The decimal fraction is used when you need to perform operations on non-integer numbers. This may seem irrational. But this type of numbers greatly facilitates the mathematical operations that must be performed with them. This understanding comes with time, when their writing becomes familiar, and reading does not cause difficulties, and the rules of decimal fractions are mastered. Moreover, all actions are repeated already known, which are learned with 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 and the answer is one and possibly zeros. In other words, if the denominator is 10, 100, 1000, and so on, it is more convenient to rewrite the number using a comma. Then before it will be located whole part, and then - fractional. Moreover, the record 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 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:
Simplify 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 just to correlate the numbers that are in the same positions, while with ordinary fractions one would have to bring them to a common denominator.
Simplification of calculations.
Calculators are not designed for the introduction of ordinary fractions, they use for all operations decimal notation numbers.
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 exceptions are fractions without an integer value, then when reading you need to say “zero integers”.
For example, 45/1000 should be pronounced as forty five thousandths, while 0.045 will sound like zero point forty-five thousandths.
A mixed number with an integer part equal to 7 and a fraction of 17/100, which will be written as 7.17, in both cases will be read as seven point seventeen hundredths.
The role of digits in the notation of fractions
It is true to note the discharge - this is what mathematics requires. Decimals and their meaning can change significantly if you write a digit in the wrong place. However, this has been true before.
To read the digits of the integer part of a decimal fraction, you just need to use the rules known for natural numbers. And on the right side they are mirrored and read differently. If "tens" sounded in the whole part, then after the decimal point it will be already "tenths".
This can be clearly seen in this table.
| Class | thousands | units | , | fraction | |||||||
| discharge | hundred | dec. | units | hundred | dec. | units | tenth | hundredth | thousandth | ten thousandth | |
How to 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 simple. To do this, it is enough to rewrite all its constituent parts in a different way. The following points will help with this:
write the numerator of the fraction a little aside, 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 correctly count the numbers - you need to move it as many positions as there are zeros in the denominator;
if there are not enough of them, then zeros should appear in empty positions;
zeros that were at the end of the numerator are no longer needed, and they can be crossed out;
add an integer part before the comma, if it was not there, then zero will also appear here.
Attention. You can not cross out zeros that are surrounded by other numbers.
About how to be in a situation where the denominator contains a number not only from one and zeros, how to convert a fraction to a decimal, you can read a little lower. 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 ten to any power.
If such an operation cannot be done.
How to check it? You need to factorize 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 other prime numbers appear, then the result will be infinite. It is customary to round such a decimal fraction for ease of use in mathematical operations. This will be discussed a little lower.
Studying how such decimal fractions are obtained, Grade 5. Examples will be very helpful here.
Let the denominators contain numbers: 40, 24 and 75. Decomposition into prime factors for them it will be:
- 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 a final fraction.
Algorithm for converting an ordinary 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 to these numbers so many 2 and 5 that they become an equal number. They will give the value of the additional multiplier.
Multiply the denominator and numerator by this number. The result is an ordinary fraction, under the line of which there is 10 to some extent.
If in a task these actions are performed with mixed number, then it must first be represented as an improper fraction. And only then act according to the described scenario.
Representation of a common fraction as a rounded decimal
This way of how to convert a fraction to a decimal will seem even easier to someone. Because it doesn't have a lot of action. 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 should be used.
First, write down the whole part and put a comma after it. If the fraction is correct, write zero.
Then it is necessary to perform the division of the numerator by the denominator. So that they have the same number of digits. That is, assign the required number of zeros to the right of the numerator.
Perform division in a column until the required number of digits is dialed. For example, if you need to round up to hundredths, then there should be 3 of them in the answer. In general, there should be one more digits than you need to get in the end.
Record 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 must be increased by one, discarding the last one.
Return from decimal to ordinary
In mathematics, there are problems when it is more convenient to represent decimal fractions in the form of ordinary ones, 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 integer part, if it is equal to zero, then nothing needs to be written;
draw a fractional line;
above it, write the numbers from the right side, if the first are zeros, then they must 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 common fraction.
