Operations with ordinary fractions
Let's remember how to perform simple calculations with ordinary fractions. To multiply fractions, you need to multiply their numerators and write the result in the numerator, and then multiply the denominators and write the result in the denominator:
If the numerator and denominator of a fraction are divisible by the same number, then each of them is usually divided by it and this is called “reducing the fraction”: 
Sometimes reduction is performed while multiplying fractions:
If the fractions are mixed (with the integer part highlighted), then they need to be converted to ordinary fractions (consisting only of a numerator and a denominator). To do this, the whole part is multiplied by the denominator, the numerator is added and the result is written into the numerator, but the denominator is left the same: 
To convert an improper fraction (the numerator is greater than the denominator) to a mixed fraction (select the whole part), you need to divide the numerator by the denominator with the remainder. Then the incomplete quotient will be the whole part, the remainder will be the numerator, and the denominator will remain the same: 
To convert a common fraction to a decimal, you need to bring this denominator to the denominator of 10, 100, 1000, etc.:
Or divide the numerator by the denominator: 
To multiply a common fraction by a decimal, you need to either convert the common fraction to a decimal, or the decimal to a common: 
To divide a number by a common fraction, you need to swap the numerator with the denominator in this fraction and multiply the number by the resulting fraction: 
To write an integer as a fraction, you need to write it with a denominator of 1: 
To add fractions with the same denominators, you need to add their numerators and write the numerator of the new fraction, leaving the denominator the same. To subtract fractions with like denominators, you need to subtract their numerators and write the numerator of the new fraction, leaving the denominator the same. If fractions have an integer part, then you must first add or subtract the integer parts: 
You can add fractions with different denominators in the following way:
multiply the numerator and denominator of each fraction by additional factors so that the new denominator is equal to the least common multiple of the denominators of the original fractions. Let's add the resulting fractions with the same denominator:
It happens that the fractional part of the subtrahend is less than the minuend, then we must take 1 from the whole part of the minuend:
Actions with decimals:
Rule 1 : To add (subtract) decimal fractions, you need to: 1) write them one under the other so that the comma is under the comma (if the fractions have a different number of decimal places, then they need to be equalized using zeros; 2) perform addition (subtraction) , ignoring the comma; 3) put a comma under the comma in the answer (if necessary, discard the zeros after the decimal point).
4,12
3,78
7,90=7,9
12,76 0
8,674
4,086
Rule 2 : To multiply two decimal fractions, you need to 1) perform the multiplication without paying attention to the commas; 2) separate with a comma as many digits on the right as there are after the decimal point in both factors together.
0,671
2,9
6039
1342
1,9459
2,35
1,4
940
235
3,290=3,29
Rule 3 : To divide a number by a decimal fraction, you need to: 1) make a natural number from the decimal fraction by moving the decimal point to the right; 2) in the dividend, move the comma to the same number of digits as in the decimal fraction (or add 0 if it is a natural number); 3) after this, divide by a natural number (remembering to put a comma in the quotient where the division of the integer part ends)..
Rule 4. To multiply a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point in this fraction to the right by as many digits as there are zeros in the factor after the one.
7.567·10 =75.67
3.1 100 = 310
23.981 100 = 23981
To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point in this fraction to the left by as many digits as there are zeros in the factor after the one.
7, 567 : 10 = 0,7567
3,1 : 100 = 0, 031
2,398 : 1000 = 0, 002398
We will devote this material to such an important topic as decimal fractions. First, let's define the basic definitions, give examples and dwell on the rules of decimal notation, as well as what the digits of decimal fractions are. Next, we highlight the main types: finite and infinite, periodic and non-periodic fractions. In the final part we will show how the points corresponding to fractional numbers are located on the coordinate axis.
What is decimal notation of fractional numbers
The so-called decimal notation of fractional numbers can be used for both natural and fractional numbers. It looks like a set of two or more numbers with a comma between them.
The decimal point is needed to separate the whole part from the fractional part. As a rule, the last digit of a decimal fraction is not a zero, unless the decimal point appears immediately after the first zero.
What are some examples of fractional numbers in decimal notation? This could be 34, 21, 0, 35035044, 0, 0001, 11,231,552, 9, etc.
In some textbooks you can find the use of a period instead of a comma (5. 67, 6789. 1011, etc.). This option is considered equivalent, but it is more typical for English-language sources.
