1 What are ordinary fractions. Types of fractions.
A fraction always means some part of a whole. The fact is that it is not always possible to convey the quantity in natural numbers, that is, to recalculate: 1,2,3, etc. How, for example, to designate half a watermelon or a quarter of an hour? This is why fractional numbers, or fractions, appeared.

To begin with, it must be said that in general there are two types of fractions: ordinary fractions and decimal fractions. Ordinary fractions are written like this:
Decimals are written differently:

Ordinary fractions consist of two parts: at the top is the numerator, at the bottom is the denominator. The numerator and denominator are separated by a fractional bar. So remember:

Every fraction is part of a whole. The whole is usually taken 1 (unit). The denominator of a fraction shows how many parts the whole is divided into ( 1 ), and the numerator is how many parts were taken. If we cut the cake into 6 identical pieces (in mathematics they say shares ), then each part of the cake will be equal to 1/6. If Vasya ate 4 pieces, then he ate 4/6.

On the other hand, a fractional bar is nothing more than a division sign. Therefore, a fraction is a quotient of two numbers - the numerator and the denominator. In the text of problems or in recipes for dishes, fractions are usually written like this: 2/3, 1/2, etc. Some fractions got own name, for example, 1/2 - "half", 1/3 - "third", 1/4 - "quarter"
Now let's figure out what types of ordinary fractions are.

2 Types of ordinary fractions

There are three types of common fractions: regular, improper, and mixed:

Proper fraction

If the numerator is less than the denominator, then such a fraction is called correct, For example: A proper fraction is always less than 1.

Improper fraction

If the numerator is greater than or equal to the denominator, the fraction is called wrong, For example:

An improper fraction is greater than one (if the numerator is greater than the denominator) or equal to one (if the numerator is equal to the denominator)

mixed fraction

If the fraction is a whole number ( whole part) and a proper fraction (fractional part), then such a fraction is called mixed, For example:

A mixed fraction is always greater than one.

3 Fraction conversions

In mathematics, ordinary fractions often have to be converted, that is, a mixed fraction must be turned into an improper one and vice versa. This is necessary to perform some operations, such as multiplication and division.

So, any mixed fraction can be converted to an improper. To do this, the integer part is multiplied by the denominator and the numerator of the fractional part is added. The resulting amount is taken as the numerator, and the denominator is left the same, for example:

Any improper fraction can be converted into a mixed fraction. To do this, divide the numerator by the denominator (with a remainder). The resulting number will be the integer part, and the remainder will be the numerator of the fractional part, for example:

At the same time, they say: “We singled out the whole part from an improper fraction.”

There is one more rule to remember: Any whole number can be represented as a common fraction with denominator 1, For example:

Let's talk about how to compare fractions.

4 Fraction Comparison

There are several options when comparing fractions: It is easy to compare fractions with same denominators, much more difficult if the denominators are different. There is also a comparison mixed fractions. But don't worry, now we'll take a closer look at each option and learn how to compare fractions.

Comparing fractions with the same denominators

Of two fractions with the same denominator but different numerators, the fraction with the larger numerator is larger, for example:

Comparing fractions with the same numerator

Of two fractions with the same numerators, but different denominators the larger is the fraction whose denominator is smaller, for example:

Comparison of mixed and improper fractions with correct fractions

An improper or mixed fraction is always greater than a proper fraction, for example:

Comparing two mixed fractions

When comparing two mixed fractions, the fraction with the larger integer part is greater, for example:

If the integer parts of mixed fractions are the same, the fraction with the larger fractional part is greater, for example:

Comparing fractions with different numerators and denominators

It is impossible to compare fractions with different numerators and denominators without converting them. First, the fractions must be brought to the same denominator, and then their numerators should be compared. The larger fraction is the one with the larger numerator. But how to bring fractions to the same denominator, we will consider in the next two sections of the article. First, we will consider the basic property of a fraction and the reduction of fractions, and then directly reducing fractions to the same denominator.

5 Basic property of a fraction. Fraction reduction. The concept of GCD.

Remember: You can only add, subtract, and compare fractions that have the same denominators.. If the denominators are different, then first you need to bring the fractions to the same denominator, that is, transform one of the fractions in such a way that its denominator becomes the same as that of the second fraction.

