The word “fractions” gives many people goosebumps. Because I remember school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. What if you treated problems involving proper and improper fractions like a puzzle? After all, many adults solve digital and Japanese crosswords. We figured out the rules, and that’s it. It's the same here. One has only to delve into the theory - and everything will fall into place. And the examples will turn into a way to train your brain.
What types of fractions are there?
Let's start with what it is. A fraction is a number that has some part of one. It can be written in two forms. The first one is called ordinary. That is, one that has a horizontal or slanted line. It is equivalent to the division sign.
In this notation, the number above the line is called the numerator, and the number below it is called the denominator.
Among ordinary fractions, proper and improper fractions are distinguished. For the former, the absolute value of the numerator is always less than the denominator. The wrong ones are called that because they have everything the other way around. The value of a proper fraction is always less than one. While the incorrect one is always greater than this number.
There are also mixed numbers, that is, those that have an integer and a fractional part.
The second type of notation is a decimal fraction. There is a separate conversation about her.
How are improper fractions different from mixed numbers?
In essence, nothing. These are just different recordings of the same number. Improper fractions easily become mixed numbers after simple steps. And vice versa.
It all depends on specific situation. Sometimes it is more convenient to use an improper fraction in tasks. And sometimes it is necessary to translate it into mixed number and then the example will be solved very easily. Therefore, what to use: improper fractions, mixed numbers, depends on the observation skills of the person solving the problem.
The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second one is always less than one.
How to 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 the following algorithm:
- multiply the denominator by the whole part;
- add the value of 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 to 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 by improper fractions. The algorithm of actions will be as follows:
- divide the numerator by the denominator to obtain the remainder;
- write the quotient in place of the whole part of the mixed one;
- 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 remainder 6; the answer will be 5 whole and 6/14; the fractional part in this example needs to be reduced by 2, resulting in 3/7; the final answer is 5 point 3/7.
108/54; after division, the quotient of 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer will be an integer - 2.
How to turn a whole number into an improper fraction?
There are situations when such action is necessary. To obtain 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 below it.
The simplest option is when the denominator is equal to one. Then you don't need to multiply anything. It is enough to simply write the integer given in the example, and place one under the line.
Example: Make 5 an improper fraction with a denominator of 3. Multiplying 5 by 3 gives 15. This number will be the denominator. The answer to the task is a fraction: 15/3.
Two approaches to solving problems with different numbers
The example requires calculating the sum and difference, as well as the product and quotient of two numbers: 2 integers 3/5 and 14/11.
In the first approach the mixed number will be represented as an improper fraction.
After performing the steps described above, you will get the following value: 13/5.
In order to find out the sum, you need to reduce the fractions to the same denominator. 13/5 after multiplying by 11 becomes 143/55. And 14/11 after multiplying by 5 will look like: 70/55. To calculate the sum, you only need to add the numerators: 143 and 70, and then write down the answer with one denominator. 213/55 - this improper fraction is 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 reduce them to a common denominator. It is enough to multiply the numerators and denominators in pairs. The answer will be: 182/55.
The same goes for division. To solve correctly, you need to replace division with multiplication and invert 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 part of 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. In the first approach the fraction was 213/55. You can check its correctness by converting it to a mixed number. After dividing 213 by 55, the quotient is 3 and the remainder is 48. It is easy to see that the answer is correct.
When subtracting, the “+” sign is replaced by “-”. 2 - 1 = 1, 33/55 - 15/55 = 18/55. To check, the answer from the previous approach needs to be converted into a mixed number: 73 is divided by 55 and the quotient is 1 and the remainder is 18.
To find the product and quotient, it is inconvenient to use mixed numbers. It is always recommended to move on to improper fractions here.
Proper fraction
Quarters
- Orderliness. a And b there is a rule that allows one to uniquely identify one and only one of three relationships 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 relationship as two non-negative numbers and ; if suddenly a non-negative, but b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
Adding Fractions
- Addition operation. For any rational numbers a And b there is a so-called summation rule c. Moreover, the number itself c called amount numbers a And b and is denoted by , and the process of finding such a number is called summation. The summation rule has next view:
.
- Multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which assigns them some rational number c. Moreover, the number itself c called work numbers a And b and is denoted by , and the process of finding such a number is also called multiplication. The multiplication rule looks like this:
.
- Transitivity of the order relation. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
- Associativity of addition. The order in which three rational numbers are added does not affect the result.
