A legal fraction is greater than or less than one. What is a legal fraction? Right and wrong fraction: rules. What is a fraction
We encounter fractions in life much earlier than they begin to study them in school. If we cut a whole apple in half, we get ½ part of the fruit. Cut it again - it will be ¼. These are fractions. And everything, it would seem, is simple. For an adult. For a child (and this topic begins to be studied at the end of elementary school), abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must explain in an accessible way what a correct fraction and an incorrect one, ordinary and decimal ones are, what operations can be performed with them and, most importantly, why all this is needed.
What are the fractions
Acquaintance with a new topic at school begins with ordinary fractions. They are easy to recognize by the horizontal line dividing two numbers - above and below. The top is called the numerator, the bottom is called the denominator. There is also a lowercase version of writing incorrect and regular ordinary fractions - separated by a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to apply a "two-story" form of record. Why? Because it is more convenient. We will be convinced of this a little later.
In addition to the common ones, there are also decimal fractions. It is very simple to distinguish between them: if in one case a horizontal or slash is used, then in the other - a comma separating the sequences of numbers. Let's see an example: 2.9; 163.34; 1.953. We deliberately used a semicolon as a separator to delimit numbers. The first of them will read like this: "two whole, nine tenths."
New concepts
Let's go back to ordinary fractions. They are of two types.
The definition of a correct fraction is as follows: it is such a fraction, the numerator of which is less than the denominator. Why is it important? We'll see now!
You have several apples, split into halves. In total - 5 parts. How do you say: do you have "two and a half" or "five second" apples? Of course, the first option sounds more natural, and we will use it when talking with friends. But if you need to calculate how many fruits each will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from the point of view of mathematics, this will be clearer.
So, for the naming of correct and incorrect fractions, the rule is as follows: if an integer part can be distinguished in a fraction (14/5, 2/1, 173/16, 3/3), then it is incorrect. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.
Basic property of a fraction
If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value will not change. Imagine: the cake was cut into 4 equal parts and you were given one. They cut the same cake into eight pieces and gave you two. Is it all the same? After all, ¼ and 2/8 are one and the same!
Abbreviation
The authors of problems and examples in mathematics textbooks often try to confuse students by offering cumbersome fractions in writing, which can actually be abbreviated. Here's an example of a correct fraction: 167/334, which seemingly looks very "scary". But in fact, we can write it as ½. The number 334 is divisible by 167 without a remainder - by doing this, we get 2.
Mixed numbers
An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and recorded at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on.
To bring out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division above, above the line, and the whole part before the expression. Thus, we get two structural parts: whole units + regular fractions.
You can also carry out the reverse operation - for this you need to multiply the whole part by the denominator and add the resulting value to the numerator. Nothing complicated.
Multiplication and division
Oddly enough, multiplying fractions is easier than adding. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2 * 3/3 * 5 = 2/5.
With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) = 7 * 15/8 * 14 = 15/16.
Adding fractions
What if you want to add, or have different numbers in the denominator? Doing the same as with multiplication will not work - here you should understand the definition of a correct fraction and its essence. It is necessary to bring the terms to common denominator, that is, the same numbers should appear at the bottom of both fractions.
To do this, you should use the basic property of the fraction: multiply both sides by the same number. For example, 2/5 + 1/10 = (2 * 2) / (5 * 2) + 1/10 = 5/10 = ½.
How to choose which denominator to bring the terms to? This should be the minimum multiple of both numbers in the denominators of fractions: for 1/3 and 1/9, this will be 9; for ½ and 1/7 - 14, because there is no smaller value divisible by 2 and 7 without remainder.
Using
What are improper fractions for? After all, it is much more convenient to immediately select the whole part, to get mixed number- and that's the end! It turns out that if you need to multiply or divide two fractions, it is more profitable to use the wrong ones.
Let's take the following example: (2 + 3/17) / (37/68).
