How to solve decimals. Decimal fractions. Decimal concept
Read also
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 in decimal form. 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 to ordinary fractions if whole part absent, 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 100. That is given examples the answers will be numbers: 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 looks like this 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 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 most simple option turns out to be a number whose denominator contains 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 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 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 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 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, it can get very big common fraction and decimal notation will allow you to calculate the task faster and easier. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.
DECIMALS. OPERATIONS ON DECIMALS
(summarizing lesson)
Tumysheva Zamira Tansykbaevna, mathematics teacher, gymnasium school No. 2
Khromtau city, Aktobe region, Republic of Kazakhstan
This lesson development is intended as a generalization lesson for the chapter “Actions on decimals.” It can be used in both 5th and 6th grades. The lesson is conducted in a playful way.
Decimal fractions. Operations with decimal fractions.(summarizing lesson)
Target:
Practicing skills in addition, subtraction, multiplication and division of decimals by natural numbers and decimals
Creating conditions for skills development independent work, self-control and self-esteem, development of intellectual qualities: attention, imagination, memory, ability to analyze and generalize
Vaccinate cognitive interest to the subject and develop self-confidence
LESSON PLAN:
1. Organizational part.
3. The topic and purpose of our lesson.
4. Game “To the cherished flag!”
5. Game "Number Mill".
6. Lyrical digression.
8. Game “Encryption” (work in pairs)
9. Summing up.
10. Homework.
1. Organizational part. Hello. Have a seat.
2. Review of the rules for performing arithmetic operations with decimals.
Rule for adding and subtracting decimals:
1) equalize the number of decimal places in these fractions;
2) write one below the other so that the comma is under the comma;
3) without noticing the comma, perform the action (addition or subtraction), and put a comma under the commas as a result.
3,455 + 0,45 = 3,905 3,5 + 4 = 7,5 15 – 7,88 = 7,12 4,57 - 3,2 = 1,37
3,455 + 3,5 _15,00 _ 4,57
0,450 4,0 7,88 3,20
3,905 7,5 7,12 1,37
When adding and subtracting, natural numbers are written as a decimal fraction with decimal places equal to zero
Rule for multiplying decimals:
1) without paying attention to the comma, multiply the numbers;
2) in the resulting product, separate as many digits from right to left with a comma as there are in decimal fractions separated by a comma.
When multiplying a decimal fraction by digit units (10, 100, 1000, etc.), the decimal point is moved to the right by as many numbers as there are zeros in the digit unit
4
17.25 4 = 69
x 1 7.2 5
4
6 9,0 0
15.256 100 = 1525.6
.5 · 0.52 = 2.35X 0.5 2
4,5
2 7 0
2 0 8__
2,3 5 0
When multiplying, natural numbers are written as natural numbers.
The rule for dividing decimal fractions by a natural number:
1) divide the whole part of the dividend, put a comma in the quotient;
2) continue division.
When dividing, we add only one number from the dividend to the remainder.
If in the process of dividing a decimal fraction there remains a remainder, then by adding the required number of zeros to it, we will continue division until the remainder is zero.
15,256: 100 = 0,15256
0,25: 1000 = 0,00025
When dividing a decimal fraction into digit units (10, 100, 1000, etc.), the comma is moved to the left by as many numbers as there are zeros in the digit unit.
18,4: 8 = 2,3
_ 18,4 І_8_
16 2,3
2 4
2 4
22,2: 25 = 0,88
22,2 І_25_
0 0,888
22 2
20 0
2 20
2 00
200
200
3,56: 4 = 0,89
3,56 І_4_
0 0,89
3 5
3 2
36
When dividing, natural numbers are written as natural numbers.
The rule for dividing decimals by decimals is:
1) move the comma in the divisor to the right so that we get a natural number;
2) move the comma in the dividend to the right as many numbers as were moved in the divisor;
3) divide the decimal fraction by a natural number.
3,76: 0,4 = 9, 4
_ 3,7,6 І_0,4,_
3 6 9, 4
1 6
1 6
0
Game “To the cherished flag!”
Rules of the game: From each team, one student is called to the board and performs an oral count from the bottom step. The person who solves one example marks the answer in the table. Then he is replaced by another team member. There is an upward movement - towards the coveted flag. Students in the field orally review their players' performance. If the answer is incorrect, another team member comes to the board to continue solving the problems. Team captains call students to work at the board. The team that reaches the flag first with the fewest number of students wins.
