Do you want to feel like a sapper? Then this lesson is for you! Because now we will study fractions - they are so simple and harmless mathematical objects which surpass the rest of the algebra course in their ability to "endure the brain".

The main danger of fractions is that they occur in real life. In this they differ, for example, from polynomials and logarithms, which can be passed and easily forgotten after the exam. Therefore, the material presented in this lesson, without exaggeration can be called explosive.

A numeric fraction (or simply a fraction) is a pair of integers written through a slash or horizontal bar.

Fractions written through a horizontal bar:

The same fractions written with a slash:
5/7; 9/(−30); 64/11; (−1)/4; 12/1.

Usually fractions are written through a horizontal line - it's easier to work with them, and they look better. The number written on top is called the numerator of the fraction, and the number written on the bottom is called the denominator.

Any whole number can be represented as a fraction with a denominator of 1. For example, 12 = 12/1 is the fraction from the above example.

In general, you can put any whole number in the numerator and denominator of a fraction. The only restriction is that the denominator must be different from zero. Remember the good old rule: “You can’t divide by zero!”

If the denominator is still zero, the fraction is called indefinite. Such a record does not make sense and cannot participate in calculations.

Basic property of a fraction

Fractions a /b and c /d are called equal if ad = bc.

From this definition it follows that the same fraction can be written in different ways. For example, 1/2 = 2/4 because 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4 because 1 4 ≠ 3 5.

A reasonable question arises: how to find all fractions equal to a given one? We give the answer in the form of a definition:

The main property of a fraction is that the numerator and denominator can be multiplied by the same number other than zero. This will result in a fraction equal to the given one.

This is very important property- remember it. With the help of the basic property of a fraction, many expressions can be simplified and shortened. In the future, it will constantly “emerge” in the form various properties and theorems.

Incorrect fractions. Selection of the whole part

If the numerator less than the denominator, such a fraction is called proper. Otherwise (that is, when the numerator is greater than or at least equal to the denominator), the fraction is called an improper fraction, and an integer part can be distinguished in it.

The integer part is written as a large number in front of the fraction and looks like this (marked in red):

To isolate the whole part in an improper fraction, you need to follow three simple steps:

1. Find how many times the denominator fits in the numerator. In other words, find the maximum integer that, when multiplied by the denominator, will still be less than the numerator (in the extreme case, equal). This number will be whole part, so we write it in front;
2. Multiply the denominator by the integer part found in the previous step, and subtract the result from the numerator. The resulting "stub" is called the remainder of the division, it will always be positive (in extreme cases, zero). We write it down in the numerator of the new fraction;
3. We rewrite the denominator unchanged.

Well, is it difficult? At first glance, it may be difficult. But it takes a little practice - and you will do it almost verbally. For now, take a look at the examples:

Task. Select the whole part in the given fractions:

In all examples, the integer part is highlighted in red, and the remainder of the division is in green.

Pay attention to the last fraction, where the remainder of the division turned out to be zero. It turns out that the numerator is completely divided by the denominator. This is quite logical, because 24: 6 \u003d 4 is a harsh fact from the multiplication table.

If everything is done correctly, the numerator of the new fraction will necessarily be less than the denominator, i.e. fraction becomes correct. I also note that it is better to highlight the whole part at the very end of the task, before writing the answer. Otherwise, you can significantly complicate the calculations.

Transition to improper fraction

There is also an inverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because improper fractions are much easier to work with.

The transition to an improper fraction is also done in three steps:

1. Multiply the integer part by the denominator. The result can be quite big numbers, but we should not be embarrassed;
2. Add the resulting number to the numerator of the original fraction. Write the result in the numerator of an improper fraction;
3. Rewrite the denominator - again, no change.

Here are specific examples:

Task. Convert to an improper fraction:

For clarity, the integer part is again highlighted in red, and the numerator of the original fraction is in green.

Consider the case when the numerator or denominator of a fraction contains a negative number. For example:

In principle, there is nothing criminal in this. However, working with such fractions can be inconvenient. Therefore, in mathematics it is customary to take out minuses as a fraction sign.

