11.3 Commutative Properties

Commutative Properties describe how addition and multiplication of real numbers produce the same result regardless of the order in which the two numbers are combined, and how this order-independence fails to hold for subtraction and division.

Commutativity of Addition

Commutative Property of Addition

The commutative property of addition states that two real numbers can be added in either order without changing the sum, so that the first and second addend can be swapped freely with no effect on the result.

Addend Order Reversal

Reversing the order of two addends is always permitted under this property, and this reversal can be applied to any pair of terms joined by addition within a larger expression, not only to a simple two-number sum standing alone.

Signed Addition Commutativity

The commutative property of addition holds regardless of the signs of the two numbers involved, so that reversing the order of a positive and a negative addend, or of two negative addends, still leaves the sum unchanged.

Commutativity of Multiplication

Commutative Property of Multiplication

The commutative property of multiplication states that two real numbers can be multiplied in either order without changing the product, so that the first and second factor can be swapped freely with no effect on the result.

Factor Order Reversal

Reversing the order of two factors is always permitted under this property, and this reversal can be applied to any pair of factors within a larger product, not only to a simple two-number multiplication standing alone.

Signed Multiplication Commutativity

The commutative property of multiplication holds regardless of the signs of the two numbers involved, so that reversing the order of a positive and a negative factor, or of two negative factors, still leaves the product unchanged.

Operations That Are Not Commutative

Subtraction Noncommutativity

Subtraction is not commutative, meaning that reversing the order of the two numbers being subtracted generally changes the result; the minuend and subtrahend cannot be swapped without altering the value of the expression, except in the special case where the two numbers are equal.

Division Noncommutativity

Division is likewise not commutative, meaning that reversing the order of the dividend and divisor generally changes the result; swapping these two numbers produces the reciprocal of the original quotient rather than the same value, except in the special case where the two numbers are equal.

Confirming Noncommutativity

Numerical Counterexample to False Commutativity

A single numerical example in which reversing the order of two numbers under subtraction or division changes the result is sufficient to confirm that these operations are not commutative in general, since a true commutative property must hold for every possible pair of numbers, and one disagreeing pair disproves it entirely.