Fractions have one important property also called basic property of a fraction:

If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the value of the fraction will not change:

Thanks to this property, we can reduce fractions:

To reduce a fraction means to divide both the numerator and the denominator by the same number.(see example just above). When we reduce a fraction, we can describe our actions as follows:

More often, in a notebook, a fraction is reduced like this:

But remember: only multipliers can be reduced. If the numerator or denominator is the sum or difference, the terms cannot be reduced. Example:

We need to convert the sum to a multiplier first:

Sometimes, when working with big numbers, in order to reduce the fraction, it is convenient to find greatest common factor of numerator and denominator (gcd)

Greatest Common Divisor (GCD) several numbers - this is the largest natural number by which these numbers are divisible without a remainder.

In order to find the GCD of two numbers (for example, the numerator and denominator of a fraction), you need to expand both numbers into prime factors, note the same factors in both expansions, and multiply these factors. The resulting product will be GCD. For example, we need to reduce a fraction:

Find the GCD of numbers 96 and 36:

The GCD shows us that both the numerator and the denominator have a factor12, and we can easily reduce the fraction.

Sometimes, to bring fractions to the same denominator, it is enough to reduce one of the fractions. But more often it is necessary to select additional factors for both fractions. Now we will look at how this is done. So:

6 How to bring fractions to the same denominator. Least common multiple (LCM).

When we reduce fractions to the same denominator, we select for the denominator a number that would be divisible by both the first and the second denominator (that is, it would be a multiple of both denominators, expressed mathematical language). And it is desirable that this number be as small as possible, so it is more convenient to count. So we have to find the LCM of both denominators.

Least common multiple of two numbers (LCM) is the smallest natural number that is divisible by both of these numbers without a remainder. Sometimes the LCM can be found orally, but more often, especially when working with large numbers, you have to find the LCM in writing, using the following algorithm:

In order to find the LCM of several numbers, you need:

1. Decompose these numbers into prime factors
2. Take the largest expansion, and write these numbers as a product
3. Select in other expansions the numbers that do not occur in the largest expansion (or occur in it lesser number times), and add them to the work.
4. Multiply all the numbers in the product, this will be the LCM.

For example, let's find the LCM of numbers 28 and 21:

But back to our fractions. After we have selected or calculated in writing the LCM of both denominators, we must multiply the numerators of these fractions by additional multipliers. You can find them by dividing the LCM by the denominator of the corresponding fraction, for example:

Thus, we reduced our fractions to one denominator - 15.

7 Addition and subtraction of fractions

Adding and subtracting fractions with the same denominators

To add fractions with the same denominators, you need to add their numerators, and leave the denominator the same, for example:

To subtract fractions with the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:

Addition and subtraction of mixed fractions with the same denominators

To add mixed fractions, you need to add their whole parts separately, and then add their fractional parts, and write the result as a mixed fraction:

If, when adding the fractional parts, an improper fraction is obtained, we select the integer part from it and add it to the integer part, for example:

Subtraction is carried out in the same way: the integer part is subtracted from the integer, and the fractional part is subtracted from the fractional part:

If the fractional part of the subtrahend is greater than the fractional part of the minuend, we “take” one from the integer part, turning the minuend into an improper fraction, and then proceed as usual:

Similarly subtract a fraction from a whole number:

How to add a whole number and a fraction

In order to add a whole number and a fraction, you just need to add this number before the fraction, and you get a mixed fraction, for example:

If we add a whole number and a mixed fraction, we add this number to the integer part of the fraction, for example:

Addition and subtraction of fractions with different denominators.

In order to add or subtract fractions with different denominators, you must first bring them to the same denominator, and then proceed as when adding fractions with the same denominators (add the numerators):

When subtracting, we proceed in the same way:

If we work with mixed fractions, we reduce their fractional parts to the same denominator and then subtract as usual: the integer part from the integer, and the fractional part from the fractional part:

8 Multiplication and division of fractions.

Multiplying and dividing fractions is much easier than adding and subtracting because you don't have to bring them to the same denominator. Remember simple rules multiplication and division of fractions:

Before multiplying the numbers in the numerator and denominator, it is desirable to reduce the fraction, that is, get rid of same multipliers in the numerator and denominator, as in our example.

To divide a fraction by a natural number, you need to multiply the denominator by this number, and leave the numerator unchanged:

For example:

Division of a fraction by a fraction

To divide one fraction into another, you need to multiply the dividend by the reciprocal of the divisor (the reciprocal). What is this reciprocal?

