A proper fraction is always greater or less than 1. Proper and improper fractions

Improper fraction

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1. Orderliness. a and b there is a rule that allows you to uniquely identify between them one and only one of the three relations: “< », « >' 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, and b- negative, then a > b. style="max-width: 98%; height: auto; width: auto;" src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">

   

   summation of fractions

2. addition operation. For any rational numbers a and b there is a so-called summation rule c. However, 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 has the following form: .
3. 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. However, the number itself c called work 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: .
4. Transitivity of the order relation. For any triple of rational numbers a , b and c if a less b and b less c, then a less c, what if a equals b and b equals c, then a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
5. Associativity of addition. The order in which three rational numbers are added does not affect the result.
6. The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
8. Commutativity of multiplication. By changing the places of rational factors, the product does not change.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
12. Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
13. 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. max-width: 98% height: auto; width: auto;" src="/pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
14. Axiom of Archimedes. 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/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

All other properties inherent in rational numbers are not singled out as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proved on the basis of the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense here to cite just a few of them.

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Set countability

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 suffices to give an algorithm that enumerates rational numbers, that is, establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms is as follows. An infinite table of ordinary fractions is compiled, on each i-th line in each j th column of which is a fraction. For definiteness, it is assumed that the rows and columns of this table are numbered from one. Table cells are denoted , where i- the row number of the table in which the cell is located, and j- column number.

The resulting table is managed by a "snake" according to the following formal algorithm.

These rules are searched from top to bottom and the next position is selected by the first match.

In the process of such a bypass, each new rational number is assigned to the next natural number. That is, fractions 1 / 1 are assigned the number 1, fractions 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 unity of the greatest common divisor of the numerator and denominator of the fraction.

Following this algorithm, one 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, simply by 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 bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.

Insufficiency 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 arbitrarily small quantities can be measured. This fact creates a deceptive impression that rational numbers can measure any geometric distances in general. It is easy to show that this is not true.

It is known from the Pythagorean theorem 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., a number whose square is 2.

We encounter fractions in life much earlier than they begin to study at school. If you cut a whole apple in half, then we get a piece of fruit - ½. Cut it again - it will be ¼. This is what fractions are. And everything, it would seem, is simple. For an adult. For a child (and they begin to study this topic at the end of elementary school), abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must explain in an accessible way what a proper fraction and improper, ordinary and decimal are, what operations can be performed with them and, most importantly, why all this is needed.

What are fractions

Acquaintance with a new topic at school begins with ordinary fractions. They are easy to recognize by the horizontal line separating the two numbers - above and below. The top is called the numerator, the bottom is called the denominator. There is also a lower case spelling of 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 apply the "two-story" form of the record. Why? Yes, because it is more convenient. A little later we will verify this.

In addition to ordinary, there are also decimal fractions. It is very easy to distinguish between them: if in one case a horizontal or slash is used, then in the other - a comma separating sequences of numbers. Let's see an example: 2.9; 163.34; 1.953. We deliberately used the semicolon as a delimiter to delimit the numbers. The first of them will be read like this: "two whole, nine tenths."

New concepts

Let's go back to ordinary fractions. They are of two kinds.

The definition of a proper fraction is as follows: it is such a fraction, the numerator of which is less than the denominator. Why is it important? Now we'll see!

You have several apples cut into halves. In total - 5 parts. How do you say: you have "two and a half" or "five second" apples? Of course, the first option sounds more natural, and when talking with friends, we will use it. But if you need to calculate how much fruit 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 naming proper and improper fractions, the rule is as follows: if an integer part (14/5, 2/1, 173/16, 3/3) can be distinguished in a fraction, 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 they gave you one. The same cake was cut into eight pieces and given you two. Isn't it all the same? After all, ¼ and 2/8 are the same thing!

Reduction

Authors of problems and examples in math textbooks often try to confuse students by offering fractions that are cumbersome to write and can actually be reduced. Here is an example of a proper fraction: 167/334, which, it would seem, looks very "scary". But in fact, we can write it as ½. The number 334 is divisible by 167 without a remainder - having done 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 above, 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 reverse operation - for 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 them. 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) \u003d 7 * 15 / 8 * 14 \u003d 15/16.

Addition of fractions

What if you need to perform addition or if they have different numbers in the denominator? It will not work in the same way as with multiplication - here one 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 same numbers should appear at the bottom of both fractions.

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 bring the terms to? This must be the smallest multiple of both denominators: 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 for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and that's it! 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 we write the result of the addition in the first brackets 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 reduce the 37 in the numerator and denominator, and finally divide the top and bottom parts 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, and the answer contains only one digit. This often happens in mathematics. The main thing is not to be afraid and follow simple rules.

Common Mistakes

When exercising, the student can easily make one of the popular mistakes. Usually they occur due to inattention, and sometimes due to the fact that the studied material has not yet been properly deposited in the head.

Often the sum of the numbers in the numerator causes a desire to reduce its individual components. Suppose, in the example: (13 + 2) / 13, written without brackets (with a horizontal line), many students, due to inexperience, cross out 13 from above and below. 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 get the number 2 in the answer. But when adding, no operations with one of the terms are allowed, only with the entire sum.