What can you do with decimals?
In mathematics, this will be certain actions with decimal fractions 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 integer part. If they turn out to be equal, then they switch to the fractional one and compare them in the same way by digits. The number with the largest digit in the highest order will be the answer.
Adding and subtracting decimals
These are perhaps the simplest steps. Because they are performed according to the rules for natural numbers.
So, in order to add decimal fractions, they need to be written one under the other, placing commas in a column. With such a record, integer 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 down the comma. 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 add zeros.
Subtraction works in the same way. And here the rule applies, which describes the possibility of taking a unit from the highest digit. If the reduced fraction has fewer digits after the decimal point than the subtrahend, then zeros are simply assigned to it.
The situation is a little more complicated with tasks where you need to perform multiplication and division of decimal fractions.
How to multiply decimal in different examples?
The rule for multiplying decimal fractions by a natural number is as follows:
write them down in a column, ignoring the comma;
multiply as if they were natural;
separate with a comma as many digits as there were in the fractional part of the original number.
A special case is an example in which a natural number is equal to 10 to any power. Then, to get an answer, you just need to move the comma to the right by as many positions as there are zeros in another factor. In other words, when multiplying by 10, the comma shifts by one digit, by 100 - there will be two of them, and so on. If there are not enough digits in the fractional part, then you need to write zeros in empty positions.
The rule that is used when in the task you need to multiply decimal fractions by another of the same number:
write them down one under the other, ignoring the commas;
multiply as if they were natural numbers;
separate with a comma as many digits as there were in the fractional parts of both original fractions together.
As a special case, examples are distinguished in which one of the factors is equal to 0.1 or 0.01 and so on. In them, you need to move the comma to the left by the number of digits in the presented factors. That is, if multiplied by 0.1, then the comma is shifted by one position.
How to divide a decimal fraction in different tasks?
The division of 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;
divide according to the usual rule until the whole part ends;
put a comma in the answer;
continue dividing the fractional component until the remainder is zero;
if necessary, you can assign the desired number of zeros.
If the integer part is equal to zero, then it will not be in the answer either.
Separately, there is a division into numbers equal to ten, one hundred, and so on. In such problems, you need to move the comma to the left by the number of zeros in the divisor. It happens that there are not enough digits in the integer part, then zeros are used instead. It can be seen that this operation is similar to multiplying by 0.1 and similar numbers.
To perform division of 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 comma and in the divisible by the same number of digits;
follow the previous scenario.
The division by 0.1 is highlighted; 0.01 and other similar numbers. In such examples, the comma is shifted to the right by the number of digits in the fractional part. If they are over, then you need to assign the missing number of zeros. It is worth noting that this action repeats the division by 10 and similar numbers.
Conclusion: it's all about practice
Nothing in learning is easy or effortless. It takes time and practice to master new material reliably. Mathematics is no exception.
So that the topic of decimal fractions does not cause difficulties, you need to solve as many examples as possible with them. After all, there was a time when the addition of natural numbers was confusing. And now everything is fine.
Therefore, to paraphrase a well-known phrase: decide, decide and decide again. Then tasks with such numbers will be performed 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. The same is true in mathematical examples: after going along the same path several times, then you will no longer think about where to turn.
Farafonova Natalia Igorevna
After passing the topic "Actions with decimal fractions" to practice counting skills and check the assimilation of the material, you can conduct individual work with students on cards. Each student must complete all tasks without errors. Many options are presented for each action, this allows each student to solve the task for each action with decimal fractions several times and achieve an error-free result or complete the task with a minimum number of errors. Since each student performs an individual task, the teacher has the opportunity, as the completed tasks are presented to him, to discuss them personally with each student. If the student made mistakes, the teacher corrects them and offers to do the task from another option. So, until the student completes the entire task or most of it without errors. Cards are best done on colored paper.
At the last stage of work, we can offer to solve an example containing several actions.
For each correctly completed option, regardless of the attempt at which the task was completed correctly, students can be given an excellent mark, you can set an average mark, after completing all the work, at the discretion of the teacher.
Adding decimals.
