Finite and infinite decimal fractions. Decimals, definitions, notation, examples, operations with decimals. Operations with decimals
Of the many fractions found in arithmetic, those that have 10, 100, 1000 in the denominator - in general, any power of ten - deserve special attention. These fractions have a special name and notation.
A decimal is any number fraction whose denominator is a power of ten.
Examples of decimal fractions:
Why was it necessary to separate out such fractions at all? Why do they need their own recording form? There are at least three reasons for this:
- Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, reduce the fractions to a common denominator. In decimal fractions, nothing like this is required;
- Reduce computation. Decimal fractions are added and multiplied by own rules, and after a little training you will work with them much faster than with regular ones;
- Ease of recording. Unlike ordinary fractions, decimals are written on one line without loss of clarity.
Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you ask for change in the store in the amount of 2/3 of a ruble :)
Rules for writing decimal fractions
The main advantage of decimal fractions is convenient and visual notation. Namely:
Decimal notation is a form of writing decimal fractions where the integer part is separated from the fractional part by a regular period or comma. In this case, the separator itself (period or comma) is called a decimal point.
For example, 0.3 (read: “zero pointers, 3 tenths”); 7.25 (7 whole, 25 hundredths); 3.049 (3 whole, 49 thousandths). All examples are taken from the previous definition.
In writing, a comma is usually used as a decimal point. Here and further throughout the site, the comma will also be used.
To write an arbitrary decimal fraction in this form, you need to follow three simple steps:
- 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 moved, and after it there are zeros at the end of the entry, they must be crossed out.
It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, to the left of any number you can assign any number of zeros without harm to your health. It's ugly, but sometimes useful.
At first glance, this algorithm may seem quite complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:
Task. For each fraction, indicate its decimal notation:
The numerator of the first fraction is: 73. We shift the decimal point by one place (since the denominator is 10) - we get 7.3.
Numerator of the second fraction: 9. We shift the decimal point by two places (since the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange entry like “.09”.
The numerator of the third fraction is: 10029. We shift the decimal point by three places (since the denominator is 1000) - we get 10.029.
The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10,500. There are extra zeros at the end of the number. Cross them out and we get 10.5.
Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as was done in the last example. However, you should never do this with zeros inside a number (which are surrounded by other numbers). That's why we got 10.029 and 10.5, and not 1.29 and 1.5.
So, we figured out the definition and form of writing decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.
Conversion from fractions to decimals
Consider a simple numerical fraction of the form a /b. You can use the basic property of a fraction and multiply the numerator and denominator by such a number that the bottom turns out to be a power of ten. But before you do, read the following:
There are denominators that cannot be reduced to powers of ten. Learn to recognize such fractions, because they cannot be worked with using the algorithm described below.
That's it. Well, how do you understand whether the denominator is reduced to a power of ten or not?
The answer is simple: factor the denominator into prime factors. If the expansion contains only factors 2 and 5, this number can be reduced to a power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the power of ten.
Task. Check whether the indicated fractions can be represented as decimals:
Let us write out and factor the denominators of these fractions:
20 = 4 · 5 = 2 2 · 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.
12 = 4 · 3 = 2 2 · 3 - there is a “forbidden” factor 3. The fraction cannot be represented as a decimal.
640 = 8 · 8 · 10 = 2 3 · 2 3 · 2 · 5 = 2 7 · 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction can be represented as a decimal.
48 = 6 8 = 2 3 2 3 = 2 4 3. The factor 3 “surfaced” again. Present it in the form decimal it is forbidden.
So, we’ve sorted out the denominator - now let’s look at the entire algorithm for moving to decimal fractions:
- Factor the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
- Count how many twos and fives are present in the expansion (there will be no other numbers there, remember?). Choose an additional factor such that the number of twos and fives is equal.
- Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.
Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest multiplier of all possible.
And one more thing: if the original fraction contains an integer part, be sure to convert this fraction to an improper fraction - and only then apply the described algorithm.