Definition of decimals
Based on the above concept of decimal notation, we can formulate the following definition of decimal fractions:
Definition 1
Decimals represent fractional numbers in decimal notation.
Why do we need to write fractions in this form? It gives us some advantages over ordinary ones, for example, a more compact notation, especially in cases where the denominator contains 1000, 100, 10, etc., or a mixed number. For example, instead of 6 10 we can specify 0.6, instead of 25 10000 - 0.0023, instead of 512 3 100 - 512.03.
How to correctly represent ordinary fractions with tens, hundreds, thousands in the denominator in decimal form will be discussed in a separate material.
How to read decimals correctly
There are some rules for reading decimal notations. Thus, those decimal fractions to which their regular ordinary equivalents correspond are read almost the same, but with the addition of the words “zero tenths” at the beginning. Thus, the entry 0, 14, which corresponds to 14,100, is read as “zero point fourteen hundredths.”
If a decimal fraction can be associated with a mixed number, then it is read in the same way as this number. So, if we have the fraction 56, 002, which corresponds to 56 2 1000, we read this entry as “fifty-six point two thousandths.”
The meaning of a digit in a decimal fraction depends on where it is located (the same as in the case of natural numbers). So, in the decimal fraction 0.7, seven is tenths, in 0.0007 it is ten thousandths, and in the fraction 70,000.345 it means seven tens of thousands of whole units. Thus, in decimal fractions there is also the concept of place value.
The names of the digits located before the decimal point are similar to those that exist in natural numbers. The names of those located after are clearly presented in the table:
Let's look at an example.
Example 1
We have the decimal fraction 43,098. She has a four in the tens place, a three in the units place, a zero in the tenths place, 9 in the hundredths place, and 8 in the thousandths place.
It is customary to distinguish the ranks of decimal fractions by precedence. If we move through the numbers from left to right, then we will go from the most significant to the least significant. It turns out that hundreds are older than tens, and parts per million are younger than hundredths. If we take the final decimal fraction that we cited as an example above, then the highest, or highest, place in it will be the hundreds place, and the lowest, or lowest, place will be the 10-thousandth place.
Any decimal fraction can be expanded into individual digits, that is, presented as a sum. This action is performed in the same way as for natural numbers.
Example 2
Let's try to expand the fraction 56, 0455 into digits.
We will get:
56 , 0455 = 50 + 6 + 0 , 4 + 0 , 005 + 0 , 0005
If we remember the properties of addition, we can represent this fraction in other forms, for example, as the sum 56 + 0, 0455, or 56, 0055 + 0, 4, etc.
What are trailing decimals?
All the fractions we talked about above are finite decimals. This means that the number of digits after the decimal point is finite. Let's derive the definition:
Definition 1
Trailing decimals are a type of decimal fraction that has a finite number of decimal places after the decimal sign.
Examples of such fractions can be 0, 367, 3, 7, 55, 102567958, 231 032, 49, etc.
Any of these fractions can be converted either to a mixed number (if the value of their fractional part is different from zero) or to an ordinary fraction (if the integer part is zero). We have devoted a separate article to how this is done. Here we’ll just point out a couple of examples: for example, we can reduce the final decimal fraction 5, 63 to the form 5 63 100, and 0, 2 corresponds to 2 10 (or any other fraction equal to it, for example, 4 20 or 1 5.)
But the reverse process, i.e. writing a common fraction in decimal form may not always be possible. So, 5 13 cannot be replaced by an equal fraction with the denominator 100, 10, etc., which means that a final decimal fraction cannot be obtained from it.
Main types of infinite decimal fractions: periodic and non-periodic fractions
We indicated above that finite fractions are so called because they have a finite number of digits after the decimal point. However, it may well be infinite, in which case the fractions themselves will also be called infinite.
Definition 2
Infinite decimal fractions are those that have an infinite number of digits after the decimal point.
Obviously, such numbers simply cannot be written down in full, so we indicate only part of them and then add an ellipsis. This sign indicates an infinite continuation of the sequence of decimal places. Examples of infinite decimal fractions include 0, 143346732…, 3, 1415989032…, 153, 0245005…, 2, 66666666666…, 69, 748768152…. etc.
The “tail” of such a fraction may contain not only seemingly random sequences of numbers, but also a constant repetition of the same character or group of characters. Fractions with alternating numbers after the decimal point are called periodic.
Definition 3
Periodic decimal fractions are those infinite decimal fractions in which one digit or a group of several digits is repeated after the decimal point. The repeating part is called the period of the fraction.