1 option
7,468 + 2,85
9,6 + 0,837
38,64 + 8,4
3,9 + 26,117
Option 2
19,45 + 34,8
4,9 + 0,716
75,86 + 4,2
5,6 + 44,408
3 option
24,38 + 7,9
6,5 + 0,952
48,59 + 1,8
35,906 + 2,8
4 option
7,6 + 319,75
888,99 + 4,5
64,15 + 18,9
4,5 + 0,738
5 option
7,62 + 8,9
25,38 + 0,09
12,842 + 8,6
412 + 78,83
6 option
70,7 + 3,8645
3,65 + 0,89
61,22 + 31.719
12,842 + 8,6
Answers: 1st option: 10.318; 10.437; 47.04; 30.017;
Option 2: 54.25; 5.616; 80.06; 50.008;
3rd option: 32.28; 7.452; 50.19; 38.706;
4th option: 327.35; 893.49; 83.05; 5.238;
5th option: 16.52; 25.47; 21.442; 490.83;
6 option: 74.5645; 4.54; 92.939; 21.442;
Subtraction of decimals.
1 option
26,38 - 9,69
41,12 - 8,6
5,2 - 3,445
7 - 0,346
Option 2
47,62 - 8,78
54,06 - 9,1
7,1 - 6,346
3 - 1,551
3 option
50,41 - 9,62
72,03 - 6,3
9,2 - 5,453
4 - 2,662
4 option
60,01 - 8,364
123,61 - 69,8
8,7 - 4,915
10 - 3,817
5 option
6,52 - 3,8
7,41 - 0,758
67,351 - 9,7
22 - 0,618
6 option
4,5 - 0,496
61,3 - 20,3268
24,7 - 15,276
50 - 2,38
Answers: 1 option: 16.69; 32.52; 1.755; 6.654;
Option 2: 38.84; 44.96; 0.754; 1.449;
3rd option: 40.79; 65.73; 3.747; 1.338;
4th option: 51.646; 53.81; 3.785; 6.183;
5th option: 2.72; 6.652; 57.651; 21.382;
6 option: 4.004; 40.9732; 9.424; 47.62;
Multiplying decimals.
1 option
7.4 3.5
20.2 3.04
0.68 0.65
2.5 840
Option 2
2.8 9.7
6.05 7.08
0.024 0.35
560 3.4
3 option
6.8 5.9
6.06 8.05
0.65 0.014
720 4.6
4 option
34.7 8.4
9.06 7.08
0.038 0.29
3.6 540
5 option
62.4 2.5
0.038 9
1.8 0.009
4.125 0.16
6 option
0.28 45
20.6 30.5
2.3 0.0024
0.0012 0.73
7 option
68 0.15
0.08 0.012
1.4 1.04
0.32 2.125
8 option
4.125 0.16
0.0012 0.73
1.4 1.04
720 4.6
Answers: 1st option: 25.9; 61.408; 0.442; 2100;
Option 2: 27.16; 42.834; 0.0084; 1904;
3rd option: 40.12; 48.783; 0.0091; 3312;
4th option: 291.48; 64.1448; 0.01102; 1944;
5th option: 156; 0.342; 0.0162; 0.66;
6 option: 12.6; 628.3; 0.00552; 0.000876;
7 option: 10.2; 0.00096; 1.456; 0.68;
8 option: 0.66; 0.000876; 1.456; 3312;
Dividing a decimal by a natural number.