Task. Convert these numerical fractions to decimals:
Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, the fraction can be represented as a decimal. The expansion contains two twos and not a single five, so the additional factor is 5 2 = 25. With it, the number of twos and fives will be equal. We have:
Now let's look at the second fraction. To do this, note that 24 = 3 8 = 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.
The last two fractions have denominators 5 (prime number) and 20 = 4 · 5 = 2 2 · 5 respectively - only twos and fives are present everywhere. Moreover, in the first case, “for complete happiness” a factor of 2 is not enough, and in the second - 5. We get:
Conversion from decimals to common fractions
The reverse conversion - from decimal to regular notation - is much simpler. There are no restrictions or special checks here, so you can always convert a decimal fraction to the classic “two-story” fraction.
The translation algorithm is as follows:
- Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing is not to overdo it and do not cross out the inner zeros surrounded by other numbers;
- Count how many decimal places there are after the decimal point. Take the number 1 and add as many zeros to the right as there are characters you count. This will be the denominator;
- Actually, write down the fraction whose numerator and denominator we just found. If possible, reduce it. If the original fraction contained an integer part, we will now get an improper fraction, which is very convenient for further calculations.
Task. Convert decimal fractions to ordinary fractions: 0.008; 3.107; 2.25; 7,2008.
Cross out the zeros on the left and the commas - we get the following numbers (these will be the numerators): 8; 3107; 225; 72008.
In the first and second fractions there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.
Finally, let's combine the numerators and denominators into ordinary fractions:
As can be seen from the examples, the resulting fraction can very often be reduced. Let me note once again that any decimal fraction can be represented as an ordinary fraction. The reverse conversion may not always be possible.
| Ending decimals |
| Multiplying and dividing decimals by 10, 100, 1000, 10000, etc. |
| Converting a trailing decimal to a fraction |
Decimals are divided into the following three classes: finite decimals, infinite periodic decimals, and infinite non-periodic decimals.
Ending decimals
Definition . Final decimal fraction (decimal fraction) called a fraction or mixed number having a denominator of 10, 100, 1000, 10000, etc.
For example,
Decimal fractions also include those fractions that can be reduced to fractions having a denominator of 10, 100, 1000, 10000, etc., using the basic property of fractions.
For example,
Statement . An irreducible simple fraction or an irreducible mixed non-integer number is a finite decimal fraction if and only if the factorization of their denominators into prime factors contains only the numbers 2 and 5 as factors, and in arbitrary powers.
For decimal fractions there is special recording method , using a comma. To the left of the decimal point the whole part of the fraction is written, and to the right is the numerator of the fractional part, before which such a number of zeros are added so that the number of digits after the decimal point is equal to the number of zeros in the denominator of the decimal fraction.
For example,
Note that the decimal fraction will not change if you add several zeros to the right or left of it.
For example,
3,14 = 3,140 =
=
3,1400 = 003,14 .
The numbers before the decimal point (to the left of the decimal point) in decimal notation of final decimal fraction, form a number called whole part of the decimal.
The numbers after the decimal point (to the right of the decimal point) in the decimal notation of the final decimal fraction are called decimals.
A final decimal has a finite number of decimal places. Decimals form fractional part of a decimal.
Multiplying and dividing decimals by 10, 100, 1000, etc.
In order to multiply a decimal by 10, 100, 1000, 10000, etc., enough move comma to the right by 1, 2, 3, 4, etc. decimal places respectively.
fractional number.
Decimal notation of a fractional number is a set of two or more digits from $0$ to $9$, between which there is a so-called \textit (decimal point).
Example 1
For example, $35.02$; $100.7$; $123\456.5$; $54.89$.
The leftmost digit in the decimal notation of a number cannot be zero, the only exception being when the decimal point is immediately after the first digit $0$.
Example 2
For example, $0.357$; $0.064$.
Often the decimal point is replaced with a decimal point. For example, $35.02$; $100.7$; $123\456.5$; $54.89$.