It would seem that there is nothing to cut at all. But what if you write the result of the addition in the first parentheses as an improper fraction? Look: (37/17) / (37/68)
Now everything falls into place! Let's write an example in such a way that everything becomes obvious: (37 * 68) / (17 * 37).
Reduce 37 in the numerator and denominator, and finally divide the top and bottom by 17. Do you remember the basic rule for correct and improper fractions? We can multiply and divide them by any number if we do it simultaneously for the numerator and denominator.
So, we get the answer: 4. The example looked complicated, and the answer contains only one number. It happens so often in mathematics. The main thing is not to be afraid and follow simple rules.
Common mistakes
When exercising, a student can easily make one of the popular mistakes. Usually they occur due to carelessness, and sometimes - due to the fact that the studied material has not yet been properly deposited in the head.
Often, the sum of numbers in the numerator makes you want to reduce its individual components. For example, in the example: (13 + 2) / 13, written without brackets (with a horizontal line), many students, due to inexperience, cross out 13 at the top and bottom. But this should not be done in any case, because this is a gross mistake! If, instead of addition, there was a multiplication sign, we would receive the number 2. But when performing addition, no operations with one of the terms are allowed, only with the entire sum as a whole.
Also, guys often make mistakes when dividing fractions. Let's take two regular irreducible fractions and divide by each other: (5/6) / (25/33). The student may confuse and write the resulting expression as (5 * 25) / (6 * 33). But this would happen with multiplication, but in our case everything will be somewhat different: (5 * 33) / (6 * 25). We shorten what is possible, and in the answer we will see 11/10. The resulting incorrect fraction is written as a decimal - 1.1.
Brackets
Remember that in any mathematical expression, the order of actions is determined by the precedence of the operation signs and the presence of parentheses. All other things being equal, the sequence of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.
After all, This is the result of dividing one number by another. If they are not completely divisible, it turns out to be a fraction - that's all.
How to write a fraction on a computer
Since standard tools do not always allow you to create a fraction consisting of two "tiers", students sometimes go to various tricks. For example, they copy the numerators and denominators into the "Paint" graphics editor and glue them together, drawing a horizontal line between them. Of course, there is a simpler option, which, by the way, provides a lot of additional features that will be useful to you in the future.
Open Microsoft Word. One of the panels at the top of the screen is called "Insert" - click it. On the right, in the side where the icons for closing and minimizing the window are located, there is a button "Formula". This is exactly what we need!
If you use this function, a rectangular area will appear on the screen, in which you can use any mathematical signs that are not on the keyboard, as well as write fractions in the classical form. That is, dividing the numerator and denominator with a horizontal bar. You may even be surprised that such a correct fraction is so easy to write down.
Study math
If you are in grades 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) Will be required in many school subjects. Practically in any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, you cannot do without fractions. Soon you will learn how to calculate everything in your head, without even writing down expressions on paper, but more and more complex examples will appear. Therefore, learn what a correct fraction is and how to work with it, keep up with curriculum, do your homework on time and then you will succeed.
At the word "fractions" goosebumps run for many. Because I recall the school and the tasks that were solved in mathematics. This was a duty to be fulfilled. But what if we treat tasks containing right and wrong fractions like a puzzle? After all, many adults solve digital and Japanese crosswords. Sorted out the rules, that's all. It's the same here. One has only to delve into the theory - and everything will fall into place. And examples will turn into a way to train your brain.
What kinds of fractions are there?
For a start, about what it is. A fraction is a number that has a fraction of one. It can be written in two forms. The first is called ordinary. That is, one that has a horizontal or oblique line. It equates to the division sign.
In such a record, the number above the dash is called the numerator, and below it, the denominator.
Among the ordinary ones, correct and incorrect fractions are distinguished. For the former, the modulo numerator is always less than the denominator. The wrong ones are called so because they have the opposite. A correct fraction is always less than one. While the wrong one is always greater than this number.