This is very easy to do if you remember the rules:

1. Plus times minus equals minus. Therefore, if the numerator is a negative number, and the denominator is positive (or vice versa), feel free to cross out the minus and put it in front of the whole fraction;
2. "Two negatives make an affirmative". When the minus is in both the numerator and the denominator, we simply cross them out - no additional action is required.

Of course, these rules can also be applied to reverse direction, i.e. you can add a minus under the fraction sign (most often - in the numerator).

We deliberately do not consider the case of “plus on plus” - with him, I think, everything is clear anyway. Let's take a look at how these rules work in practice:

Task. Take out the minuses of the four fractions written above.

Pay attention to the last fraction: it already has a minus sign in front of it. However, it is “burned” according to the rule “minus times minus gives plus”.

Also, do not move minuses in fractions with a highlighted integer part. These fractions are first converted to improper ones - and only then they begin to calculate.

We will begin our consideration of this topic by studying the concept of a fraction as a whole, which will give us a more complete understanding of the meaning of an ordinary fraction. Let's give the main terms and their definition, study the topic in a geometric interpretation, i.e. on the coordinate line, and also define a list of basic actions with fractions.

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Shares of the whole

Imagine an object consisting of several, completely equal parts. For example, it can be an orange, consisting of several identical slices.

Definition 1

Share of a whole or share is each of the equal parts that make up the whole object.

Obviously, the shares can be different. To clearly explain this statement, imagine two apples, one of which is cut into two equal parts, and the second into four. It is clear that the size of the resulting shares for different apples will vary.

The shares have their own names, which depend on the number of shares that make up the whole subject. If an item has two parts, then each of them will be defined as one second part of this item; when an object consists of three parts, then each of them is one-third, and so on.

Definition 2

Half- one second part of the subject.

Third- one third of the subject.

Quarter- one fourth of the subject.

To shorten the record, the following notation for shares was introduced: half - 1 2 or 1 / 2 ; third - 1 3 or 1 / 3 ; one fourth share 1 4 or 1/4 and so on. Entries with a horizontal bar are used more often.

The concept of a share naturally expands from objects to magnitudes. So, you can use fractions of a meter (one third or one hundredth) to measure small objects, as one of the units of length. Shares of other quantities can be applied in a similar way.

Common fractions, definition and examples

Common fractions used to describe the number of shares. Consider a simple example that will bring us closer to the definition of an ordinary fraction.

Imagine an orange, consisting of 12 slices. Each share will then be - one twelfth or 1 / 12. Two shares - 2/12; three shares - 3 / 12, etc. All 12 parts or an integer would look like this: 12 / 12 . Each of the entries used in the example is an example of a common fraction.

Definition 3

Common fraction is a record of the form m n or m / n , where m and n are any natural numbers.

According to this definition, examples of ordinary fractions can be entries: 4 / 9, 1134, 91754. And these entries: 11 5 , 1 , 9 4 , 3 are not ordinary fractions.

Numerator and denominator

Definition 4

numerator common fraction m n or m / n is a natural number m .

denominator common fraction m n or m / n is a natural number n .

Those. the numerator is the number above the bar of an ordinary fraction (or to the left of the slash), and the denominator is the number below the bar (to the right of the slash).

What is the meaning of the numerator and denominator? The denominator of an ordinary fraction indicates how many shares one item consists of, and the numerator gives us information about how many such shares are considered. For example, the common fraction 7 54 indicates to us that a certain object consists of 54 shares, and for consideration we took 7 such shares.

Natural number as a fraction with denominator 1

The denominator of a common fraction can be equal to one. In this case, it is possible to say that the object (value) under consideration is indivisible, is something whole. Numerator in like a fraction will indicate how many such items are taken, i.e. an ordinary fraction of the form m 1 makes sense natural number m . This statement serves as a justification for the equality m 1 = m .

Let's write the last equality like this: m = m 1 . It will give us the opportunity to use any natural number in the form of an ordinary fraction. For example, the number 74 is an ordinary fraction of the form 74 1 .

Definition 5

Any natural number m can be written as an ordinary fraction, where the denominator is one: m 1 .

In turn, any ordinary fraction of the form m 1 can be represented by a natural number m .

