Fractions of a unit and is represented as \frac(a)(b).

Numerator of fraction (a)- the number located above the fraction line and showing the number of shares into which the unit was divided.

Fraction denominator (b)- the number located under the line of the fraction and showing how many parts the unit is divided into.

Hide Show

The main property of a fraction

If ad=bc then two fractions \frac(a)(b) And \frac(c)(d) are considered equal. For example, the fractions will be equal \frac35 And \frac(9)(15), since 3 \cdot 15 = 15 \cdot 9 , \frac(12)(7) And \frac(24)(14), since 12 \cdot 14 = 7 \cdot 24 .

From the definition of equality of fractions it follows that the fractions will be equal \frac(a)(b) And \frac(am)(bm), since a(bm)=b(am) is a clear example of the use of the combinative and commutative properties of multiplication natural numbers In action.

Means \frac(a)(b) = \frac(am)(bm)- this is what it looks like main property of a fraction.

In other words, we get a fraction equal to the given one by multiplying or dividing the numerator and denominator of the original fraction by the same natural number.

Reducing a fraction is the process of replacing a fraction in which the new fraction is equal to the original one, but with a smaller numerator and denominator.

It is customary to reduce fractions based on the basic property of the fraction.

For example, \frac(45)(60)=\frac(15)(20)(numerator and denominator are divided by the number 3); the resulting fraction can again be reduced by dividing by 5, that is \frac(15)(20)=\frac 34.

Irreducible fraction is a fraction of the form \frac 34, where the numerator and denominator are mutual prime numbers. The main purpose of reducing a fraction is to make the fraction irreducible.

Reducing fractions to a common denominator

Let's take two fractions as an example: \frac(2)(3) And \frac(5)(8) with different denominators 3 and 8. In order to bring these fractions to a common denominator, we first multiply the numerator and denominator of the fraction \frac(2)(3) by 8. We get the following result: \frac(2 \cdot 8)(3 \cdot 8) = \frac(16)(24). Then we multiply the numerator and denominator of the fraction \frac(5)(8) by 3. As a result we get: \frac(5 \cdot 3)(8 \cdot 3) = \frac(15)(24). So, the original fractions are reduced to a common denominator 24.

Arithmetic operations on ordinary fractions

Addition of ordinary fractions

a) If the denominators are the same, the numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As you can see in the example:

\frac(a)(b)+\frac(c)(b)=\frac(a+c)(b);

b) For different denominators, fractions are first reduced to a common denominator, and then the numerators are added according to rule a):

\frac(7)(3)+\frac(1)(4)=\frac(7 \cdot 4)(3)+\frac(1 \cdot 3)(4)=\frac(28)(12) +\frac(3)(12)=\frac(31)(12).

Subtracting fractions

a) If the denominators are the same, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same:

\frac(a)(b)-\frac(c)(b)=\frac(a-c)(b);

b) If the denominators of the fractions are different, then first the fractions are brought to a common denominator, and then the actions are repeated as in point a).

Multiplying common fractions

Multiplying fractions obeys the following rule:

\frac(a)(b) \cdot \frac(c)(d)=\frac(a \cdot c)(b \cdot d),

that is, they multiply the numerators and denominators separately.

For example:

\frac(3)(5) \cdot \frac(4)(8) = \frac(3 \cdot 4)(5 \cdot 8)=\frac(12)(40).

Dividing fractions

Fractions are divided in the following way:

\frac(a)(b) : \frac(c)(d)= \frac(ad)(bc),

that is, a fraction \frac(a)(b) multiplied by a fraction \frac(d)(c).

Example: \frac(7)(2) : \frac(1)(8)=\frac(7)(2) \cdot \frac(8)(1)=\frac(7 \cdot 8)(2 \cdot 1 )=\frac(56)(2).

Reciprocal numbers

If ab=1 , then the number b is reciprocal number for the number a.

Example: for the number 9 the reciprocal is \frac(1)(9), because 9\cdot\frac(1)(9)=1, for the number 5 - \frac(1)(5), because 5\cdot\frac(1)(5)=1.

Decimals

Decimal called a proper fraction whose denominator is 10, 1000, 10\,000, ..., 10^n.

For example: \frac(6)(10)=0.6;\enspace \frac(44)(1000)=0.044.

Irregular numbers with a denominator of 10^n or mixed numbers are written in the same way.

For example: 5\frac(1)(10)=5.1;\enspace \frac(763)(100)=7\frac(63)(100)=7.63.

Any number can be represented as a decimal fraction common fraction with a denominator that is a divisor of a certain power of 10.

Example: 5 is a divisor of 100, so it is a fraction \frac(1)(5)=\frac(1 \cdot 20)(5 \cdot 20)=\frac(20)(100)=0.2.

Arithmetic operations on decimals

Adding Decimals

To add two decimal fractions, you need to arrange them so that there are identical digits under each other and a comma under the comma, and then add the fractions like ordinary numbers.

Subtracting Decimals

It is performed in the same way as addition.

Multiplying Decimals

When multiplying decimal numbers It is enough to multiply the given numbers, not paying attention to commas (like natural numbers), and in the resulting answer, a comma on the right separates as many digits as there are after the decimal point in both factors in total.