1 option
62,5: 25
0,5: 25
9,6: 12
1,08: 8
Option 2
0,28: 7
0,2: 4
16,9: 13
22,5: 15
3 option
0,75: 15
0,7: 35
1,6: 8
0,72: 6
4 option
2,4: 6
1,5: 75
0,12: 4
1,69: 13
5 option
3,5: 175
1,8: 24
10,125: 9
0,48: 16
6 option
0,35: 7
1,2: 3
0,2: 5
7,2: 144
7 option
151,2: 63
4,8: 32
0,7: 25
2,3: 40
8 option
397,8: 78
5,2: 65
0,9: 750
3,4: 80
9 option
478,8: 84
7,3: 4
0,6: 750
5,7: 80
10 option
699,2: 92
1,8: 144
0,7: 875
6,3: 24
Answers: 1 option: 2.5; 0.02; 0.8; 0.135;
Option 2: 0.04; 0.05; 1.3; 1.5;
3rd option: 0.05; 0.02; 0.2; 0.12;
4th option: 0.4; 0.02; 0.03; 0.13;
5th option: 0.02; 0.075; 1.125; 0.03;
6 option: 0.05; 0.4; 0.04; 0.05;
7 option: 2.4; 0.15; 0.28; 0.0575;
8 option: 5.1; 0.08; 0.0012; 0.0425;
9 option: 5.7; 1.825; 0.0008; 0.07125;
10 option: 7.6; 0.0125; 0.0008; 0.2625;
Division by decimal.
1 option
32: 1,25
54: 12,5
6: 125
Option 2
50,02: 6,1
34,2: 9,5
67,6: 6,5
3 option
2,8036: 0,4
3,1: 0,025
0,0008: 0,16
4 option
4: 32
303: 75
687,4: 10
1,59: 100
5 option
5: 16
336: 35
412,5: 10
24,3: 100
6 option
41,82: 6,8
73,44: 3,6
7,2: 0,045
32,89: 4,6
Answers: 1st option: 25.6; 4.32; 0.048;
Option 2: 8.2; 3.6; 10.4;
3rd option: 7.009; 124; 0.005;
4th option: 0.125; 4.04; 68.74; 0.0159;
5th option: 0.3125; 9.6; 41.25; 0.243;
6 option: 6.15; 20.4; 160; 7.15;
Joint actions with decimal fractions.
824,72 - 475: (0,071 + 0,929) + 13,8
(7.351 + 12.649) 105 - 95.48 - 4.52
(3.82 - 1.084 + 12.264) (4.27 + 1.083 - 3.353) + 83
278 - 16,7 - (15,75 + 24,328 + 39,2)
57.18 42 - 74.1: 13 + 21.35: 7
(18.8: 16 + 9.86 3) 40 - 12.73
(2 - 0.25 0.8) : (0.16: 0.5 - 0.02)
(3,625 + 0,25 + 2,75) : (28,75 + 92,25 - 15) : 0,0625
Answers: 1) 363.52; 2) 2000; 3) 113; 4) 182.022; 5) 2398.91; 6) 1217.47; 7) 6; 8) 1.
Fraction- a number that consists of an integer number of fractions of one and is represented as: a / b
Fraction numerator (a)- the number above the line of the fraction and showing the number of shares into which the unit was divided.
Fraction denominator (b)- the number under the line of the fraction and showing how many shares the unit was divided.
2. Bringing fractions to a common denominator
3. Arithmetic operations on ordinary fractions
3.1. Addition of ordinary fractions
3.2. Subtraction of ordinary fractions
3.3. Multiplication of ordinary fractions
3.4. Division of ordinary fractions
4. Reciprocal numbers
5. Decimals
6. Arithmetic operations on decimal fractions
6.1. Adding decimals
6.2. Subtraction of decimals
6.3. Decimal multiplication
6.4. Decimal division
#one. Basic property of a fraction
If the numerator and denominator of a fraction are multiplied or divided by the same number that is not equal to zero, then a fraction equal to the given one will be obtained.
3/7=3*3/7*3=9/21 i.e. 3/7=9/21
a/b=a*m/b*m - this is how the main property of a fraction looks like.
In other words, we get a fraction equal to the given one by multiplying or dividing the numerator and denominator of the original fraction by the same natural number.
If ad=bc, then two fractions a/b =c /d are considered equal.
For example, the fractions 3/5 and 9/15 will be equal, since 3*15=5*9, that is, 45=45
Fraction reduction is the process of replacing a fraction, in which the new fraction is equal to the original, but with a smaller numerator and denominator.
It is customary to reduce fractions based on the main property of a fraction.
