How to determine proper or improper fraction. Proper fraction

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 recording is decimal. There is a separate conversation about her.

How are improper fractions different from mixed numbers?

In essence, nothing. It's simple different entry the same number. Improper fractions become easy after simple steps. mixed numbers. 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 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 one improper fraction problem answer.

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. For the right decision 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.

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 is 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.

Improper 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 relation 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 side For a 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 is created ordinary fractions, 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. length of the hypotenuse of an isosceles right triangle with a unit leg is equal to, i.e., a 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.

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 relation 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 the following form: .
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. The same rational number can be added to the left and right sides of a rational inequality. /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

Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">

Countability of a set

Numbering of rational 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.

Lack of rational numbers

The hypotenuse of such a triangle cannot be expressed by any rational number

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 , 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.

What types of fractions are there?

In this notation, the number above the line is called the numerator, and the number below it is called the denominator.

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.

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.

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.

To find the product and quotient, it is inconvenient to use mixed numbers. It is always recommended to move on to improper fractions here.