Let's multiply 2.7 by 1.3. We have 27 \cdot 13=351 . We separate two digits on the right with a comma (the first and second numbers have one digit after the decimal point; 1+1=2). As a result, we get 2.7 \cdot 1.3=3.51.

If the resulting result contains fewer digits than need to be separated by a comma, then the missing zeros are written in front, for example:

To multiply by 10, 100, 1000, you need to move the decimal point 1, 2, 3 digits to the right (if necessary, a certain number of zeros are assigned to the right).

For example: 1.47\cdot 10\,000 = 14,700.

Decimal division

Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number. The comma in the quotient is placed after the division of the whole part is completed.

If the integer part of the dividend less than divisor, then the answer turns out to be zero integers, for example:

Let's look at dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. First of all, let's multiply the dividend and divisor of the fraction by 100, that is, move the decimal point to the right in the dividend and divisor by as many digits as there are in the divisor after the decimal point (in this example, two). Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:

It happens that the final result is not always obtained decimal when dividing one number by another. The result is an infinite decimal fraction. In such cases, we move on to ordinary fractions.

2.8: 0.09= \frac(28)(10) : \frac (9)(100)= \frac(28 \cdot 100)(10 \cdot 9)=\frac(280)(9)= 31\frac(1)(9).

Lesson content

Adding fractions with like denominators

There are two types of addition of fractions:

1. Adding fractions with like denominators
2. Adding fractions with different denominators

First, let's learn the addition of fractions with like denominators. Everything is simple here. To add fractions with the same denominators, you need to add their numerators and leave the denominator unchanged. For example, let's add the fractions and . Add the numerators and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into four parts. If you add pizza to pizza, you get pizza:

Example 2. Add fractions and .

The answer turned out to be an improper fraction. When the end of the task comes, it is customary to get rid of improper fractions. To get rid of an improper fraction, you need to select the whole part of it. In our case, the whole part is easily isolated - two divided by two equals one:

This example can be easily understood if we remember about a pizza that is divided into two parts. If you add more pizza to the pizza, you get one whole pizza:

Example 3. Add fractions and .

Again, we add up the numerators and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into three parts. If you add more pizza to the pizza, you get pizza:

Example 4. Find the value of an expression

This example is solved in exactly the same way as the previous ones. The numerators must be added and the denominator left unchanged:

Let's try to depict our solution using a drawing. If you add pizzas to a pizza and add more pizzas, you get 1 whole pizza and more pizzas.

As you can see, there is nothing complicated about adding fractions with the same denominators. It is enough to understand the following rules:

1. To add fractions with the same denominator, you need to add their numerators and leave the denominator unchanged;

Adding fractions with different denominators

Now let's learn how to add fractions with different denominators. When adding fractions, the denominators of the fractions must be the same. But they are not always the same.

For example, fractions can be added because they have same denominators.

But fractions cannot be added right away, since these fractions different denominators. In such cases, fractions must be reduced to the same (common) denominator.

There are several ways to reduce fractions to the same denominator. Today we will look at only one of them, since the other methods may seem complicated for a beginner.

The essence of this method is that first the LCM of the denominators of both fractions is searched. The LCM is then divided by the denominator of the first fraction to obtain the first additional factor. They do the same with the second fraction - the LCM is divided by the denominator of the second fraction and a second additional factor is obtained.

The numerators and denominators of the fractions are then multiplied by their additional factors. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.

Example 1. Let's add the fractions and

First of all, we find the least common multiple of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 2. The least common multiple of these numbers is 6

LCM (2 and 3) = 6

Now let's return to fractions and . First, divide the LCM by the denominator of the first fraction and get the first additional factor. LCM is the number 6, and the denominator of the first fraction is the number 3. Divide 6 by 3, we get 2.

The resulting number 2 is the first additional multiplier. We write it down to the first fraction. To do this, make a small oblique line over the fraction and write down the additional factor found above it:

We do the same with the second fraction. We divide the LCM by the denominator of the second fraction and get the second additional factor. LCM is the number 6, and the denominator of the second fraction is the number 2. Divide 6 by 2, we get 3.

The resulting number 3 is the second additional multiplier. We write it down to the second fraction. Again, we make a small oblique line over the second fraction and write down the additional factor found above it:

Now we have everything ready for addition. It remains to multiply the numerators and denominators of the fractions by their additional factors:

Look carefully at what we have come to. We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to add such fractions. Let's take this example to the end:

This completes the example. It turns out to add .

Let's try to depict our solution using a drawing. If you add pizza to a pizza, you get one whole pizza and another sixth of a pizza:

Reducing fractions to the same (common) denominator can also be depicted using a picture. Reducing the fractions and to a common denominator, we got the fractions and . These two fractions will be represented by the same pieces of pizza. The only difference will be that this time they will be divided into equal shares (reduced to the same denominator).

The first drawing represents a fraction (four pieces out of six), and the second drawing represents a fraction (three pieces out of six). Adding these pieces we get (seven pieces out of six). This fraction is improper, so we highlighted the whole part of it. As a result, we got (one whole pizza and another sixth pizza).