There are also mixed numbers, that is, those that have whole and fractional parts.
The second type of notation is a decimal fraction. It's a separate conversation about her.
How do improper fractions differ from mixed numbers?
At its core, nothing. They are simply different entries for the same number. Irregular fractions easily become mixed numbers after simple actions. And vice versa.
It all depends on the specific situation. Sometimes it is more convenient to use the wrong fraction in tasks. And sometimes it is necessary to translate it into a mixed number and then the example will be solved very easily. Therefore, what to use: improper fractions, mixed numbers, depends on the observantness of the problem solver.
The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second is always less than one.
How do I represent a mixed number as an improper fraction?
If you need to perform any action with several numbers that are written in different types, then you need to make them the same. One method is to represent numbers as improper fractions.
For this purpose, you will need to perform actions according to the following algorithm:
- multiply the denominator by an integer part;
- add the numerator to the result;
- write the answer above the line;
- leave the denominator the same.
Here are examples of how to write improper fractions from mixed numbers:
- 17 ¼ = (17 x 4 + 1): 4 = 69/4;
- 39 ½ = (39 x 2 + 1): 2 = 79/2.
How do I write an improper fraction as a mixed number?
The next technique is the opposite of the one discussed above. That is, when all mixed numbers are replaced with improper fractions. The algorithm of actions will be as follows:
- divide the numerator by the denominator to get the remainder;
- write down the quotient in place of the whole part of the mixed;
- the remainder should be placed above the line;
- the divisor will be the denominator.
Examples of such a transformation:
76/14; 76:14 = 5 with a remainder of 6; the answer is 5 integers and 6/14; the fractional part in this example needs to be reduced by 2, it turns out 3/7; the final answer is 5 point 3/7.
108/54; after division, the quotient is 2 without a remainder; this means that not all irregular fractions can be represented as a mixed number; the answer is the whole - 2.
How to convert an integer to an improper fraction?
There are situations when such an action is also necessary. To get improper fractions with a known denominator, you will need to perform the following algorithm:
- multiply an integer by the desired denominator;
- write this value above the line;
- place the denominator under it.
The simplest option is when the denominator is equal to one... Then you don't need to multiply anything. It is enough just to write the integer, which is given in the example, and place the unit under the line.
Example: 5 make an improper fraction with denominator 3. After multiplying 5 by 3, you get 15. This number will be the denominator. The answer to the problem is a fraction: 15/3.
Two approaches to solving problems with different numbers
In the example, you need to calculate the sum and difference, as well as the product and the quotient of two numbers: 2 integers 3/5 and 14/11.
In the first approach the mixed number will be presented as an improper fraction.
After completing the steps described above, you get the following value: 13/5.
In order to find out the amount, you need to bring the fractions to the same denominator. 13/5 after multiplying by 11 becomes 143/55. And 14/11 after multiplying by 5 will take the form: 70/55. To calculate the sum, you just need to add the numerators: 143 and 70, and then write down the answer with one denominator. 213/55 is an incorrect fraction the answer to the problem.
When finding the difference, the same numbers are subtracted: 143 - 70 = 73. The answer will be a fraction: 73/55.
When multiplying 13/5 and 14/11, you do not need to bring to a common denominator. It is enough to multiply the numerators and denominators in pairs. The answer is 182/55.
The same is with division. For the correct solution, you need to replace division with multiplication and flip the divisor: 13/5: 14/11 = 13/5 x 11/14 = 143/70.
In the second approach an improper fraction becomes a mixed number.
After performing the actions of the algorithm, 14/11 will turn into a mixed number with an integer part of 1 and a fractional 3/11.
When calculating the sum, you need to add the whole and fractional parts separately. 2 + 1 = 3, 3/5 + 3/11 = 33/55 + 15/55 = 48/55. The final answer is 3 point 48/55. The first round was 213/55. You can check the correctness by converting it to a mixed number. After dividing 213 by 55, you get the quotient 3 and the remainder 48. It is easy to see that the answer is correct.
