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 is made up 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 a situation where its tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. She is well divided into three. But 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 a unit. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written on top (left) is called the numerator. What is below (right) is the denominator.
In fact, the fractional line is a sign of division. That is, the numerator can be called divisible, and the denominator - the divisor.
What fractions exist?
In mathematics, there are only two types: ordinary and decimal fractions. Pupils get acquainted with the first in the elementary grades, calling them simply “fractions”. The second will be recognized in the 5th grade. It is then that 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 record and is separated from the integer by a comma. For example, 4.7. Students need to clearly understand 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 the opposite direction. There are rules that allow you to write a decimal fraction with an ordinary fraction.
What subspecies have the indicated types of fractions?
It is better to start in chronological order, as they are being studied. The first are ordinary fractions. 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.
Contractible / irreducible. It can be both right and wrong. Another thing is important, whether the numerator with the denominator has common factors. If any, then they are supposed to separate both parts of the fraction, that is, reduce it.
Mixed. An integer is assigned to its usual correct (incorrect) fractional part. Moreover, it always stands on the left.
Compound. It is formed from two fractions divided into each other. That is, it has three fractional features at once.
Decimal fractions have only two subspecies:
final, that is, one in which the fractional part is bounded (has an end);
infinite - a number whose digits after the decimal point do not end (they can be written indefinitely).
How to convert decimal to ordinary?
If this is a finite number, then an association based on the rule is applied - as I hear, I write. That is, you need to correctly read it and write it, but without a comma, but with a fractional line.
As a hint about the required denominator, you need to remember that it is always one and several zeros. The last need to write as many digits in the fractional part of the considered number.
How to convert decimal fractions to ordinary, if their integer part is absent, 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 fractional parts. For the first number, the denominator will be 10, for the second - 100. That is, the indicated examples with answers will have the numbers: 9/10, 5/100. Moreover, the latter can be reduced by 5. Therefore, the result for it is to write 1/20.
How to make a decimal fraction ordinary if its integer part is nonzero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its meaning is written. In the first case it is 5, in the second - 13. Then you need to move on to the fractional part. They are supposed to carry out the same operation. The first number appears 23/100, the second - 108/100000. The second value needs to be reduced again. In response, such mixed fractions are obtained: 5 23/100 and 13 27/25000.
How to convert infinite decimal to ordinary?
If it is non-periodic, then such an operation will not succeed. This fact is due to the fact that each decimal fraction is always converted either to the final or to the periodic.
The only thing allowed to 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 ordinary. But the reverse process: decimal conversion will never give an initial meaning. That is, infinite non-periodic fractions are not converted into ordinary fractions. This must be remembered.
How to write an infinite periodic fraction in the form of an ordinary fraction?
In these numbers, one or more digits always appear after the decimal point, which are repeated. They are called the period. For example, 0.3 (3). Here is "3" in the period. They belong to the class of 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 begins immediately from the comma. In the second, the fractional part begins with any numbers, and then the repetition begins.
The rule according to which an infinite decimal is to be written as an ordinary fraction will be different for the two types of numbers indicated. It is quite simple to write pure periodic fractions with ordinary fractions. As with the finite ones, they need to be converted: the period is written in the numerator, and the denominator will be the number 9, repeated as many times as the digits the period contains.
For example, 0, (5). The number does not have an integer part, so immediately you need to start fractional. Write 5 in the numerator, and one in the denominator 9. That is, the answer will be 5/9.
The rule about how to write ordinary decimal periodic fraction, which is mixed.
Count the digits of the fractional part before the period. They will indicate the number of zeros in the denominator.
Look at the length of the period. So many 9 will have a denominator.
Write the denominator: first nine, then zeros.
To determine the numerator, you need to record the difference of two numbers. All digits after the decimal point will be reduced, along with the period. Subtracted - it’s without a period.
For example, 0.5 (8) - write down the periodic decimal fraction as an ordinary fraction. In the fractional part before the period, there is one digit. So zero will be one. In the period, too, only one figure is 8. That is, nine is one. 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. The answer, for example, would have to be written 53/90.
How are ordinary fractions converted to decimal?
The simplest option is a number in the denominator of which is the number 10, 100 and so on. Then the denominator is simply discarded, and a comma is placed between the fractional and the 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. Not only the denominator, but also the numerator relies on the same number to multiply.
For all other cases, a simple rule comes in handy: divide the numerator by the denominator. In this case, two answers can be obtained: the final or periodic decimal fraction.
Common Fraction Actions
Addition and Subtraction
Students get to know them earlier than others. And at first fractions have the same denominators, and then they are different. General rules can be reduced to such a plan.
Find the smallest common multiple of denominators.
Record additional factors to all common fractions.
Multiply the numerators and denominators by their specific factors.
Add (subtract) the numerators of the fractions, and leave the common denominator unchanged.
If the numerator of the lesser is less than the subtracted one, then we need to find out whether we have a mixed number or the right fraction.
In the first case, the integer part must be occupied by one. Add the denominator to the numerator of the fraction. And then do the subtraction.
In the second, it is necessary to apply the rule of subtracting from a smaller number a larger one. That is, subtract the module to be reduced from the module subtracted, and put a “-" sign in response.
Closely look at the result of addition (subtraction). If you get the wrong fraction, then you are supposed to select the whole part. That is, divide the numerator by the denominator.
Multiplication and Division
Fractions do not need to be reduced to a common denominator. This simplifies the execution of actions. But they are still supposed 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 the numerators.
Multiply the denominators.
If a contractible fraction is obtained, then it is supposed to simplify it again.
When dividing, you must first replace the division by multiplication, and the divisor (second fraction) - by the inverse fraction (interchange the numerator and denominator).
Then act as if multiplying (starting from point 1).
In tasks where you need to multiply (divide) by an integer, the latter is supposed to be written as an incorrect fraction. That is, with the denominator 1. Then act as described above.
Decimal Actions
Addition and Subtraction
Of course, you can always turn a decimal fraction into an ordinary fraction. And act according to the plan already described. But sometimes it’s 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 to it.
Record fractions so that the comma is under the comma.
Add (subtract) as natural numbers.
Carry a 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.
To multiply, you need to write fractions one below the other, not paying attention to commas.
Multiply like natural numbers.
Put a comma in the answer, counting as many digits from the right end of the answer as there 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 there are in the fractional part of the divider.
Multiply the dividend by the same number.
Divide the decimal by a natural number.
Put a comma in the answer at the moment when the division of the whole part ends.
What if in one example there are both kinds of fractions?
Yes, in mathematics there are often examples in which you need to perform actions on ordinary and decimal fractions. In such tasks, two solutions are possible. We need to objectively weigh the numbers and choose the optimal one.
The first way: to represent ordinary decimal
It is suitable if the division or translation results in finite fractions. If at least one number gives the 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: to write decimal fractions with ordinary
This technique is convenient if there are 1-2 digits in the part after the decimal point. If there are more of them, you can get a very large ordinary fraction and decimal records will allow you to count the task faster and easier. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.