Please note that we have described this example too detailed. IN educational institutions It’s not customary to write in such detail. You need to be able to quickly find the LCM of both denominators and additional factors to them, as well as quickly multiply the found additional factors by your numerators and denominators. If we were at school, we would have to write this example as follows:

But there is also back side medals. If you do not take detailed notes in the first stages of studying mathematics, then questions of the sort begin to appear. “Where does that number come from?”, “Why do fractions suddenly turn into completely different fractions? «.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

1. Find the LCM of the denominators of fractions;
2. Divide the LCM by the denominator of each fraction and obtain an additional factor for each fraction;
3. Multiply the numerators and denominators of fractions by their additional factors;
4. Add fractions that have the same denominators;
5. If the answer turns out to be an improper fraction, then select its whole part;

Example 2. Find the value of an expression .

Let's use the instructions given above.

Step 1. Find the LCM of the denominators of the fractions

Find the LCM of the denominators of both fractions. The denominators of fractions are the numbers 2, 3 and 4

Step 2. Divide the LCM by the denominator of each fraction and get an additional factor for each fraction

Divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 2. Divide 12 by 2, we get 6. We got the first additional factor 6. We write it above the first fraction:

Now we divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 3. Divide 12 by 3, we get 4. We get the second additional factor 4. We write it above the second fraction:

Now we divide the LCM by the denominator of the third fraction. LCM is the number 12, and the denominator of the third fraction is the number 4. Divide 12 by 4, we get 3. We get the third additional factor 3. We write it above the third fraction:

Step 3. Multiply the numerators and denominators of the fractions by their additional factors

We multiply the numerators and denominators by their additional factors:

Step 4. Add fractions with the same denominators

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. All that remains is to add these fractions. Add it up:

The addition didn't fit on one line, so we moved the remaining expression to the next line. This is allowed in mathematics. When an expression does not fit on one line, it is moved to the next line, and it is necessary to put an equal sign (=) at the end of the first line and at the beginning of the new line. The equal sign on the second line indicates that this is a continuation of the expression that was on the first line.

Step 5. If the answer turns out to be an improper fraction, then select the whole part of it

Our answer turned out to be an improper fraction. We have to highlight a whole part of it. We highlight:

We received an answer

Subtracting fractions with like denominators

There are two types of subtraction of fractions:

1. Subtracting fractions with like denominators
2. Subtracting fractions with different denominators

First, let's learn how to subtract fractions with like denominators. Everything is simple here. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, but leave the denominator the same.

For example, let's find the value of the expression . To solve this example, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged. Let's do this:

This example can be easily understood if we remember the pizza, which is divided into four parts. If you cut pizzas from a pizza, you get pizzas:

Example 2. Find the value of the expression.

Again, from the numerator of the first fraction, subtract the numerator of the second fraction, and leave the denominator unchanged:

This example can be easily understood if we remember the pizza, which is divided into three parts. If you cut pizzas from a pizza, you get pizzas:

Example 3. Find the value of an expression

This example is solved in exactly the same way as the previous ones. From the numerator of the first fraction you need to subtract the numerators of the remaining fractions:

As you can see, there is nothing complicated about subtracting fractions with the same denominators. It is enough to understand the following rules:

1. To subtract another from one fraction, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator unchanged;
2. If the answer turns out to be an improper fraction, then you need to highlight the whole part of it.

Subtracting fractions with different denominators

For example, you can subtract a fraction from a fraction because the fractions have the same denominators. But you cannot subtract a fraction from a fraction, since these fractions have different denominators. In such cases, fractions must be reduced to the same (common) denominator.

The common denominator is found using the same principle that we used when adding fractions with different denominators. First of all, find the LCM of the denominators of both fractions. Then the LCM is divided by the denominator of the first fraction and the first additional factor is obtained, which is written above the first fraction. Similarly, the LCM is divided by the denominator of the second fraction and a second additional factor is obtained, which is written above the second fraction.

The fractions are then multiplied by their additional factors. As a result of these operations, fractions that had different denominators are converted into fractions that have the same denominators. And we already know how to subtract such fractions.

Example 1. Find the meaning of the expression:

These fractions have different denominators, so you need to reduce them to the same (common) denominator.

First we find the LCM of the denominators of both fractions. The denominator of the first fraction is the number 3, and the denominator of the second fraction is the number 4. The least common multiple of these numbers is 12

LCM (3 and 4) = 12

Now let's return to fractions and

Let's find an additional factor for the first fraction. To do this, divide the LCM by the denominator of the first fraction. LCM is the number 12, and the denominator of the first fraction is the number 3. Divide 12 by 3, we get 4. Write a four above the first fraction:

We do the same with the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 12, and the denominator of the second fraction is the number 4. Divide 12 by 4, we get 3. Write a three over the second fraction:

Now we are ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same denominators. And we already know how to subtract such fractions. Let's take this example to the end:

We received an answer

Let's try to depict our solution using a drawing. If you cut pizza from a pizza, you get pizza

This is the detailed version of the solution. If we were at school, we would have to solve this example shorter. Such a solution would look like this:

Reducing fractions to a common denominator can also be depicted using a picture. Reducing these fractions to a common denominator, we got the fractions and . These fractions will be represented by the same pizza slices, but this time they will be divided into equal shares (reduced to the same denominator):

The first picture shows a fraction (eight pieces out of twelve), and the second picture shows a fraction (three pieces out of twelve). By cutting three pieces from eight pieces, we get five pieces out of twelve. The fraction describes these five pieces.

