Closure and Commutative Properties of Real Numbers: Meaning and Examples
Closure and Commutative Properties of Real Numbers: Real numbers possess particular properties used in different operations in Mathematics, such as addition, subtraction, multiplication, and division. It is important to have a good understanding of these Mathematical operations in order to make calculations easier for us. The commutative property involves working with the arithmetic operations of addition and multiplication. This means that switching the sequence or position of two numbers when adding or multiplying them will not change the final output.
The closure property can be explained as the closure of a set of numbers by using arithmetic operations and executed and closed by these operations. In this article, we will explain to you the properties of real numbers, namely the closure property and commutative property of real numbers, and their meaning, along with some examples.
What is the Closure Property of Real Numbers?
According to the closure property of real numbers, a set of numbers when closed under any Mathematical operation like addition, subtraction, multiplication, and division, and done on any two numbers of that set having the answer that is another number from that same set. If there are two or more real numbers and we perform any one of the operations of addition, subtraction, multiplication, or division on them and the result is also a real number, then we say that the set of real numbers is closed for that operation.
Let us take an example. Assume x, y, and z to be real numbers. Now let us look at the closure property with respect to these three real numbers.
x + y = z, for y ≠ 0, will always be real for any real values of x and y.
3+4 = 7. These are all a set of real numbers. Hence, the set of real numbers for the operation of addition is closed.
x × y = z. The values of x and y will always be real for any real values of x and y.
4 × 2 = 8. Hence, the set of real numbers is closed for the operation of multiplication.
x – y = z, will always be real for any values of x and y.
8 – 4 = 4. Hence, the set of real numbers for the operation of subtraction is closed.
x ÷ y = z, where y ≠ 0, will always be real for any real values of x and y.
7 ÷ 10 = 7/10, which is a real number. Hence, the set of real numbers for the operation of division is closed, except for when divided by 0.
What is the Commutative Property of Real Numbers?
The origin of the word commutative is from the word commute, which means moving around. Thus, it involves the movement of numbers around. In this section, we will learn about the commutative properties of the different operations for a set of real numbers. The commutative law means that you can swap numbers over and still get the same result when you add or when you multiply two real numbers.
For unknowns x and y, x × y = y × x. Let us take the area of a rectangle as an example. For a rectangle with the shorter side, x = 5, and the longer side, y = 10, we have:
x × y = 5 × 10 = 50.
At the same time, y × x = 10 × 5 = 50.
Therefore, x × y = y × x
For unknown, x and y, x + y = y + x.
For unknowns x and y, x + y = y + x. We know that the perimeter of a rectangle = 2.(x + y)
Consider a rectangle with the shorter side, x = 7, and the longer side, y = 9, the perimeter of the rectangle = 2.(x + y) = 2.(7 + 9) = 2.(16) = 32.
At the same time, 2.(y + x) = 2.(9 + 7) = 2.(16) = 32.
Therefore, x + y = y + x.
Note: There is an important point to be noted here, which is that commutative law is not applicable to division and subtraction. Let us find out why this is the case.
Let x/y = z. For x = 21 and y = 7, we get z = 21/7 = 3.
In this way, let y/x = w. Therefore, w = 7/21 = 1/3.
Hence, we can see that z ≠ w. Therefore, x/y ≠ y/x.
Let x – y = z and y – x = w.
For x = 17 and y = 9, z = 17 – 9 = 8. Also, w = 9 – 17 = -8.
Thus, z ≠ w.
Therefore, x – y ≠ y – x
We hope that this article on the closure and commutative properties of real numbers has proven to be helpful to you and has provided a great insight into the topic. Please remember that understanding the basic principles behind any topic from any chapter or subject is the most important aspect of learning the concepts. Also, practice is the best way to master Mathematics.
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