Game "Number Mill"
Rules of the game: The mill circles contain numbers. The arrows connecting the circles indicate actions. The task is to perform sequential actions, moving along the arrow from the center to the outer circle. By performing sequential actions along the indicated route, you will find the answer in one of the circles below. The result of performing actions on each arrow is recorded in the oval next to it.
Lyrical digression.
Lifshitz's poem "Three Tenths"
Who is this
From the briefcase
Throws it in frustration
Hateful problem book,
Pencil case and notebooks
And he puts in his diary.
Without blushing,
Under an oak sideboard.
To lie under the sideboard?..
Please meet:
Kostya Zhigalin.
Victim of eternal nagging, -
He failed again.
And hisses
To disheveled
Looking at the problem book:
I'm just unlucky!
I'm just a loser!
What is the reason
His grievances and annoyance?
That the answer didn't add up
Only three tenths.
This is a mere trifle!
And to him, of course,
Find fault
Strict
Marya Petrovna.
Three tenths...
Tell me about this mistake -
And, perhaps, on their faces
You will see a smile.
Three tenths...
And yet about this mistake
I ask you
Listen to me
No smile.
If only, building your house.
The one you live in.
Architect
A little bit
Wrong
In counting, -
What would happen?
Do you know, Kostya Zhigalin?
This house
Would have turned
Into a pile of ruins!
You step onto the bridge.
It is reliable and durable.
Don't be an engineer
Accurate in his drawings, -
Would you, Kostya,
Having fallen
into the cold river
I wouldn't say thank you
That man!
Here's the turbine.
She has a shaft
Wasted by turners.
If only the turner
In progress
Wasn't very accurate -
It would happen, Kostya,
Big misfortune:
The turbine would be blown apart
Into small pieces!
Three tenths -
And the walls
Are being built
Koso!
Three tenths -
And they will collapse
Cars
Off the slope!
Make a mistake
Only three tenths
Pharmacy, -
The medicine will become poison
Will kill a man!
We smashed and drove
Fascist gang.
Your father served
Battery command.
He made a mistake when he arrived
At least three tenths, -
The shells wouldn't have reached me
Damned fascists.
Think about it
My friend, coolly
And tell me.
Wasn't it right?
Marya Petrovna?
Honestly
Just think about it, Kostya.
You won't lie down for long
To the diary under the buffet!
Test work on the topic “Decimals” (mathematics -5)
9 slides will appear on the screen in sequence. Students write down the option number and answers to the question in their notebooks. For example, Option 2
1. C; 2. A; and so on.
QUESTION 1
Option 1
When multiplying a decimal fraction by 100, you need to move the decimal point in this fraction:
A. to the left by 2 digits; B. to the right by 2 digits; C. do not change the place of the comma.
Option 2
When multiplying a decimal fraction by 10, you need to move the decimal point in this fraction:
A. to the right by 1 digit; B. to the left by 1 digit; C. do not change the place of the comma.
QUESTION 2
Option 1
The sum 6.27+6.27+6.27+6.27+6.27 as a product is written as follows:
A. 6.27 5; V. 6.27 · 6.27; P. 6.27 · 4.
Option 2
The sum 9.43+9.43+9.43+9.43 as a product is written as follows:
A. 9.43 · 9.43; V. 6 · 9.43; P. 9.43 · 4.
QUESTION 3
Option 1
In the product 72.43·18 after the decimal point there will be:
Option 2
In the product 12.453 35 after the decimal point there will be:
A. 2 digits; B. 0 digits; C. 3 digits.
QUESTION 4
Option 1
In the quotient 76.4: 2 after the decimal point it will be:
A. 2 digits; B. 0 digits; C. 1 digit.
Option 2
In the quotient 95.4: 6 after the decimal point it will be:
A. 1 digit; B. 3 digits; C. 2 digits.
QUESTION 5
Option 1
Find the value of the expression 34.5: x + 0.65· y, with x=10 y=100:
A. 35.15; V. 68.45; pp. 9.95.
Option 2
Find the value of the expression 4.9 x +525:y, with x=100 y=1000:
A. 4905.25; V. 529.9; pp. 490.525.
QUESTION 6
Option 1
The area of a rectangle with sides 0.25 and 12 cm is
A. 3; V. 0.3; P. 30.
Option 2
The area of a rectangle with sides 0.5 and 36 cm is equal to
A. 1.8; V. 18; S. 0.18.
QUESTION 7
Option 1
Two students left the school at the same time in opposite directions. The speed of the first student is 3.6 km/h, the speed of the second is 2.56 km/h. After 3 hours the distance between them will be equal:
A. 6.84 km; E. 18.48 km; N. 3.12 km
Option 2
Two cyclists left the school at the same time in opposite directions. The speed of the first is 11.6 km/h, the speed of the second is 13.06 km/h. After 4 hours the distance between them will be equal:
A. 5.84 km; E. 100.8 km; N. 98.64 km
Option 1
Option 2
Check your answers. Put “+” for a correct answer and “-” for an incorrect answer.
Game "Encryption"
Rules of the game: Each desk is given a card with a task that has a letter code. After completing the steps and receiving the result, write down the letter code of your card under the number corresponding to your answer.
As a result, we get the following sentence:
6,8
420
21,6
420
306
65,8
21,6
Summing up the lesson.
Grades for the test work are announced.
Homework No. 1301, 1308, 1309
Thank you for your attention!!!
The decimal is used when you need to perform operations with non-integer numbers. This may seem irrational. But this type of numbers greatly simplifies the mathematical operations that need to be performed with them. This understanding comes over time, when writing them becomes familiar, and reading them does not cause difficulties, and the rules of decimal fractions have been mastered. Moreover, all actions repeat already known ones, which have been learned from natural numbers. You just need to remember some features.
Decimal definition
A decimal is a special representation of a non-integer number with a denominator that is divisible by 10, giving the answer as one and possibly zeros. In other words, if the denominator is 10, 100, 1000, and so on, then it is more convenient to rewrite the number using a comma. Then the whole part will be located before it, and then the fractional part. Moreover, the recording of the second half of the number will depend on the denominator. The number of digits that are in the fractional part must be equal to the digit of the denominator.
The above can be illustrated with these numbers:
9/10=0,9; 178/10000=0,0178; 3,05; 56 003,7006.
Reasons for using decimals
Mathematicians needed decimals for several reasons:
Simplifying recording. Such a fraction is located along one line without a dash between the denominator and numerator, while clarity does not suffer.
Simplicity in comparison. It is enough to simply correlate numbers that are in the same positions, while with ordinary fractions you would have to reduce them to a common denominator.
Simplify calculations.
Calculators are not designed to accept fractions; they use decimal notation for all operations.
How to read such numbers correctly?
The answer is simple: just like an ordinary mixed number with a denominator that is a multiple of 10. The only exception is fractions without an integer value, then when reading you need to pronounce “zero integers.”
For example, 45/1000 should be pronounced as forty-five thousandths, at the same time 0.045 will sound like zero point forty five thousandths.
A mixed number with an integer part of 7 and a fraction of 17/100, which would be written as 7.17, would in both cases be read as seven point seventeen.
The role of digits in writing fractions
Correctly marking the rank is what mathematics requires. Decimals and their meaning can change significantly if you write the digit in the wrong place. However, this was true before.
To read the digits of the whole part of a decimal fraction, you simply need to use the rules known for natural numbers. And on the right side they are mirrored and read differently. If the whole part sounded “tens”, then after the decimal point it will be “tenths”.
This can be clearly seen in this table.
| Class | thousands | units | , | fraction | |||||||
| discharge | cell | dec. | units | cell | dec. | units | tenth | hundredth | thousandth | ten-thousandth | |
How to correctly write a mixed number as a decimal?
If the denominator contains a number equal to 10 or 100, and others, then the question of how to convert a fraction to a decimal is not difficult. To do this, it is enough to rewrite all its components differently. The following points will help with this:
write the numerator of the fraction a little to the side, at this moment the decimal point is located on the right, after the last digit;
move the comma to the left, the most important thing here is to count the numbers correctly - you need to move it by as many positions as there are zeros in the denominator;
if there are not enough of them, then there should be zeros in the empty positions;
the zeros that were at the end of the numerator are now not needed and can be crossed out;
Before the comma, add the whole part; if it was not there, then there will also be zero here.
Attention. You cannot cross out zeros that are surrounded by other numbers.
