Introduction

The term "commutative" comes from the word "commute," which means "to move around." As a result, the commutative property is concerned with moving the numbers around. So, if changing the order of the operands does not affect the result of the arithmetic operation, that arithmetic operation is commutative.

Other properties of numbers include the associative property, the distributive property, and the identity property. They are not the same as the commutative property of numbers.

For example, 4 + 5 equals 9, and 5 + 4 equals 9. The order in which two numbers are added does not affect the sum. The same idea applies to multiplication. The commutative property does not apply to subtraction and division because the results are completely different when the numbers are ordered differently. In this tutorial, we will discuss the commutative property.

The word "commutative," which means "to move around," derives from the verb "commute." The commutative condition, therefore, cares about changing the positions of the integers. Therefore, an arithmetic operation is commutative if altering the operands' sequence has no impact on the computation's outcome.

The associative, proportional, and identity qualities are additional characteristics of numbers. They are distinct from a number's common characteristics.

Commutative Law

Only addition and multiplication processes are subject to the commutative law in mathematics. The commutative principle does not apply to subtraction and division because the outcomes are entirely different when the numbers are arranged differently.

The commutative principle states that no matter how a and b are arranged, their addition and multiplication produce the same outcome if a and b are any two integers. It can be written symbolically as:

For example, 6 +3 = 9, and 3+6 is also equal to 9. The total is unaffected by the arrangement of the two integers as they are added. The same principle holds for multiplying. 

Mathematical Operations

Subtraction

The subtraction method is used to determine the difference between two numbers. 

The minus symbol (-) stands for subtraction. It is the conjugate of the second term and is virtually similar to addition but opposite to it.

The word is added to the negative term in subtraction. This process is primarily used to calculate the number of things left after some are taken away.

Rules of Subtraction 

• The sum of the two positive numbers also contains a positive integer.

• The sum of the other two negative numbers also contains a negative value.

• To subtract integers, add positive and negative integers, and add positive and negative integers, use the value of the largest integer number.

Addition

• Addition is the action of placing objects (numbers) together. 

• The adding procedure is symbolized by the "+" sign. It involves combining two or more integers to create a single expression. Furthermore, it doesn't matter what sequence the procedures are done in.

• It indicates that adding is a commutative operation. You can use any kind of integer, including fractions, decimals, and real and complex values.

Rules of Addition

• A positive integer also exists in the total of two positive integers.

• A negative number also exists in the total of two other negative integers.

• Use the sign of the biggest integer number to subtract the integers while adding the positive and negative integers.

Multiplication

• Repeated addition is another name for multiplication. 

• The phrase is denoted by "x". To create a singular value, it can also be mixed with two or more other values. 

• The processes of multiplying incorporate the multiplicand and multiplier. The product is the outcome of combining the multiplicand by the factor.

Rules of Multiplications

• A positive integer is produced when two positive numbers are multiplied.

• Multiplying two negative integers yields a positive integer.

• A positive and a negative number combine to form a negative integer.

Division 

• The division symbol ‘÷’ is typically used to indicate the opposite of multiplication. 

• Its two-component terms, dividend, and divisor, are multiplied together to yield a single-term number.

• If the yield is greater than the divisor, the outcome is greater than one; otherwise, the result is less than one.

Rules of Division:

• Divide two positive integers to get a positive integer.

• A positive integer is obtained by dividing two negative numbers.

• When two numbers with different signs are divided the resulting number is negative.

Solved Examples

1. Let us assume a situation where we have a class of 15 students, comprising 12 girls and 3 boys. The maths teacher introduces the class to commutative property by undertaking an experiment.

A teacher asks two students to give the total number of boys and girls. 

The first student informs the following data 

no of boys =3 and no of girls = 12

total students = 3+12=15 students

Then, the Second student gives the following data 

no of girls = 12 and no of boys =3

total students = 12+3 =15 students

So from both scenarios, the teacher says that adding 3 with 12 gives the same result when 12 is added to 3 which is 15.

So the commutative property is shown in this way.

The given equation is 12+3=3+12.

RHS=3+12

RHS=15

and LHS=12+3

LHS=51

Since LHS=RHS

It follows the commutative property.

2. Show that the following concept follows the commutative property.

 Imagine yourself in a shop where you need to buy 3 packets of eggs and each pack contains 6 eggs.

So the total number of eggs is given by the product 3 x 6 = 18 eggs 

So if you find the product by multiplying 6 by 3 or 3 by 6 the product will be the same so the given example follows the commutative property.

The given equation is 63=36.

RHS=36

RHS=18

and LHS=63

LHS=18

Since LHS=RHS

It follows the commutative property.

Summary

The addition and multiplying arithmetic processes are addressed by the commutative property. This implies that the outcome of adding or multiplying two numbers remains unchanged if the sequence or position of the numbers is changed.  The sum is unaffected by the sequence in which two integers are added. The idea behind multiplying is the same. For subtraction and division, the commutative condition does not apply because the outcomes are entirely different when the numbers are changed in sequence.

Frequently Asked Questions

1. Can the commutative property apply to three numbers?

The order of three integers is described by the associative property rather than the commutative property. The commutative principle says that we can change the positions of the two numbers in an addition or multiplication operation without altering the result.

2. What distinguishes commutative property from the associative property?

According to the associative characteristic of addition, you can group additional numbers in various methods without changing the result. According to the commutative property of addition, you can rearrange the addends without altering the result.

3. What distinguishes commutative property from the distributive property?

The Distributive Property does not have two variations (one for addition and another for multiplication), unlike the Associative and Commutative Properties. Instead, the same concept applies to addition and multiplication. This is accurate according to the Distributive Property because they dispersed through the brackets.