Speaking of mathematics, one cannot help but recall fractions. Their study is given a lot of attention and time. Remember how many examples you had to solve in order to learn certain rules of working with fractions, how you memorized and applied the basic property of a fraction. How many nerves were spent finding a common denominator, especially if the examples had more than two terms!
Let's remember what it is and a little refresh in memory the basic information and rules for working with fractions.
Fraction Definition
Let's start with the most important thing - definitions. A fraction is a number that consists of one or more parts of a unit. A fractional number is written in the form of two numbers separated by a horizontal or a slash. In this case, the upper (or first) is called the numerator, and the lower (second) is called the denominator.
It is worth noting that the denominator shows how many parts the unit is divided, and the numerator shows the number of shares or parts taken. Often fractions, if they are correct, are less than one.
Now let's look at the properties of these numbers and the basic rules that are used when working with them. But before we analyze such a concept as “the main property of a rational fraction”, we will talk about the types of fractions and their features.
What are the fractions
There are several types of such numbers. First of all, these are ordinary and decimal. The first ones are the form of recording a rational number that we have already indicated using a horizontal or slash. The second kind of fractions is indicated with the help of the so-called positional notation, when the integer part of the number is indicated first, and then, after the decimal point, the fractional part is indicated.
It is worth noting that in mathematics, both decimal and ordinary fractions are equally used. The main property of the fraction is valid only for the second option. In addition, in ordinary fractions, the correct and incorrect numbers are distinguished. For the former, the numerator is always less than the denominator. We also note that such a fraction is less than unity. In the wrong fraction, on the contrary, the numerator is larger than the denominator, and it itself is greater than unity. Moreover, an integer can be extracted from it. In this article we will consider only ordinary fractions.
Fraction Properties
Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers were no exception. They have one important feature, with the help of which one or another operation can be performed on them. What is the main property of the fraction? The rule says that if its numerator and denominator are multiplied or divided by the same rational number, we will get a new fraction, the value of which will be equal to the original value. That is, multiplying the two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, and they will be equal.
Based on this property, you can reduce fractions, as well as select common denominators for a particular pair of numbers.
Operations
Despite the fact that fractions seem more complicated to us, compared with prime numbers, you can also perform basic mathematical operations with them, such as addition and subtraction, multiplication and division. In addition, there is such a specific action as the reduction of fractions. Naturally, each of these actions is performed according to certain rules. The knowledge of these laws facilitates the work with fractions, makes it easier and more interesting. That is why we will further consider the basic rules and the algorithm of actions when working with such numbers.
But before talking about such mathematical operations as addition and subtraction, we will analyze such an operation as reduction to a common denominator. This is where the knowledge of what the basic property of the fraction exists is just what will come in handy for us.
Common denominator
In order to bring the number to a common denominator, you first need to find the least common multiple for the two denominators. That is the smallest number that is simultaneously divided by both denominators without a remainder. The easiest way to choose the NOC (the smallest common multiple) is to write the numbers that are multiple for one denominator, then for the second and find the same number among them. In the event that the NOC is not found, that is, these numbers do not have a common multiple, they should be multiplied, and the resulting value should be considered as the NOC.
So, we found the NOC, now we need to find an additional factor. To do this, you need to alternately divide the NOC into the denominators of the fractions and write down the resulting number over each of them. Next, multiply the numerator and denominator by the obtained additional factor and write the results in the form of a new fraction. If you doubt that the number you received is equal to the former, remember the basic property of the fraction.
Addition
Now we pass directly to the mathematical operations on fractional numbers. Let's start with the simplest one. There are several options for adding fractions. In the first case, both numbers have the same denominator. In this case, it remains only to add the numerators together. But the denominator does not change. For example, 1/5 + 3/5 = 4/5.
If fractions have different denominators, you should bring them to a common one and only then add them. How to do this, we have discussed a little higher. In this situation, the basic property of the fraction will come in handy for you. The rule allows you to bring the numbers to a common denominator. In this case, the value will not change in any way.
Alternatively, it may happen that the fraction is mixed. Then you should first add together the whole parts, and then the fractional ones.
Multiplication
Multiplication of fractions does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of the fraction. It is enough to first multiply the numerators and denominators among themselves. In this case, the product of the numerators will become the new numerator, and the denominators will become the new denominator. As you can see, nothing complicated.
The only thing that is required of you is knowledge of the multiplication table, as well as attentiveness. In addition, after receiving the result, you should definitely check whether this number can be reduced or not. About how to reduce fractions, we will talk a little later.
Subtraction
Performing the subtraction of fractions, one should be guided by the same rules as with addition. So, in numbers with the same denominator, it is enough to subtract the subtractor from the numerator of the decremented. If fractions have different denominators, you should bring them to a common one and then perform this operation. As in the similar case with addition, you will need to use the main property of the algebraic fraction, as well as the skills in finding the LCL and common divisors for fractions.
Division
And the last, most interesting operation when working with such numbers is division. It is quite simple and does not cause special difficulties even for those who are poorly versed in how to work with fractions, in particular, perform addition and subtraction operations. When dividing, a rule such as multiplication by the inverse fraction applies. The main property of the fraction, as in the case of multiplication, will not be used for this operation. Let's analyze in more detail.
When dividing numbers, the dividend remains unchanged. The divisor fraction turns into the opposite, that is, the numerator with the denominator swaps. After this, the numbers are multiplied among themselves.
Abbreviation
So, we have already sorted out the definition and structure of fractions, their types, the rules of operations on given numbers, and found out the basic property of an algebraic fraction. Now let's talk about such an operation as reduction. The reduction of a fraction is the process of its transformation - dividing the numerator and denominator by the same number. Thus, the fraction is reduced, without changing its properties.
Usually, when performing a mathematical operation, you should carefully look at the result obtained in the end and find out whether it is possible to reduce the fraction obtained or not. Remember that the fractional number that does not require reduction is always written in the final result.
Other operations
Finally, we note that we have listed far from all operations on fractional numbers, mentioning only the most famous and necessary. Fractions can also be equalized, converted to decimals, and vice versa. But in this article we did not consider these operations, since in mathematics they are carried out much less frequently than those that we presented above.
conclusions
We talked about fractional numbers and operations with them. We also examined the main property of the fraction, the reduction of fractions. But note that all these issues were examined by us in passing. We gave only the most famous and used rules, gave the most important, in our opinion, advice.
This article is intended to refresh the information you have forgotten about fractions, rather than give new information and "hammer" your head with endless rules and formulas that, most likely, will never come in handy.
We hope that the material presented in the article simply and concisely has become useful to you.