- Presence of zero. There is a rational number 0 that preserves every other rational number when added.
- The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
- Commutativity of multiplication. Changing the places of rational factors does not change the product.
- Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
- Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
- Presence of reciprocal numbers. Any rational number has an inverse rational number, which when multiplied by gives 1.
- Distributivity of multiplication relative to addition. The multiplication operation is coordinated with the addition operation through the distribution law:
- Connection of the order relation with the operation of addition. To the left and right parts rational inequality you can add the same rational number. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
- Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum exceeds a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">
Additional properties
All other properties inherent in rational numbers are not distinguished as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proven based on the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense to list only a few of them here.
Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">
Countability of a set
Numbering of rational numbers
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 enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers.
The simplest of these algorithms looks like this. An endless table of ordinary fractions is compiled, on each i-th line in each j the 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 denoted by , where i- the number of the table row in which the cell is located, and j- column number.
The resulting table is traversed using a “snake” according to the following formal algorithm.
These rules are searched from top to bottom and the next position is selected based on the first match.
In the process of such a traversal, each new rational number is associated with another natural number. That is, the fraction 1/1 is assigned to the number 1, the fraction 2/1 to the number 2, etc. It should be noted that only irreducible fractions are numbered. A formal sign of irreducibility is that the greatest common divisor of the numerator and denominator of the fraction is equal to one.
Following this algorithm, we can enumerate all positive rational numbers. 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 to each rational number its opposite. That. 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 about the countability of the set of rational numbers may cause some confusion, since at first glance it seems 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 cannot be expressed by any rational number
Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates the misleading impression that rational numbers can be used to measure any geometric distances. It is easy to show that this is not true.
From the Pythagorean theorem we know that the hypotenuse of a right triangle is expressed as the square root of the sum of the squares of its legs. That. the length of the hypotenuse of an isosceles right triangle with a unit leg is equal to , i.e., the number whose square is 2.
If we assume that a number can be represented by some rational number, then there is such an integer m and such a natural number n, that , and the fraction is irreducible, i.e. numbers m And n- mutually simple.
If , then 
, i.e. m 2 = 2n 2. Therefore, the number m 2 is even, but the product of two odd numbers is odd, which means that the number itself m also even. So there is a natural number k, such that the number m can be represented in the form m = 2k. Number square m In this sense m 2 = 4k 2, but on the other hand m 2 = 2n 2 means 4 k 2 = 2n 2, or n 2 = 2k 2. As shown earlier for the number m, this means that the number n- even as m. But then they are not relatively prime, since both are bisected. The resulting contradiction proves that it is not a rational number.
The word “fractions” gives many people goosebumps. Because I remember school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. What if you treated problems involving proper and improper fractions like a puzzle? After all, many adults solve digital and Japanese crosswords. We figured out the rules, and that’s it. It's the same here. One has only to delve into the theory - and everything will fall into place. And the examples will turn into a way to train your brain.
What types of fractions are there?
Let's start with what it is. A fraction is a number that has some part of one. It can be written in two forms. The first one is called ordinary. That is, one that has a horizontal or slanted line. It is equivalent to the division sign.
In this notation, the number above the line is called the numerator, and the number below it is called the denominator.
Among ordinary fractions, proper and improper fractions are distinguished. For the former, the absolute value of the numerator is always less than the denominator. The wrong ones are called that because they have everything the other way around. The value of a proper fraction is always less than one. While the incorrect one is always greater than this number.
There are also mixed numbers, that is, those that have an integer and a fractional part.
The second type of notation is a decimal fraction. There is a separate conversation about her.
How are improper fractions different from mixed numbers?
In essence, nothing. These are just different recordings of the same number. Improper fractions easily become mixed numbers after simple steps. And vice versa.
It all depends on the specific situation. Sometimes it is more convenient to use an improper fraction in tasks. And sometimes it is necessary to convert 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 observation skills of the person solving the problem.
The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second one is always less than one.
How to represent a mixed number as an improper fraction?
If you need to perform any action with several numbers that are written in different forms, 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 the following algorithm:
- multiply the denominator by the whole part;
- add the value of 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 to 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 by improper fractions. The algorithm of actions will be as follows:
- divide the numerator by the denominator to obtain the remainder;
- write the quotient in place of the whole part of the mixed one;
- 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 remainder 6; the answer will be 5 whole and 6/14; the fractional part in this example needs to be reduced by 2, resulting in 3/7; the final answer is 5 point 3/7.