You can read below about what to do in a situation where the denominator has a number not only consisting of ones and zeros, and how to convert a fraction to a decimal. This important information, which is definitely worth checking out.
How to convert a fraction to a decimal if the denominator is an arbitrary number?
There are two options here:
When the denominator can be represented as a number that is equal to ten to any power.
If such an operation cannot be performed.
How can I check this? You need to factor the denominator. If only 2 and 5 are present in the product, then everything is fine, and the fraction is easily converted to a final decimal. Otherwise, if 3, 7 and others appear prime numbers, then the result will be endless. It is customary to round such a decimal fraction for ease of use in mathematical operations. This will be discussed a little below.
Explores how decimals are made, 5th grade. Examples here will be very helpful.
Let the denominators be the numbers: 40, 24 and 75. Decomposition into prime factors for them it will be like this:
- 40=2·2·2·5;
- 24=2·2·2·3;
- 75=5·5·3.
In these examples, only the first fraction can be represented as the final fraction.
Algorithm for converting a common fraction to a final decimal
Check the factorization of the denominator into prime factors and make sure that it will consist of 2 and 5.
Add as many 2s and 5s to these numbers so that there are an equal number of them. They will give the value of the additional multiplier.
Multiply the denominator and numerator by this number. The result will be an ordinary fraction, under the line of which there is 10 to some extent.
If in the problem these actions are performed with a mixed number, then it must first be represented in the form improper fraction. And only then act according to the described scenario.
Representing a fraction as a rounded decimal
This method of converting a fraction to a decimal may seem even easier to some. Because it doesn't have large quantity actions. You just need to divide the numerator by the denominator.
Any number with a decimal part to the right of the decimal point can be assigned an infinite number of zeros. This property is what you need to take advantage of.
First, write down the whole part and put a comma after it. If the fraction is correct, write zero.
Then you need to divide the numerator by the denominator. So that they have the same number of digits. That is, add to the right of the numerator required quantity zeros.
Fulfill long division until the required number of digits is dialed. For example, if you need to round to hundredths, then the answer should be 3. In general, there should be one more number than you need to get in the end.
Write down the intermediate answer after the decimal point and round according to the rules. If the last digit is from 0 to 4, then you just need to discard it. And when it is equal to 5-9, then the one in front of it needs to be increased by one, discarding the last one.
Return from decimal to common fraction
In mathematics, there are problems when it is more convenient to represent decimal fractions in the form of ordinary fractions, in which there is a numerator with a denominator. You can breathe a sigh of relief: this operation is always possible.
For this procedure you need to do the following:
write down the whole part, if it is equal to zero, then there is no need to write anything;
draw a fraction line;
above it, write down the numbers from the right side, if the zeros come first, then they need to be crossed out;
Under the line, write a unit with as many zeros as there are digits after the decimal point in the original fraction.
That's all you need to do to convert a decimal to a fraction.
What can you do with decimals?
In mathematics, these will be certain operations with decimals that were previously performed for other numbers.
They are:
comparison;
addition and subtraction;
multiplication and division.
The first action, comparison, is similar to how it was done for natural numbers. To determine which is greater, you need to compare the digits of the whole part. If they turn out to be equal, then they move on to the fractional and also compare them by digits. The number with the largest digit in the most significant digit will be the answer.
Adding and subtracting decimals
These are perhaps the most simple steps. Because they are carried out according to the rules for natural numbers.
So, in order to add decimal fractions, they need to be written one below the other, placing commas in a column. With this notation, whole parts appear to the left of the commas, and fractional parts to the right. And now you need to add the numbers bit by bit, as is done with natural numbers, moving the comma down. You need to start adding from the smallest digit of the fractional part of the number. If there are not enough numbers in the right half, then zeros are added.
The same applies to subtraction. And here there is a rule that describes the possibility of taking a unit from the highest rank. If the fraction being reduced has fewer digits after the decimal point than the fraction being subtracted, then zeros are simply added to it.
The situation is a little more complicated with tasks where you need to multiply and divide decimal fractions.
How to multiply a decimal fraction in different examples?
The rule for multiplying decimal fractions by a natural number is:
write them down in a column, ignoring the comma;
multiply as if they were naturals;
Separate with a comma as many digits as there were in the fractional part of the original number.