For instance, 45/60=15/ 20 =9/12=3/4 (the numerator and denominator are divisible by 3, by 5 and by 15).
irreducible fraction is a fraction of the form 3/4 , where the numerator and denominator are relatively prime numbers. The main purpose of fraction reduction is to make the fraction irreducible.
2. Reducing fractions to a common denominator
To bring two fractions to a common denominator:
1) decompose the denominator of each fraction into prime factors;
2) multiply the numerator and denominator of the first fraction by the missing ones
factors from the expansion of the second denominator;
3) multiply the numerator and denominator of the second fraction by the missing factors from the first decomposition.
Examples: Reduce fractions to a common denominator.
Let's decompose the denominators into prime factors: 18=3∙3∙2, 15=3∙5
We multiplied the numerator and denominator of the fraction by the missing factor 5 from the second decomposition.
numerator and denominator of the fraction by the missing factors 3 and 2 from the first expansion.
= , 90 is the common denominator of fractions .
3. Arithmetic operations on ordinary fractions
3.1. Addition of ordinary fractions
a) With the same denominators, the numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As seen in the example:
a/b+c/b=(a+c)/b ;
b) With different denominators, the fractions are first reduced to a common denominator, and then the numerators are added according to the rule a):
7/3+1/4=7*4/12+1*3/12=(28+3)/12=31/12
3.2. Subtraction of ordinary fractions
a) With the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same:
a/b-c/b=(a-c)/b ;
b) If the denominators of the fractions are different, then first the fractions are reduced to a common denominator, and then repeat the steps as in paragraph a).
3.3. Multiplication of ordinary fractions
Multiplication of fractions obeys the following rule:
a/b*c/d=a*c/b*d,
that is, multiply the numerators and denominators separately.
For instance:
3/5*4/8=3*4/5*8=12/40.
3.4. Division of ordinary fractions
Fractions are divided in the following way:
a/b:c/d=a*d/b*c,
that is, the fraction a / b is multiplied by the reciprocal of the given one, that is, it is multiplied by d / c.
Example: 7/2:1/8=7/2*8/1=56/2=28
4. Reciprocal numbers
If a*b=1, then the number b is reverse number for number a .
Example: for the number 9, the reverse is 1/9 , since 9*1/9 = 1 , for the number 5 - the reciprocal of 1/5 , because 5* 1/5 = 1 .
5. Decimals
Decimal is a proper fraction whose denominator is 10, 1000, 10000, …, 10^n 1 0 , 1 0 0 0 , 1 0 0 0 0 , . . . , 1 0 n .
For example: 6/10 =0,6; 44/1000=0,044 .
In the same way, incorrect ones are written with a denominator 10^n or mixed numbers.
For example: 51/10= 5,1; 763/100=7,63
In the form of a decimal fraction, any ordinary fraction with a denominator that is a divisor of a certain power of the number 10 is represented.
a denominator, which is a divisor of a certain power of the number 10.
Example: 5 is a divisor of 100, so a fraction 1/5=1 *20/5*20=20/100=0,2 0 = 0 , 2 .
6. Arithmetic operations on decimal fractions
6.1. Adding decimals
To add two decimal fractions, you need to arrange them so that the same digits and a comma under a comma appear under each other, and then add the fractions as ordinary numbers.
6.2. Subtraction of decimals
It works in the same way as addition.
6.3. Decimal multiplication
When multiplying decimal numbers, it is enough to multiply the given numbers, ignoring the commas (as natural numbers), and in the received answer, the comma on the right separates as many digits as there are after the decimal point in both factors in total.
Let's do the multiplication of 2.7 by 1.3. We have 27\cdot 13=351 2 7 ⋅ 1 3 = 3 5 1 . We separate two digits with a comma on the right (the first and second numbers have one digit after the decimal point; 1+1=2 1 + 1 = 2 ). As a result, we get 2.7\cdot 1.3=3.51 2 , 7 ⋅ 1 , 3 = 3 , 5 1 .
If the result is fewer digits than it is necessary to separate with a comma, then the missing zeros are written in front, for example:
To multiply by 10, 100, 1000, in a decimal fraction, move the comma 1, 2, 3 digits to the right (if necessary, a certain number of zeros are assigned to the right).