Fraction bar as division sign

The representation used above this subject how n shares is nothing but a division into n equal parts. When an object is divided into n parts, we have the opportunity to divide it equally between n people - everyone gets their share.

In the case when we initially have m identical items(each is divided into n parts), then these m items can be equally divided among n people, giving each of them one share from each of the m items. In this case, each person will have m shares 1 n , and m shares 1 n will give an ordinary fraction m n . Therefore, the common fraction m n can be used to represent the division of m items among n people.

The resulting statement establishes a connection between ordinary fractions and division. And this relationship can be expressed as follows : it is possible to mean the line of a fraction as a sign of division, i.e. m/n=m:n.

With the help of an ordinary fraction, we can write the result of dividing two natural numbers. For example, dividing 7 apples by 10 people will be written as 7 10: each person will get seven tenths.

Equal and unequal common fractions

The logical action is to compare ordinary fractions, because it is obvious that, for example, 1 8 of an apple is different from 7 8 .

The result of comparing ordinary fractions can be: equal or unequal.

Definition 6

Equal Common Fractions are ordinary fractions a b and c d , for which the equality is true: a d = b c .

Unequal common fractions- ordinary fractions a b and c d , for which the equality: a · d = b · c is not true.

Example equal fractions: 1 3 and 4 12 - since the equality 1 · 12 = 3 · 4 is fulfilled.

In the case when it turns out that fractions are not equal, it is usually also necessary to find out which of the given fractions is less and which is greater. To answer these questions, ordinary fractions are compared, leading them to common denominator and then comparing the numerators.

Fractional numbers

Each fraction is a record of a fractional number, which in fact is just a “shell”, visualization semantic load. But still, for convenience, we combine the concepts of a fraction and a fractional number, simply speaking - a fraction.

All fractional numbers, like any other number, have their own unique location on coordinate beam: there is a one-to-one correspondence between fractions and points of the coordinate ray.

In order to find a point on the coordinate ray, denoting the fraction m n , it is necessary to postpone m segments in the positive direction from the origin of coordinates, the length of each of which will be 1 n a fraction of a unit segment. Segments can be obtained by dividing a single segment into n identical parts.

As an example, let's denote the point M on the coordinate ray, which corresponds to the fraction 14 10 . The length of the segment, the ends of which is the point O and the nearest point marked with a small stroke, is equal to 1 10 fractions of the unit segment. The point corresponding to the fraction 14 10 is located at a distance from the origin of coordinates at a distance of 14 such segments.

If the fractions are equal, i.e. they correspond to the same fractional number, then these fractions serve as coordinates of the same point on the coordinate ray. For example, the coordinates in the form of equal fractions 1 3 , 2 6 , 3 9 , 5 15 , 11 33 correspond to the same point on the coordinate ray, located at a distance of a third of the unit segment, postponed from the origin in the positive direction.

The same principle works here as with integers: on a horizontal, right-directed coordinate ray, the point corresponding to a large fraction will be located to the right of the point corresponding to lesser fraction. And vice versa: the point, the coordinate of which is the smaller fraction, will be located to the left of the point, which corresponds to the larger coordinate.

Proper and improper fractions, definitions, examples

The division of fractions into proper and improper is based on the comparison of the numerator and denominator within the same fraction.

Definition 7

Proper fraction is an ordinary fraction in which the numerator is less than the denominator. That is, if the inequality m< n , то обыкновенная дробь m n является правильной.

Not proper fraction is a fraction whose numerator is greater than or equal to the denominator. That is, if the inequality undefined is true, then the ordinary fraction m n is improper.

Here are some examples: - proper fractions:

Example 1

5 / 9 , 3 67 , 138 514 ;

Improper fractions:

Example 2

13 / 13 , 57 3 , 901 112 , 16 7 .

It is also possible to give a definition of proper and improper fractions, based on the comparison of a fraction with a unit.

Definition 8

Proper fraction is an ordinary fraction less than one.

Improper fraction is a common fraction equal to or greater than one.

For example, the fraction 8 12 is correct, because 8 12< 1 . Дроби 53 2 и 14 14 являются неправильными, т.к. 53 2 >1 , and 14 14 = 1 .

