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 may be switched.

For example x + y = y + x

Commutative Axiom for Multiplication: The order of factors in a multiplication expression may be switched.

For example x * y = y * x

To remember the commutative axioms, it might be helpful to think about the word commute which means to switch places between home and work (or home and school). In the example above, you can see the 4 and the 2 commuting or switching places.

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 (x * y) * z = x * (y * z)

To remember the associative axioms, it might be helpful to think about the word associate, which as a verb means to interact with a group (maybe you associate with a certain group of friends!). The parentheses are grouping operators, that is, they form groups of numbers and operations. You can see in the example above, the 3 can associate with either the 2 or the 4, but the value of each side is still a product of 24.

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.

For example 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.

For example xyz = x(yz) = z(yx) = y(zx)