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Commutative Property of Multiplication You Have Probably Forgotten

Omkar Phatak Apr 23, 2019
Multiplication as a mathematical operation has many properties, prime amongst which is the commutative property, explained further in this write-up.
One of the most basic of mathematical operations, multiplication is one of the most essential subjects of study. Addition, multiplication, subtraction, and division are the four operations, which need to be a part of your mathematical arsenal. The aim of this post is to bring multiplication operation under the scanner and explain its commutative property.

Definition

When you multiply two numbers and change the order, does the end product change? Or in other words, does order matter when multiplying numbers? That's the question which this property deals with. In simple words, the property is stated as follows -

'Irrespective of the order in which you multiply the numbers, the end product remains the same.'
In terms of an equation, it states that, for any two variables or numbers - a and b,

(a x b) = (b x a)

It could be restated in the following way - for the variables or numbers p,q, and r,

(p x q) = r = (q x p)
This is perhaps the most important property of multiplication which you need to know about. The term 'Commutative' itself means 'independent of factor order'. Multiplying numbers is actually carrying out addition in another way.
Since addition shows the property of commutativity, multiplication, as a form of addition, inherits it. Let us take a look at some examples that demonstrate the implications of this property.

Examples

  • m x n = n x m
  • 5 x 8 = 8 x 5 = 40
  • 3/5 x 2/9 = 2/9 x 3/5 = 2/15
  • a x b x c = b x c x a = c x a x b = b x a x c
  • 2.3 x 10 = 10 x 2.3 = 23
  • 7.8 x 2/39 = 2/39 x 7.8 = 0.4
Thus, the commutative property makes it possible to factorize a number in any order. As discussed before, if you delve into it deeply, you will realize that multiplication is actually addition in another form.
That's the reason why multiplication and addition have this property, while subtraction and division don't have it. In the latter two operations, order matters. To sum it all up, this property lets you multiply numbers in any order.