If we flip the fraction, that is, swap the numerator and denominator, we get the reciprocal. The product of a fraction and its reciprocal gives one. In mathematics, such numbers are called mutually reciprocal numbers:

For example, numbers are mutually inverse, since

Thus, we return to the division of a fraction by a fraction:

To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor:

For example:

When dividing mixed fractions, just as when multiplying, you must first convert them to improper fractions:

When multiplying and dividing fractions by integers integers , you can also represent these numbers as fractions with a denominator 1 .

And at dividing a whole number by a fraction represent this number as a fraction with a denominator 1 :

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 the exact value of the quantity. You can do the same with fractions. 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. This value 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 through the separator ":" and continue the usual division.

To get the final 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 recording 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 wrong way: 1 2/10 kg 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.

"Fractional" mathematics for children

Let's agree right away that a fraction is a part of a whole, less than one. Into how many parts will we divide the whole? And this is how we agree. What will be considered a unit? Same as we agree. That's how accommodating they are, these fractions. And you also need to remember one thing: the number into how many parts we decided to divide the whole is the denominator, how many of these parts we took is the numerator.

For example, here is a story. There are 3 apples on the grass, the hedgehog took only 2. For the whole (one), we will take all the apples - the entire crop. But we have 3 of them, which means that our crop is divided into 3 parts. 3 is the denominator. The whole crop (unit) is 3/3 and each apple is 1/3 of the crop. Since the hedgehog took 2 apples, it means that he took 2/3 of the crop!

And you can take Lego, such a designer loved by many children. We have long noticed that all its elements are different in size, right? And on every detail different amount dots - "pimples". Let's count - here is one, two, four, six and even eight.

Let's consider a lego "brick" with eight points as a whole (one). First, let's compare it with others. How many Lego pieces with 4 dots do you need to take to make our “brick” unit? That's right, two. So, one detail with 4 points is 1/2 of our "one". And how many details with two points do you need to take to get the whole? That's right, four. So, one such detail is 1/4. And a detail with one point is 1/8, because such details will be needed as many as 8 pieces to make a whole. Now the task is more complicated: we have an element with six points. It fits 3 "quarters", and if you add one more to it, you get a whole (one). So, here is the first example ready: 3/4+1/4=4/4 or 1 (if the numerator and denominator are equal, then this is one!)

This is far from the only experiment that can be done with Lego. With fractions, you can agree on a lot. But what if we are the same, we will consider not quarters, but eighths? And the denominator will be 8? We look at the picture: the unit is a “brick” with eight points. 1/2 is 4/8, and 1/4=2/8. And this is a story about how you can reduce fractions. But this topic can really wait a bit!

Examples with fractions are one of the basic elements of mathematics. There are many different types equations with fractions. Below is detailed instructions by solving examples of this type.

How to solve examples with fractions - general rules

To solve examples with fractions of any type, whether it be addition, subtraction, multiplication or division, you need to know the basic rules:

• In order to add fractional expressions with the same denominator (the denominator is the number at the bottom of the fraction, the numerator at the top), you need to add their numerators, and leave the denominator the same.
• In order to subtract from one fractional expression the second (with the same denominator), you need to subtract their numerators, and leave the denominator the same.
• In order to add or subtract fractional expressions with different denominators, you need to find the smallest common denominator.
• In order to find a fractional product, you need to multiply the numerators and denominators, while, if possible, reduce.
• To divide a fraction by a fraction, you need to multiply the first fraction by the reversed second.

How to solve examples with fractions - practice

Rule 1, example 1:

Calculate 3/4 +1/4.

According to Rule 1, if fractions of two (or more) have the same denominator, you just need to add their numerators. We get: 3/4 + 1/4 = 4/4. If a fraction has the same numerator and denominator, the fraction will be 1.

Answer: 3/4 + 1/4 = 4/4 = 1.

Rule 2, example 1:

Calculate: 3/4 - 1/4

Using rule number 2, to solve this equation, you need to subtract 1 from 3, and leave the denominator the same. We get 2/4. Since two 2 and 4 can be reduced, we reduce and get 1/2.

Answer: 3/4 - 1/4 = 2/4 = 1/2.

Rule 3, Example 1

Calculate: 3/4 + 1/6

Solution: Using the 3rd rule, we find the least common denominator. The least common denominator is the number that is divisible by the denominators of all fractional expressions example. Thus, we need to find such a minimum number that will be divisible by both 4 and 6. This number is 12. We write 12 as the denominator. We divide 12 by the denominator of the first fraction, we get 3, we multiply by 3, we write 3 in the numerator *3 and + sign. We divide 12 by the denominator of the second fraction, we get 2, we multiply 2 by 1, we write 2 * 1 in the numerator. So, we got a new fraction with a denominator equal to 12 and a numerator equal to 3*3+2*1=11. 11/12.

