How to Determine if a Mathematical System is Commutative (Table Method)

The commutative property of a mathematical operation states that you get the same answer regardless of the order the two arguments are given in. In other words, if * is a commutative operator, then a*b = b*a. Addition and multiplication are examples of operations that have the commutative property. Exponentiation and subtraction are examples which don't (2^5=32, but 5^2=25; 4-1=-3, but 1-4=+3). This article will help you determine if the system is commutative given a table for that operation.


  1. Image titled Determine if a Mathematical System is Commutative (Table Method) Step 1
    Obtain the mathematical system table you need to interpret. It'll work for all mathematical systems, but for you to understand, think about a set of multiplication tables from your childhood. Organize the table so that a*b is the value in a's row and b's column. Also make sure that the numbers written down the leftmost column appear in the same order as the ones reading across the top row.
  2. Image titled Determine if a Mathematical System is Commutative (Table Method) Step 3
    Look down the length of the downward sloping diagonal. These don't matter for commutativity because a*a always equals a*a (in the reverse order).
  3. 3
    Find the reflective objects across each of the table's diagonals. For every entry not on the main diagonal, find its reflection across that diagonal. This pair of entries correspond to a*b and b*a for some arguments a and b. If the operation is commutative, these must be equal. Equivalently, the table entries must be symmetric under reflection across the main diagonal.
  4. 4
    Look for symmetry in the table. If the table is the required symmetry, then it describes a commutative operation. If any entry does not match its opposite across the diagonal, than those table entries represent a counterexample where a*b does not equal b*a and the operation is not commutative.


  • It is common for non-commutative operators to have some situations where a*b = b*a, such as 2^4=4^2. An operation is not commutative unless a*b always equals b*a.

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Categories: Mathematics