Axioms of Algebra An Axiom is a mathematical statement that is assumed to be true. There are four rearrangement axioms and two rearrangement properties of algebra. Addition has the commutative axiom, associative axiom, and rearrangement property. Multiplication has the commutative axiom, associative axiom, and rearrangement property. Commutative Axiom for Addition: The order of addends in an addition expression does not matter. For example: x + y = y + x Commutative Axiom for Multiplication: The order of factors in a multiplication expression does not matter. For example: xy = yx Associative Axiom for Addition: In an addition expression it does not matter how the addends are grouped. For example: (x + y) + z = x + (y + z) Associative Axiom for Multiplication: In a multiplication expression it does not matter how the factors are grouped. For example: (xy)z = x(yz) Rearrangement Property of Addition: The addends in an addition expression may be arranged and grouped in any order. This is a combination of the associative and commutative axioms. e.g. x + y + z = x + (y + z) = (x + y) + z = z + (y + x) = y + (z + x) Rearrangement Property of Multiplication: The factors in a multiplication expression may be arranged and grouped in any order. This is a combination of the associative and commutative axioms. e.g. xyz = x(yz) = z(yx) = y(zx) Return to Top