For example, for the fraction 3, 444444.... the period will be the number 4, and for 76, 134134134134... - the group 134.
What is the minimum number of characters that can be left in the notation of a periodic fraction? For periodic fractions, it will be enough to write the entire period once in parentheses. So, fraction 3, 444444…. It would be correct to write it as 3, (4), and 76, 134134134134... – as 76, (134).
In general, entries with several periods in brackets will have exactly the same meaning: for example, the periodic fraction 0.677777 is the same as 0.6 (7) and 0.6 (77), etc. Records of the form 0, 67777 (7), 0, 67 (7777), etc. are also acceptable.
To avoid mistakes, we introduce uniformity of notation. Let's agree to write down only one period (the shortest possible sequence of numbers), which is closest to the decimal point, and enclose it in parentheses.
That is, for the above fraction, we will consider the main entry to be 0, 6 (7), and, for example, in the case of the fraction 8, 9134343434, we will write 8, 91 (34).
If the denominator of an ordinary fraction contains prime factors that are not equal to 5 and 2, then when converted to decimal notation, they will result in infinite fractions.
In principle, we can write any finite fraction as a periodic one. To do this, we just need to add an infinite number of zeros to the right. What does it look like in recording? Let's say we have the final fraction 45, 32. In periodic form it will look like 45, 32 (0). This action is possible because adding zeros to the right of any decimal fraction gives us the result of a fraction equal to it.
Special attention should be paid to periodic fractions with a period of 9, for example, 4, 89 (9), 31, 6 (9). They are an alternative notation for similar fractions with a period of 0, so they are often replaced when writing with fractions with a zero period. In this case, one is added to the value of the next digit, and (0) is indicated in parentheses. The equality of the resulting numbers can be easily verified by representing them as ordinary fractions.
For example, the fraction 8, 31 (9) can be replaced with the corresponding fraction 8, 32 (0). Or 4, (9) = 5, (0) = 5.
Infinite decimal periodic fractions are classified as rational numbers. In other words, any periodic fraction can be represented as an ordinary fraction, and vice versa.
There are also fractions that do not have an endlessly repeating sequence after the decimal point. In this case, they are called non-periodic fractions.
Definition 4
Non-periodic decimal fractions include those infinite decimal fractions that do not contain a period after the decimal point, i.e. repeating group of numbers.
Sometimes non-periodic fractions look very similar to periodic ones. For example, 9, 03003000300003 ... at first glance seems to have a period, but a detailed analysis of the decimal places confirms that this is still a non-periodic fraction. You need to be very careful with such numbers.
Non-periodic fractions are classified as irrational numbers. They are not converted to ordinary fractions.
Basic operations with decimals
The following operations can be performed with decimal fractions: comparison, subtraction, addition, division and multiplication. Let's look at each of them separately.
Comparing decimals can be reduced to comparing fractions that correspond to the original decimals. But infinite non-periodic fractions cannot be reduced to this form, and converting decimal fractions into ordinary fractions is often a labor-intensive task. How can we quickly perform a comparison action if we need to do this while solving a problem? It is convenient to compare decimal fractions by digit in the same way as we compare natural numbers. We will devote a separate article to this method.
To add some decimal fractions with others, it is convenient to use the column addition method, as for natural numbers. To add periodic decimal fractions, you must first replace them with ordinary ones and count according to the standard scheme. If, according to the conditions of the problem, we need to add infinite non-periodic fractions, then we need to first round them to a certain digit, and then add them. The smaller the digit to which we round, the higher the accuracy of the calculation will be. For subtraction, multiplication and division of infinite fractions, pre-rounding is also necessary.
Finding the difference between decimal fractions is the inverse of addition. Essentially, using subtraction we can find a number whose sum with the fraction we are subtracting will give us the fraction we are minimizing. We will talk about this in more detail in a separate article.
Multiplying decimal fractions is done in the same way as for natural numbers. The column calculation method is also suitable for this. We again reduce this action with periodic fractions to the multiplication of ordinary fractions according to the rules already studied. Infinite fractions, as we remember, must be rounded before calculations.
The process of dividing decimals is the inverse of multiplying. When solving problems, we also use columnar calculations.
You can establish an exact correspondence between the final decimal fraction and a point on the coordinate axis. Let's figure out how to mark a point on the axis that will exactly correspond to the required decimal fraction.