A special case is the example in which a natural number is equal to 10 to any power. Then to get the answer you just need to move the decimal point to the right by as many positions as there are zeros in the other factor. In other words, when multiplied by 10, the decimal point moves by one digit, by 100 - there will already be two of them, and so on. If there are not enough numbers in the fractional part, then you need to write zeros in the empty positions.
The rule that is used when a task requires multiplying decimal fractions by another same number:
write them down one after another, not paying attention to commas;
multiply as if they were natural;
Separate with a comma as many digits as there were in the fractional parts of both original fractions together.
A special case are examples in which one of the multipliers is equal to 0.1 or 0.01 and so on. In them you need to move the decimal point to the left by the number of digits in the presented factors. That is, if it is multiplied by 0.1, then the decimal point is shifted by one position.
How to divide a decimal fraction in different tasks?
Dividing decimal fractions by a natural number is performed according to the following rule:
write them down for division in a column as if they were natural ones;
divide according to the usual rule until the whole part is over;
put a comma in the answer;
continue dividing the fractional component until the remainder is zero;
if necessary, you can add the required number of zeros.
If the integer part is equal to zero, then it will not be in the answer either.
Separately, there is division into numbers equal to ten, hundred, and so on. In such problems, you need to move the decimal point to the left by the number of zeros in the divisor. It happens that there are not enough numbers in a whole part, then zeros are used instead. You can see that this operation is similar to multiplying by 0.1 and similar numbers.
To divide decimals, you need to use this rule:
turn the divisor into a natural number, and to do this, move the comma in it to the right to the end;
move the decimal point in the dividend by the same number of digits;
act according to the previous scenario.
Stands out division by 0.1; 0.01 and others similar numbers. In such examples, the decimal point is shifted to the right by the number of digits in the fractional part. If they run out, then you need to add the missing number of zeros. It is worth noting that this action repeats division by 10 and similar numbers.
Conclusion: It's all about practice
Nothing in learning comes easy or without effort. Reliably mastering new material takes time and practice. Mathematics is no exception.
To ensure that the topic about decimal fractions does not cause difficulties, you need to solve as many examples with them as possible. After all, there was a time when adding natural numbers was a dead end. And now everything is fine.
Therefore, to paraphrase famous phrase: decide, decide and decide again. Then tasks with such numbers will be completed easily and naturally, like another puzzle.
By the way, puzzles are difficult to solve at first, and then you need to do the usual movements. It’s the same in mathematical examples: having walked along the same path several times, then you will no longer think about where to turn.
This article is about decimals. Here we will deal with decimal notation fractional numbers, we introduce the concept of a decimal fraction and give examples of decimal fractions. Next we’ll talk about the digits of decimal fractions and give the names of the digits. After this, we will focus on infinite decimal fractions, let's talk about periodic and non-periodic fractions. Next we list the basic operations with decimal fractions. In conclusion, let us establish the position of decimal fractions on the coordinate beam.
Page navigation.
Decimal notation of a fractional number
Reading Decimals
Let's say a few words about the rules for reading decimal fractions.
Decimal fractions, which correspond to proper ordinary fractions, are read in the same way as these ordinary fractions, only “zero integer” is first added. For example, the decimal fraction 0.12 corresponds to the common fraction 12/100 (read “twelve hundredths”), therefore, 0.12 is read as “zero point twelve hundredths”.
Decimal fractions that correspond to mixed numbers are read exactly the same as these mixed numbers. For example, the decimal fraction 56.002 corresponds to a mixed number, so the decimal fraction 56.002 is read as “fifty-six point two thousandths.”
Places in decimals
In writing decimal fractions, as well as in writing natural numbers, the meaning of each digit depends on its position. Indeed, the number 3 in the decimal fraction 0.3 means three tenths, in the decimal fraction 0.0003 - three ten thousandths, and in the decimal fraction 30,000.152 - three tens of thousands. So we can talk about decimal places, as well as about the digits in natural numbers.
The names of the digits in the decimal fraction up to the decimal point completely coincide with the names of the digits in natural numbers. And the names of the decimal places after the decimal point can be seen from the following table.
For example, in the decimal fraction 37.051, the digit 3 is in the tens place, 7 is in the units place, 0 is in the tenths place, 5 is in the hundredths place, and 1 is in the thousandths place.