108/54; after division, the quotient of 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer will be an integer - 2.
How to turn a whole number into an improper fraction?
There are situations when such action is necessary. To obtain 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 below it.
The simplest option is when the denominator is equal to one. Then you don't need to multiply anything. It is enough to simply write the integer given in the example, and place one under the line.
Example: Make 5 an improper fraction with a denominator of 3. Multiplying 5 by 3 gives 15. This number will be the denominator. The answer to the task is a fraction: 15/3.
Two approaches to solving problems with different numbers
The example requires calculating the sum and difference, as well as the product and quotient of two numbers: 2 integers 3/5 and 14/11.
In the first approach the mixed number will be represented as an improper fraction.
After performing the steps described above, you will get the following value: 13/5.
In order to find out the sum, you need to reduce the fractions to the same denominator. 13/5 after multiplying by 11 becomes 143/55. And 14/11 after multiplying by 5 will look like: 70/55. To calculate the sum, you only need to add the numerators: 143 and 70, and then write down the answer with one denominator. 213/55 - this improper fraction is 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 reduce them to a common denominator. It is enough to multiply the numerators and denominators in pairs. The answer will be: 182/55.
The same goes for division. To solve correctly, you need to replace division with multiplication and invert 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 part of 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. In the first approach the fraction was 213/55. You can check its correctness by converting it to a mixed number. After dividing 213 by 55, the quotient is 3 and the remainder is 48. It is easy to see that the answer is correct.
When subtracting, the “+” sign is replaced by “-”. 2 - 1 = 1, 33/55 - 15/55 = 18/55. To check, the answer from the previous approach needs to be converted into a mixed number: 73 is divided by 55 and the quotient is 1 and the remainder is 18.
To find the product and quotient, it is inconvenient to use mixed numbers. It is always recommended to move on to improper fractions here.
We come across fractions in life much earlier than we begin studying them at school. If we cut a whole apple in half, we get ½ of the fruit. Let's cut it again - it will be ¼. These are fractions. And everything seemed simple. For an adult. For the child (and this topic begins to be studied at the end junior school) abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must clearly explain what a proper fraction and an improper fraction, an ordinary and a decimal are, what operations can be performed with them and, most importantly, what all this is needed for.
What are fractions?
Getting to know new topic at school it starts with ordinary fractions. They are easily recognized by the horizontal line separating the two numbers - above and below. The top one is called the numerator, the bottom one is the denominator. There is also a lowercase option for writing improper and proper ordinary fractions - through 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 use a “two-story” entry form. Why? Yes, because it is more convenient. We'll see this a little later.
In addition to ordinary fractions, there are also decimal fractions. It is very simple to distinguish them: if in one case a horizontal or slash is used, in the other a comma is used to separate sequences of numbers. Let's look at an example: 2.9; 163.34; 1.953. We intentionally used a semicolon as a separator to delimit the numbers. The first of them will read like this: “two point nine.”
New concepts
Let's return to ordinary fractions. They come in two types.
The definition of a proper fraction is as follows: it is a fraction whose numerator is less than its denominator. Why is it important? We'll see now!
You have several apples, halved. Total - 5 parts. How would you say: do you have “two and a half” or “five and a half” apples? Of course, the first option sounds more natural, and we will use it when talking with friends. But if we need to calculate how many fruits each person will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from a mathematical point of view, this will be more clear.
So, for naming proper and improper fractions, the rule is this: if a whole part can be distinguished in a fraction (14/5, 2/1, 173/16, 3/3), then it is improper. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.
The main property of a fraction
If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value does not change. Imagine: they cut the cake into 4 equal parts and gave you one. They cut the same cake into eight pieces and gave you two. Does it really matter? After all, ¼ and 2/8 are the same thing!
Reduction
Authors of problems and examples in mathematics textbooks often seek to confuse students by offering fractions that are cumbersome to write but can actually be abbreviated. Here is an example of a proper fraction: 167/334, which, it would seem, looks very “scary”. But we can actually write it as ½. The number 334 is divisible by 167 without a remainder - after performing this operation, 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 written 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 take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division on top, above the line, and the whole part - before the expression. Thus, we get two structural parts: whole units + proper fraction.
You can also carry out the inverse operation - to do this, you need to multiply the integer 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 to do if you need to perform addition or their denominator is different numbers? It will not work to do the same as with multiplication - here you should understand the definition of a proper fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the lower part of both fractions must have the same numbers.