We have already studied how to construct points corresponding to ordinary fractions, but decimal fractions can be reduced to this form. For example, the common fraction 14 10 is the same as 1, 4, so the corresponding point will be removed from the origin in the positive direction by exactly the same distance:
You can do without replacing the decimal fraction with an ordinary one, but use the method of expansion by digits as a basis. So, if we need to mark a point whose coordinate will be equal to 15, 4008, then we will first present this number as the sum 15 + 0, 4 +, 0008. To begin with, let’s set aside 15 whole unit segments in the positive direction from the beginning of the countdown, then 4 tenths of one segment, and then 8 ten-thousandths of one segment. As a result, we get a coordinate point that corresponds to the fraction 15, 4008.
For an infinite decimal fraction, it is better to use this method, since it allows you to get as close as you like to the desired point. In some cases, it is possible to construct an exact correspondence to an infinite fraction on the coordinate axis: for example, 2 = 1, 41421. . . , and this fraction can be associated with a point on the coordinate ray, distant from 0 by the length of the diagonal of the square, the side of which will be equal to one unit segment.
If we find not a point on the axis, but a decimal fraction corresponding to it, then this action is called the decimal measurement of a segment. Let's see how to do this correctly.
Let's say we need to get from zero to a given point on the coordinate axis (or get as close as possible in the case of an infinite fraction). To do this, we gradually postpone unit segments from the origin until we get to the desired point. After whole segments, if necessary, we measure tenths, hundredths and smaller fractions so that the match is as accurate as possible. As a result, we received a decimal fraction that corresponds to a given point on the coordinate axis.
Above we showed a drawing with point M. Look at it again: to get to this point, you need to measure one unit segment and four tenths of it from zero, since this point corresponds to the decimal fraction 1, 4.
If we cannot get to a point in the process of decimal measurement, then it means that it corresponds to an infinite decimal fraction.
If you notice an error in the text, please highlight it and press Ctrl+Enter
When adding decimal fractions, you need to write them one under the other so that the same digits are under each other, and the comma is under the comma, and add the fractions the same way you add natural numbers. Let's add, for example, the fractions 12.7 and 3.442. The first fraction contains one decimal place, and the second contains three. To perform addition, we transform the first fraction so that there are three digits after the decimal point: , then
The subtraction of decimal fractions is performed in the same way. Let's find the difference between the numbers 13.1 and 0.37:
When multiplying decimal fractions, it is enough to multiply the given numbers, not paying attention to commas (like natural numbers), and then, as a result, separate as many digits from the right with a comma as there are after the decimal point in both factors in total.
For example, let's multiply 2.7 by 1.3. We have. We use a comma to separate two digits on the right (the sum of the digits of the factors after the decimal point is two). As a result, we get 2.7 1.3 = 3.51.
If the product contains fewer digits than must be separated by a comma, then the missing zeros are written in front, for example:
Let's consider multiplying a decimal fraction by 10, 100, 1000, etc. Let's say we need to multiply the fraction 12.733 by 10. We have . Separating three digits to the right with a comma, we get But. Means,
12,733 10=127.33. Thus, multiplying a decimal fraction by 10 is reduced to moving the decimal point one digit to the right.
In general, to multiply a decimal fraction by 10, 100, 1000, you need to move the decimal point in this fraction 1, 2, 3 digits to the right, adding, if necessary, a certain number of zeros to the fraction on the right). For example,
Dividing a decimal fraction by a natural number is performed in the same way as dividing a natural number by a natural number, and the comma in the quotient is placed after the division of the integer part is completed. Let us divide 22.1 by 13:
If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:
Let us now consider dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. To do this, in both the dividend and the divisor, move the comma to the right by as many digits as there are after the decimal point in the divisor (in this example, two). In other words, if we multiply the dividend and the divisor by 100, the quotient will not change. Then you need to divide the fraction 257.6 by the natural number 112, i.e. the problem reduces to the case already considered:
To divide a decimal fraction by, you need to move the decimal point in this fraction to the left (and, if necessary, add the required number of zeros to the left). For example, .
Just as division is not always feasible for natural numbers, it is not always feasible for decimal fractions. For example, let's divide 2.8 by 0.09.
