Already in primary school students are dealing with fractions. And then they appear in every topic. It is impossible to forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are simple, the main thing is to understand everything in order.

Why are fractions needed?

The world around us consists of whole objects. Therefore, there is no need for shares. But everyday life constantly pushes people to work with parts of objects and things.

For example, chocolate consists of several slices. Consider the situation where its tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It will be well divided into three. But the five will not be able to give a whole number of slices of chocolate.

By the way, these slices are already fractions. And their further division leads to the appearance of more complex numbers.

What is a "fraction"?

This is a number consisting of parts of one. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written on the top (left) is called the numerator. The one on the bottom (right) is the denominator.

In fact, the fractional bar turns out to be a division sign. That is, the numerator can be called a dividend, and the denominator can be called a divisor.

What are the fractions?

In mathematics, there are only two types of them: ordinary and decimal fractions. Schoolchildren get acquainted with the first ones in the elementary grades, calling them simply “fractions”. The second learn in the 5th grade. That's when these names appear.

Common fractions are all those that are written as two numbers separated by a bar. For example, 4/7. Decimal is a number in which the fractional part has a positional notation and is separated from the integer with a comma. For example, 4.7. Students need to be clear that the two examples given are completely different numbers.

Every simple fraction can be written as a decimal. This statement is almost always true in reverse as well. There are rules that allow you to write a decimal fraction as an ordinary fraction.

What subspecies do these types of fractions have?

It is better to start in chronological order, as they are being studied. Common fractions come first. Among them, 5 subspecies can be distinguished.

Correct. Its numerator is always less than the denominator.

Wrong. Its numerator is greater than or equal to the denominator.

Reducible / irreducible. It can be either right or wrong. Another thing is important, whether the numerator and denominator have common factors. If there are, then they are supposed to divide both parts of the fraction, that is, to reduce it.

Mixed. An integer is assigned to its usual correct (incorrect) fractional part. And it always stands on the left.

Composite. It is formed from two fractions divided into each other. That is, it has three fractional features at once.

Decimals have only two subspecies:

final, that is, one in which the fractional part is limited (has an end);

infinite - a number whose digits after the decimal point do not end (they can be written endlessly).

How to convert decimal to ordinary?

If this is a finite number, then an association based on the rule is applied - as I hear, so I write. That is, you need to read it correctly and write it down, but without a comma, but with a fractional line.

As a hint about the required denominator, remember that it is always a one and a few zeros. The latter need to be written as many as the digits in the fractional part of the number in question.

How to convert decimal fractions to ordinary ones if their whole part is missing, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. It remains to write down only the fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter turns out to be possible to reduce by 5. Therefore, the result for it must be written 1/20.

How to make an ordinary fraction from a decimal if its integer part is different from zero? For example, 5.23 or 13.00108. Both examples read the integer part and write its value. In the first case, this is 5, in the second - 13. Then you need to move on to the fractional part. With them it is necessary to carry out the same operation. The first number has 23/100, the second has 108/100000. The second value needs to be reduced again. The response is like this mixed fractions: 5 23/100 and 13 27/25000.

How to convert an infinite decimal to a common fraction?

If it is non-periodic, then such an operation cannot be carried out. This fact is due to the fact that each decimal fraction is always converted to either final or periodic.

The only thing that is allowed to be done with such a fraction is to round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal - will never give the initial value. That is, endless non-periodic fractions are not converted to ordinary. This must be remembered.

How to write an infinite periodic fraction in the form of an ordinary?

In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called periods. For example, 0.3(3). Here "3" in the period. They are classified as rational, as they can be converted into ordinary fractions.

Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second, the fractional part begins with any numbers, and then the repetition begins.

The rule by which you need to write an infinite decimal in the form of an ordinary fraction will be different for these two types of numbers. It is quite easy to write pure periodic fractions as ordinary fractions. As with the final ones, they need to be converted: write the period into the numerator, and the number 9 will be the denominator, repeating as many times as there are digits in the period.

For example, 0,(5). The number does not have an integer part, so you need to immediately proceed to the fractional part. Write 5 in the numerator, and write 9 in the denominator. That is, the answer will be the fraction 5/9.

A rule on how to write a common decimal fraction that is a mixed fraction.

