Algebra is an extension of arithmetic that includes the use of letters to represent unknown numerical values. This permits a problem to be solved in a general form and for the solution of a specific problem to be obtained more efficiently by substituting the variables into an algebraic solution as opposed to re-solving the problem with different numbers on every occasion.
Rules associated with the properties and manipulation of algebraic quantities (Dolciani et al. 1967) are given below.
Axioms of Equality (for real numbers A, B, and C)
Reflexive property: A = A.
Symmetric property: if A = B, then B = A.
Transitive property: if A = B and B = C, then A = C.
Axioms of Addition (for real numbers A, B, and C)
Associative Rule: (A + B) + C = A + (B + C).
Existence of Identity: There is a unique number, zero, such that 0 + A = A and A + 0 = A.
Existence of Inverses: For each real number A, there is a number -A such that A + (-A) = 0 and (-A) + A = 0.
Commutativity: For all real numbers A and B, A + B = B + A.
Axioms of Multiplication (for real numbers A, B, and C)
Associative Rule: (A * B) * C = A * (B * C)
Existence of Identity: There exists a unique number, 1, such that for every real number A, 1 * A = A and A * 1 = A.
Existence of Inverse: For every real number A (except 0), there is an element 1/A such that A * (1 / A) = 1 and (1 / A) * A = 1.
Note: This property is the basis of the rule that division by zero is undefined.
Commutativity: A * B = B * A.
Distributive: A * (B + C) = (A * B) + (B * C) and (B + C) * A = (B * A) + (C * A).
Boolean Algebra
Boolean algebra involves the use of logic conditions in which the result of some operation is assigned only one of two values, true or false. In terms of binary computer code, the conditions equate to 1 for true and 0 for false. Computer programmers make extensive use of Boolean algebra in writing computer programs and testing for differing conditions that may exist at the time a given part of the program is executed. Boolean algebra is beyond the scope of this topic.