Example 2. Find the value of an expression

These fractions have different denominators, so first you need to reduce them to the same (common) denominator.

Let's find the LCM of the denominators of these fractions.

The denominators of the fractions are the numbers 10, 3 and 5. The least common multiple of these numbers is 30

LCM(10, 3, 5) = 30

Now we find additional factors for each fraction. To do this, divide the LCM by the denominator of each fraction.

Let's find an additional factor for the first fraction. LCM is the number 30, and the denominator of the first fraction is the number 10. Divide 30 by 10, we get the first additional factor 3. We write it above the first fraction:

Now we find an additional factor for the second fraction. Divide the LCM by the denominator of the second fraction. LCM is the number 30, and the denominator of the second fraction is the number 3. Divide 30 by 3, we get the second additional factor 10. We write it above the second fraction:

Now we find an additional factor for the third fraction. Divide the LCM by the denominator of the third fraction. LCM is the number 30, and the denominator of the third fraction is the number 5. Divide 30 by 5, we get the third additional factor 6. We write it above the third fraction:

Now everything is ready for subtraction. It remains to multiply the fractions by their additional factors:

We came to the conclusion that fractions that had different denominators turned into fractions that had the same (common) denominators. And we already know how to subtract such fractions. Let's finish this example.

The continuation of the example will not fit on one line, so we move the continuation to the next line. Don't forget about the equal sign (=) on the new line:

The answer turned out to be a regular fraction, and everything seems to suit us, but it is too cumbersome and ugly. We should make it simpler. What can be done? You can shorten this fraction.

To reduce a fraction, you need to divide its numerator and denominator by (GCD) of the numbers 20 and 30.

So, we find the gcd of numbers 20 and 30:

Now we return to our example and divide the numerator and denominator of the fraction by the found gcd, that is, by 10

We received an answer

Multiplying a fraction by a number

To multiply a fraction by a number, you need to multiply the numerator of the given fraction by that number and leave the denominator the same.

Example 1. Multiply a fraction by the number 1.

Multiply the numerator of the fraction by the number 1

The recording can be understood as taking half 1 time. For example, if you take pizza once, you get pizza

From the laws of multiplication we know that if the multiplicand and the factor are swapped, the product will not change. If the expression is written as , then the product will still be equal to . Again, the rule for multiplying a whole number and a fraction works:

This notation can be understood as taking half of one. For example, if there is 1 whole pizza and we take half of it, then we will have pizza:

Example 2. Find the value of an expression

Multiply the numerator of the fraction by 4

The answer was an improper fraction. Let's highlight the whole part of it:

The expression can be understood as taking two quarters 4 times. For example, if you take 4 pizzas, you will get two whole pizzas

And if we swap the multiplicand and the multiplier, we get the expression . It will also be equal to 2. This expression can be understood as taking two pizzas from four whole pizzas:

Multiplying fractions

To multiply fractions, you need to multiply their numerators and denominators. If the answer turns out to be an improper fraction, you need to highlight the whole part of it.

Example 1. Find the value of the expression.

We received an answer. It is advisable to reduce this fraction. The fraction can be reduced by 2. Then the final solution will take the following form:

The expression can be understood as taking a pizza from half a pizza. Let's say we have half a pizza:

How to take two thirds from this half? First you need to divide this half into three equal parts:

And take two from these three pieces:

We'll make pizza. Remember what pizza looks like when divided into three parts:

One piece of this pizza and the two pieces we took will have the same dimensions:

In other words, we're talking about about the same size pizza. Therefore the value of the expression is

Example 2. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer was an improper fraction. Let's highlight the whole part of it:

Example 3. Find the value of an expression

Multiply the numerator of the first fraction by the numerator of the second fraction, and the denominator of the first fraction by the denominator of the second fraction:

The answer turned out to be a regular fraction, but it would be good if it was shortened. To reduce this fraction, you need to divide the numerator and denominator of this fraction by the greatest common divisor (GCD) of the numbers 105 and 450.

So, let’s find the gcd of numbers 105 and 450:

Now we divide the numerator and denominator of our answer by the gcd that we have now found, that is, by 15

Representing a whole number as a fraction

Any whole number can be represented as a fraction. For example, the number 5 can be represented as . This will not change the meaning of five, since the expression means “the number five divided by one,” and this, as we know, is equal to five:

Reciprocal numbers

Now we will get acquainted with very interesting topic in mathematics. It's called "reverse numbers".

Definition. Reverse to numbera is a number that, when multiplied bya gives one.

Let's substitute in this definition instead of the variable a number 5 and try to read the definition:

Reverse to number 5 is a number that, when multiplied by 5 gives one.

Is it possible to find a number that, when multiplied by 5, gives one? It turns out it is possible. Let's imagine five as a fraction:

Then multiply this fraction by itself, just swap the numerator and denominator. In other words, let’s multiply the fraction by itself, only upside down:

What will happen as a result of this? If we continue to solve this example, we get one:

This means that the inverse of the number 5 is the number , since when you multiply 5 by you get one.

