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 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 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.
| 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.
Topic: Decimal fractions. Adding and subtracting decimals
Lesson: Decimal notation of fractional numbers
The denominator of a fraction can be expressed by any natural number. Fractional numbers in which the denominator is expressed as 10; 100; 1000;…, where n, we agreed to write it without a denominator. Any fractional number whose denominator is 10; 100; 1000, etc. (that is, a one followed by several zeros) can be represented in decimal notation (as a decimal). First write the whole part, then the numerator of the fractional part, and the whole part is separated from the fraction by a comma.
For example,
If an entire part is missing, i.e. If the fraction is proper, then the whole part is written as 0.
To write a decimal correctly, the numerator of the fraction must have as many digits as there are zeros in the fraction.
1. Write as a decimal.
2. Represent a decimal as a fraction or mixed number.
3. Read the decimals.
12.4 - 12 point 4;
0.3 - 0 point 3;
1.14 - 1 point 14 hundredths;
2.07 - 2 point 7 hundredths;
0.06 - 0 point 6 hundredths;
0.25 - 0 point 25;
1.234 - 1 point 234 thousandths;
1.230 - 1 point 230 thousandths;
1.034 - 1 point 34 thousandths;
1.004 - 1 point 4 thousandths;
1.030 - 1 point 30 thousandths;
0.010101 - 0 point 10101 millionths.
4. Move the comma in each digit 1 place to the left and read the numbers.
34,1; 310,2; 11,01; 10,507; 2,7; 3,41; 31,02; 1,101; 1,0507; 0,27.
5. Move the comma in each number 1 place to the right and read the resulting number.
1,37; 0,1401; 3,017; 1,7; 350,4; 13,7; 1,401; 30,17; 17; 3504.
6. Express in meters and centimeters.
3.28 m = 3 m + .
7. Express in tons and kilograms.
24.030 t = 24 t.
8. Write the quotient as a decimal fraction.
1710: 100 = 
;
64: 10000 = 
803: 100 = 
407: 10 = 
9. Express in dm.
5 dm 6 cm = 5 dm + 
;
9 mm = 
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 this number as a decimal fraction.
Solution. Divide the numerator of each fraction into a column by its denominator: A) divide 6 by 25; b) divide 2 by 3; V) divide 1 by 2, and then add the resulting fraction to one - the integer part of this mixed number.
Irreducible ordinary fractions whose denominators do not contain prime factors other than 2 And 5 , are written as a final decimal fraction.
IN example 1 when A) denominator 25=5·5; when V) the denominator is 2, so we get the final decimals of 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a finite decimal.
Is it possible, without long division, to convert into a decimal fraction such an ordinary fraction, the denominator of which does not contain other divisors other than 2 and 5? Let's figure it out! What fraction is called a decimal and is written without a fraction bar? Answer: fraction with denominator 10; 100; 1000, etc. And each of these numbers is a product equal number of twos and fives. In fact: 10=2 ·5 ; 100=2 ·5 ·2 ·5 ; 1000=2 ·5 ·2 ·5 ·2 ·5 etc.
Consequently, the denominator of an irreducible ordinary fraction will need to be represented as the product of “twos” and “fives”, and then multiplied by 2 and (or) 5 so that the “twos” and “fives” become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. To ensure that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which we multiplied the denominator.
Express the following common fractions as decimals:
Solution. Each of these fractions is irreducible. Let's factor the denominator of each fraction into prime factors.
20=2·2·5. Conclusion: one “A” is missing.
8=2·2·2. Conclusion: three “A”s are missing.
25=5·5. Conclusion: two “twos” are missing.
Comment. In practice, they often do not use factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is one 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 be obtained, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.
When V)(example 2) from 25 you get 100 if you multiply by 4. This means that the numerator 8 must be multiplied by 4.
An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodic as a decimal. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosed in parentheses.
When b)(example 1) there is only one repeating digit and is equal to 6. Therefore, our result 0.66... will be written like this: 0,(6) . They read: zero point, six in period.
If there are one or more non-repeating digits between the decimal point and the first period, then such a periodic fraction is called a mixed periodic fraction.
An irreducible common fraction whose denominator is together with others multiplier contains multiplier 2 or 5 , becomes mixed periodic fraction.
Write numbers as decimals.
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
Decimals, which correspond to regular fractions, are read in the same way as ordinary fractions, only the phrase “zero integers” is added in front. For example, the common fraction $\frac(25)(100)$ (read “twenty-five hundredths”) corresponds to the decimal fraction $0.25$ (read “zero point twenty-five hundredths”).
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 proper common fraction $\frac(5)(10)$ (or any fraction which is equal to it, for example, $\frac(1)(2)$ 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 same number of zeros in the denominator.
The essence of “preliminary preparation” of proper ordinary fractions for conversion to decimal fractions is adding such a number of zeros to the left in the numerator 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 $127.56$.