Look at the length of the period. So much 9 will have a denominator.

Write down the denominator: first nines, then zeros.

To determine the numerator, you need to write the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtractable - it is without a period.

For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period is one digit. So zero will be one. There is also only one digit in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.

To determine the numerator from 58, you need to subtract 5. It turns out 53. For example, you will have to write 53/90 as an answer.

How are common fractions converted to decimals?

The simplest option is a number whose denominator is the number 10, 100, and so on. Then the denominator is simply discarded, and between the fractional and whole parts a comma is placed.

There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. Only it is necessary to multiply not only the denominator, but also the numerator by the same number.

For all other cases, a simple rule will come in handy: divide the numerator by the denominator. In this case, you may get two answers: a final or a periodic decimal fraction.

Operations with common fractions

Addition and subtraction

Students get to know them earlier than others. And at first the fractions have the same denominators, and then different. General rules can be reduced to such a plan.

Find the least common multiple of the denominators.

Write additional factors to all ordinary fractions.

Multiply the numerators and denominators by the factors defined for them.

Add (subtract) the numerators of fractions, and leave the common denominator unchanged.

If the numerator of the minuend is less than the subtrahend, then you need to find out whether we have a mixed number or a proper fraction.

In the first case, the integer part needs to take one. Add a denominator to the numerator of a fraction. And then do the subtraction.

In the second - it is necessary to apply the rule of subtraction from a smaller number to a larger one. That is, subtract the modulus of the minuend from the modulus of the subtrahend, and put the “-” sign in response.

Look carefully at the result of addition (subtraction). If you get an improper fraction, then it is supposed to select the whole part. That is, divide the numerator by the denominator.

Multiplication and division

For their implementation, fractions do not need to be reduced to a common denominator. This makes it easier to take action. But they still have to follow the rules.

When multiplying ordinary fractions, it is necessary to consider the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.

Multiply numerators.

Multiply the denominators.

If you get a reducible fraction, then it is supposed to be simplified again.

When dividing, you must first replace division with multiplication, and the divisor (second fraction) with a reciprocal (swap the numerator and denominator).

Then proceed as in multiplication (starting from point 1).

In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written as an improper fraction. That is, with a denominator of 1. Then proceed as described above.



Operations with decimals

Addition and subtraction

Of course, you can always turn a decimal into a common fraction. And act according to the already described plan. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.

Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Assign the missing number of zeros in it.

Write fractions so that the comma is under the comma.

Add (subtract) like natural numbers.

Remove the comma.

Multiplication and division

It is important that you do not need to append zeros here. Fractions are supposed to be left as they are given in the example. And then go according to plan.

For multiplication, you need to write fractions one under the other, not paying attention to commas.

Multiply like natural numbers.

Put a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.

To divide, you must first convert the divisor: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.

Multiply the dividend by the same number.

Divide a decimal by a natural number.

Put a comma in the answer at the moment when the division of the whole part ends.



What if there are both types of fractions in one example?

Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. There are two possible solutions to these problems. You need to objectively weigh the numbers and choose the best one.

First way: represent ordinary decimals

It is suitable if, when dividing or converting, final fractions are obtained. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you do not like working with ordinary fractions, you will have to count them.

The second way: write decimal fractions as ordinary

This technique is convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, a very large ordinary fraction can turn out and decimal entries will allow you to calculate the task faster and easier. Therefore, it is always necessary to soberly evaluate the task and choose the simplest solution method.

To write a rational number m / n as a decimal fraction, you need to divide the numerator by the denominator. In this case, the quotient is written as a finite or infinite decimal fraction.

Write the given number as a decimal.

Solution. Divide the numerator of each fraction by its denominator: A) divide 6 by 25; b) divide 2 by 3; V) divide 1 by 2, and then add the resulting fraction to unity - the integer part of this mixed number.

Irreducible ordinary fractions whose denominators contain no prime divisors other than 2 And 5 , are written as a final decimal fraction.

IN example 1 when A) denominator 25=5 5; when V) the denominator is 2, so we got the final decimals 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a final decimal.

