Commutative Property - Definition, Formula, Examples
Study Abroad5 Feb, 2026
The commutative property is a basic rule of math that states that the order of the two numbers being added or the two numbers being multiplied doesn't make a difference to the final outcome. In fact, the term commutative property is coined from the term “commute,” which means to move around.
For example, consider your commute. Whether it is from Home to Office or Office to Home, your total commuting distance stays the same. Something like that happens in math. We use a similar logic to two operations in math – Addition, as well as Multiplicat
Understanding the Commutative Property of Addition and Multiplication
It is a kind of "permission slip" that helps to simplify, in general, calculations. It allows numbers in the process of calculation to "travel" (or change their positions) within an expression without changing the validity of the answer. Simple as it sounds, this can be considered one of the basic pillars of arithmetic and algebra.
The Commutative Property of Addition
The commutative property of addition states that for any two real numbers, the sum remains the same regardless of the order in which they are added. If you have two numbers, let’s call them a and b, the result of adding them stays consistent even if you swap their places.
The Formula: a + b = b + a
This property is incredibly useful for mental math. If you find it easier to add a smaller number to a larger one, you can rearrange them without worrying about the outcome. For instance, if you are given 3 + 15, you can think of it as 15 + 3. Both will result in 18.
Read More - Closure Property – Definition, Formula, Addition, Multiplication, Division
The Commutative Property of Multiplication
Similarly, the commutative property of multiplication explains that when you multiply two numbers, the product is the same no matter which number comes first. This rule applies to integers, fractions, decimals, and even complex algebraic variables.
The Formula: a * b = b * a
A very common example of a commutative property of multiplication is the times table that we first learn in lower education. For example, 4 * 5 will give 20, and 5 * 4 will also give 20. The table for this has symmetry because of this property. It is also why you only need to memorize half of the multiplication table!
Commutative Property Examples and Operations
A way to fully understand this idea is to view it in terms of several types of numbers. Whether you are working with basic integers or advanced decimals, the rules for this idea are the same.
Addition Examples:
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Integer Example: 8 + 6 = 14 and 6 + 8 = 14. Therefore, 8 + 6 = 6 + 8.
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Decimal Example: 1.5 + 2.5 = 4.0 and 2.5 + 1.5 = 4.0. Therefore, 1.5 + 2.5 = 2.5 + 1.5.
Multiplication Examples:
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Integer Example: 7 * 3 = 21 and 3 * 7 = 21. Therefore, 7 * 3 = 3 * 7.
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Fraction Example: (1/2) * (1/4) = 1/8 and (1/4) * (1/2) = 1/8.
Non-Commutative Operations: Subtraction and Division
It is vital to note that the commutative property does not apply to all mathematical operations. Specifically, subtraction and division are "non-commutative." This means that the order of numbers is extremely important, and changing that order will lead to a different result.
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Subtraction: 10 - 5 = 5, but 5 - 10 = -5. Since 5 is not equal to -5, the property fails.
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Division: 12 / 3 = 4, but 3 / 12 = 0.25. Since 4 is not equal to 0.25, the property fails.
Read More - Properties of Integrals: Rules, Formulae & Uses
Practical Applications and Problem Solving
The commutative property isn't just a theoretical rule; it’s a tool used to solve problems more efficiently. By recognizing that order doesn't matter in addition and multiplication, you can group numbers together to make them easier to handle.
Simplified Calculations
If you are faced with a string of numbers to add, such as 2 + 17 + 8, the commutative property allows you to move the 8 next to the 2.
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Original: 2 + 17 + 8
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Rearranged: 2 + 8 + 17
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Simplified: 10 + 17 = 27
By "commuting" the 8, you created a "friendly" number (10), making the mental math much faster.
Commutative Property in Algebra
In algebra, we often deal with variables like x and y. Thanks to this property, we know that:
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x + y is the same as y + x
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xy (which means x times y) is the same as yx
This allows mathematicians to standardize the way equations are written, usually putting variables in alphabetical order (like 3ab instead of 3ba), even though both mean the same thing.
Why Understanding This Helps Beyond the Classroom
As we discuss this in "a" and "b," what it is really all about is efficiency. It shows that certain processes have flexibility. In computer coding, it is helpful to realize that a process is commutative. In physics, it is helpful for building formulas for force and velocity.
In the everyday world, the realization that “the order doesn't change the outcome” alone can eliminate the stress of a particular task. Regardless of the order you put the milk and the cereal in the bowl, the result happens to be the same thing. In mathematics, having this information is the starting point to achieving mathematical fluency.
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FAQs related to Commutative Property
What is the simple definition of the commutative property?
It is a math rule stating that the order in which you add or multiply numbers does not change the result. For example, 2 + 3 is the same as 3 + 2, and 4 * 5 is the same as 5 * 4.
Does the commutative property apply to division?
What is a good commutative property of multiplication example?
Think of a rug that is 3 feet by 5 feet (3 * 5 = 15 sq ft). If you rotate it 90 degrees, it becomes 5 feet by 3 feet (5 * 3 = 15 sq ft). The total area of the rug hasn't changed, just its orientation.
How is the commutative property different from the associative property?
The commutative property is about the order of the numbers (a + b = b + a). The associative property is about how numbers are grouped using parentheses, such as a + (b + c) = (a + b) + c.