The reciprocal of a number can also be found for any other integer.

You can also find the reciprocal of any other fraction. To do this, just turn it over.

Dividing a fraction by a number

Let's say we have half a pizza:

Let's divide it equally between two. How much pizza will each person get?

It can be seen that after dividing half the pizza, two equal pieces were obtained, each of which constitutes a pizza. So everyone gets a pizza.

Division of fractions is done using reciprocals. Reciprocal numbers allow you to replace division with multiplication.

To divide a fraction by a number, you need to multiply the fraction by the inverse of the divisor.

Using this rule, we will write down the division of our half of the pizza into two parts.

So, you need to divide the fraction by the number 2. Here the dividend is the fraction and the divisor is the number 2.

To divide a fraction by the number 2, you need to multiply this fraction by the reciprocal of the divisor 2. The reciprocal of the divisor 2 is the fraction. So you need to multiply by

Definition of a common fraction

Definition 1

Common fractions are used to describe the number of parts. Let's look at an example that can be used to define a common fraction.

The apple was divided into $8$ shares. In this case, each share represents one-eighth of a whole apple, i.e. $\frac(1)(8)$. Two shares are denoted by $\frac(2)(8)$, three shares by $\frac(3)(8)$, etc., and $8$ shares by $\frac(8)(8)$ . Each of the entries presented is called ordinary fraction.

Let's give general definition ordinary fraction.

Definition 2

Common fraction is called a notation of the form $\frac(m)(n)$, where $m$ and $n$ are any natural numbers.

You can often find the following notation for a common fraction: $m/n$.

Example 1

Examples of common fractions:

\[(3)/(4), \frac(101)(345),\ \ (23)/(5), \frac(15)(15), (111)/(81).\]

Note 1

Numbers $\frac(\sqrt(2))(3)$, $-\frac(13)(37)$, $\frac(4)(\frac(2)(7))$, $\frac( 2,4)(8,3)$ are not ordinary fractions, because do not fit the above definition.

Numerator and denominator

A common fraction consists of a numerator and a denominator.

Definition 3

Numerator An ordinary fraction $\frac(m)(n)$ is a natural number $m$, which shows the number of equal parts taken from a single whole.

Definition 4

Denominator An ordinary fraction $\frac(m)(n)$ is a natural number $n$, which shows how many equal parts the whole whole is divided into.

Picture 1.

The numerator is located above the fraction line, and the denominator is located below the fraction line. For example, the numerator of the common fraction $\frac(5)(17)$ is the number $5$, and the denominator is the number $17$. The denominator shows that the item is divided into $17$ shares, and the numerator shows that $5$ such shares were taken.

Natural number as a fraction with denominator 1

The denominator of a common fraction can be one. In this case, the object is considered to be indivisible, i.e. represents a single whole. The numerator of such a fraction shows how many whole objects are taken. An ordinary fraction of the form $\frac(m)(1)$ has the meaning of a natural number $m$. Thus, we obtain the well-founded equality $\frac(m)(1)=m$.

If we rewrite the equality in the form $m=\frac(m)(1)$, then it will make it possible to represent any natural number $m$ as an ordinary fraction. For example, the number $5$ can be represented as a fraction $\frac(5)(1)$, the number $123\456$ can be represented as a fraction $\frac(123\456)(1)$.

Thus, any natural number $m$ can be represented as an ordinary fraction with a denominator $1$, and any ordinary fraction of the form $\frac(m)(1)$ can be replaced by a natural number $m$.

Fractional bar as a division sign

Representing an object in the form of $n$ parts is a division into $n$ equal parts. After dividing an item into $n$ shares, it can be divided equally between $n$ people - each will receive one share.

Let there be $m$ identical objects divided into $n$ parts. These $m$ items can be divided equally among $n$ people by giving each person one share of each of the $m$ items. In this case, each person will receive $m$ shares of $\frac(1)(n)$, which give the common fraction $\frac(m)(n)$. We find that the common fraction $\frac(m)(n)$ can be used to denote the division of $m$ items between $n$ people.

The connection between ordinary fractions and division is expressed in the fact that the fraction bar can be understood as a division sign, i.e. $\frac(m)(n)=m:n$.

An ordinary fraction makes it possible to write down the result of dividing two natural numbers for which a whole division is not performed.

Example 2

For example, the result of dividing $7$ apples by $9$ people can be written as $\frac(7)(9)$, i.e. everyone will receive seven-ninths of an apple: $7:9=\frac(7)(9)$.

Equal and unequal fractions, comparison of fractions

The result of comparing two ordinary fractions can be either their equality or their non-equality. When ordinary fractions are equal, they are called equal; otherwise, ordinary fractions are called unequal.

equal, if the equality $a\cdot d=b\cdot c$ is true.

The ordinary fractions $\frac(a)(b)$ and $\frac(c)(d)$ are called unequal, if the equality $a\cdot d=b\cdot c$ does not hold.

Example 3

Find out whether the fractions $\frac(1)(3)$ and $\frac(2)(6)$ are equal.

The equality is satisfied, which means that the fractions $\frac(1)(3)$ and $\frac(2)(6)$ are equal: $\frac(1)(3)=\frac(2)(6)$.

