Commutative Property of Multiplication

The commutative property in multiplication states that changing the order of two numbers does not change the result. The term “commutative” is derived from the word “commute”, which means to switch places or interchange. In arithmetic, both addition and multiplication follow the commutative property.

Multiplication is one of the basic operations of mathematics and can be understood as repeated addition. Multiplication represents the sum of a number taken n times.

For example: 5 × 3 means add 5 three times: 5 + 5 + 5 = 15.

The numbers that are multiplied together are called factors, and the answer is called the product. Multiplication is extensively used in our daily life, such as in shopping, construction, and finance.
 

According to the commutative law of multiplication, when two or more numbers are multiplied, the result remains the same even if the order of the numbers is changed. Here, the order refers to the way the numbers are arranged in the multiplication expression. For example, 5 × 7 = 35, and 7 × 5 = 35. Hence, according to the commutative property of multiplication, 5 × 7 = 7 × 5. Here, even though the order is changed, the result remains the same.

The formula for the commutative property of multiplication indicates that the order of the numbers being multiplied does not affect the product obtained. All real numbers exhibit the commutative property of multiplication. The commutative property of multiplication can be mathematically expressed as:

A × B = B × A. 

The commutative property applies to two numbers at a time; for multiple numbers, re-grouping is required (associativity).

 For example, (5 × 4) × (3 × 5) = (3 × 5) × (4 × 5) = 300.

According to the commutative property of multiplication, changing the order of numbers being multiplied will not change the product. This is a basic property that applies to whole numbers, fractions, and decimals, simplifies computation, and is consistent with algebraic thinking. Here are a handful of steps to help with your understanding:
 

• Learn and memorize the basic formula A × B = B × A. This simple way of thinking helps you internalize the fact that the order of factors does not change the product.
   
• Use a range of numerical values that include positive numbers, negative numbers, fractions, and decimal values. A variety of samples will help you see the commutative property in action.
   
• To help with your visualization of multiplication, create a visual with an area model or arrays to help you see the multiplication as a rectangular area. A rectangle will help show the reorder of factors yields the same area or product.
   
• In problem-solving, trying to strategically rearrange the order of factors is a good way to both simply problems and think in your head when calculating problems.
   
• The commutative property only applies to addition and multiplication, so we would want to clarify what its scope is not applicable, which would be subtraction and division.

The commutative property of multiplication has numerous applications. Let us explore how the commutative property of multiplication is used in different areas:
 

• Shopping and Total Cost Calculation:
  
  When multiplying a quantity by a price to find the total cost, the order in which they are multiplied does not affect the total cost. This illustrates the commutative property, simplifying calculations while shopping.
   
• Arranging Objects in Rows and Columns:
  
  For seating arrangements, or stacking items, commutative property helps in organizing objects. It ensures that the order of objects does not affect the arrangement outcome.
   
• Construction and Area Calculation:
  
  We use this property when calculating area. For example, the area of a rectangular room is calculated by multiplying its length and width, mathematically expressed as: Area = Length × Width. This property helps in material estimation and space planning in construction projects.
   

• Recipes and Cooking:
  
  When adjusting a recipe, we often increase one or more of the ingredients in the recipe by multiplying them by a specific factor or scaling it. The order in which we do this does not matter due to the commutative property of multiplication, so it is faster and more flexible to do it in the way described above.
   

• Counting and Grouping:
  
  Multiplication's commutative property aids counting and grouping since it shows that the order of the factors does not affect the total. For example, 4 boxes with 6 pencils each equal the same total as 6 boxes with 4 pencils each. This shows that changing the order in which we counted (factors) did not change the total.