Already in elementary school, students are faced 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 do we need fractions?
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 five cannot be given a whole number of pieces of chocolate.
By the way, these slices are already fractions. And their further division leads to more complex numbers.
What is a "fraction"?
This is a number consisting of parts of one. Outwardly, it looks like two numbers separated byhorizontal or slash. This feature is called fractional. The number written on the top (left) is called the numerator. The one below (on the 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 fractions exist?
There are only two types of them in mathematics: 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.
Ordinary fractions - 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.
Each 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.
Which subtypes do these types of fractions have?
Better 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. Her numerator is greater than or equal to the denominator.
- Reducible/irreducible. She might be likeright and 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 contains three fractional features at once.
Decimal fractions have only two subtypes:
- final, that is, one whose 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 a decimal to a common fraction?
If this is a finite number, then the 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 one and some 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. At the first numberthe denominator will be equal to 10, the second one will have 100. That is, the indicated examples will have numbers as answers: 9/10, 5/100. Moreover, the latter can be reduced by 5. Therefore, the result for it should 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 appears 23/100, the second - 108/100000. The second value needs to be reduced again. The answer is 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 performed. This fact is due to the fact that each decimal fraction is always converted to either final or periodic.
The only thing you can do 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, infinite non-periodic fractions are not converted to ordinary fractions. This is something to remember.
How to write an infinite periodic fraction as a common fraction?
In these numbers, after the decimal point, one or more digits always appear, which are repeated. They are called periods. For example, 03(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 some numbers, and then the repetition begins.
The rule according to which you need to write an infinite decimal as 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 9 in the denominator. That is, the answer will be the fraction 5/9.
The rule on how to write an ordinary decimal periodic fraction that is mixed.
- Count the fractional digits up to the period. They will indicate the number of zeros in the denominator.
- View 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 down 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 isone 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, in the denominator you need to write 90.
To determine the numerator from 58, you need to subtract 5. It turns out 53. For example, the answer will have to be written 53/90.
How do you convert common fractions 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 a comma is placed between the fractional and integer parts.
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 multiplication is required not only for the denominator, but also for the numerator by the same number.
For all other cases, a simple rule is useful: divide the numerator by the denominator. In this case, you may get two answers: a final or a periodic decimal fraction.
Actions with common fractions
Addition and subtraction
Students get to know them before others. And at first the fractions have the same denominators, and then different. The general rules can be reduced to this plan.
- Find the least common multiple of the denominators.
- Record additional factors to all common fractions.
- Multiply the numerators and denominators by the factors defined for them.
- Add (subtract) the numerators of fractions, and leave the common denominator withoutchanges.
- If the numerator of the minuend is less than the subtrahend, then you need to find out if we have a mixed number or a proper fraction.
- In the first case, the integer part must 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.
- Carefully look 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 numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.
- Multiply numerators.
- Multiply denominators.
- If the result is a reduced 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 step 1).
- In tasks where you need to multiply (divide) by an integer, the lastshould be written as an improper fraction. That is, with a denominator of 1. Then proceed as described above.
Decimal operations
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 adding and subtracting them 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's important that you don't add zeros here. Fractions are supposed to be left as they are given in the example. And then go according to plan.
- For multiplication, write the fractions one below the other, ignoring the 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 integer part is over.
What if there are both kinds 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 division or conversion results in finite fractions. 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.
Second way: write decimal fractions as common fractions
This technique is convenient if there are 1-2 digits 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, you should always soberly evaluate the task and choose the simplest solution method.