This example can be considered using apples: one of two identical apples is divided into three equal shares, the second into $6$ shares. It can be seen that two-sixths of an apple constitutes a $\frac(1)(3)$ share.

Example 4

Check whether the ordinary fractions $\frac(3)(17)$ and $\frac(4)(13)$ are equal.

Let's check whether the equality $a\cdot d=b\cdot c$ holds:

\ \

The equality does not hold, which means that the fractions $\frac(3)(17)$ and $\frac(4)(13)$ are not equal: $\frac(3)(17)\ne \frac(4)(13) $.

By comparing two common fractions and finding that they are not equal, you can find out which is larger and which is smaller than the other. To do this, use the rule for comparing ordinary fractions: you need to bring the fractions to a common denominator and then compare their numerators. Whichever fraction has a larger numerator, that fraction will be the larger one.

Fractions on a coordinate ray

All fractional numbers that correspond to ordinary fractions can be displayed on a coordinate ray.

To mark a point on a coordinate ray that corresponds to the fraction $\frac(m)(n)$, it is necessary to plot $m$ segments from the origin of coordinates in the positive direction, the length of which is $\frac(1)(n)$ a fraction of a unit segment . Such segments are obtained by dividing a unit segment into $n$ equal parts.

To display on a coordinate ray a fractional number, you need to divide a single segment into parts.

Figure 2.

Equal fractions are described by the same fractional number, i.e. equal fractions represent the coordinates of the same point on the coordinate ray. For example, the coordinates $\frac(1)(3)$, $\frac(2)(6)$, $\frac(3)(9)$, $\frac(4)(12)$ describe the same the same point on the coordinate ray, since all written fractions are equal.

If a point is described by a coordinate with a larger fraction, then it will be located to the right on a horizontal coordinate ray directed to the right from the point whose coordinate is a smaller fraction. For example, because the fraction $\frac(5)(6)$ is greater than the fraction $\frac(2)(6)$, then the point with coordinate $\frac(5)(6)$ is located to the right of the point with coordinate $\frac(2) (6)$.

Likewise, a point with a smaller coordinate will lie to the left of a point with a larger coordinate.

Fraction- a form of representing a number in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.

Correct A fraction is called in which the modulus of the numerator is greater than the modulus of the denominator. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.

The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:

The main property of a fraction

If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,

Reducing fractions to a common denominator

To bring two fractions to a common denominator, you need:

1. Multiply the numerator of the first fraction by the denominator of the second
2. Multiply the numerator of the second fraction by the denominator of the first
3. Replace the denominators of both fractions with their product

Operations with fractions

Addition. To add two fractions you need

1. Add the new numerators of both fractions and leave the denominator unchanged

Example:

Subtraction. To subtract one fraction from another, you need

1. Reduce fractions to a common denominator
2. Subtract the numerator of the second from the numerator of the first fraction, and leave the denominator unchanged

Example:

Multiplication. To multiply one fraction by another, multiply their numerators and denominators:

Division. To divide one fraction by another, multiply the numerator of the first fraction by the denominator of the second, and multiply the denominator of the first fraction by the numerator of the second:

This article is about common fractions. Here we will introduce the concept of a fraction of a whole, which will lead us to the definition of a common fraction. Next we will dwell on the accepted notation for ordinary fractions and give examples of fractions, let’s say about the numerator and denominator of a fraction. After this, we will give definitions of proper and improper, positive and negative fractions, and also consider the position of fractional numbers on the coordinate ray. In conclusion, we list the main operations with fractions.

Page navigation.

Shares of the whole

First we introduce concept of share.

Let's assume that we have some object made up of several absolutely identical (that is, equal) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange consisting of several equal slices. Each of these equal parts that make up the whole object is called parts of the whole or simply shares.

Note that the shares are different. Let's explain this. Let us have two apples. Cut the first apple into two equal parts, and the second into 6 equal parts. It is clear that the share of the first apple will be different from the share of the second apple.

Depending on the number of shares that make up the whole object, these shares have their own names. Let's sort it out names of beats. If an object consists of two parts, any of them is called one second part of the whole object; if an object consists of three parts, then any of them is called one third part, and so on.

One second share has a special name - half. One third is called third, and one quarter part - a quarter.

For the sake of brevity, the following were introduced: beat symbols. One second share is designated as or 1/2, one third share is designated as or 1/3; one fourth share - like or 1/4, and so on. Note that the notation with a horizontal bar is used more often. To reinforce the material, let’s give one more example: the entry denotes one hundred and sixty-seventh part of the whole.

The concept of share naturally extends from objects to quantities. For example, one of the measures of length is the meter. To measure lengths shorter than a meter, fractions of a meter can be used. So you can use, for example, half a meter or a tenth or thousandth of a meter. The shares of other quantities are applied similarly.

Common fractions, definition and examples of fractions

To describe the number of shares we use common fractions. Let us give an example that will allow us to approach the definition of ordinary fractions.

Let the orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . We denote two beats as , three beats as , and so on, 12 beats we denote as . Each of the given entries is called an ordinary fraction.

Now let's give a general definition of common fractions.