Answer: 11/12

Rule 3, Example 2:

Calculate 3/4 - 1/6. This example is very similar to the previous one. We do all the same actions, but in the numerator instead of the + sign, we write the minus sign. We get: 3*3-2*1/12 = 9-2/12 = 7/12.

Answer: 7/12

Rule 4, Example 1:

Calculate: 3/4 * 1/4

Using the fourth rule, we multiply the denominator of the first fraction by the denominator of the second and the numerator of the first fraction by the numerator of the second. 3*1/4*4 = 3/16.

Answer: 3/16

Rule 4, Example 2:

Calculate 2/5 * 10/4.

This fraction can be reduced. In the case of a product, the numerator of the first fraction and the denominator of the second and the numerator of the second fraction and the denominator of the first are reduced.

2 is reduced from 4. 10 is reduced from 5. we get 1 * 2/2 = 1 * 1 = 1.

Answer: 2/5 * 10/4 = 1

Rule 5, Example 1:

Calculate: 3/4: 5/6

Using the 5th rule, we get: 3/4: 5/6 = 3/4 * 6/5. We reduce the fraction according to the principle of the previous example and get 9/10.

Answer: 9/10.

How to Solve Fraction Examples - Fractional Equations

Fractional equations are examples where the denominator contains an unknown. In order to solve such an equation, you need to use certain rules.

Consider an example:

Solve equation 15/3x+5 = 3

Recall that you cannot divide by zero, i.e. the denominator value must not be zero. When solving such examples, this must be indicated. To do this, there is ODZ (range of acceptable values).

So 3x+5 ≠ 0.
Hence: 3x ≠ 5.
x ≠ 5/3

For x = 5/3, the equation simply has no solution.

By specifying the ODZ, in the best possible way decide given equation will get rid of the fractions. For this, we first imagine all fractional values as a fraction, in this case the number 3. We get: 15/(3x+5) = 3/1. To get rid of fractions, you need to multiply each of them by the smallest common denominator. In this case, that would be (3x+5)*1. Sequencing:

1. Multiply 15/(3x+5) by (3x+5)*1 = 15*(3x+5).
2. Expand the brackets: 15*(3x+5) = 45x + 75.
3. We do the same with right side equations: 3*(3x+5) = 9x + 15.
4. Equate left and right side: 45x + 75 = 9x +15
5. Move x's to the left, numbers to the right: 36x = -50
6. Find x: x = -50/36.
7. We reduce: -50/36 = -25/18

Answer: ODZ x ≠ 5/3. x = -25/18.

How to solve examples with fractions - fractional inequalities

Fractional inequalities of the type (3x-5)/(2-x)≥0 are solved using the numerical axis. Consider this example.

Sequencing:

• Equate the numerator and denominator to zero: 1. 3x-5=0 => 3x=5 => x=5/3
  2. 2-x=0 => x=2
• We draw a numerical axis, painting the resulting values ​​​​on it.
• Draw a circle under the value. The circle is of two types - filled and empty. The filled circle means that given value included in the range of solutions. An empty circle indicates that this value is not included in the range of solutions.
• Since the denominator cannot be zero, there will be an empty circle under the 2nd.

• To determine the signs, we substitute any number greater than two into the equation, for example 3. (3 * 3-5) / (2-3) \u003d -4. the value is negative, so we write a minus over the area after the deuce. Then we substitute any value of the interval from 5/3 to 2 instead of x, for example 1. The value is again negative. We write minus. We repeat the same with the area up to 5/3. We substitute any number less than 5/3, for example 1. Minus again.

• Since we are interested in x values, at which the expression will be greater than or equal to 0, and there are no such values ​​(cons everywhere), this inequality has no solution, i.e. x = Ø (empty set).

Answer: x = Ø

A part of a unit or several of its parts is called a simple or ordinary fraction. Quantity equal parts, into which the unit is divided, is called the denominator, and the number of parts taken is called the numerator. The fraction is written as:

In this case, a is the numerator, b is the denominator.

If the numerator less than the denominator, then the fraction less than 1 is called proper fraction. If the numerator is greater than the denominator, then the fraction is greater than 1, then the fraction is called an improper fraction.

If the numerator and denominator of a fraction are equal, then the fraction is equal.

