Commutative property

Introduction

In mathematics, a binary operation is commutative if the result is unaffected by the order of the operands. Many binary operations have it as a fundamental property, and many mathematical arguments rely on it. The property is best known for its name, which says something like “3 + 4 Equals 4 + 3″ or ”2 × 8 = 8 × 2″, but it can also be utilized in more complicated circumstances. There is a need for the word since some operations, such as division and subtraction, do not have it (for example,”3 − 5 ≠ 5 − 3”); these operations are referred to as non-commutative operations because they are not commutative. For many years, the concept that simple operations like multiplication and addition are commutative was taken for granted.

Formula for Commutative Property

If two numbers a and b are given, the commutative property of numbers formula is as follows:

a + b = b + a

a × b = b × a

a – b ≠ b – a

a ÷ b ≠ b ÷ a

The commutative property formula asserts that when addition or multiplying two integers, the order in which they are added or multiplied has no effect on the result. However, when subtracting or dividing any two real numbers, the sequence of the numbers is crucial, and it cannot be modified.

Commutative property logic

In mathematics, commutative law logic refers to one of two laws relating to number operations of addition and multiplication: a + b = b + a and ab = ba. Rearranging the terms or components has no effect on any finite sum or product, according to these principles.

Rearranging the terms or components has no effect on any finite sum or product, according to these principles. While commutativity holds for many systems, such as real or complex numbers, commutativity of multiplication is invalid for others, such as the system of  n × n matrices or the system of quaternions. 

Vector multiplication (to generate the cross product) is commutative i.e., a·b = b·a, whereas scalar multiplication (to give the dot product) is not. 

Multiplication of conditionally convergent series does not always follow the commutative law. Also see distributive law and associative law.

Commutative property to rewrite an expression

Consider the addition of two numbers, such as 5 and 3.

5+3         3+5

8                8

The outcome is the same.5+3 =3+8

It’s worth noting that the order in which we add things doesn’t matter. When multiplication, the same is true. When multiplying 5 and 3, the same holds true.

3×5             5×3               

15                      15 

.The results are the same as before. 3×5  =   5×3               

It makes no difference in which sequence we multiply.

The commutative properties of addition and multiplication are demonstrated in these examples.

COMMUTATIVE PROPERTIES

Commutative Property of Addition: if a & b are real numbers, then

a+b=b+a 

Commutative Property of Multiplication: if a & b are real numbers, then

ab=ba 

Example of Commutative property

Example 1: John’s mother questioned if the commutative property applies to the addition of two natural integers. Can you assist John in determining whether or not it is commutative?

Solution: We know that the commutative property of addition asserts that changing the order of the addends has no effect on the sum’s value. 

Take any two natural numbers, such as 2 and 5, and multiply them together to get 

2 + 5 = 7 = 5 + 2. 

As a result, the commutative property is demonstrated by the addition of two natural integers.

Conclusion

In this artcle we learn, the purpose of commutative property is that the numbers we work with can be moved or exchanged from their original positions without affecting the result. The feature only applies to addition and multiplication, not division or subtraction. The simplest of multiplication properties is the commutative property. It has an intuitive logic and an instant impact: it decreases the amount of independent basic multiplication facts that must be memorized.