For instance: 1.47 \cdot 10,000 = 14,700 1 , 4 7 ⋅ 1 0 0 0 0 = 1 4 7 0 0 .
6.4. Decimal division
Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number. A comma in the private is placed after the division of the integer part is completed.
If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:
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Consider dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. First of all, we multiply the dividend and the divisor of the fraction by 100, that is, we move the comma to the right in the dividend and divisor by as many characters as there are in the divisor after the decimal point (in this example, two). Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:
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It happens that the final decimal fraction is not always obtained when dividing one number by another. The result is an infinite decimal. In such cases, go to ordinary fractions.
For example, 2.8: 0.09= 28/10: 9/100= 28*100/10*9=2800/90=280/9= 31 1/9 .
In mathematics different types numbers have been studied since its inception. There are a large number of sets and subsets of numbers. Among them are integers, rational, irrational, natural, even, odd, complex and fractional. Today we will analyze information about the last set - fractional numbers.
Definition of fractions
Fractions are numbers consisting of a whole part and fractions of a unit. Just like integers, there are an infinite number of fractional numbers between two integers. In mathematics, operations with fractions are performed, as with integers and natural numbers. It is quite simple and can be learned in a couple of lessons.
The article presents two types
Common fractions
Ordinary fractions are the integer part a and two numbers written through the fractional bar b/c. Common fractions can be extremely handy if the fractional part cannot be represented in rational decimal form. In addition, it is more convenient to perform arithmetic operations through a fractional line. Top part called the numerator, the bottom - the denominator.
Actions with ordinary fractions: examples
Basic property of a fraction. At multiplying the numerator and denominator by the same number that is not zero, the result is a number equal to the given one. This property of a fraction helps to bring a denominator for addition (this will be discussed below) or reduce a fraction, making it more convenient for counting. a/b = a*c/b*c. For example, 36/24 = 6/4 or 9/13 = 18/26
Reduction to a common denominator. To bring the denominator of a fraction, you need to represent the denominator in the form of factors, and then multiply by the missing numbers. For example, 7/15 and 12/30; 7/5*3 and 12/5*3*2. We see that the denominators differ by two, so we multiply the numerator and denominator of the first fraction by 2. We get: 14/30 and 12/30.
Compound fractions- ordinary fractions with a highlighted integer part. (A b/c) To represent a compound fraction as a common fraction, multiply the number in front of the fraction by the denominator and then add it to the numerator: (A*c + b)/c.
Arithmetic operations with fractions
It will not be superfluous to consider the well-known arithmetic operations only when working with fractional numbers.
Addition and subtraction. Adding and subtracting fractions is just as easy as whole numbers, with the exception of one difficulty - the presence of a fractional bar. When adding fractions with the same denominator, it is necessary to add only the numerators of both fractions, the denominators remain unchanged. For example: 5/7 + 1/7 = (5+1)/7 = 6/7
If the denominators of two fractions are different numbers, you first need to bring them to a common one (as discussed above). 1/8 + 3/2 = 1/2*2*2 + 3/2 = 1/8 + 3*4/2*4 = 1/8 + 12/8 = 13/8. Subtraction occurs according to exactly the same principle: 8/9 - 2/3 \u003d 8/9 - 6/9 \u003d 2/9.
Multiplication and division. Actions with fractions by multiplication occur according to the following principle: numerators and denominators are multiplied separately. In general terms, the multiplication formula looks like this: a/b *c/d = a*c/b*d. In addition, as you multiply, you can reduce the fraction by eliminating the same factors from the numerator and denominator. In another language, the numerator and denominator are divisible by the same number: 4/16 = 4/4*4 = 1/4.