1. If the numerator can be divided by the denominator, then this fraction is equal to the quotient of division:

If the division is performed with a remainder, then this improper fraction can be represented by a mixed number, for example:

Then 9 is an incomplete quotient (the integer part of the mixed number),
1 - remainder (numerator of the fractional part),
5 is the denominator.

To convert a mixed number to a fraction, multiply the integer part of the mixed number by the denominator and add the numerator of the fractional part.

The result obtained will be the numerator of an ordinary fraction, and the denominator will remain the same.

Actions with fractions

Fraction expansion. The value of a fraction does not change if its numerator and denominator are multiplied by the same non-zero number.
for example:

Fraction reduction. The value of a fraction does not change if its numerator and denominator are divided by the same non-zero number.
for example:

Fraction comparison. Of two fractions with the same numerator, the larger one is the one with the smaller denominator:

Of two fractions with the same denominators, the one with the larger numerator is greater:

To compare fractions whose numerators and denominators are different, it is necessary to expand them, that is, to bring them to common denominator. Consider, for example, the following fractions:

Addition and subtraction of fractions. If the denominators of fractions are the same, then in order to add the fractions, it is necessary to add their numerators, and in order to subtract the fractions, it is necessary to subtract their numerators. The resulting sum or difference will be the numerator of the result, while the denominator will remain the same. If the denominators of the fractions are different, you must first reduce the fractions to a common denominator. When added mixed numbers their integer and fractional parts are added separately. When subtracting mixed numbers, you must first convert them to the form of improper fractions, then subtract from one another, and then again bring the result, if necessary, to the form of a mixed number.

Multiplication of fractions. To multiply fractions, you need to multiply their numerators and denominators separately and divide the first product by the second.

Division of fractions. To divide a number by a fraction, you need to multiply that number by its reciprocal.

Decimal is the result of dividing one by ten, one hundred, one thousand, etc. parts. First, the integer part of the number is written, then the decimal point is placed on the right. The first digit after the decimal point means the number of tenths, the second - the number of hundredths, the third - the number of thousandths, etc. The numbers after the decimal point are called decimal places.

For example:

Decimal Properties

Properties:

• The decimal fraction does not change if zeros are added to the right: 4.5 = 4.5000.
• The decimal fraction does not change if the zeros located at the end of the decimal fraction are removed: 0.0560000 = 0.056.
• The decimal increases at 10, 100, 1000, and so on. times, if you move the decimal point to one, two, three, etc. positions to the right: 4.5 45 (the fraction has increased 10 times).
• The decimal is reduced by 10, 100, 1000, etc. times, if you move the decimal point to one, two, three, etc. positions to the left: 4.5 0.45 (the fraction has decreased 10 times).

A periodic decimal contains an infinitely repeating group of digits called a period: 0.321321321321…=0,(321)

Operations with decimals

Adding and subtracting decimals is done in the same way as adding and subtracting whole numbers, you just need to write the corresponding decimal places one under the other.
For example:

Multiplication of decimal fractions is carried out in several stages:

• We multiply decimals as integers, without taking into account the decimal point.
• The rule applies: the number of decimal places in the product is equal to the sum of the decimal places in all factors.

for example:

The sum of the numbers of decimal places in the factors is: 2+1=3. Now you need to count 3 digits from the end of the resulting number and put a decimal point: 0.675.

Division of decimals. Dividing a decimal by an integer: if the dividend less divisor, then you need to write zero in the integer part of the quotient and put a decimal point after it. Then, without taking into account the decimal point of the dividend, add the next digit of the fractional part to its integer part and again compare the resulting integer part of the dividend with the divisor. If the new number is again less than the divisor, the operation must be repeated. This process is repeated until the resulting dividend is greater than the divisor. After that, division is performed as for integers. If the dividend is greater than or equal to the divisor, first we divide its integer part, write the result of the division in the quotient and put a decimal point. After that, the division continues, as in the case of integers.

Dividing one decimal fraction into another: first, the decimal points in the dividend and divisor are transferred by the number of decimal places in the divisor, that is, we make the divisor an integer, and the actions described above are performed.

In order to turn decimal into an ordinary one, it is necessary to take the number after the decimal point as the numerator, and take the k-th power of ten as the denominator (k is the number of decimal places). The non-zero integer part is preserved in the common fraction; the zero integer part is omitted.
For example:

In order to turn common fraction to decimal, it is necessary to divide the numerator by the denominator in accordance with the rules of division.

A percentage is a hundredth of a unit, for example: 5% means 0.05. A ratio is the quotient of dividing one number by another. Proportion is the equality of two ratios.