Places in decimal fractions also differ in precedence. If in writing a decimal fraction we move from digit to digit from left to right, then we will move from seniors To junior ranks. For example, the hundreds place is older than the tenths place, and the millions place is lower than the hundredths place. In a given final decimal fraction, we can talk about the major and minor digits. For example, in decimal fraction 604.9387 senior (highest) the place is the hundreds place, and junior (lowest)- digit of ten thousandths.
For decimal fractions, expansion into digits takes place. It is similar to expansion by digits of natural numbers. For example, the expansion into decimal places of 45.6072 is as follows: 45.6072=40+5+0.6+0.007+0.0002. And the properties of addition from the decomposition of a decimal fraction into digits allow you to move on to other representations of this decimal fraction, for example, 45.6072=45+0.6072, or 45.6072=40.6+5.007+0.0002, or 45.6072= 45.0072+0.6.
Ending decimals
Up to this point, we have only talked about decimal fractions, in the notation of which there is a finite number of digits after the decimal point. Such fractions are called finite decimals.
Definition.
Ending decimals- These are decimal fractions, the records of which contain a finite number of characters (digits).
Here are some examples of final decimal fractions: 0.317, 3.5, 51.1020304958, 230,032.45.
However, not every fraction can be represented as a final decimal. For example, the fraction 5/13 cannot be replaced by an equal fraction with one of the denominators 10, 100, ..., therefore, cannot be converted into a final decimal fraction. We will talk more about this in the theory section, converting ordinary fractions to decimals.
Infinite Decimals: Periodic Fractions and Non-Periodic Fractions
In writing a decimal fraction after the decimal point, one can assume the possibility of an infinite number of digits. In this case, we will come to consider the so-called infinite decimal fractions.
Definition.
Infinite decimals- these are decimal fractions, the recording of which contains infinite set numbers
It is clear that we cannot write down infinite decimal fractions in full form, so in their recording we limit ourselves to only a certain finite number of digits after the decimal point and put an ellipsis indicating an infinitely continuing sequence of digits. Here are some examples of infinite decimal fractions: 0.143940932…, 3.1415935432…, 153.02003004005…, 2.111111111…, 69.74152152152….
If you look closely at the last two infinite decimal fractions, then in the fraction 2.111111111... the endlessly repeating number 1 is clearly visible, and in the fraction 69.74152152152..., starting from the third decimal place, a repeating group of numbers 1, 5 and 2 is clearly visible. Such infinite decimal fractions are called periodic.
Definition.
Periodic decimals(or simply periodic fractions) are endless decimal fractions, in the recording of which, starting from a certain decimal place, some number or group of numbers is endlessly repeated, which is called period of the fraction.
For example, the period of the periodic fraction 2.111111111... is the digit 1, and the period of the fraction 69.74152152152... is a group of digits of the form 152.
For infinite periodic decimal fractions, a special form of notation is adopted. For brevity, we agreed to write down the period once, enclosing it in parentheses. For example, the periodic fraction 2.111111111... is written as 2,(1) , and the periodic fraction 69.74152152152... is written as 69.74(152) .
It is worth noting that for the same periodic decimal fraction you can specify different periods. For example, the periodic decimal fraction 0.73333... can be considered as a fraction 0.7(3) with a period of 3, and also as a fraction 0.7(33) with a period of 33, and so on 0.7(333), 0.7 (3333), ... You can also look at the periodic fraction 0.73333 ... like this: 0.733(3), or like this 0.73(333), etc. Here, in order to avoid ambiguity and discrepancies, we agree to consider the shortest of all as the period of the decimal fraction possible sequences repeating digits, and starting from the position closest to the decimal point. That is, the period of the decimal fraction 0.73333... will be considered a sequence of one digit 3, and the periodicity starts from the second position after the decimal point, that is, 0.73333...=0.7(3). Another example: the periodic fraction 4.7412121212... has a period of 12, the periodicity starts from the third digit after the decimal point, that is, 4.7412121212...=4.74(12).
Infinite decimal periodic fractions are obtained by converting into decimal fractions ordinary fractions whose denominators contain prime factors other than 2 and 5.