Farafonova Natalia Igorevna
After completing the topic “Actions with decimal fractions”, to practice counting skills and check the mastery of the material, you can conduct individual work with students using cards. Each student must complete tasks for all activities without errors. There are many options for each action, this gives each student the opportunity to solve the task for each action with decimals several times and achieve an error-free result or complete the task with a minimum number of errors. Since each student completes an individual task, the teacher has the opportunity, as completed tasks are presented to him, to discuss them personally with each student. If a student makes mistakes, the teacher corrects them and offers to do the task from a different option. So, until the student completes the entire task or most of it without errors. It is better to make cards on colored paper.
At the last stage of work, you can propose solving an example containing several actions.
For each error-free option, regardless of which attempt the task was completed correctly, students can be given an excellent mark, or an average grade can be given 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
Option 3
24,38 + 7,9
6,5 + 0,952
48,59 + 1,8
35,906 + 2,8
Option 4
7,6 + 319,75
888,99 + 4,5
64,15 + 18,9
4,5 + 0,738
Option 5
7,62 + 8,9
25,38 + 0,09
12,842 + 8,6
412 + 78,83
Option 6
70,7 + 3,8645
3,65 + 0,89
61,22 + 31.719
12,842 + 8,6
Answers: Option 1: 10.318; 10.437; 47.04; 30.017;
Option 2: 54.25; 5.616; 80.06; 50.008;
Option 3: 32.28; 7.452; 50.19; 38.706;
Option 4: 327.35; 893.49; 83.05; 5.238;
Option 5: 16.52; 25.47; 21.442; 490.83;
Option 6: 74.5645; 4.54; 92.939; 21.442;
Subtracting 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
Option 3
50,41 - 9,62
72,03 - 6,3
9,2 - 5,453
4 - 2,662
Option 4
60,01 - 8,364
123,61 - 69,8
8,7 - 4,915
10 - 3,817
Option 5
6,52 - 3,8
7,41 - 0,758
67,351 - 9,7
22 - 0,618
Option 6
4,5 - 0,496
61,3 - 20,3268
24,7 - 15,276
50 - 2,38
Answers: Option 1: 16.69; 32.52; 1.755; 6.654;
Option 2: 38.84; 44.96; 0.754; 1.449;
Option 3: 40.79; 65.73; 3.747; 1.338;
Option 4: 51.646; 53.81; 3.785; 6.183;
Option 5: 2.72; 6.652; 57.651; 21.382;
Option 6: 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
Option 3
6.8 5.9
6.06 8.05
0.65 0.014
720 4.6
Option 4
34.7 8.4
9.06 7.08
0.038 0.29
3.6 540
Option 5
62.4 2.5
0.038 9
1.8 0.009
4.125 0.16
Option 6
0.28 45
20.6 30.5
2.3 0.0024
0.0012 0.73
Option 7
68 0.15
0.08 0.012
1.4 1.04
0.32 2.125
Option 8
4.125 0.16
0.0012 0.73
1.4 1.04
720 4.6
Answers: Option 1: 25.9; 61.408; 0.442; 2100;
Option 2: 27.16; 42.834; 0.0084; 1904;
Option 3: 40.12; 48.783; 0.0091; 3312;
Option 4: 291.48; 64.1448; 0.01102; 1944;
Option 5: 156; 0.342; 0.0162; 0.66;
Option 6: 12.6; 628.3; 0.00552; 0.000876;
Option 7: 10.2; 0.00096; 1.456; 0.68;
Option 8: 0.66; 0.000876; 1.456; 3312;
Dividing a decimal fraction 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
Option 3
0,75: 15
0,7: 35
1,6: 8
0,72: 6
Option 4
2,4: 6
1,5: 75
0,12: 4
1,69: 13
Option 5
3,5: 175
1,8: 24
10,125: 9
0,48: 16
Option 6
0,35: 7
1,2: 3
0,2: 5
7,2: 144
Option 7
151,2: 63
4,8: 32
0,7: 25
2,3: 40
Option 8
397,8: 78
5,2: 65
0,9: 750
3,4: 80
Option 9
478,8: 84
7,3: 4
0,6: 750
5,7: 80
Option 10
699,2: 92
1,8: 144
0,7: 875
6,3: 24
Answers: Option 1: 2.5; 0.02; 0.8; 0.135;
Option 2: 0.04; 0.05; 1.3; 1.5;
Option 3: 0.05; 0.02; 0.2; 0.12;
Option 4: 0.4; 0.02; 0.03; 0.13;
Option 5: 0.02; 0.075; 1.125; 0.03;
Option 6: 0.05; 0.4; 0.04; 0.05;
Option 7: 2.4; 0.15; 0.28; 0.0575;
Option 8: 5.1; 0.08; 0.0012; 0.0425;
Option 9: 5.7; 1.825; 0.0008; 0.07125;
Option 10: 7.6; 0.0125; 0.0008; 0.2625;
Division by decimal fraction.