Is it possible, without dividing into a column, to convert such an ordinary fraction into a decimal fraction, the denominator of which does not contain other divisors, except 2 and 5? Let's figure it out! What fraction is called decimal and is written without a fractional line? Answer: a fraction with a denominator of 10; 100; 1000 etc. And each of these numbers is a product equal number of twos and fives. Actually: 10=2 5 ; 100=2 5 2 5 ; 1000=2 5 2 5 2 5 etc.

Therefore, the denominator of an irreducible ordinary fraction will need to be represented as a product of twos and fives, and then multiplied by 2 and (or) 5 so that the twos and fives become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. So that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which the denominator was multiplied.

Express the following fractions as a decimal:

Solution. Each of these fractions is irreducible. Let us decompose the denominator of each fraction into prime factors.

20=2 2 5. Conclusion: one "five" is missing.

8=2 2 2. Conclusion: there are not enough three "fives".

25=5 5. Conclusion: two "twos" are missing.

Comment. In practice, they often do not use the factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is a unit with zeros (10 or 100 or 1000, etc.). And then the numerator is multiplied by the same number.

So, in case A)(example 2) from the number 20 you can get 100 by multiplying by 5, therefore, you need to multiply the numerator and denominator by 5.

When b)(example 2) from the number 8, the number 100 will not work, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.

When V)(example 2) out of 25 you get 100 when multiplied by 4. This means that the numerator 8 must also be multiplied by 4.

An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodical decimal fraction. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosing it in parentheses.

When b)(example 1 ) the repeated digit is one and equals 6. Therefore, our result 0.66... ​​will be written like this: 0,(6) . They read: zero integers, six in the period.

If there is one or more non-recurring digits between the comma and the first period, then such a periodic fraction is called a mixed periodic fraction.

An irreducible common fraction whose denominator together with others multiplier contains multiplier 2 or 5 , becomes mixed periodic fraction.

Write the number as a decimal.

Decimal fraction- variety fractions, which has a "round" number in the denominator: 10, 100, 1000, etc., for example, fraction 5/10 has a decimal notation of 0.5. Based on this principle, fraction can be presented in form decimal fractions.

Instruction

Suppose we need to imagine form decimal fraction 18/25.
First you need to make sure that one of the "round" numbers appears in the denominator: 100, 1000, etc. To do this, you need to multiply the denominator by 4. But by 4, you will need to multiply both the numerator and the denominator.

Multiplying the numerator and denominator fractions 18/25 times 4 is 72/100. This is being recorded fraction in decimal form so: 0.72.

A fraction in mathematics is a rational number equal to one or more parts into which a unit is divided. In this case, the record of the fraction must contain an indication of two numbers: one of them indicates exactly how many shares the unit was divided into when creating this fraction, and the other - how many of these shares include a fractional number. If these two numbers are written as a numerator and denominator separated by a bar, then this recording format is called an “ordinary” fraction. However, there is another format for writing fractions, called "decimal".

The three-story form of writing numbers, in which the denominator is located above the numerator, and there is also a separating line between them, is not always convenient. Especially this inconvenience began to manifest itself with the mass distribution personal computers. The decimal form of representation of fractions is devoid of this drawback - it is not required to indicate the numerator in it, since by definition it is always equal to ten to a negative power. Therefore, a fractional number can be written in one line, although its length in most cases will be much larger than the length of the corresponding ordinary fraction.

Another advantage of writing numbers in decimal format is that they are much easier to compare. Since the denominator of each digit of two such numbers is the same, it is enough to compare only two digits of the corresponding digits, while when comparing ordinary fractions, both the numerator and the denominator of each of them must be taken into account. This advantage is important not only for humans, but also for computers - comparing numbers in decimal format is easy enough to program.

There are centuries-old rules for addition, multiplication, and other mathematical operations that allow you to perform calculations on paper or in your mind with numbers in decimal format. This is another advantage of this format over ordinary fractions. Although with the development of computer technology, when the calculator is even in the watch, it is becoming less and less noticeable.

The described advantages of the decimal format for recording fractional numbers show that its main purpose is to simplify the work with mathematical quantities. This format also has disadvantages - for example, to write periodic fractions to a decimal fraction, you also have to add a number in brackets, and irrational numbers in decimal format always have an approximate value. However, at the current level of development of people and their technologies, it is much more convenient to use than the usual format for recording fractions.