For example:

The main property of the proportion: the product of the extreme members of the proportion is equal to the product of its middle members, that is, 5x30 = 6x25. Two mutually dependent quantities are called proportional if the ratio of their quantities remains unchanged (proportionality coefficient).

Thus, the following arithmetic operations are revealed.
For example:

The set of rational numbers includes positive and negative numbers (whole and fractional) and zero. More precise definition rational numbers, accepted in mathematics, the following: a number is called rational if it can be represented as an ordinary irreducible fraction of the form:, where a and b are integers.

For negative number absolute value(modulus) is a positive number obtained by changing its sign from "-" to "+"; for positive number and zero is the number itself. To designate the modulus of a number, two straight lines are used, inside which this number is written, for example: |–5|=5.

Absolute value properties

Let the modulus of a number be given , for which the properties are valid:

A monomial is the product of two or more factors, each of which is either a number, or a letter, or the power of a letter: 3 x a x b. The coefficient is most often called only a numerical factor. Monomials are said to be similar if they are the same or differ only in coefficients. The degree of a monomial is the sum of the exponents of all its letters. If there are similar ones among the sum of monomials, then the sum can be reduced to more plain sight: 3 x a x b + 6 x a \u003d 3 x a x (b + 2). This operation is called coercion of like terms or parentheses.

The polynomial is algebraic sum monomial. The degree of a polynomial is the largest of the degrees of the monomials included in the given polynomial.

There are the following formulas for abbreviated multiplication:

Factoring methods:

An algebraic fraction is an expression of the form , where A and B can be a number, a monomial, a polynomial.

If two expressions (numeric and alphabetic) are connected by the sign "=", then they are said to form equality. Any true equality that is valid for all admissible numerical values letters included in it is called an identity.

An equation is a literal equality that is valid for certain values letters included in it. These letters are called unknowns (variables), and their values, at which the given equation becomes an identity, are called the roots of the equation.

Solving an equation means finding all its roots. Two or more equations are said to be equivalent if they have the same roots.

• zero was the root of the equation;
• the equation only had finite number roots.

Main types of algebraic equations:

The linear equation has ax + b = 0:

• if a x 0, there is a single root x = -b/a;
• if a = 0, b ≠ 0, no roots;
• if a = 0, b = 0, the root is any real number.

Equation xn = a, n N:

• if n - odd number, has a real root equal to a/n for any a;
• if n is an even number, then for a 0, then it has two roots.

Main identical transformations: replacement of one expression by another, identically equal to it; transfer of the terms of the equation from one side to the other with opposite signs; multiplication or division of both parts of the equation by the same expression (number) other than zero.

A linear equation with one unknown is an equation of the form: ax+b=0, where a and b are known numbers, and x is an unknown quantity.

Systems of two linear equations with two unknowns have the form:

Where a, b, c, d, e, f are given numbers; x, y are unknown.

Numbers a, b, c, d - coefficients for unknowns; e, f - free members. The solution to this system of equations can be found by two main methods: the substitution method: from one equation we express one of the unknowns through the coefficients and the other unknown, and then we substitute it into the second equation, solving the last equation, we first find one unknown, then we substitute the found value into the first equation and find the second unknown; method of adding or subtracting one equation from another.

Operations with roots:

Arithmetic the root of the nth degree from a non-negative number a is called a non-negative number, nth power which is equal to a. algebraic root nth degree from given number the set of all roots from this number is called.

Irrational numbers, unlike rational ones, cannot be represented as an ordinary irreducible fraction of the form m/n, where m and n are integers. These are numbers of a new type that can be calculated with any precision, but cannot be replaced rational number. They can appear as a result of geometric measurements, for example: the ratio of the length of the diagonal of a square to the length of its side is equal.

There is a quadratic equation algebraic equation second degree ax2+bx+c=0, where a, b, c - given numeric or alphabetic coefficients, x - unknown. If we divide all the terms of this equation by a, as a result we get x2+px+q=0 - the reduced equation p=b/a, q=c/a. Its roots are found by the formula:

If b2-4ac>0, then there are two different root, b2- 4ac=0, then there are two equal root; b2-4ac Equations containing modules

Main types of equations containing modules:
1) |f(x)| = |g(x)|;
2) |f(x)| = g(x);
3) f1(x)|g1(x)| + f2(x)|g2(x)| + … + fn(x)|gn(x)| =0, n N, where f(x), g(x), fk(x), gk(x) are given functions.