Decimal definition
Definition 1
Decimals-- these are fractional numbers that are represented in decimal notation.
For example, $121.05; $67.9$; $345.6700$.
Decimals are used to more compactly write proper fractions, the denominators of which are the numbers $10$, $100$, $1\000$, etc. and mixed numbers, the denominators of the fractional part of which are the numbers $10$, $100$, $1\000$, etc.
For example, the common fraction $\frac(8)(10)$ can be written as a decimal $0.8$, and the mixed number $405\frac(8)(100)$ can be written as a decimal $405.08$.
Reading Decimals
Decimal fractions, which correspond to regular fractions, are read the same as ordinary fractions, only the phrase “zero integer” is added in front. For example, the common fraction $\frac(25)(100)$ (read “twenty-five hundredths”) corresponds to the decimal fraction $0.25$ (read “zero point twenty-five hundredths”).
Decimal fractions that correspond to mixed numbers are read the same way as mixed numbers. For example, the mixed number $43\frac(15)(1000)$ corresponds to the decimal fraction $43.015$ (read “forty-three point fifteen thousandths”).
Places in decimals
In writing a decimal fraction, the meaning of each digit depends on its position. Those. in decimal fractions the concept also applies category.
Places in decimal fractions up to the decimal point are called the same as places in natural numbers. The decimal places after the decimal point are listed in the table:
Picture 1.
Example 3
For example, in the decimal fraction $56.328$, the digit $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenths place, $2$ is in the hundredths place, $8$ is in the thousandths place.
Places in decimal fractions are distinguished by precedence. When reading a decimal fraction, move from left to right - from senior rank to younger.
Example 4
For example, in the decimal fraction $56.328$, the most significant (highest) place is the tens place, and the low (lowest) place is the thousandths place.
A decimal fraction can be expanded into digits similar to the digit decomposition of a natural number.
Example 5
For example, let's break down the decimal fraction $37.851$ into digits:
$37,851=30+7+0,8+0,05+0,001$
Ending decimals
Definition 2
Ending decimals are called decimal fractions, the records of which contain a finite number of characters (digits).
For example, $0.138$; $5.34$; $56.123456$; $350,972.54.
Any finite decimal fraction can be converted to a fraction or a mixed number.
Example 6
For example, the final decimal fraction $7.39$ corresponds to the fractional number $7\frac(39)(100)$, and the final decimal fraction $0.5$ corresponds to the correct common fraction$\frac(5)(10)$ (or any fraction that is equal to it, such as $\frac(1)(2)$ or $\frac(10)(20)$.
Converting a fraction to a decimal
Converting fractions with denominators $10, 100, \dots$ to decimals
Before converting some proper fractions to decimals, they must first be “prepared.” The result of such preparation should be the same number of digits in the numerator and the number of zeros in the denominator.
The essence of " preliminary preparation» converting regular fractions to decimals - adding such a number of zeros to the left in the numerator so that the total number of digits becomes equal to the number of zeros in the denominator.
Example 7
For example, let's prepare the fraction $\frac(43)(1000)$ for conversion to a decimal and get $\frac(043)(1000)$. And the ordinary fraction $\frac(83)(100)$ does not need any preparation.
Let's formulate rule for converting a proper common fraction with a denominator of $10$, or $100$, or $1\000$, $\dots$ into a decimal fraction:
write $0$;
after it put a decimal point;
write down the number from the numerator (along with added zeros after preparation, if necessary).
Example 8
Convert the proper fraction $\frac(23)(100)$ to a decimal.
Solution.
The denominator contains the number $100$, which contains $2$ and two zeros. The numerator contains the number $23$, which is written with $2$.digits. This means that there is no need to prepare this fraction for conversion to a decimal.
Let's write $0$, put a decimal point and write down the number $23$ from the numerator. We get the decimal fraction $0.23$.
Answer: $0,23$.
