Speaking of mathematics, it is impossible not to remember 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 for working with fractions, how you memorized and applied the main property of a fraction. How many nerves were spent to find a common denominator, especially if there were more than two terms in the examples!
Let's remember what it is and refresh our memory a little about the basic information and rules for working with fractions.
Definition of fractions
Let's start with the most important thing - definitions. A fraction is a number that consists of one or more unit parts. A fractional number is written as two numbers separated by a horizontal or 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 into, and the numerator shows the number of shares or parts taken. Often fractions, if 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", let's talk about the types of fractions and their features.
What are fractions
There are several types of such numbers. First of all, these are ordinary and decimal. The first ones represent the type of recording of a rational number already indicated by us using a horizontal or slash. The second type of fractions is indicated using 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.
Here it is worth noting that in mathematics both decimal and ordinary fractions are used equally. The main property of the fraction is valid only for the second option. In addition, in ordinary fractions, right and wrong numbers are distinguished. For the former, the numerator is always less than the denominator. Note also that such a fraction is less than unity. In an improper fraction, on the contrary, the numerator is greater than the denominator, and it itself is greater than one. In this case, an integer can be extracted from it. In this article, we will consider only ordinary fractions.
Properties of fractions
Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers are no exception. They have one important feature, with the help of which it is possible to carry out certain operations on them. What is the main property of a 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 two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, while they will be equal.
Based on this property, you can reduce fractions, as well as select common denominators for a particular pair of numbers.
Despite the fact that fractions seem to us more complex than prime numbers, they can also perform basic mathematical operations, 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. Knowing these laws makes it easier to work with fractions, making it easier and more interesting. That is why further we will 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, let's analyze such an operation as reduction to a common denominator. This is where the knowledge of what basic property of a fraction exists will come in handy.
In order to reduce a number to a common denominator, you first need to find the least common multiple of the two denominators. That is, the smallest number that is simultaneously divisible by both denominators without a remainder. The easiest way to pick up NOC(least common multiple) - write out in a line the numbers that are multiples for one denominator, then for the second and find a matching number among them. In the event that the LCM 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 LCM.
So, we have found the LCM, now we need to find an additional multiplier. To do this, you need to alternately divide the LCM into denominators of fractions and write down the resulting number over each of them. Next, multiply the numerator and denominator by the resulting additional factor and write the results as a new fraction. If you doubt that the number you received is equal to the previous one, remember the basic property of the fraction.
Now let's go directly to mathematical operations on fractional numbers. Let's start with the simplest. 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 the fractions have different denominators, you should bring them to a common one and only then perform addition. How to do this, we have discussed with you a little higher. In this situation, the main property of the fraction will come in handy. The rule will allow you to bring the numbers to a common denominator. This will not change the value 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 of fractions does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of a fraction. It is enough to first multiply the numerators and denominators together. In this case, the product of the numerators will become the new numerator, and the product of 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. We will talk about how to reduce fractions a little later.
When subtracting fractions, you should be guided by the same rules as when adding. So, in numbers with the same denominator, it is enough to subtract the numerator of the subtrahend from the numerator of the minuend. In the event that the fractions have different denominators, you should bring them to a common one and then perform this operation. As with addition, you will need to use the basic property of an algebraic fraction, as well as skills in finding the LCM and common factors for fractions.
And the last, most interesting operation when working with such numbers is division. It is quite simple and does not cause any particular difficulties even for those who do not understand how to work with fractions, especially to perform addition and subtraction operations. When dividing, such a rule applies as multiplication by a reciprocal fraction. The main property of a fraction, as in the case of multiplication,will not be used for this operation. Let's take a closer look.
When dividing numbers, the dividend remains unchanged. The divisor is reversed, i.e. the numerator and denominator are reversed. After that, the numbers are multiplied with each other.
So, we have already analyzed the definition and structure of fractions, their types, the rules of operations on these numbers, found out the main property of an algebraic fraction. Now let's talk about such an operation as reduction. Reducing a fraction is the process of converting it - 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 resulting fraction or not. Remember that the final result is always written as a fractional number that does not require reduction.
Finally, we note that we have not listed all operations on fractional numbers, mentioning only the most famous and necessary ones. Fractions can also be compared, 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 have given above.
We talked about fractional numbers and operations with them. We also disassembled the main property of a fraction,reduction of fractions. But we note that all these questions were considered by us in passing. We have given 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 "fill" your head with endless rules and formulas, which, most likely, will not be useful to you.
We hope that the material presented in the article simply and concisely has become useful for you.