The voiced definition of ordinary fractions allows us to give examples of common fractions: 5/10, , 21/1, 9/4, . And here are the records do not fit the stated definition of ordinary fractions, that is, they are not ordinary fractions.

Numerator and denominator

For convenience, ordinary fractions are distinguished numerator and denominator.

Definition.

Numerator ordinary fraction (m/n) is a natural number m.

Definition.

Denominator common fraction (m/n) is a natural number n.

So, the numerator is located above the fraction line (to the left of the slash), and the denominator is located below the fraction line (to the right of the slash). For example, let's take the common fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.

It remains to discuss the meaning contained in the numerator and denominator of an ordinary fraction. The denominator of a fraction shows how many parts one object consists of, and the numerator, in turn, indicates the number of such parts. For example, the denominator 5 of the fraction 12/5 means that one object consists of five shares, and the numerator 12 means that 12 such shares are taken.

Natural number as a fraction with denominator 1

The denominator of a common fraction can be equal to one. In this case, we can consider that the object is indivisible, in other words, it represents something whole. The numerator of such a fraction indicates how many whole objects are taken. Thus, an ordinary fraction of the form m/1 has the meaning of a natural number m. This is how we substantiated the validity of the equality m/1=m.

Let's rewrite the last equality as follows: m=m/1. This equality allows us to represent any natural number m as an ordinary fraction. For example, the number 4 is the fraction 4/1, and the number 103,498 is equal to the fraction 103,498/1.

So, any natural number m can be represented as an ordinary fraction with a denominator of 1 as m/1, and any ordinary fraction of the form m/1 can be replaced by a natural number m.

Fraction bar as a division sign

Representing the original object in the form of n shares is nothing more than division into n equal parts. After an item is divided into n shares, we can divide it equally among n people - each will receive one share.

If we initially have m identical objects, each of which is divided into n shares, then we can equally divide these m objects between n people, giving each person one share from each of the m objects. In this case, each person will have m shares of 1/n, and m shares of 1/n gives the common fraction m/n. Thus, the common fraction m/n can be used to denote the division of m items between n people.

This is how we got an explicit connection between ordinary fractions and division (see the general idea of ​​​​dividing natural numbers). This connection is expressed as follows: the fraction line can be understood as a division sign, that is, m/n=m:n.

Using an ordinary fraction, you can write the result of dividing two natural numbers for which a whole division cannot be performed. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, everyone will get five-eighths of an apple: 5:8 = 5/8.

Equal and unequal fractions, comparison of fractions

A fairly natural action is comparing fractions, because it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as another 1/6 of this apple.

As a result of comparing two ordinary fractions, one of the results is obtained: the fractions are either equal or unequal. In the first case we have equal common fractions, and in the second – unequal ordinary fractions. Let us give a definition of equal and unequal ordinary fractions.

Definition.

equal, if the equality a·d=b·c is true.

Definition.

Two common fractions a/b and c/d not equal, if the equality a·d=b·c is not satisfied.

Here are some examples of equal fractions. For example, the common fraction 1/2 is equal to the fraction 2/4, since 1·4=2·2 (if necessary, see the rules and examples of multiplying natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second is cut into 4 parts. It is obvious that two quarters of an apple equals 1/2 share. Other examples of equal common fractions are the fractions 4/7 and 36/63, and the pair of fractions 81/50 and 1,620/1,000.

But ordinary fractions 4/13 and 5/14 are not equal, since 4·14=56, and 13·5=65, that is, 4·14≠13·5. Other examples of unequal common fractions are the fractions 17/7 and 6/4.

If, when comparing two common fractions, it turns out that they are not equal, then you may need to find out which of these common fractions less different, and which one - more. To find out, the rule for comparing ordinary fractions is used, the essence of which is to bring the compared fractions to a common denominator and then compare the numerators. Detailed information on this topic is collected in the article comparison of fractions: rules, examples, solutions.

Fractional numbers

Each fraction is a notation fractional number. That is, a fraction is just a “shell” of a fractional number, its appearance, and all semantic load is contained precisely in the fractional number. However, for brevity and convenience, the concepts of fraction and fractional number are combined and simply called fraction. Here it is appropriate to paraphrase a well-known saying: we say a fraction - we mean a fractional number, we say a fractional number - we mean a fraction.

Fractions on a coordinate ray

All fractional numbers corresponding to ordinary fractions have their own unique place on, that is, there is a one-to-one correspondence between the fractions and the points of the coordinate ray.

In order to get to the point on the coordinate ray corresponding to the fraction m/n, you need to set aside m segments from the origin in the positive direction, the length of which is 1/n fraction of a unit segment. Such segments can be obtained by dividing a unit segment into n equal parts, which can always be done using a compass and a ruler.

For example, let's show point M on the coordinate ray, corresponding to the fraction 14/10. The length of a segment with ends at point O and the point closest to it, marked with a small dash, is 1/10 of a unit segment. The point with coordinate 14/10 is removed from the origin at a distance of 14 such segments.

Equal fractions correspond to the same fractional number, that is, equal fractions are the coordinates of the same point on the coordinate ray. For example, the coordinates 1/2, 2/4, 16/32, 55/110 correspond to one point on the coordinate ray, since all the written fractions are equal (it is located at a distance of half a unit segment laid out from the origin in the positive direction).

