Despite the fact that mathematics seems to be a difficult science for most people, this is far from the case. Many mathematical operations are fairly easy to understand, especially if you know the rules and formulas. So, knowing the multiplication table, you can quickly multiply large numbers in the mind . The main thing is to constantly train and do not forget the rules of multiplication. The same can be said of division.
Let's look at the division of integers, fractional and negative. Recall the basic rules, techniques and methods.
Division operation
Let's start with the very definition and name of the numbers that participate in this operation. This will greatly facilitate the further presentation and perception of information.
Division is one of four basic mathematical operations. Her study begins in elementary school. It is then that children are shown the first example of dividing a number by a number, the rules are explained.
Two numbers are involved in the operation: the dividend and the divisor. The first is the number that is divided, the second is the number that is divided. The result of division is quotient.
There are several notations for recording this operation: “:”, “/” and a horizontal line - a fractional record when the dividend is at the top and the divisor is below.
rules
When studying a particular mathematical operation, the teacher must familiarize students with the basic rules that you should know. True, they are not always remembered as well as we would like. That is why we decided to refresh four fundamental rules in your memory a little.
The basic rules for dividing numbers, which you should always remember:
1. It is impossible to divide by zero. This rule should be remembered first.
2. You can divide zero by any number, but in the end there will always be zero.
3. If the number is divided by one, we get the same number.
4. If the number is divided by itself, we get a unit.
As you can see, the rules are quite simple and easy to remember. Although some may forget such a simple rule as the impossibility of dividing by zero, or else confuse the division of zero by a number with it.
Signs of divisibility by number
One of the most useful rules is a feature that determines the possibility of dividing a natural number by another without a remainder. So, they distinguish signs of divisibility by 2, 3, 5, 6, 9, 10. Let us consider them in more detail. They greatly facilitate the operation of numbers. We also give for each rule an example of dividing a number by a number.
These rules-signs are quite widely used by mathematicians.
Sign of divisibility by 2
The easiest sign to remember. A number that ends in an even digit (2, 4, 6, 8) or 0 is always divided into two integers. Pretty simple to remember and use. So, the number 236 ends in an even number, which means it is divided into two completely.
Check: 236: 2 = 118. Indeed, 236 is divisible by 2 with no remainder.
This rule is best known not only for adults, but also for children.
Sign of divisibility by 3
How to correctly divide numbers by 3? Remember the following rule.
A number is divided by 3 completely if the sum of its digits is a multiple of three. For example, take the number 381. The sum of all the numbers will be 12. This number is a multiple of three, which means it is divided by 3 without a remainder.
Also check this example. 381: 3 = 127, then everything is true.
Sign of divisibility of numbers by 5
Everything is also simple here. You can divide by 5 without a remainder only those numbers that end in 5 or 0. For example, take numbers like 705 or 800. The first ends with 5, the second ends with zero, therefore they are both divided by 5. This is one of the simplest rules, which allows you to quickly divide by a single number 5.
Let's check this symptom with the following examples: 405: 5 = 81; 600: 5 = 120. As you can see, the sign is valid.
Divisibility by 6
If you want to know if a number is divisible by 6, then you first need to find out if it is divisible by 2, and then by 3. If so, then the number can be divided by 6 without a remainder. For example, the number 216 is divisible by 2 , since it ends with an even digit, and 3, since the sum of the digits is 9.
Check: 216: 6 = 36. An example shows that this symptom is valid.
Divisibility by 9
We will also talk about how to divide numbers by 9. Those natural numbers whose sum of digits is a multiple of 9. are divided by a given number. Similar to the rule of division by 3. For example, the number 918. Add all the numbers and get 18 - a multiple of 9. Hence, it is divisible by 9 without a remainder.
We will solve this example for verification: 918: 9 = 102.
Divisibility by 10
The last sign worth knowing. Only those numbers ending in 0 are divisible by 10. This pattern is quite simple and easy to remember. So, 500: 10 = 50.
That’s all the main signs. Remembering them, you can make your life easier. Of course, there are other numbers for which there are signs of divisibility, but you and I have identified only the main ones.
Division table
In mathematics, there is not only a multiplication table, but also a division table. Having learned it, you can easily perform operations. In fact, the division table is a multiplication table on the contrary. To compose it yourself is not difficult. To do this, rewrite each row from the multiplication table as follows:
1. Put the product of the number in first place.
2. Put a division sign and write the second factor from the table.
3. After the equal sign, write the first factor.
For example, take the following row from the multiplication table: 2 * 3 = 6. Now we rewrite it according to the algorithm and get: 6 ÷ 3 = 2.
Quite often, children are asked to compose a table on their own, thus developing their memory and attention.
If you don’t have time to write it, then you can use the article in this article.
Division types
Let's talk a little about the types of division.
To begin with, we can distinguish the division of integers and fractional numbers. Moreover, in the first case, we can talk about operations with integers and decimal fractions, and in the second, only about fractional numbers. In this case, the dividend or divisor, and both at the same time, can be fractional. This separation is due to the fact that operations on fractions differ from operations on integers.
Next, we will talk about fractionation in more detail.
Based on the numbers that participate in the operation, two types of division can be distinguished: single-digit numbers and multi-digit ones. The simplest is the division by a single digit. Here you will not need to carry out cumbersome calculations. In addition, the division table can help. Dividing into other - two-, three-digit numbers is harder.
Consider examples for these types of division:
14: 7 = 2 (division by a single number).
240: 12 = 20 (division by a two-digit number).
45387: 123 = 369 (division by a three-digit number).
The last division can be distinguished, in which positive and negative numbers participate. When working with the latter, you should know the rules by which a result is assigned a positive or negative value.
When dividing numbers with different signs (the dividend is a positive number, the divisor is negative, or vice versa), we get a negative number. When dividing numbers with one sign (both the dividend and the divisor are positive or vice versa), we get a positive number.
Consider the following examples for clarity:
21: (- 7) = -3
-36: 6 = (-6)
-48: (-8) = 6.
Fraction division
So, we have examined the basic rules, given an example of dividing a number by a number, now we’ll talk about how to correctly perform the same operations with fractions.
Despite the fact that the division of fractions at first seems rather difficult, in reality, working with them is not so difficult. The division of the fraction is performed in almost the same way as the multiplication, but with one difference.
In order to divide the fraction, you must first multiply the numerator of the dividend by the denominator of the divisor and fix the result as a numerator of the quotient. Then multiply the denominator of the dividend by the numerator of the divisor and write the result as the denominator of the quotient.
It can be made easier. Rewrite the fraction of the divisor by interchanging the numerator with the denominator, and then multiply the resulting numbers.
For example, we divide two fractions: 4/5: 3/9. To begin with, we flip the divider, we get 9/3. Now multiply the fractions: 4/5 * 9/3 = 36/15.
As you can see, everything is quite easy and no more complicated than dividing by a single digit. Examples of actions with fractions are solved simply if you do not forget this rule.
conclusions
Division is one of the mathematical operations that every child learns in elementary school. There are certain rules that you should know, techniques that facilitate the implementation of this operation. Division happens with the remainder and without, there is a division of negative and fractional numbers.
Remembering the features of this mathematical operation is quite easy. We have examined the most important points, examined more than one example of dividing a number by a number, and even talked about how to work with fractional numbers.
If you want to improve your knowledge of mathematics, we advise you to remember these simple rules. In addition, we can advise you to develop memory and mental counting skills by performing mathematical dictations or simply trying to calculate the verbally quotient of two random numbers. Believe me, these skills will never be superfluous.