To divide one ordinary fraction by another, you need to change the numerator and denominator of the divisor and perform the multiplication of two fractions, according to the principle discussed earlier: 5/11: 25/11 = 5/11 * 11/25 = 5*11/11*25 = 1/5
Decimals
Decimals are the more popular and commonly used version. fractional numbers. They are easier to write down in a line or present on a computer. The structure of the decimal fraction is as follows: first the whole number is written, and then, after the decimal point, the fractional part is written. At their core, decimal fractions are compound fractions, but their fractional part is represented by a number divided by a multiple of 10. Hence their name. Operations with decimal fractions are similar to operations with integers, since they are also written in the decimal number system. Also, unlike ordinary fractions, decimals can be irrational. This means that they can be infinite. They are written as 7,(3). The following entry is read: seven whole, three tenths in the period.
Basic operations with decimal numbers
Addition and subtraction of decimal fractions. Performing actions with fractions is no more difficult than with whole natural numbers. The rules are exactly the same as those used when adding or subtracting natural numbers. They can also be considered a column in the same way, but if necessary, replace the missing places with zeros. For example: 5.5697 - 1.12. In order to perform a column subtraction, you need to equalize the number of numbers after the decimal point: (5.5697 - 1.1200). So, numerical value not change and it will be possible to count in a column.
Operations with decimal fractions cannot be performed if one of them has irrational view. To do this, you need to convert both numbers to ordinary fractions, and then use the techniques described earlier.
Multiplication and division. Multiplying decimals is similar to multiplying natural numbers. They can also be multiplied by a column, simply ignoring the comma, and then separated by a comma in the final value the same number of digits as the sum after the decimal point was in two decimal fractions. For example, 1.5 * 2.23 = 3.345. Everything is very simple, and should not cause difficulties if you have already mastered the multiplication of natural numbers.
Division also coincides with the division of natural numbers, but with a slight digression. To divide by a decimal number in a column, you must discard the comma in the divisor, and multiply the dividend by the number of digits after the decimal point in the divisor. Then perform division as with natural numbers. With incomplete division, you can add zeros to the dividend on the right, also adding a zero after the decimal point.
Examples of actions with decimal fractions. Decimals are a very handy tool for arithmetic counting. They combine the convenience of natural, whole numbers and the precision of common fractions. In addition, it is quite simple to convert one fraction to another. Operations with fractions are no different from operations with natural numbers.
- Addition: 1.5 + 2.7 = 4.2
- Subtraction: 3.1 - 1.6 = 1.5
- Multiplication: 1.7 * 2.3 = 3.91
- Division: 3.6: 0.6 = 6
In addition, decimals are suitable for representing percentages. So, 100% = 1; 60% = 0.6; and vice versa: 0.659 = 65.9%.
That's all you need to know about fractions. The article considered two types of fractions - ordinary and decimal. Both are fairly easy to calculate, and if you have a complete mastery of natural numbers and operations with them, you can safely start learning fractional ones.
Of the many fractions found in arithmetic, those with 10, 100, 1000 in the denominator deserve special attention - in general, any power of ten. These fractions have a special name and notation.
A decimal is any number whose denominator is a power of ten.
Decimal examples:
Why was it necessary to isolate such fractions at all? Why do they need their own entry form? There are at least three reasons for this:
- Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, bring the fractions to a common denominator. In decimal fractions, none of this is required;
- Reduction of calculations. Decimals add and multiply according to their own rules, and with a little practice you will be able to work with them much faster than with ordinary ones;
- Ease of recording. Unlike ordinary fractions, decimals are written in one line without loss of clarity.
Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you demand change in the amount of 2/3 rubles in a store :)
Rules for writing decimal fractions
The main advantage of decimal fractions is a convenient and visual notation. Namely:
Decimal notation is a form of decimal notation where the integer part is separated from the fractional part using a regular dot or comma. In this case, the separator itself (dot or comma) is called the decimal point.
For example, 0.3 (read: “zero integer, 3 tenths”); 7.25 (7 integers, 25 hundredths); 3.049 (3 integers, 49 thousandths). All examples are taken from the previous definition.
In writing, a comma is usually used as a decimal point. Here and below, the comma will also be used throughout the site.
To write an arbitrary decimal fraction in the specified form, you need to follow three simple steps:
- Write out the numerator separately;
- Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
- If the decimal point has shifted, and after it there are zeros at the end of the record, they must be crossed out.
