In mathematics, there are several discussions about arithmetic operations. The arithmetic operations in mathematics generally have several properties in them. Starting from commutative, associative, and also distributive properties. Then, what is meant by commutative property? What is the definition of associative property? What is the meaning of the distributive property? What is an example of the commutative property? Where all three apply to the arithmetic operating system of division, subtraction, addition, and also multiplication. To work on various kinds of problems related to commutative properties, there are many different ways or methods.
These three properties are indeed different, because they have the goal of making learning arithmetic operations easier for basic education. In learning mathematics, generally teachers will use more integers so that solving problems given to students does not cause them difficulties. In material for mathematical arithmetic operations, there will usually be an explanation of the meaning of commutative properties, overall understanding, examples of problems, and so on.
As we already understand, mathematical operations such as multiplication, subtraction, addition, and division have various properties in them. This property is useful in learning mathematics that uses more integers. However, you need to know that the three properties above have different ways of working and methods for each arithmetic operation.
In this article, we will discuss what the commutative property is, along with a brief explanation of the other two properties, such as associative and distributive properties. To be clearer, you can read the article below:
Definition of Commutative Properties
In general, Mathematics has basic arithmetic operations such as subtraction, addition, division, and multiplication. The arithmetic operations apply to the form of algebraic numbers, fractions, and so on. Because, its use is very broad, therefore the way to do it in each arithmetic operation also varies depending on the form of the number. But on the other hand, there are also several properties that are used in each arithmetic operation such as commutative properties, associative properties, and also distributive properties.
As we discussed earlier, that mathematical arithmetic operations have several properties in them. Starting from commutative, distributive, and associative. In the following, there are several explanations regarding the notion of commutative properties, distributive properties, and associative properties along with examples of problems.
Commutative or Exchange Properties
The first thing we will discuss is the definition of the commutative property. So, commutative is the property of arithmetic operations that is used to swap the location of two numbers so that the resulting value is the same. The commutative property can also be called the commutative law. The following is the commutative property written by the formula:
a + b = b + a = c
a and b are two numbers to be operated on
c is the result of arithmetic operations
note: The commutative nature of arithmetic operations has a provision that even if the numbers being counted have their positions interchanged, the results will be the same.
The commutative property basically exists in the arithmetic operations of multiplication and addition. This is because the concepts in this nature fulfill the conditions in the arithmetic operation. The following is a full explanation:
The Commutative Property of Addition
After explaining the meaning of the commutative property, then we will discuss the application of the commutative property in addition operations. The following is the addition formula using the commutative property:
a + b = b + a = c
So that we can better understand the above formula, here are some examples of commutative problems in addition. For more details, see examples of addition problems that use the commutative property below:
a. 4 + 5 = 5 + 4 = 9, where 4 + 5 = 9 and 5 + 4 = 9
b. 7 + 8 = 8 + 7 = 15, where 7 + 8 = 15 and 8 + 7 = 15
The Commutative Property of Multiplication
The operation to calculate multiplication also uses the commutative property in it. The following is a formula for the commutative property that uses the multiplication arithmetic operation:
a × b = b × a = c
In order to better understand the above formula, in the following there will be an example of a problem regarding the commutative property in multiplication. An example of a multiplication problem that uses the commutative property is as follows:
a. 2 x 3 = 3 x 2 = 6, where 2 x 3 = 6 and 3 x 2 = 6
b. 4 x 5 = 5 x 4 = 20, where 4 x 5 = 20 and 5 x 4 = 20
As previously discussed, that in the sense of the commutative property above it only applies to addition and multiplication arithmetic operations. Therefore, division as well as subtraction of integers will not apply the commutative property. This is because in this operation there are unequal values if the numbers are exchanged. For example, as below:
a. 5 – 3 = 2 is different from 3 – 5 = (-2)
b. 9 : 3 = 3 in contrast to 3 : 9 = 0.33
Examples of Commutative Properties in Addition and Multiplication
The following are some examples of commutative property problems in addition and multiplication:
1. Examples of Commutative Properties in the Addition of Positive/Negative Integers
Below is an example of a commutative problem in adding positive or negative integers. Check out the full explanation to make it easier to understand.
a. Examples of Commutative Properties in Adding Positive Integers with Positive
a + b = b + a
2 + 3 = 3 + 2
2 + 3 = 6 and 3 + 2 = 6
In the pattern mentioned above, whether the numbers 2 or 3 are in front or behind, then the result of two plus three or three plus two is equal to six.
b. Examples of Commutative Properties in Addition of Positive and Negative Integers
a + b = b + a
4 + (-6) = -6 + 4
4 + (-6)= -2 and -6 + 4= -2
c. Examples of Commutative Properties in Adding Negative Integers with Negatives
a + b = b + a
-2 + -5 = -5 + -2
-2 + -5= -7 and -5 + -2 also = -7
2. Examples of Commutative Properties in Multiplication of Positive and Negative Numbers
The following are examples of questions about the commutative property of multiplying positive and negative numbers. For more details, see the full explanation below.
a. Examples of Commutative Properties in Multiplying Positive Integers with Positives
axb = bxa
4 x 5 = 5 x 4
4 x 5 = 5 + 5 + 5 + 5 = 20
5 x 4 = 4 + 4 + 4 + 4 + 4 = 20
4 x 5 = 20 and 5 x 4 also equals 20
In the pattern mentioned above, whether the number 4 or 5 is in front or behind, then the result of four times five or five times four is equally twenty
b. Examples of Commutative Properties in Multiplying Positive and Negative Integers
axb = bxa
2 x -5 = -5 x 2
2 x -5 = -10 and -5 x 2 also = -10
c. Examples of Commutative Properties in Multiplying Negative and Negative Integers
axb = bxa
-3 x -4 = -4 x -3
-3 x -4 = 12 and -4 x -3 also = 12
Why doesn’t the commutative property apply to subtraction and division?
We cannot apply the commutative property to subtraction and division. Because if the commutative nature of arithmetic operations, subtraction or division is applied, the results will not be the same.
Here’s the proof:
1. The commutative property formula cannot be applied to subtraction arithmetic operations because a – b ≠ b – a ( a minus b the result is not the same as b minus a)
a – b ≠ b – a
10 – 5 ≠ 5 – 10
10 – 5 = 5, while 5 – 10 = -5
Until here, explain that the results of 10-5 are not the same as the results of 5-10
2. The commutative property formula cannot be applied to division arithmetic operations because a : b ≠ b : a ( a divided by b the result is not the same as b divided by a)
a : b ≠ b : a
20 : 4 ≠ 4 : 20
20 : 4 = 5, while 4 : 20 = 0.2
Mathematics is a basic science that plays a very important role in the development of science and technology and advances human thinking. The presence of this book is expected to add to the reference and become a reference for students in particular and those who are interested in mathematics in general.
This book briefly presents theories and mathematical problem solving related to: number systems, graphs, functions, limits, derivatives (differentials), use of derivatives, transcendent functions, integrals, integration techniques, use of integrals, conic sections and polar coordinates, derivative in n-dimensional space, integral in n-dimensional space,
This book is different from other Applied Mathematics books because this book has advantages in its study. The theory given is brief and concise and accompanied by examples and solutions that are complete and complete.