Let's go a little deeper into thinking why fractions in which the numerator is greater than or equal to the denominator are called "improper".

Consider the improper fraction 8 8: it tells us that 8 parts of an object consisting of 8 parts are taken. Thus, from the available eight shares, we can compose a whole object, i.e. the given fraction 8 8 essentially represents the whole object: 8 8 \u003d 1. Fractions in which the numerator and denominator are equal fully replace the natural number 1.

Consider also fractions in which the numerator exceeds the denominator: 11 5 and 36 3 . It is clear that the fraction 11 5 indicates that we can make two whole objects out of it and there will still be one fifth of it. Those. fraction 11 5 is 2 objects and another 1 5 from it. In turn, 36 3 is a fraction, which essentially means 12 whole objects.

These examples make it possible to conclude that improper fractions can be replaced with natural numbers (if the numerator is divisible by the denominator without a remainder: 8 8 \u003d 1; 36 3 \u003d 12) or the sum of a natural number and a proper fraction (if the numerator is not divisible by the denominator without a remainder: 11 5 = 2 + 1 5). This is probably why such fractions are called "improper".

Here, too, we encounter one of the most important number skills.

Definition 9

Extracting the integer part from an improper fraction is an improper fraction written as the sum of a natural number and a proper fraction.

Also note that there is strong relationship between improper fractions and mixed numbers.

Positive and negative fractions

Above we said that each ordinary fraction corresponds to a positive fractional number. Those. ordinary fractions are positive fractions. For example, fractions 5 17 , 6 98 , 64 79 are positive, and when it is necessary to emphasize the “positiveness” of a fraction, it is written using a plus sign: + 5 17 , + 6 98 , + 64 79 .

If we assign a minus sign to an ordinary fraction, then the resulting record will be a record of a negative fractional number, and in this case we are talking about negative fractions. For example, - 8 17 , - 78 14 etc.

Positive and negative fractions m n and - m n are opposite numbers. For example, the fractions 7 8 and - 7 8 are opposite.

Positive fractions, like any positive numbers in general, they mean an addition, a change in the direction of increase. In turn, negative fractions correspond to consumption, a change in the direction of decrease.

If we consider the coordinate line, we will see that negative fractions are located to the left of the reference point. The points to which the fractions correspond, which are opposite (m n and - m n), are located at the same distance from the origin of the coordinates O, but along different sides from her.

Here we also separately talk about fractions written in the form 0 n . Such a fraction is equal to zero, i.e. 0 n = 0 .

Summarizing all of the above, we come to the most important concept rational numbers.

Definition 10

Rational numbers is a set of positive fractions, negative fractions and fractions of the form 0 n .

Actions with fractions

Let's list the basic operations with fractions. In general, their essence is the same as the corresponding operations with natural numbers

1. Fraction Comparison - this action we reviewed above.
2. Addition of fractions - the result of adding ordinary fractions is an ordinary fraction (in a particular case, reduced to a natural number).
3. Subtraction of fractions is an action, the opposite of addition, when one known fraction and given amount fractions is determined by the unknown fraction.
4. Multiplication of fractions - this action can be described as finding a fraction from a fraction. The result of multiplying two ordinary fractions is an ordinary fraction (in a particular case, equal to a natural number).
5. Division of fractions - action, reciprocal of multiplication, when we determine the fraction by which it is necessary to multiply the given one in order to get famous work two fractions.

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The numerator, and that by which it is divided is the denominator.

To write a fraction, first write its numerator, then draw a horizontal line under this number, and write the denominator under the line. The horizontal line separating the numerator and denominator is called a fractional bar. Sometimes it is depicted as an oblique "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction "two-thirds" will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3, you can find: ⅔.

To calculate the product of fractions, first multiply the numerator of one fractions to another numerator. Write the result to the numerator of the new fractions. Then multiply the denominators as well. Specify the final value in the new fractions. For example, 1/3? 1/5 = 1/15 (1 × 1 = 1; 3 × 5 = 15).