Example 9
Write the proper fraction $\frac(351)(100000)$ as a decimal.
Solution.
The numerator of this fraction contains $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction must be prepared for conversion to a decimal. To do this, you need to add $5-3=2$ zeros to the left in the numerator: $\frac(00351)(100000)$.
Now we can form the desired decimal fraction. To do this, write down $0$, then add a comma and write down the number from the numerator. We get the decimal fraction $0.00351$.
Answer: $0,00351$.
Let's formulate rule for converting improper fractions with denominators $10$, $100$, $\dots$ into decimal fractions:
write down the number from the numerator;
Use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.
Example 10
Convert the improper fraction $\frac(12756)(100)$ to a decimal.
Solution.
Let's write down the number from the numerator $12756$, then separate the $2$ digits on the right with a decimal point, because the denominator of the original fraction $2$ is zero. We get the decimal fraction $127.56$.
There is another idea rational number 1/2, different from representations of the form 2/4, 3/6, 4/8, etc. We mean representation as a decimal fraction 0.5. Some fractions have finite decimal representations, e.g.
while the decimal representations of other fractions are infinite:
These infinite decimals can be obtained from the corresponding rational fractions by dividing the numerator by the denominator. For example, in the case of the fraction 5/11, dividing 5.000... by 11 gives 0.454545...
Which rational fractions have finite decimal representations? Before answering this question in general, let's look at a specific example. Let's take, say, the final decimal fraction 0.8625. We know that
and that any finite decimal fraction can be written as a rational decimal fraction with a denominator equal to 10, 100, 1000, or some other power of 10.
Reducing the fraction on the right to an irreducible fraction, we get
The denominator of 80 is obtained by dividing 10,000 by 125 - the greatest common divisor of 10,000 and 8625. Therefore, the prime factorization of the number 80, like the number 10,000, includes only two prime factors: 2 and 5. If we did not start with 0, 8625, and with any other finite decimal fraction, then the resulting irreducible rational fraction would also have this property. In other words, the expansion of the denominator b into prime factors could only include prime numbers 2 and 5, since b is a divisor of some power of 10, and . This circumstance turns out to be decisive, namely, the following general statement holds:
An irreducible rational fraction has a finite decimal representation if and only if the number b has no prime factors of 2 and 5.
Note that b does not have to have both numbers 2 and 5 among its prime factors: it can be divisible by only one of them or not be divisible by them at all. For example,
here b is equal to 25, 16 and 1, respectively. What is significant is that b has no other divisors other than 2 and 5.
The above sentence contains the expression if and only if. So far we have proved only the part that relates to turnover only then. It was we who showed that the decomposition of a rational number into a decimal fraction will be finite only in the case when b has no prime factors other than 2 and 5.
(In other words, if b is divisible by a prime number other than 2 and 5, then the irreducible fraction has no finite decimal expression.)
The then part of the sentence states that if the integer b has no prime factors other than 2 and 5, then the irreducible rational fraction can be represented by a finite decimal fraction. To prove this, we must take an arbitrary irreducible rational fraction, for which b has no prime factors other than 2 and 5, and make sure that the corresponding decimal fraction is finite. Let's look at an example first. Let
To obtain the decimal expansion, we transform this fraction into a fraction whose denominator is an integer power of ten. This can be achieved by multiplying the numerator and denominator by:
The above reasoning can be extended to the general case as follows. Suppose b is of the form , where the type is non-negative integers (i.e., positive numbers or zero). Two cases are possible: either less than or equal (this condition is written), or greater (which is written). When we multiply the numerator and denominator of the fraction by
Since the integer is not negative (that is, positive or equal to zero), then , and therefore a is a positive integer. Let's put it. Then
Already in primary school students encounter 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 first meet in primary 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's 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 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 number 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 a common 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, you would have to write the answer 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 between the fractional and whole in parts a comma is added.
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 such a 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 common multiplier, 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 an integer, the latter should be written in the form 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 convert the divisor: make it 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.