answers: a final or a periodic decimal fraction.
Operations with common fractions
Addition and subtraction
Students get to know them earlier than others. And at first the fractions have the same denominators, and then different. General rules can be reduced to such a plan.
Find the least common multiple of the denominators.
Write additional factors to all ordinary fractions.
Multiply the numerators and denominators by the factors defined for them.
Add (subtract) the numerators of fractions, and leave the common denominator unchanged.
If the numerator of the minuend is less than the subtrahend, then you need to find out whether we have a mixed number or a proper fraction.
In the first case, the integer part needs to take one. Add a denominator to the numerator of a fraction. And then do the subtraction.
In the second - it is necessary to apply the rule of subtraction from a smaller number to a larger one. That is, subtract the modulus of the minuend from the modulus of the subtrahend, and put the “-” sign in response.
Look carefully at the result of addition (subtraction). If you get an improper fraction, then it is supposed to select the whole part. That is, divide the numerator by the denominator.
Multiplication and division
For their implementation, fractions do not need to be reduced to a common denominator. This makes it easier to take action. But they still have to follow the rules.
When multiplying ordinary fractions, it is necessary to consider the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.
Multiply numerators.
Multiply the denominators.
If you get a reducible fraction, then it is supposed to be simplified again.
When dividing, you must first replace division with multiplication, and the divisor (second fraction) with a reciprocal (swap the numerator and denominator).
Then proceed as in multiplication (starting from point 1).
In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written as an improper fraction. That is, with a denominator of 1. Then proceed as described above.
Operations with decimals
Addition and subtraction
Of course, you can always turn a decimal into a common fraction. And act according to the already described plan. 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. Assign the missing number of zeros in it.
Write fractions so that the comma is under the comma.
Add (subtract) like natural numbers.
Remove the comma.
Multiplication and division
It is important that you do not need to append zeros here. Fractions are supposed to be left as they are given in the example. And then go according to plan.
For multiplication, you need to write fractions one under the other, not paying attention to commas.
Multiply like natural numbers.
Put 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 convert 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 by a natural number.
Put a comma in the answer at the moment when the division of the whole part ends.
What if there are both types of fractions in one example?
Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. There are two possible solutions to these problems. You need to objectively weigh the numbers and choose the best one.
First way: represent ordinary decimals
It is suitable if, when dividing or converting, final fractions are obtained. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to count them.
The second way: write decimal fractions as ordinary
This technique is convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, a very large ordinary fraction can turn out and decimal entries will allow you to calculate the task faster and easier. Therefore, it is always necessary to soberly evaluate the task and choose the simplest solution method.
