Axioms of Algebra This lesson is a review of both the basic axioms of algebra and the rearrangement axioms and properties of algebra. An Axiom is a mathematical statement that is assumed to be true. There are five basic axioms of algebra. The axioms are the reflexive axiom, symmetric axiom, transitive axiom, additive axiom and multiplicative axiom. Reflexive Axiom: A number is equal to itelf. (e.g a = a). This is the first axiom of equality. It follows Euclid's Common Notion One: "Things equal to the same thing are equal to each other." Symmetric Axiom: Numbers are symmetric around the equals sign. If a = b then b = a. This is the second axiom of equality It follows Euclid's Common Notion One: "Things equal to the same thing are equal to each other." Transitive Axiom: If a = b and b = c then a = c. This is the third axiom of equality. It follows Euclid's Common Notion One: "Things equal to the same thing are equal to each other." Additive Axiom: If a = b and c = d then a + c = b + d. If two quantities are equal and an equal amount is added to each, they are still equal. Multiplicative Axiom: If a=b and c = d then ac = bd. Since multiplication is just repeated addition, the multiplicative axiom follows from the additive axiom. 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