Subtraction replaces the + sign with -. 2 - 1 = 1.33/55 - 15/55 = 18/55. To check, the answer from the previous approach must be converted into a mixed number: 73 is divided by 55 and the quotient is 1 and the remainder is 18.
It is inconvenient to use mixed numbers to find the work and the quotient. It is always recommended to go to the wrong fractions here.
Improper fraction
Quarters
- Orderliness. a and b there is a rule that makes it possible to unambiguously identify one and only one of the three relations between them: “<
», « >"Or" = ". This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and; two non-positive numbers a and b are related by the same relation as two non-negative numbers and; if suddenly a is nonnegative and b- negative, then a > b... style = "max-width: 98%; height: auto; width: auto;" src = "/ pictures / wiki / files / 57 /.png" border = "0">
Summation of fractions
- Addition operation. For any rational numbers a and b there is a so-called summation rule c... Moreover, the number itself c called sum numbers a and b and is denoted, and the process of finding such a number is called summation... The summation rule is as follows:
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- Multiplication operation. For any rational numbers a and b there is a so-called multiplication rule, which puts them in correspondence with some rational number c... Moreover, the number itself c called product numbers a and b and is denoted, and the process of finding such a number is also called multiplication... The multiplication rule is as follows:
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- Transitivity of the order relation. For any triple of rational numbers a , b and c if a a less b and b less c then a less c, what if a equally b and b equally c then a equally c... 6435 "> Commutativity of addition. The sum does not change from the change of places of rational terms.
- Addition associativity. The order of addition of the three rational numbers does not affect the result.
- The presence of zero. There is a rational number 0 that preserves any other rational number when summed up.
- The presence of opposite numbers. Any rational number has an opposite rational number, which, when added together, gives 0.
- Commutativity of multiplication. The product does not change from a change in the places of the rational factors.
- Associativity of multiplication. The order in which the three rational numbers are multiplied does not affect the result.
- Unit availability. There is a rational number 1 that preserves any other rational number when multiplied.
- Reverse numbers. Any rational number has an inverse rational number, which, when multiplied by, gives 1.
- Distributivity of multiplication relative to addition. The operation of multiplication is consistent with the operation of addition by means of the distribution law:
- The relationship of the order relation with the addition operation. The same rational number can be added to the left and right sides of a rational inequality. max-width: 98%; height: auto; width: auto; "src =" / pictures / wiki / files / 51 /.png "border =" 0 ">
- Archimedes' axiom. Whatever the rational number a, you can take so many units that their sum will exceed a... style = "max-width: 98%; height: auto; width: auto;" src = "/ pictures / wiki / files / 55 /.png" border = "0">
Additional properties
All other properties inherent in rational numbers are not singled out as the main ones, because, generally speaking, they no longer rely directly on the properties of integers, but can be proved based on the given basic properties or directly by the definition of a certain mathematical object. There are a lot of such additional properties. It makes sense to cite only a few of them here.
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Countability of a set
Rational numbering
To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it is enough to give an algorithm that numbers rational numbers, that is, it establishes a bijection between the sets of rational and natural numbers.
The simplest of these algorithms is as follows. An endless table of ordinary fractions is compiled, for each i-th line in each j-th column of which the fraction is located. For definiteness, it is assumed that the rows and columns of this table are numbered starting from one. Table cells are designated, where i is the row number of the table in which the cell is located, and j- column number.
The resulting table is bypassed by the "snake" according to the following formal algorithm.
These rules are viewed from top to bottom and the next position is selected on the first match.
In the process of such a traversal, each new rational number is associated with the next natural number. That is, the fraction 1/1 is assigned the number 1, the fraction 2/1 - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to one of the greatest common divisor of the numerator and denominator of the fraction.
Following this algorithm, all positive rational numbers can be enumerated. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers by simply assigning the opposite to each rational number. T. about. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.