It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, any number of zeros can be assigned to the left of any number without harm to health. It's ugly, but sometimes useful.
At first glance, this algorithm may seem rather complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:
Task. For each fraction, indicate its decimal notation:
The numerator of the first fraction: 73. We shift the decimal point by one sign (because the denominator is 10) - we get 7.3.
The numerator of the second fraction: 9. We shift the decimal point by two digits (because the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange notation like “.09”.
The numerator of the third fraction: 10029. We shift the decimal point by three digits (because the denominator is 1000) - we get 10.029.
The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10.500. There are extra zeros at the end of the number. We cross them out - we get 10.5.
Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as is done in the last example. However, in no case should you do this with zeros that are inside the number (which are surrounded by other digits). That is why we got 10.029 and 10.5, and not 1.29 and 1.5.
So, we figured out the definition and form of recording decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.
Change from fractions to decimals
Consider a simple numerical fraction of the form a / b . You can use the basic property of a fraction and multiply the numerator and denominator by such a number that you get a power of ten below. But before doing so, please read the following:
There are denominators that are not reduced to the power of ten. Learn to recognize such fractions, because they cannot be worked with according to the algorithm described below.
That's it. Well, how to understand whether the denominator is reduced to the power of ten or not?
The answer is simple: factorize the denominator into prime factors. If only factors 2 and 5 are present in the expansion, this number can be reduced to the power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the degree of ten.
Task. Check if the specified fractions can be represented as decimals:
We write out and factorize the denominators of these fractions:
20 \u003d 4 5 \u003d 2 2 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.
12 \u003d 4 3 \u003d 2 2 3 - there is a "forbidden" factor 3. The fraction cannot be represented as a decimal.
640 \u003d 8 8 10 \u003d 2 3 2 3 2 5 \u003d 2 7 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction is represented as a decimal.
48 \u003d 6 8 \u003d 2 3 2 3 \u003d 2 4 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.
So, we figured out the denominator - now we will consider the entire algorithm for switching to decimal fractions:
- Factorize the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
- Count how many twos and fives are present in the decomposition (there will be no other numbers there, remember?). Choose such an additional multiplier so that the number of twos and fives is equal.
- Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.
Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest such factor from all possible ones.
And one more thing: if there is an integer part in the original fraction, be sure to convert this fraction to an improper one - and only then apply the described algorithm.
Task. Convert these numbers to decimals:
Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, a fraction can be represented as a decimal. There are two twos and no fives in the expansion, so the additional factor is 5 2 = 25. The number of twos and fives will be equal to it. We have:
Now let's deal with the second fraction. To do this, note that 24 \u003d 3 8 \u003d 3 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.
The last two fractions have denominators 5 (a prime number) and 20 = 4 5 = 2 2 5 respectively - only twos and fives are present everywhere. At the same time, in the first case, “for complete happiness”, there is not enough multiplier 2, and in the second - 5. We get:
Switching from decimals to ordinary
The reverse conversion - from decimal notation to normal - is much easier. There are no restrictions and special checks, so you can always convert a decimal fraction into a classic "two-story" one.
The translation algorithm is as follows:
- Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing - do not overdo it and do not cross out the internal zeros surrounded by other numbers;
- Calculate how many digits are in the original decimal fraction after the decimal point. Take the number 1 and add as many zeros to the right as you counted the characters. This will be the denominator;
- Actually, write down the fraction whose numerator and denominator we just found. Reduce if possible. If there was an integer part in the original fraction, now we will get an improper fraction, which is very convenient for further calculations.
Task. Convert decimals to ordinary: 0.008; 3.107; 2.25; 7,2008.
We cross out the zeros on the left and the commas - we get the following numbers (these will be numerators): 8; 3107; 225; 72008.
In the first and second fractions after the decimal point there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; one hundred; 10000.
Finally, let's combine the numerators and denominators into ordinary fractions:
As can be seen from the examples, the resulting fraction can very often be reduced. Once again, I note that any decimal fraction can be represented as an ordinary one. The reverse transformation is not always possible.