To divide one fraction by another, first multiply the numerator of the first by the denominator of the second. Do the same with the second fraction (divisor). Or, before performing all the steps, first “flip” the divisor, if it’s more convenient for you: the denominator should be in place of the numerator. Then multiply the denominator of the dividend by the new denominator of the divisor and multiply the numerators. For example, 1/3: 1/5 = 5/3 = 1 2/3 (1 × 5 = 5; 3 × 1 = 3).

Sources:

• Basic tasks for fractions

Fractional numbers allow you to express in different form exact value quantities. With fractions, you can perform the same mathematical operations as with integers: subtraction, addition, multiplication, and division. To learn how to decide fractions, it is necessary to remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations after execution, they require reduction of the fractional part of the result.

You will need

• - calculator

Instruction

Look carefully at the numbers. If there are decimals and irregulars among the fractions, it is sometimes more convenient to first perform actions with decimals, and then convert them to the wrong form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part stands out, lead to the wrong form by multiplying it by the denominator and adding the numerator to the result. Given values will become the new numerator fractions. To extract the whole part from the initially incorrect fractions, divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division becomes the new numerator, the denominator fractions while not changing. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation separately of integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 + (8/12 + 9/12) = 3 + 17/12 = 3 + 1 5/12 = 4 5 /12.

Rewrite them with a ":" separator and continue ordinary division.

To receive end result reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integer numbers above and below the line.

note

Don't do arithmetic with fractions that have different denominators. Choose a number such that when the numerator and denominator of each fraction are multiplied by it, as a result, the denominators of both fractions are equal.

Helpful advice

When writing fractional numbers, the dividend is written above the line. This quantity is referred to as the numerator of a fraction. Under the line, the divisor, or denominator, of the fraction is written. For example, one and a half kilograms of rice in the form of a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, it is called a decimal fraction. In this case, the numerator (dividend) is written to the right of the whole part separated by a comma: 1.5 kg of rice. For the convenience of calculations, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the numerator and denominator values ​​by dividing them by a single whole number. AT this example dividing by 2 is possible. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to do arithmetic with are in the same form.

Speaking of mathematics, one cannot help but remember fractions. Their study is given a lot of attention and time. Remember how many examples you had to solve in order to learn certain rules for working with fractions, how you memorized and applied the main property of a fraction. How many nerves were spent to find a common denominator, especially if there were more than two terms in the examples!

Let's remember what it is, and refresh our memory a little about the basic information and rules for working with fractions.

Definition of fractions

Let's start with the most important thing - definitions. A fraction is a number that consists of one or more unit parts. A fractional number is written as two numbers separated by a horizontal or slash. In this case, the upper (or first) is called the numerator, and the lower (second) is called the denominator.

It is worth noting that the denominator shows how many parts the unit is divided into, and the numerator shows the number of shares or parts taken. Often fractions, if they are correct, are less than one.

Now let's look at the properties of these numbers and the basic rules that are used when working with them. But before we analyze such a thing as "basic property rational fraction Let's talk about the types of fractions and their features.

What are fractions

There are several types of such numbers. First of all, these are ordinary and decimal. The first are the type of record already indicated by us using a horizontal or slash. The second type of fractions is indicated using the so-called positional notation, when the integer part of the number is indicated first, and then, after the decimal point, the fractional part is indicated.

It is worth noting here that in mathematics both decimal and ordinary fractions are used equally. The main property of the fraction is valid only for the second option. In addition, in ordinary fractions, correct and wrong numbers. For the former, the numerator is always less than the denominator. Note also that such a fraction is less than unity. In an improper fraction, on the contrary, the numerator is greater than the denominator, and it itself is greater than one. In this case, an integer can be extracted from it. In this article, we will consider only ordinary fractions.

Fraction properties

Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers are no exception. They have one important feature, with the help of which it is possible to carry out certain operations on them. What is the main property of a fraction? The rule says that if its numerator and denominator are multiplied or divided by the same rational number, we get a new fraction, the value of which will be equal to the value of the original. That is, multiplying the two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, while they will be equal.

Based on this property, you can reduce fractions, as well as select common denominators for a particular pair of numbers.