To do this, you should use the basic property of a fraction: multiply both parts 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 reduce the terms to? This must be the minimum number that is a multiple of both numbers in the denominators of the fractions: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because there is no smaller value divisible by 2 and 7 without a remainder.
Usage
What are improper fractions used for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and be done with it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use irregular 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 we write the addition result in the first parentheses as an improper fraction? Look: (37/17) / (37/68)
Now everything falls into place! Let's write the example in such a way that everything becomes obvious: (37*68) / (17*37).
Let's cancel 37 in the numerator and denominator and finally divide the top and bottom by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide them by any number as long as we do it for the numerator and denominator at the same time.
So, we get the answer: 4. The example looked complicated, but the answer contains only one number. This happens often in mathematics. The main thing is not to be afraid and follow simple rules.
Common Mistakes
When implementing, a student can easily make one of the common mistakes. Usually they occur due to inattention, and sometimes due to the fact that the material studied has not yet been properly stored in the head.
Often the sum of numbers in the numerator makes you want to reduce its individual components. Let’s say in the example: (13 + 2) / 13, written without parentheses (with a horizontal line), many students, due to inexperience, cross out 13 above and below. But this should not be done under any circumstances, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are allowed, only with the entire sum.
Guys also often make mistakes when dividing fractions. Let's take two proper irreducible fractions and divide by each other: (5/6) / (25/33). The student can mix it up 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 reduce what is possible, and the answer will be 11/10. We write the resulting improper fraction as a decimal - 1.1.
Brackets
Remember that in any mathematical expression the order of operations is determined by the precedence of the operation signs and the presence of parentheses. All other things being equal, the order 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 evenly divided, it becomes a fraction - that's all.
How to write a fraction on a computer
Because the standard means It is not always possible to create a fraction consisting of two “tiers”; students sometimes resort to various tricks. For example, copy the numerators and denominators into graphics editor"Paint" and glue them together, drawing between them horizontal line. Of course, there is a simpler option, which, by the way, provides a lot of additional features, which 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, on the side where the close and minimize window icons are located, there is a “Formula” button. 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 symbols that are not on the keyboard, as well as write fractions in classic form. That is, dividing the numerator and denominator with a horizontal line. You might even be surprised that such a proper fraction is so easy to write.
Learn 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. In almost 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 to calculate everything in your mind, without even writing down expressions on paper, but more and more complex examples. Therefore, learn what a proper fraction is and how to work with it, keep up with curriculum, do your homework on time and you will succeed.
Simple mathematical rules and techniques, if they are not used constantly, are forgotten most quickly. Terms disappear from memory even faster.
One of these simple actions– converting an improper fraction into a proper or, in other words, a mixed fraction.
Improper fraction
An improper fraction is one in which the numerator (the number above the line) is greater than or equal to the denominator (the number below the line). This fraction is obtained by adding fractions or multiplying a fraction by a whole number. According to the rules of mathematics, such a fraction must be converted into a proper one.
Proper fraction
It is logical to assume that all other fractions are called proper. A strict definition is that a fraction whose numerator is less than its denominator is called proper. A fraction that has an integer part is sometimes called a mixed fraction.
Converting an improper fraction to a proper fraction
- First case: the numerator and denominator are equal to each other. The result of converting any such fraction is one. It doesn't matter if it's three-thirds or one hundred and twenty-five one hundred and twenty-fifths. Essentially, such a fraction denotes the action of dividing a number by itself.
- Second case: the numerator is greater than the denominator. Here you need to remember the method of dividing numbers with a remainder.
To do this, you need to find the number closest to the numerator value, which is divisible by the denominator without a remainder. For example, you have the fraction nineteen thirds. The closest number that can be divided by three is eighteen. That's six. Now subtract the resulting number from the numerator. We get one. This is the remainder. Write down the result of the conversion: six whole and one third.
But before reducing the fraction to the right kind, you need to check whether it can be shortened.
You can reduce a fraction if the numerator and denominator have a common factor. That is, a number by which both are divisible without a remainder. If there are several such divisors, you need to find the largest one.
For example, all even numbers have such a common divisor - two. And the fraction sixteen-twelfths has one more common divisor - four. This is the greatest divisor. Divide the numerator and denominator by four. Result of reduction: four thirds. Now, as a practice, convert this fraction to a proper fraction.