Children often make mistakes when dividing fractions. Let's take two regular irreducible fractions and divide by each other: (5/6) / (25/33). The student can confuse and write the resulting expression as (5*25) / (6*33). But this would have happened with multiplication, and in our case everything will be a little different: (5 * 33) / (6 * 25). We reduce what is possible, and in the answer we will see 11/10. We write the resulting improper fraction as a decimal - 1.1.

Parentheses

Remember that in any mathematical expression, the order of operations is determined by the precedence of operation signs and the presence of brackets. 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.

It is the result of dividing one number by another. If they do not divide completely, it turns out 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 for various tricks. For example, they copy the numerators and denominators into the Paint editor and glue them together, drawing a horizontal line between them. Of course, there is a simpler option, which, by the way, also 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, on the side where the icons for closing and minimizing the window 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 the classic form. That is, separating the numerator and denominator with a horizontal line. You may even be surprised that such a proper fraction is so easy to write down.

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, fractions cannot be dispensed with. Soon you will learn to calculate everything in your mind, without even writing expressions on paper, but more and more complex examples will appear. Therefore, learn what a proper fraction is and how to work with it, keep up with the curriculum, do your homework on time, and then you will succeed.

At the word "fractions" many goosebumps run. Because I remember the school and the tasks that were solved in mathematics. This was a duty that had to be fulfilled. But what if we treat tasks containing proper and improper fractions as a puzzle? After all, many adults solve digital and Japanese crosswords. Understand the rules and that's it. 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 the brain.

What types of fractions are there?

Let's start with what it is. A fraction is a number that has some fraction of one. It can be written in two forms. The first is called ordinary. That is, one that has a horizontal or oblique stroke. It equates to the division sign.

In such a notation, the number above the dash is called the numerator, and below it is called the denominator.

Among ordinary fractions, right and wrong fractions are distinguished. For the former, the modulo numerator is always less than the denominator. The wrong ones are called that because they have the opposite. The value of a proper 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 an integer and a fractional part.

The second type of notation is decimal. About her separate conversation.

What is the difference between improper fractions and mixed numbers?

Basically, nothing. It's just a different notation of the same number. Improper fractions after simple operations easily become mixed numbers. And vice versa.

It all depends on the specific situation. Sometimes in tasks it is more convenient to use an improper fraction. 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 observation of the solver of the problem.

The mixed number is also compared with the sum of the integer part and the fractional part. Moreover, the second is always less than unity.

How to represent a mixed number as an improper fraction?

If you want to perform some 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 follow the following algorithm:

• multiply the denominator by the integer 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 \u003d (17 x 4 + 1): 4 \u003d 69/4;
• 39 ½ \u003d (39 x 2 + 1): 2 \u003d 79/2.

How to write an improper fraction as a mixed number?

The next method 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 the quotient in place of the integer 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, you get 3/7; the final answer is 5 whole 3/7.

108/54; after division, the quotient 2 is obtained without a remainder; this means that not all improper fractions can be represented as a mixed number; the answer is an integer - 2.

How do you turn an integer into an improper fraction?

There are situations when such action is necessary. To get improper fractions with a predetermined denominator, you will need to perform the following algorithm:

• multiply an integer by the desired denominator;
• write this value above the line;
• place a denominator below it.

The simplest option is when the denominator is equal to one. Then there is no need to multiply. It is enough just to write an integer, which is given in the example, and place a unit under the line.

Example: Make 5 an improper fraction with a denominator of 3. After multiplying 5 by 3, you get 15. This number will be the denominator. The answer to the task is a fraction: 15/3.

Two approaches to solving tasks with different numbers

In the example, it is required to calculate 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 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 multiplied by 11 becomes 143/55. And 14/11 after multiplying by 5 will take the form: 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, these same numbers are subtracted: 143 - 70 = 73. The answer is a fraction: 73/55.

When multiplying 13/5 and 14/11, you do not need to reduce to a common denominator. Just multiply the numerators and denominators in pairs. The answer will be: 182/55.

Likewise with division. For the correct solution, you need to replace division with multiplication and flip the divisor: 13/5: 14/11 \u003d 13/5 x 11/14 \u003d 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 integer and fractional parts separately. 2 + 1 = 3, 3/5 + 3/11 = 33/55 + 15/55 = 48/55. The final answer is 3 whole 48/55. In the first approach there was a fraction 213/55. You can check the 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, you need to convert it to a mixed number: 73 is divided by 55 and you get a quotient of 1 and a remainder of 18.

To find the product and the quotient, it is inconvenient to use mixed numbers. Here it is always recommended to switch to improper fractions.

They are divided into right and wrong.

Proper fractions

Proper fraction is an ordinary fraction whose numerator is less than the denominator.

To find out if a fraction is correct, you need to compare its terms with each other. The terms of the fraction are compared according to the rule for comparing natural numbers.

Example. Consider a fraction:

Example:

Translation rules and additional examples can be found in the topic Converting an improper fraction to a mixed number. You can also use the online calculator to convert an improper fraction to a mixed number.

Comparison of proper and improper fractions

Any improper ordinary fraction is greater than a proper one, since a proper fraction is always less than one, and an improper one is greater than or equal to one.

Example:

Comparison rules and additional examples can be found in the Comparison of ordinary fractions topic. Also to compare fractions or check the comparison you can use