On a horizontal and right-directed coordinate ray, the point whose coordinate is the larger fraction is located to the right of the point whose coordinate is the smaller fraction. Similarly, a point with a smaller coordinate lies to the left of a point with a larger coordinate.

Proper and improper fractions, definitions, examples

Among ordinary fractions there are proper and improper fractions. This division is based on a comparison of the numerator and denominator.

Let us define proper and improper ordinary fractions.

Definition.

Proper fraction is an ordinary fraction whose numerator is less than the denominator, that is, if m

Definition.

Improper fraction is an ordinary fraction in which the numerator is greater than or equal to the denominator, that is, if m≥n, then the ordinary fraction is improper.

Here are some examples of proper fractions: 1/4, , 32,765/909,003. Indeed, in each of the written ordinary fractions the numerator is less than the denominator (if necessary, see the article comparing natural numbers), so they are correct by definition.

Here are examples of improper fractions: 9/9, 23/4, . Indeed, the numerator of the first of the written ordinary fractions is equal to the denominator, and in the remaining fractions the numerator is greater than the denominator.

There are also definitions of proper and improper fractions, based on comparison of fractions with one.

Definition.

correct, if it is less than one.

Definition.

An ordinary fraction is called wrong, if it is either equal to one or greater than 1.

So the common fraction 7/11 is correct, since 7/11<1 , а обыкновенные дроби 14/3 и 27/27 – неправильные, так как 14/3>1, and 27/27=1.

Let's think about how ordinary fractions with a numerator greater than or equal to the denominator deserve such a name - “improper”.

For example, let's take the improper fraction 9/9. This fraction means that nine parts are taken of an object that consists of nine parts. That is, from the available nine parts we can make up a whole object. That is, the improper fraction 9/9 essentially gives the whole object, that is, 9/9 = 1. In general, improper fractions with a numerator equal to the denominator denote one whole object, and such a fraction can be replaced by the natural number 1.

Now consider the improper fractions 7/3 and 12/4. It is quite obvious that from these seven third parts we can compose two whole objects (one whole object consists of 3 parts, then to compose two whole objects we will need 3 + 3 = 6 parts) and there will still be one third part left. That is, the improper fraction 7/3 essentially means 2 objects and also 1/3 of such an object. And from twelve quarter parts we can make three whole objects (three objects with four parts each). That is, the fraction 12/4 essentially means 3 whole objects.

The considered examples lead us to the following conclusion: improper fractions can be replaced either by natural numbers, when the numerator is divided evenly by the denominator (for example, 9/9=1 and 12/4=3), or by the sum of a natural number and a proper fraction, when the numerator is not evenly divisible by the denominator (for example, 7/3=2+1/3). Perhaps this is precisely what earned improper fractions the name “irregular.”

Of particular interest is the representation of an improper fraction as the sum of a natural number and a proper fraction (7/3=2+1/3). This process is called separating the whole part from an improper fraction, and deserves separate and more careful consideration.

It's also worth noting that there is a very close relationship between improper fractions and mixed numbers.

Positive and negative fractions

Each common fraction corresponds to a positive fractional number (see the article on positive and negative numbers). That is, ordinary fractions are positive fractions. For example, ordinary fractions 1/5, 56/18, 35/144 are positive fractions. When you need to highlight the positivity of a fraction, a plus sign is placed in front of it, for example, +3/4, +72/34.

If you put a minus sign in front of a common fraction, then this entry will correspond to a negative fractional number. In this case we can talk about negative fractions. Here are some examples of negative fractions: −6/10, −65/13, −1/18.

Positive and negative fractions m/n and −m/n are opposite numbers. For example, the fractions 5/7 and −5/7 are opposite fractions.

Positive fractions, like positive numbers in general, denote an addition, income, an upward change in any value, etc. Negative fractions correspond to expense, debt, or a decrease in any quantity. For example, the negative fraction −3/4 can be interpreted as a debt whose value is equal to 3/4.

On a horizontal and rightward direction, negative fractions are located to the left of the origin. The points of the coordinate line, the coordinates of which are the positive fraction m/n and the negative fraction −m/n, are located at the same distance from the origin, but on opposite sides of the point O.

Here it is worth mentioning fractions of the form 0/n. These fractions are equal to the number zero, that is, 0/n=0.

Positive fractions, negative fractions, and 0/n fractions combine to form rational numbers.

Operations with fractions

We have already discussed one action with ordinary fractions - comparing fractions - above. Four more arithmetic functions are defined operations with fractions– adding, subtracting, multiplying and dividing fractions. Let's look at each of them.

The general essence of operations with fractions is similar to the essence of the corresponding operations with natural numbers. Let's make an analogy.

Multiplying fractions can be thought of as the action of finding a fraction from a fraction. To clarify, let's give an example. Let us have 1/6 of an apple and we need to take 2/3 of it. The part we need is the result of multiplying the fractions 1/6 and 2/3. The result of multiplying two ordinary fractions is an ordinary fraction (which in a special case is equal to a natural number). Next, we recommend that you study the information in the article Multiplying Fractions - Rules, Examples and Solutions.

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5th grade. educational institutions.
• Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).