Associative Property of Multiplication

There are two sets of mathematical concepts which you will study, which are numbers or objects and operators. Every mathematical operation is a symbolic manipulation of abstract objects by certain operators. The most basic of objects are natural and whole numbers while the most basic of mathematical operations are division, multiplication, addition and subtraction. Their corresponding operators (/, x, +, - ) are applied to numbers to form mathematical expressions. A linking of such expressions according to the 'equal to (=)' relation creates an equation in mathematics. Each of the basic mathematical operations are described by certain properties, which define their usage in various circumstances. To fully understand a mathematical operation like multiplication, all its properties must be known.

I assume that you know the basic concept of multiplication and what it's all about. Still I provide a brief review here. As an example, consider the expression (2 x 2), which is 2 multiplied by 2. An equivalent way of saying the same thing would be '2 added 2 times'. So (2 x 2) is in a way, (2 + 2), which makes it '4'. Thus the product of 2 multiplied by 2 is 4. Thus one can see that multiplication is actually addition. This operation of multiplication becomes simpler when you know what are known as multiplication tables.

Definition

The associative property of multiplication deals with the way numbers are grouped together in brackets when being multiplied. Here is the property stated in its simplest form:

"Irrespective of how numbers being multiplied are grouped together, the end product of multiplication remains the same."

Such properties of mathematical operations are understood better, when presented in the form of an equation:

"a x (b x c) = (a x b) x c"

Here a, b and c are variables or numbers being multiplied with each other. As a rule in arithmetic, the expression in the bracket is always solved first and the rest of the expressions are simplified later. So in the above equation, even when (b x c) expression is calculated first or when (a x b) expression is calculated first, the end result remains the same. This may seem trivial, but it is necessary that this property be stated for completeness. Let me illustrate what this property means, with some examples, in the following section.

Examples

Actual worked examples make it easier for any student of mathematics to understand any math term or concept deeply. Here are some examples which demonstrate the property.

• 24 x (2 x 3) = (24 x 2) x 3
• (10 x 8) x 9 = 10 x (8 x 9)
• m x n x (p x q) = m x (n x p) x q
• 25 x (2 x 1) = (25 x 2) x 1

These examples actually demonstrate how you are free to group terms, according to your convenience. This is the most subtle of all multiplication properties, which allows for easy grouping of multiplicative terms, making it easier for you to calculate the end result.