The statement that the set of rational numbers is countable may cause some bewilderment, since at first glance one gets the impression that it is much more extensive than the set of natural numbers. In fact, this is not so, and there are enough natural numbers to enumerate all rational ones.
Lack of rational numbers
The hypotenuse of such a triangle is not expressed by any rational number
Rational numbers of the form 1 / n at large n you can measure arbitrarily small quantities. This fact creates the deceptive impression that any geometric distance can be measured with rational numbers. It is easy to show that this is not true.
It is known from the Pythagorean theorem that the hypotenuse of a right-angled triangle is expressed as the square root of the sum of the squares of its legs. T. about. isosceles hypotenuse length right triangle with one leg is equal, i.e., to a number whose square is 2.
Divided into right and wrong.
Correct fractions
Proper fraction is an ordinary fraction with the numerator less than the denominator.
To find out if a fraction is correct, you need to compare its members with each other. The terms of the fraction are compared according to the rule for comparing natural numbers.
Example. Consider a fraction:
| 7 |
| 8 |
Example:
| 8 | = 1 | 1 |
| 7 | 7 |
Translation rules and additional examples can be found in the topic Converting an improper fraction to a mixed number. You can also use an online calculator to convert an improper fraction to a mixed number.
Comparison of correct and improper fractions
Any irregular ordinary fraction is greater than the correct one, since a regular fraction is always less than one, and an irregular fraction is greater than or equal to one.
Example:
| 3 | > | 99 |
| 2 | 100 |
Comparison rules and more examples can be found in the Comparing Fractions topic. Also, to compare fractions or check comparisons, you can use
Right and wrong fractions repel 5th grade math students with their names. However, there is nothing wrong with these numbers. In order not to make mistakes in calculations and to dispel all the secrets associated with these numbers, we will consider the topic in detail.
What is a fraction?
A fraction is an incomplete division operation. Another option: a fraction is part of a whole. The numerator is the number of parts taken into account. The denominator is the total number of parts into which the whole is divided.
Types of fractions
The following types of fractions are distinguished:
- An ordinary fraction. This is a fraction with a lower numerator than the denominator.
- An irregular fraction with a larger numerator than the denominator.
- Mixed number that has an integer and a fractional part
- Decimal. This is a number that always has a power of 10 in the denominator. Such a fraction is written using a separating comma.
What fraction is called correct?
A regular fraction is called a regular fraction. This subspecies of fractions appeared earlier than others. Later, the types of numbers increased, new numbers and fractions were discovered and created. The first fraction is called correct, because it is it that reflects the meaning that the ancient mathematicians put into the concept of a fraction: it is part of a number. Moreover, this part is always less than the whole, that is, 1.
Why is the wrong fraction called that?
The incorrect fraction is greater than 1. That is, it no longer meets the first definition. It is no longer part of the whole. You can think of an irregular fraction as slices of several pies. After all, the pie is not always the same. However, the fraction is considered incorrect.
It is not customary to leave an incorrect fraction as a result of calculations. Better to convert it to a mixed number.
How to convert a correct fraction to a wrong one?
It is impossible to convert a correct fraction to a wrong one or vice versa. These are different categories of numbers. But some students often confuse concepts and call the conversion of an improper fraction to a mixed number the conversion of an improper fraction to a correct one.
Incorrect fractions are converted to mixed numbers quite often, as are mixed numbers to improper fractions. To convert an improper fraction to a mixed number, divide the numerator by the denominator with remainder. The remainder in this case will become the numerator of the fractional part, the quotient will become the whole part, and the denominator will remain the same.
What have we learned?
We remembered what a fraction is. They repeated all types of fractions and said which fraction is called correct. Separately, they noted why the incorrect fraction received such a name. They said that converting an incorrect fraction into a correct one, or vice versa, will not work. The last statement can be considered the rule of right and wrong fractions.
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