Here it is worth mentioning periodic fractions with a period of 9. Let us give examples of such fractions: 6.43(9) , 27,(9) . These fractions are another notation for periodic fractions with period 0, and they are usually replaced by periodic fractions with period 0. To do this, period 9 is replaced by period 0, and the value of the next highest digit is increased by one. For example, a fraction with period 9 of the form 7.24(9) is replaced by a periodic fraction with period 0 of the form 7.25(0) or an equal final decimal fraction 7.25. Another example: 4,(9)=5,(0)=5. The equality of a fraction with period 9 and its corresponding fraction with period 0 is easily established after replacing these decimal fractions with equal ordinary fractions.
Finally, let's take a closer look at infinite decimal fractions, which do not contain an endlessly repeating sequence of digits. They are called non-periodic.
Definition.
Non-recurring decimals(or simply non-periodic fractions) are infinite decimal fractions that have no period.
Sometimes non-periodic fractions have a form similar to that of periodic fractions, for example, 8.02002000200002... is a non-periodic fraction. In these cases, you should be especially careful to notice the difference.
Note that non-periodic fractions do not convert to ordinary fractions; infinite non-periodic decimal fractions represent irrational numbers.
Operations with decimals
One of the operations with decimal fractions is comparison, and the four basic arithmetic functions are also defined operations with decimals: addition, subtraction, multiplication and division. Let's consider separately each of the actions with decimal fractions.
Comparison of decimals essentially based on comparison of ordinary fractions corresponding to the decimal fractions being compared. However, converting decimal fractions into ordinary fractions is a rather labor-intensive process, and infinite non-periodic fractions cannot be represented as an ordinary fraction, so it is convenient to use a place-wise comparison of decimal fractions. Place-wise comparison of decimal fractions is similar to comparison of natural numbers. For more detailed information, we recommend studying the article: comparison of decimal fractions, rules, examples, solutions.
Let's move on to the next step - multiplying decimals. Multiplication of finite decimal fractions is carried out similarly to subtraction of decimal fractions, rules, examples, solutions to multiplication by a column of natural numbers. In the case of periodic fractions, multiplication can be reduced to multiplication of ordinary fractions. In turn, the multiplication of infinite non-periodic decimal fractions after their rounding is reduced to the multiplication of finite decimal fractions. We recommend for further study the material in the article: multiplication of decimal fractions, rules, examples, solutions.
Decimals on a coordinate ray
There is a one-to-one correspondence between points and decimals.
Let's figure out how points on the coordinate ray are constructed that correspond to a given decimal fraction.
We can replace finite decimal fractions and infinite periodic decimal fractions with equal ordinary fractions, and then construct the corresponding ordinary fractions on the coordinate ray. For example, the decimal fraction 1.4 corresponds to the common fraction 14/10, so the point with coordinate 1.4 is removed from the origin in the positive direction by 14 segments equal to a tenth of a unit segment.
Decimal fractions can be marked on a coordinate ray, starting from the decomposition of a given decimal fraction into digits. For example, let us need to build a point with coordinate 16.3007, since 16.3007=16+0.3+0.0007, then in this point you can get there by sequentially laying off from the origin 16 unit segments, 3 segments whose length is equal to a tenth of a unit segment, and 7 segments whose length is equal to a ten-thousandth of a unit segment.
This way of building decimal numbers on the coordinate ray allows you to get as close as you like to the point corresponding to an infinite decimal fraction.
Sometimes it is possible to accurately plot the point corresponding to an infinite decimal fraction. For example, 
, then this infinite decimal fraction 1.41421... corresponds to a point on the coordinate ray, distant from the origin of coordinates by the length of the diagonal of a square with a side of 1 unit segment.
The reverse process of obtaining the decimal fraction corresponding to a given point on a coordinate ray is the so-called decimal measurement of a segment. Let's figure out how it's done.
Let our task be to get from the origin to a given point on the coordinate line (or to infinitely approach it if we can’t get to it). With the decimal measurement of a segment, we can sequentially lay off from the origin any number of unit segments, then segments whose length is equal to a tenth of a unit, then segments whose length is equal to a hundredth of a unit, etc. By recording the number of segments of each length laid aside, we obtain the decimal fraction corresponding to a given point on the coordinate ray.
For example, to get to point M in the above figure, you need to set aside 1 unit segment and 4 segments, the length of which is equal to a tenth of a unit. Thus, point M corresponds to the decimal fraction 1.4.
It is clear that the points of the coordinate ray, which cannot be reached in the process of decimal measurement, correspond to infinite decimal fractions.
Bibliography.
- Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
- Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.