Operations

Although fractions seem more complex to us, they can also perform basic mathematical operations, such as addition and subtraction, multiplication and division. In addition, there is such a specific action as the reduction of fractions. Naturally, each of these actions is performed according to certain rules. Knowing these laws makes it easier to work with fractions, making it easier and more interesting. That is why further we will consider the basic rules and the algorithm of actions when working with such numbers.

But before we talk about such mathematical operations as addition and subtraction, we will analyze such an operation as reduction to a common denominator. This is where the knowledge of what basic property of a fraction exists will come in handy.

Common denominator

In order to reduce a number to a common denominator, you first need to find the least common multiple of the two denominators. I.e smallest number, which is simultaneously divisible by both denominators without a remainder. The easiest way to find the LCM (least common multiple) is to write in a line for one denominator, then for the second and find a matching number among them. In the event that the LCM is not found, that is, these numbers do not have a common multiple, they should be multiplied, and the resulting value should be considered as the LCM.

So, we have found the NOC, now we must find additional multiplier. To do this, you need to alternately divide the LCM into denominators of fractions and write down the resulting number over each of them. Next, multiply the numerator and denominator by the resulting additional factor and write the results as a new fraction. If you doubt that the number you received is equal to the previous one, remember the main property of the fraction.

Addition

Now let's go directly to mathematical operations on fractional numbers. Let's start with the simplest. There are several options for adding fractions. In the first case, both numbers have the same denominator. In this case, it remains only to add the numerators together. But the denominator does not change. For example, 1/5 + 3/5 = 4/5.

If the fractions different denominators, you should reduce them to a common one and only then perform addition. How to do this, we have discussed with you a little higher. In this situation, the main property of the fraction will come in handy. The rule will allow you to bring the numbers to a common denominator. The value will not change in any way.

Alternatively, it may happen that the fraction is mixed. Then you should first add together the whole parts, and then the fractional ones.

Multiplication

It does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of the fraction. It is enough to first multiply the numerators and denominators together. In this case, the product of the numerators will become the new numerator, and the product of the denominators will become the new denominator. As you can see, nothing complicated.

The only thing that is required of you is knowledge of the multiplication table, as well as attentiveness. In addition, after receiving the result, it is imperative to check whether it is possible to reduce given number or not. We will talk about how to reduce fractions a little later.

Subtraction

Performing should be guided by the same rules as when adding. So, in numbers with same denominator it is enough to subtract the numerator of the subtrahend from the numerator of the minuend. In the event that fractions have different denominators, you should bring them to a common one and then execute this operation. As with the analogous addition case, you will need to use the main property algebraic fraction, as well as skills in finding the NOC and common divisors for fractions.

Division

And the last, most interesting operation when working with such numbers is division. It is quite simple and does not cause any particular difficulties even for those who do not understand how to work with fractions, especially to perform addition and subtraction operations. When dividing, the rule is to multiply by reciprocal. The main property of a fraction, as in the case of multiplication, will not be used for this operation. Let's take a closer look.

When dividing numbers, the dividend remains unchanged. The divisor is reversed, i.e. the numerator and denominator are reversed. After that, the numbers are multiplied with each other.

Reduction

So, we have already examined the definition and structure of fractions, their types, the rules of operations on given numbers, and found out the main property of an algebraic fraction. Now let's talk about such an operation as reduction. Reducing a fraction is the process of converting it - dividing the numerator and denominator by the same number. Thus, the fraction is reduced without changing its properties.

Usually when making mathematical operation you should carefully look at the final result and find out whether it is possible to reduce the resulting fraction or not. Remember that in final result a fractional number that does not require reduction is always written.

Other operations

Finally, we note that we have listed far from all operations on fractional numbers, mentioning only the most famous and necessary. Fractions can also be compared, converted to decimals, and vice versa. But in this article we did not consider these operations, since in mathematics they are carried out much less frequently than those that we have given above.

findings

We have talked about fractional numbers and transactions with them. We also analyzed the main property. But we note that all these issues were considered by us in passing. We have given only the most well-known and used rules, we have given the most important, in our opinion, advice.

This article is intended to refresh the information you have forgotten about fractions rather than to give new information and hit your head endless rules and formulas that, most likely, you will never need.

We hope that the material presented in the article simply and concisely has become useful to you.