1 option
32: 1,25
54: 12,5
6: 125
Option 2
50,02: 6,1
34,2: 9,5
67,6: 6,5
Option 3
2,8036: 0,4
3,1: 0,025
0,0008: 0,16
Option 4
4: 32
303: 75
687,4: 10
1,59: 100
Option 5
5: 16
336: 35
412,5: 10
24,3: 100
Option 6
41,82: 6,8
73,44: 3,6
7,2: 0,045
32,89: 4,6
Answers: Option 1: 25.6; 4.32; 0.048;
Option 2: 8.2; 3.6; 10.4;
Option 3: 7.009; 124; 0.005;
Option 4: 0.125; 4.04; 68.74; 0.0159;
Option 5: 0.3125; 9.6; 41.25; 0.243;
Option 6: 6.15; 20.4; 160; 7.15;
Joint operations with decimals.
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.
In mathematics, different types of numbers have been studied since their 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 an integer part and fractions of a unit. Just like integers, there is an infinite number of fractions between two integers. In mathematics, operations with fractions are performed in the same way as with integers and natural numbers. It's 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 fraction line b/c. Common fractions can be extremely convenient if the fractional part cannot be represented in rational decimal form. In addition, it is more convenient to perform arithmetic operations through the fractional line. The upper part is called the numerator, the lower part is the denominator.
Operations with ordinary fractions: examples
The main 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 is an excellent way to provide a denominator for addition (this will be discussed below) or to shorten a fraction and make 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 get the denominator of a fraction, you need to present 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 the whole part highlighted. (A b/c) To represent a compound fraction as a common fraction, you need to multiply the number in front of the fraction by the denominator, and then add it with the numerator: (A*c + b)/c.
Arithmetic operations with fractions
It would be a good idea to consider well-known arithmetic operations only when working with fractional numbers.
Addition and subtraction. Adding and subtracting fractions is just as easy as adding and subtracting whole numbers, except for one difficulty - the presence of a fraction line. When adding fractions with the same denominator, you only need to add 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 number (how to do this was 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 follows exactly the same principle: 8/9 - 2/3 = 8/9 - 6/9 = 2/9.
Multiplication and division. Actions Multiplication with fractions occurs according to the following principle: numerators and denominators are multiplied separately. In general, 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 like factors from the numerator and denominator. In other words, the numerator and denominator are divided 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 multiply 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 frequently used version of fractions. It’s easier to write them down on a line or present them on a computer. The structure of a decimal is as follows: first the whole number is written, and then, after the decimal point, the fractional part is written. At their core, decimals are composite fractions, but their fractional part is represented by a number divided by a multiple of 10. This is where their name comes from. 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 endless. They are written like this: 7, (3). The following entry reads: seven point three, three tenths in a period.
Basic operations with decimal numbers
Adding and subtracting decimals. Working with fractions is no more difficult than working with whole natural numbers. The rules are absolutely similar to those used when adding or subtracting natural numbers. They can be counted as 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 column subtraction, you need to equalize the number of numbers after the decimal point: (5.5697 - 1.1200). So, the numerical value will not change and can be counted in a column.
Operations with decimal fractions cannot be performed if one of them has an irrational form. To do this, you need to convert both numbers into ordinary fractions, and then use the techniques described earlier.
Multiplication and division. Multiplying decimals is similar to multiplying natural fractions. They can also be multiplied in a column, simply, without paying attention to the comma, and then separated by a comma in the final value the same number of digits as the total 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 is also the same as division of natural numbers, but with a slight deviation. To divide by a decimal number with a column, you need to discard the decimal point 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. When dividing incompletely, you can add zeros to the dividend on the right, also adding a zero to the answer after the decimal point.
Examples of operations with decimals. Decimals are a very convenient tool for arithmetic calculations. They combine the convenience of natural numbers, whole numbers, and the precision of fractions. In addition, it is quite easy to convert some fractions to others. 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
Also, 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 examined two types of fractions - ordinary and decimal. Both are quite simple to calculate, and if you have completely mastered natural numbers and operations with them, you can safely start learning fractions.
