• 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.
• Symmetric Axiom: Numbers are symmetric around the equals sign. If a = b then b = a. This is the second axiom of equality.
• Transitive Axiom: If a = b and b = c then a = c. This is the third axiom of equality. This is directly related Euclid's Common Notion One: "Things equal to the same thing are equal to each other."
• Additive Axiom: If a = b then a + c = b + c. If two quantities are equal and an equal amount is added to each, they are still equal. This is sometimes called "the addition property of equality."
• Multiplicative Axiom: If a = b then ac = bc. Since multiplication is just repeated addition, the multiplicative axiom follows from the additive axiom. It is sometimes called "the multiplication property of equality."

Sometimes the Additive and Multiplicative Axioms are practically stated as, "always do the same thing to both sides of an equation."